Full Text
THE WILSON CHAMBER AND ITS APPLICATIONS IN PHYSICS
N. N. Das Gupta and S. K. Ghosh*)
CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
I. Physical foundations of droplet formation. § 1. Supersaturation. § 2. Theory of droplet formation. § 3. Formation of fog. § 4. Condensation on positive and negative ions. § 5. Boundary of condensation on ions and boundary of fog formation. § 6. Critical supersaturation. § 7. Growth of droplets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
II. Various types of Wilson chambers. § 1. Early designs of the Wilson chamber. § 2. Wilson chamber controlled by counters. § 3. Uncontrolled chamber with increased sensitivity time. § 4. Chambers of reduced and increased pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
III. Factors affecting the quality of tracks. § 1. Illumination. § 2. Photography. § 3. Sharpness of tracks. § 4. Distortion of tracks. § 5. Sensitivity time of the chamber. § 6. Magnets for work with the Wilson chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
IV. Physical measurements made with the aid of the Wilson chamber. § 1. Specific ionization. § 2. Momentum and curvature of tracks in a magnetic field. § 3. Range. § 4. Determination of particle mass from Wilson-chamber data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
V. Appendices. § 1. Vapor pressures of water and ethyl alcohol at various temperatures and for different compositions of the mixture. § 2. Energy, range, velocity, and value of \(H\rho\) for electrons, protons, and \(\alpha\)-particles. § 3. Dependence of the specific ionization, range, and change in momentum on particle velocity. § 4. Basic relations for Compton scattering of \(\gamma\)-rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
INTRODUCTION
In the development of modern physics the Wilson chamber, which Rutherford called “the most original and remarkable instrument in the history of science,” has played an exceptionally important role. It is also often called “the court of last resort in physics,” where conflicting theories are tested and judgment is passed. Often a single Wilson-chamber photograph is able to bring convincing clarity where much indirect evidence proves insufficient.
*) N. N. Das Gupta and S. K. Ghosh (Calcutta, India), Rev. of Modern Physics, 18, 225 (1946). Abridged translation by P. A. Cherenkov.
The history of this remarkable instrument is, in brief outline, as follows. In the earliest works of Coulier (1875), Kiessling (1884), and especially Aitken (1880–1916), attention was drawn to the little-understood role played by dust particles in the condensation of water vapor into fog droplets. Attempting to reproduce in the laboratory the conditions under which fog is formed, they found that fog, which appears upon a slight adiabatic expansion of moist, dust-containing air, no longer forms as soon as the air has been cleared of dust. In addition, it was found that fog droplets appearing in dusty air are always formed around dust particles. At the moment of their appearance, the droplets have a finite radius—of the order of the size of the dust particle—and thus do not pass through that stage of growth in which their radius would have molecular dimensions and in which, as Lord Kelvin showed, the effect of surface tension would lead to such strong evaporation that it would cause the droplets to disappear (see I, § 2).
The establishment of the fact that condensation of water vapor is connected with the presence of dust particles led for some time to the belief that in all cases the formation of fog should be attributed to dust particles. At that time C. T. R. Wilson (1897), by direct experiments, showed that under certain conditions charged ions can also take part in the formation of fog. Wilson established that in a chamber filled with dusty, moist air, a dense fog is formed already upon a very small expansion. This initial fog appears as a result of vapor condensation on dust particles, as follows from the fact that after several repeated expansions, when the air in the chamber is cleared of dust (which falls out together with the fog), no fog is formed during further small expansions. If now, after the air in the chamber has been cleared of dust, the expansion is gradually increased, i.e. the ratio of the final volume of gas in the chamber to its initial volume, then, so long as the expansion remains less than 1.25, no appearance of fog in the chamber is observed; at expansions from 1.25 to 1.37 separate fog droplets appear; and finally, at an expansion greater than 1.37, a continuous fog again forms in the chamber, the density of which increases with increasing expansion. Thus it was shown that condensation can occur even in the absence of dust particles, if the expansion becomes greater than some definite value. It remained to clarify the nature of those centers around which condensation occurs in the absence of dust particles.
Wilson soon found (1899) that if an X-ray tube, or some quantity of uranium or of any other radioactive substance, is placed near the chamber, then instead of the separate droplets that usually appear at expansions lying within the range from 1.25 to 1.37, a dense fog is formed. However, no matter how great the intensity of the radiation, fog does not form if
the expansion is less than 1.25. The intensity of the source affects only the number of droplets, but leaves unchanged the boundary at which droplets begin to appear.
J. J. Thomson (1898) showed that the centers of condensation occurring in the presence of X-rays, at expansions within the limits 1.25–1.37, are ions. Placing in a chamber containing dust-free air two parallel plates and creating a strong field between them, he showed that the dense fog formed in the chamber when X-rays pass through it disappears in the presence of an electric field. Somewhat later (1904) Wilson found that individual droplets formed in the chamber at expansions from 1.25 to 1.38 in gases not subjected to radiation decrease in number if the expansion takes place in the presence of a strong electric field. As a result of all these observations it became clear that in dust-free air, at expansions within the limits 1.25–1.38, the centers of condensation are charged ions formed in the gas under the action of radiation.
C. T. R. Wilson (1911–1912) used this phenomenon to develop a technique for photographing the tracks of $\alpha$-particles, fast electrons, X-rays, and $\gamma$-rays. His method consisted in subjecting a certain volume of air saturated with water vapor to a sudden expansion. The lowering of temperature resulting from the expansion leads to the amount of water vapor in the air being greater than corresponds to the state of saturation. The excess amount of water vapor, condensing on ions as centers of condensation, makes the path of a charged particle visible and makes it possible to photograph its track. A more complex picture arising from the interaction of particles with one another can also be photographed.
The elegance and ingenuity of this method can hardly be overestimated. Before the discovery of the Wilson chamber there was only the possibility of observing the behavior of matter as aggregates of atoms. The Wilson chamber enables us to study the behavior of individual atoms, to see and photograph the actual paths of atoms and electrons in gases and, finally, to study the complex interactions between individual atoms, nuclei, and charged particles.
The enormous possibilities of the Wilson chamber were quickly appreciated, and in the following years a number of chamber designs intended for various investigations were developed. Some of these designs will be briefly described in the second chapter. A large number of works that appeared in recent years were directed toward elucidating and improving the technique of working with the chamber. The third chapter is devoted to a review of these works. Although during the 34 years that have passed since the appearance of the chamber many improvements have been made, nevertheless the basic processes of its action—namely, the processes of formation and rapid growth of droplets in supersaturated vapor—are still not completely
have not yet been studied. In the first chapter some experimental results will be given concerning the formation of droplets and the optimum composition of liquid mixtures for the chamber.
The last chapter is devoted to a brief description of the applications of the Wilson chamber in various branches of modern physics. The applications of the Wilson chamber in physics are so numerous and diverse that it is simply impossible to use all the works relating to this question.
1. PHYSICAL FOUNDATIONS OF DROPLET FORMATION
§ 1. Supersaturation
The basic causes of cloud formation are well known. If a certain mass of air saturated with water vapor is carried upward by a convection current, it expands adiabatically, as a result of which the temperature falls. The excess quantity of water vapor is released in the form of liquid droplets, which we observe as clouds or fog. The same principle underlies the operation of the Wilson chamber. In this instrument a certain volume of a noncondensing gas, saturated with vapors (or a mixture of vapors), is subjected to a sudden expansion, which leads to the gas, now at a lower temperature, containing more vapor than in the state of saturation, i.e., to supersaturation. The excess quantity of vapor condenses on ions as centers, making visible the path of a charged particle.
Thus the condition for condensation on ions is the presence of a state of supersaturation. The supersaturation at a given moment of time may be defined as the ratio of the actual vapor density at that moment to the density of saturated vapor at the same temperature. The degree of supersaturation arising in the chamber as a result of expansion depends on various factors, namely on the nature of the noncondensing gas and of the vapors used, on the initial pressure and temperature of the gaseous mixture, and on the magnitude of the expansion. If vapors of a mixture of two liquids are used as the condensing vapors, for example water and alcohol, the supersaturation also depends on the composition of the mixture. Before passing to the critical supersaturation, on which the normal operation of the chamber depends, it is necessary to consider the influence of all these factors on the resulting supersaturation.
For simplicity we shall assume that in the chamber, immediately before expansion, there is a noncondensing gas at pressure \(P_g\) and vapor at pressure \(P_1\), occupying a volume \(V_1\) at temperature \(T_1\). Then we have
\[ P_1 V_1 = \frac{M_1}{M} R T_1, \tag{I,1} \]
where \(M_1\) is the total mass of vapor contained in the volume \(V_1\), and \(M\) is the weight of its gram-molecule.
Suppose now that a sudden expansion has occurred, changing the volume from \(V_1\) to \(V_2\). As a consequence of this expansion the temperature of the gas mixture falls from \(T_1\) to \(T'_2\), where \(T'_2\) is determined by the adiabatic relation
\[ T_1/T'_2=\left(\frac{V_2}{V_1}\right)^{k-1}, \tag{1,2} \]
where \(k\) is the ratio of the specific heats.
Immediately after the expansion, but before condensation has taken place, the initial mass of vapor \(M_1\) is distributed over the volume \(V_2\). The pressure \(P'_2\) corresponding to this volume is determined by the expression:
\[ P'_2 V_2=\frac{M_1}{M}RT'_2. \tag{1,3} \]
However, this state will not be stable, since at the temperature \(T'_2\), lower than \(T_1\), the mass of saturated vapor is smaller than at the temperature \(T_1\). Consequently, condensation of the vapor will take place, reducing the mass of vapor from \(M_1\) to \(M_2\). After the excess amount of vapor has condensed, equilibrium is reestablished, and the vapor pressure decreases to \(P_2\), corresponding to the pressure of saturated vapor at the temperature \(T_2\). Owing to the release of a certain quantity of heat during vapor condensation, the temperature \(T_2\), established after expansion and condensation, is somewhat higher than the temperature \(T'_2\) which occurs immediately after expansion.
For the vapor at the lower temperature \(T_2\), we have
\[ P_2 V_2=\frac{M_2}{M}RT_2, \tag{1,4} \]
where \(M_2\) is the mass of vapor in the volume \(V_2\) after condensation.
In the interval of time after expansion up to the moment condensation begins, when the initial mass of vapor \(M_1\) is distributed over the volume \(V_2\), the vapor density \(\rho'_2\) is equal to: \(\rho'_2=M_1/V_2\); the corresponding value of this quantity after condensation will be: \(\rho_2=M_2/V_2\). The supersaturation arising as a result of expansion is the ratio of the quantity \(\rho'_2\) to the density of saturated vapor at the lower temperature \(T_2\). Thus we may write
\[ S=\frac{\rho'_2}{\rho_2}=\frac{M_1}{M_2}=\frac{P_1V_1T_2}{P_2V_2T_1}. \tag{1,5} \]
On the other hand, from equations (1,3) and (1,4) we have
\[ S=\frac{P'_2}{P_2}\cdot\frac{T_2}{T'_2}. \tag{1,6} \]
Neglecting the small difference between \(T_2\) and \(T'_2\), we may put \(T_2 = T'_2\). Then, eliminating \(T_2/T_1\) from equation (1,5), by means of formula (1,2) we obtain
\[ S=\frac{P_1}{P_2}\left(\frac{V_1}{V_2}\right)^k =\frac{P_1}{P_2}\left(\frac{1}{1+\varepsilon}\right)^k, \tag{1,7} \]
where \(1+\varepsilon\) is the coefficient of expansion, \(P_1\) and \(P_2\) are the saturated-vapor pressures at the initial and final temperatures, and \(k\) is the ratio of the specific heats of the mixture of gases and vapors present in the chamber.
Owing to the small difference between \(T_2\) and \(T'_2\), the supersaturation obtained from formula (1,7) is somewhat smaller than its exact value, determined by expressions (1, [[unclear: formula number]]) and (1,6).
The ratio of specific heats \(k\), entering into equalities (1,2) and (1,7), refers to the mixture of all gases and vapors present in the Wilson chamber. The value of this quantity, according to Richardson (1906), is determined from the relation:
\[ \frac{1}{k-1}=\frac{1}{k_g-1}\frac{P_g}{\pi} +\frac{1}{k_v-1}\frac{P_v}{\pi}, \tag{1,8} \]
where \(P_g\) and \(P_v\) are the partial pressures of the gas and vapor, \(k_g\) and \(k_v\) are the ratios of the specific heats of each of these components, and \(\pi\) is the total pressure, equal to \(P_g+P_v\). If, together with the noncondensing gas, the vapors not of one but of several liquids are used, then instead of (1,8) we shall have:
\[ \frac{1}{k-1}= \frac{1}{k_g-1}\frac{P_g}{\pi} +\frac{1}{k'_v-1}\frac{P'_v}{\pi} +\frac{1}{k''_v-1}\frac{P''_v}{\pi} \;*). \tag{1,8a} \]
Formula (1,8) shows how the quantity \(k\), referring to the whole mixture as a whole, changes as a function of \(P_g\), \(P_v\), \(k_g\), \(k_v\), etc. Since these latter are functions of the initial temperature and pressure and depend, moreover, on the nature of the vapors and gas filling the chamber, the value of \(k\) for the gas mixture as a whole, and consequently also the supersaturation occurring at a fixed expansion, will change together with changes in these quantities.
We now proceed to a detailed consideration of all these factors.
a. Dependence of supersaturation on the nature of the gas and vapor. Table I shows how the nature of the gas affects supersaturation—
*) This follows from the fact that the work performed by the gas and vapor in adiabatic expansion is given by the expression \(R(T_1-T'_2)/(1-k)\), where \(T_1\) and \(T'_2\) are the initial and final temperatures. The work of the vapors and gas considered separately will be equal to
\[ R(T_1-T'_2)\left\{ \frac{P_g}{(1-k_g)\pi} +\frac{P'_v}{(1-k'_v)\pi} +\ldots \right\}. \]
From the equality of the expressions written, formula (1,8a) follows.
... obtained for one and the same expansion ratio, equal to 1.25.
Table I
Influence of the nature of the gas on supersaturation
| Gas | Vapors | \(k\) | \(T_2\) | \(P_2\), in mm Hg | \(S\) |
|---|---|---|---|---|---|
| A . . . . . . . | \(\mathrm{H_2O}\) | 1,66 | 252,8 | 0,77 | 15,6 |
| Air . . . . . . | \(\mathrm{H_2O}\) | 1,40 | 267,2 | 2,97 | 4,2 |
| CO . . . . . . . | \(\mathrm{H_2O}\) | 1,31 | 273,4 | 4,68 | 2,8 |
| Air . . . . . . | \(\mathrm{C_2H_5OH}\) | 1,37 | 269,8 | 9,45 | 3,41 |
In the first three rows data are given for argon, air, and carbon dioxide, used as noncondensing gases, and for water vapor. The initial temperature is \(T_1 = 20^\circ\mathrm{C}\), so that \(P_1 = 1.75\) cm Hg, and for water \(k = 1.30\). The total pressure in each case was taken equal to 1.5 atm. The value of \(k\) for the mixture of vapor and gas, calculated from formula (1,8), is given in the third column. In the fourth and fifth columns are indicated the temperature \(T_2\), established immediately after expansion, and the vapor pressure corresponding to this temperature.
The data in the last row of Table I refer to the case when, instead of water vapor, \(\mathrm{C_2H_5OH}\) vapor was used. It is easy to see that, since at one and the same temperature the vapor pressure of alcohol is higher than the vapor pressure of water, and since, moreover, the value of \(k\), equal for alcohol vapor to 1.13, is smaller than the value of \(k\) for water vapor, the value of \(k\) calculated for the air—water-vapor mixture will be greater than the analogous value for the air—alcohol-vapor mixture. Consequently, to obtain the same degree of supersaturation with an air—alcohol mixture, a greater expansion will be required than with an air—water mixture.
The table also shows that if, instead of air, a monatomic gas is used, for example argon or helium, the supersaturation increases sharply. Indeed, in a chamber with argon an expansion equal to 1.10 gives the same supersaturation \(S\) as is obtained with an expansion of 1.25 in a chamber filled with air. Since a small expansion eliminates many difficulties of a mechanical nature, monatomic gases are used in most chambers.
6. Dependence of supersaturation on the gas pressure. From formula (1,8) it is seen that when, as a result of lowering the gas pressure \(P_g\), the total pressure \(\pi\) decreases, the quantity \(P_v/\pi\) increases, while \(P_g/\pi\) remains practically unchanged, since \(P_g \gg P_v\). As a result, the value of \(k\) decreases. Therefore, for a given expansion, the drop in temperature will be smaller [see formula (1,2)], and consequently the supersaturation produced will decrease. To obtain
the former supersaturation, it will be necessary to increase the expansion. Conversely, when \(\pi\) is increased, a smaller expansion will be required in order to obtain the same supersaturation.
Table II gives data for mixtures of air—water and air—alcohol, showing how \(k\) and \(S\) change as functions of \(\pi\). The initial temperature is taken to be \(25^\circ\mathrm{C}\), and the magnitude of the expansion is \(1.2\). The values of \(k\) for air, water, and alcohol are taken to be, respectively, \(1.4\), \(1.3\), and \(1.13\).
It is immediately evident from Table II that, with a Wilson chamber, it is more advantageous to work at higher pressures. The value of \(k\) is higher for a gas than for a vapor; therefore, as the pressure is raised, the value of \(k\) for the entire system as a whole increases in comparison with its value corresponding to a lower total pressure. As a result,
Table II
Dependence of \(k\) and \(S\) on the total pressure \(\pi\)
| Total pressure \(\pi\), in cm Hg | 1140 | 330 | 114 | 76 | 38 | 20 | 10 | 6 | |
|---|---|---|---|---|---|---|---|---|---|
| Air and water vapor | \(k:\) | 1.400 | 1.399 | 1.398 | 1.396 | 1.392 | 1.385 | 1.572 | 1.354 |
| Air and water vapor | \(S:\) | 3.018 | 3.017 | 3.016 | 3.016 | 3.014 | 2.889 | 2.772 | 2.572 |
| Air and alcohol vapor | \(k:\) | 1.396 | 1.337 | 1.361 | 1.345 | 1.302 | 1.241 | 1.180 | 1.125 |
| Air and alcohol vapor | \(S:\) | 2.703 | 2.648 | 2.439 | 2.38 | 2.049 | 1.706 | 1.422 | 1.206 |
there is created the possibility of working at smaller expansions, which reduces to a minimum the possibility of the occurrence of turbulence. The higher density of the vapors also prevents the diffusion of water from the walls after expansion. And, finally, since the number of ions per centimeter of path increases with increasing pressure, the particle tracks become denser, which considerably facilitates the problem of illumination during photography.
b. Influence of the initial temperature. If, while keeping the initial volume \(V_1\) and the total pressure \(\pi\) constant, the initial temperature is raised, this will entail an increase in the quantity \(P_w/\pi\) more rapidly than \(P_g/\pi\). As a result, the value of \(k\) for the system as a whole will decrease, and, in order to obtain the same lowering of temperature, a greater expansion will be required. The corresponding numerical data for an air—alcohol mixture are given in Table III. The data of this table refer to pressures \(\pi = 114\) cm Hg and \(\pi = 38\) cm Hg and to a change of the initial temperature within the limits from 40 to \(10^\circ\mathrm{C}\). The magnitude of the expansion is taken to be \(1.2\), and the values of \(k\) for air and alcohol are the same as in Table II.
Table III
Dependence of \(k\) and \(S\) on the initial temperature
| Temperature | 40° C | 30° C | 25° C | 20° C | 10° C |
|---|---|---|---|---|---|
| 114 cm Hg \(k\) | 1.322 | 1.35 | 1.361 | 1.370 | 1.383 |
| 114 cm Hg \(S\) | 2.084 | 2.345 | 2.439 | 2.523 | 2.654 |
| 38 cm Hg \(k\) | 1.232 | 1.280 | 1.302 | 1.322 | 1.356 |
| 38 cm Hg \(S\) | 1.592 | 1.912 | 2.049 | 2.156 | 2.355 |
c. Dependence of supersaturation on the magnitude of expansion. The change in supersaturation as a function of the magnitude of expansion can be calculated by means of formulas (I,2) and (I,5). Table IV gives the results of such calculations for the mixtures air—water and air—alcohol.
Table IV
Values of supersaturation for different magnitudes of expansion
| Expansion (\(t = 20^\circ\) C) | 1.00 | 1.05 | 1.10 | 1.15 | 1.20 | 1.25 | 1.30 | 1.35 | 1.40 |
|---|---|---|---|---|---|---|---|---|---|
| Air—water \(S\) | 1.00 | 1.28 | 1.78 | 2.36 | 2.96 | 4.005 | 5.52 | 7.16 | 9.14 |
| Air—alcohol \(S\) | 1.00 | 1.29 | 1.62 | 1.84 | 2.55 | 3.25 | 4.15 | 5.32 | 6.94 |
In these calculations \(k\) was taken equal to 1.4 for the air—water system and 1.37 for the air—alcohol system. The data in the table refer to the initial temperature \(t = 20^\circ\) C, so that the partial pressure of water vapor is equal to 17.54 mm Hg and that of alcohol vapor to 44 mm Hg.
From Table IV it is seen that, at an expansion of 1.25, the supersaturation formed in the air—water vapor system is equal to 4. In order to obtain the same supersaturation when working with a mixture of air—alcohol vapor, it is necessary to raise the magnitude of expansion to 1.29.
d. Dependence of \(k\) and \(S\) on the concentration of a mixture of liquids. If a mixture of different vapors is used as the condensing vapors, then the partial pressures \(P'_v\), \(P''_v\), etc., of each component of such a mixture depend on the concentration of the corresponding liquid. Duhem and Margules (1900) showed that in a closed volume containing a mixture of two liquids, the partial vapor pressures satisfy the relation
\[ \frac{d \ln P'_v}{d \ln P''_v} = \frac{1 - x}{x}, \tag{I,9} \]
where \(x\) and \((1-x)\) are the mole fractions of the liquids used, and \(P'_v\) and \(P''_v\) are the partial pressures of their vapors. Assuming the total vapor pressure for the given composition of the mixture to be known, one may, following the method of Lewis and Merfi, solve equation (I,9) with respect to \(P'_v\) and \(P''_v\).
The dependence of the partial vapor pressures \(P'_v\) and \(P''_v\) on the concentration of the water—alcohol mixture for temperatures of 20° and 40° C is given in Table XVIII (see p. 565), borrowed from the handbook of Landolt and Bernstein. From the data of this table, using equality (I,8a), one can calculate the value \(k\) corresponding to various concentrations of the mixture of water and alcohol. The values of \(k\) calculated in this way are given in Table V (for \(t_1=20^\circ\) C).
Table V
Values of \(k\) at various relative concentrations of the water—alcohol mixture
| Content of C\(_2\)H\(_5\)OH by weight (in %) | Content of C\(_2\)H\(_5\)OH by volume (in %) | Molar content of C\(_2\)H\(_5\)OH (in %) | \(k\) |
|---|---|---|---|
| 0 | 0,00 | 0,00 | 1,398 |
| 10 | 12,31 | 4,167 | 1,393 |
| 20 | 24,00 | 8,913 | 1,339 |
| 30 | 35,12 | 14,35 | 1,386 |
| 40 | 45,71 | 20,69 | 1,384 |
| 50 | 55,80 | 28,12 | 1,382 |
| 60 | 65,46 | 36,99 | 1,381 |
| 70 | 74,66 | 47,62 | 1,379 |
| 80 | 83,47 | 61,02 | 1,377 |
| 90 | 91,91 | 77,87 | 1,374 |
| 100 | 100,00 | 100,00 | 1,370 |
The percentage content of C\(_2\)H\(_5\)OH (by volume), given in the first column of this table, was calculated from the relation:
\[ \text{volume of C}_2\text{H}_5\text{OH (in %)} = (V_{\text{alcohol}}\times 100)/(V_{\text{alcohol}}+V_{\text{water}}), \]
where \(V_{\text{alcohol}}\) and \(V_{\text{water}}\) are the true volumes of each of the liquids taken for preparing the mixture.
With the aid of equation (I,2) and the calculated values of \(k\) given in Table IV, it is possible, for a fixed value of expansion, to determine the temperature \(T_2\) established immediately after expansion, and its dependence on the component ratio of the mixture of liquids under investigation. In order to obtain the superheating corresponding to various component ratios, it is necessary to know the total vapor pressure of the mixture at temperature \(T_2\) [equation (I,5)]. However, as far as is known, experimental data on the total vapor pressure of alcohol—water mixtures for those low temperatures which pred—
are of interest, still does not exist. In this connection, the actual supersaturation formed as a result of expansion cannot be calculated.
§ 2. Theory of droplet formation
Let us now turn to a consideration of modern ideas about the process of droplet formation. The foundations of these ideas were laid by Lord Kelvin, who as early as 1870 showed that near the surface of a droplet of radius \(r\) the pressure of the saturated vapor is greater than at a plane surface of the liquid, and that the relation between these quantities is expressed by the formula
\[ \ln \frac{P_r}{P_\infty}=\frac{2\sigma}{r}\frac{M}{RT\rho}, \tag{I,10} \]
where \(\sigma\) is the surface tension, \(\rho\) the density of the liquid, \(R\) the gas constant, \(T\) the absolute temperature, and \(P_r\) and \(P_\infty\) the pressures of the saturated vapors near, respectively, the surface of the droplet and the plane surface. For water at temperature \(T=291^\circ\) abs. equation (I,10) takes the form
\[ \frac{P_r}{P_\infty}=e^{\frac{1.09\cdot 10^{-7}}{r}}, \tag{I,11} \]
whence it follows that \(P_r\) is always greater than \(P_\infty\), and, as long as \(r\) exceeds \(10^{-7}\) cm, the difference between these quantities is insignificant.
Thus, if a droplet is suspended in a space where the relative humidity is \(100\%\), it will evaporate. To prevent evaporation of the droplet, the relative humidity must be raised to \(100\,P_r/P_\infty\), i.e., it is necessary to create a supersaturation equal to \(P_r/P_\infty\).
Table VI gives the values of \(P_r/P_\infty\) calculated for various values of the radius \(r\). In Fig. 1 these results are also shown graphically (dashed line).
Table VI
Values of \(P_r/P_\infty\) for water droplets
| \(r\) in cm | \(P_r/P_\infty\) | \(r\) in cm | \(P_r/P_\infty\) | \(r\) in cm | \(P_r/P_\infty\) |
|---|---|---|---|---|---|
| \(1.0\times10^{-4}\) | 1.001 | \(7.8\times10^{-8}\) | 4.00 | \(3.9\times10^{-8}\) | 16.5 |
| \(1.0\times10^{-5}\) | 1.01 | 6.8 | 5.00 | 2.0 | 235.0 |
| \(1.0\times10^{-6}\) | 1.12 | 6.1 | 6.00 | ||
| \(2.3\times10^{-7}\) | 1.60 | 5.6 | 7.00 | ||
| \(1.9\times10^{-7}\) | 1.78 | 5.2 | 8.00 | ||
| \(1.6\times10^{-7}\) | 2.00 | 5.0 | 9.00 | ||
| \(1.0\times10^{-7}\) | 3.00 | 4.7 | 10.00 |
From the dotted curve in Fig. 1 it is seen that the supersaturation required for the formation of droplets increases rapidly as the droplet radius \(r\) decreases. For the formation of a droplet of radius \(r=2\cdot 10^{-8}\) cm, a supersaturation equal to 235 is necessary. If the supersaturation is below this
Fig. 1. Dependence of the pressure of saturated vapor near the surface of a droplet on its radius.
value, then a droplet of such small dimensions, even if it has formed, will immediately evaporate.
In air there are usually dust particles whose dimensions range from \(10^{-4}\) to \(10^{-6}\) cm. These dimensions are sufficiently large, and therefore, at very small supersaturations—of the order of 1.001 to 1.12—condensation occurs on the dust particles. This circumstance explains the phenomenon found by Aitken, and also by Wilson, consisting in the fact that in a chamber filled with air containing dust, a very small expansion is sufficient to cause the appearance of an intense fog, whereas in a chamber filled with pure air, at small expansions the formation of fog is not observed.
The dotted curve in Fig. 1 also shows that, with an increase in the droplet radius, the supersaturation necessary for maintaining its
in equilibrium with the vapor phase, decreases. However, the equilibrium state will not be stable. As the droplets increase in size, the process of their further growth becomes more rapid and continues until the supersaturation has fallen to unity. Essential for this process is the presence of centers around which the initial condensation could occur. As soon as this has happened, the droplets begin to grow rapidly to sizes visible to the eye. Their final size is determined by the degree of supersaturation and by the number of droplets formed in one cubic centimeter.
Let us turn to the explanation of the phenomenon first discovered by C. T. R. Wilson. As already indicated, this phenomenon consists in the fact that in a chamber filled with dust-free air and water vapor, during expansions from 1.25 to 1.37 condensation of water on ions takes place. Consequently, in this case the droplets that arise will possess an electric charge. Assuming that each droplet is formed on one ion, i.e. carries one elementary charge \(e\), one may, following Thomson, modify equation (I,10) in the following way:
\[ \ln \frac{P_r}{P_\infty} = \frac{M}{RT\rho} \left( \frac{2\sigma}{r} - \frac{e^2}{8\pi k r^4} \right). \tag{I,12} \]
The term on the left-hand side of equation (I,12) characterizes the excess pressure at the surface of the droplet in comparison with the pressure of saturated vapors \(P_\infty\). The two terms on the right-hand side represent the pressure due to surface tension and to the electric field. These terms have opposite signs, since the potential energy of surface tension \((4\pi r^2\sigma)\) decreases with decreasing radius \(r\), whereas the potential energy of the electric field
\[ \left(\frac{1}{2}\cdot \frac{e^2}{kr}\right) \]
increases. The first term acts in the direction of decreasing the radius of the droplet and, consequently, increasing the vapor pressure; the second term acts in the opposite direction. For this reason, the vapor pressure near the surface of a charged particle will always be lower in comparison with an uncharged particle.
From equation (I,12) we see that for
\[ \begin{aligned} r=c=\left(\frac{e^2}{16\pi k\sigma}\right)^{1/3} \qquad & P_r=P_\infty,\\ r>c \qquad & P_r>P_\infty,\\ r<c \qquad & P_r<P_\infty . \end{aligned} \]
Here \(c\) is the critical radius determining the size of the charged droplets established at the moment when the vapor pressure becomes saturated, i.e. when \(P_r=P_\infty\). In a space saturated with vapors, each gaseous ion will be surrounded by a droplet of radius \(c\).
Assuming that the droplet is formed on a single ion, i.e. that its charge is \(e=4.8\cdot 10^{-10}\), and taking the surface tension of the droplet to be equal to \(76\ \text{dyn}/\text{cm}\), which is valid for a thick layer of water, we obtain \(c=3.9\cdot 10^{-8}\ \text{cm}\).
Table VII gives the calculated values of \(P_r/P_\infty\) for charged droplets of radius \(r\). In Fig. 1 the dependence between these quantities is represented graphically (solid line). For a droplet radius \(r>10^{-7}\ \text{cm}\) the influence of the charge is negligible, and therefore the formation of either a charged or an uncharged droplet requires one and the same supersaturation. But at smaller radii there is a substantial difference. Whereas for uncharged droplets the supersaturation corresponding to equilibrium increases rapidly as the radius decreases, for charged droplets at \(r=6.5\cdot 10^{-8}\ \text{cm}\) it reaches a maximum value equal to 4.1 and then, with further decrease of the radius, falls again. Thus a charged droplet can form even in a space unsaturated with vapor (\(S<1\)), whereas the appearance of an uncharged droplet under these conditions is excluded.
Table VII
Values of \(P_r/P_\infty\) for singly charged ions
| \(r\) (in \(10^{-8}\ \text{cm}\)) | \(P_r/P_\infty\) | \(r\) (in \(10^{-8}\ \text{cm}\)) | \(P_r/P_\infty\) |
|---|---|---|---|
| 1.95 | \(10^{-18}\) | 6.45 | 4.0 |
| 3.55 | 0.37 | 7.80 | 3.71 |
| 3.90 | 1.00 | 8.80 | 3.36 |
| 4.25 | 2.00 | 11.70 | 2.62 |
| 4.65 | 3.00 | 15.6 | 2.09 |
| 5.80 | 4.00 | 19.5 | 1.81 |
| 5.85 | 4.08 | 23.4 | 1.64 |
Table VII shows that, as the amount of water vapor present in the air decreases, the radius of the droplets that can form around ions under the given conditions decreases very slowly.
Thus, for example, if the amount of water vapor present in the air at saturation is reduced to \(10^{-18}\) of this value, which corresponds to a state of almost absolute dryness, the size of the droplets will decrease by only one half: from \(r=3.9\cdot 10^{-8}\) it decreases to \(r=1.95\cdot 10^{-8}\ \text{cm}\). Since air always contains some amount of both ions and water vapor, such very small droplets are always present in it. However, these droplets cannot grow to visible sizes because, for the ascending part \(AB\) (the solid curve in Fig. 1), growth of the droplet size requires a simultaneous increase in the pressure of saturated vapors, which is physically impossible.
In the region to the right of point \(B\), an increase in the droplet size is accompanied by a decrease in the pressure of the saturating vapors. Therefore, as
only after the droplet has passed point \(B\), condensation occurs on it and its dimensions increase. Thus the region of the curve between points \(B\) and \(C\) is unstable, and a droplet, as soon as it has passed stage \(B\), soon becomes visible. To summarize, one may say that if the supersaturation exceeds 4.1, which for the air—water-vapor system corresponds to the value \(V_2/V_1 = 1.25\) (see Table IV), then, in the presence of ions, droplets can form on them and grow to visible dimensions.
§ 3. Formation of fog
Wilson observed that in a chamber filled with air and water vapor, under expansions exceeding 1.37, even in the absence of any ionizing agent, a dense fog appears, filling the entire chamber. Then, in the course of about one minute, this fog settles to the bottom, and color phenomena are observed. Under expansions less than 1.37, the drops are too large to produce color phenomena, and they are not numerous enough for a continuous fog to form. Under expansions greater than 1.44, color phenomena likewise do not occur, since the drops are too small. In this region the density of the fog increases rapidly with increasing expansion. An expansion equal to 1.37, at which a dense fog forms throughout the entire volume of the chamber, is thus the boundary for fog formation. As is evident from Table IV, the supersaturation corresponding to this expansion for the air—water-vapor system reaches approximately eight.
What is the nature of those centers on which condensation occurs at an eightfold supersaturation? According to J. J. Thomson, in a space saturated with vapor there are always the very smallest droplets of water, formed as a result of the coalescence of vapor molecules and evaporating again immediately thereafter. It may be supposed that some of these droplets at the moment of expansion act as condensation centers. Apparently, these droplets have different sizes with a sharply expressed upper limit, amounting to \(5 \cdot 10^{-8}\) cm. So long as the supersaturation does not exceed eight, condensation occurs on large droplets \((r \geq 5.2 \cdot 10^{-8}\ \text{cm})\), but as soon as the magnitude of the supersaturation exceeds this limit, condensation also begins on smaller droplets. This can explain the rapid increase in the density of the fog with increasing expansion.
In his arguments Thomson proceeded from the fact that in deriving equations (I,10) and (I,12) the surface tension of the droplets was taken to be independent of the radius, which, however, cannot be considered valid. Taking into account the dependence of the surface tension on the radius of the droplet, equation (I,12) is modified as follows:
\[ \ln \frac{P_r}{P_\infty} = \frac{M}{RT\rho} \left( \frac{2\sigma}{r} + \frac{d\sigma}{dr} - \frac{e^2}{8\pi k r^4} \right). \tag{I,13} \]
If the droplet is uncharged, \(e=0\), and we have
\[ \ln \frac{P_r}{P_\infty} = \frac{M}{RT\rho} \left( \frac{2\sigma}{r}+\frac{d\sigma}{dr} \right). \tag{I,14} \]
The exact dependence of \(\sigma\) on \(r\) is as yet unknown. Lord Rayleigh showed that, for very thin films, the surface tension is proportional to the thickness of the film. Moreover, on the basis of the experiments of Reynolds, Rücker, and Johannot, Thomson concludes that at \(r=0\) the surface tension is equal to zero; then, with increasing \(r\), it grows, reaches a maximum at some value of \(r\), and thereafter decreases again.
Starting from these assumptions, Thomson showed that the dependence of \(\ln P_r/P_\infty\), or of the equal quantity
\[ \frac{M}{RT\rho} \left( \frac{2\sigma}{r}+\frac{d\sigma}{dr} \right), \]
on the radius of the droplet \(r\) may be represented by a curve having the form shown in Fig. 2. From this transformed curve [it corresponds to equation (I,14)] it is seen that when \(r=0\), \(\ln \frac{P_r}{P_\infty}=0\); then, with increasing \(r\), this quantity increases, passes through a maximum, then, decreasing, again becomes zero and changes sign.
Fig. 2. Influence of surface tension on the pressure of saturated vapor at the surface of a droplet.
The curve also shows that, irrespective of the magnitude of the supersaturation, droplets will always exist. To determine the radius of the droplet corresponding to some supersaturation \(S\), one must draw a straight line parallel to the abscissa axis at a distance \(\ln S\) from it. The abscissa of the point of intersection of this straight line with the curve of Fig. 2 determines the size of the droplets. Since the straight line (in the region of interest to us) will always intersect the curve of Fig. 2, it follows that for any positive value of \(S\) we shall always have droplets whose size will be determined by the value of \(S\). Condensation will occur even at the weakest supersaturations.
The radius of these droplets, however, is too small for them to be visible to the naked eye. Nor can they grow to visible dimensions, since they lie on the ascending part \(OA\) of the curve under consideration. As soon as the supersaturation disappears, i.e. \(P_r\) becomes equal to or less than \(P_\infty\), the droplets evaporate. But if a droplet has passed point \(A\) (which will occur only for a sufficiently large value of …)
oversaturation), then with a further increase in its size the pressure of the saturated vapors near its surface begins to fall and, thus, conditions are created for the condensation of water vapor on it and for its growth. The region \(AC\) is unstable, and a droplet that has passed through stage \(A\) quickly grows to dimensions that make it visible. Such is Thomson’s explanation of the formation of a dense fog at eightfold oversaturation. The indicated magnitude of oversaturation is necessary in order that the droplet may pass through point \(A\). At oversaturations greater than eightfold \((S > 8)\), even the smallest condensation centers fall into the region \(AC\) and quickly turn into droplets of visible size. This explains the rapid increase, observed in this region, in the density of the fog with increasing expansion.
According to Thomson, the initial smallest droplets, which serve as condensation centers for the formation of a continuous fog, arise as a result of the union (coalescence) of water-vapor molecules. In turn, droplets of larger size are formed from the union of these droplets. Therefore there are far more droplets of small size than large ones, and for their magnitude there is a quite definite upper limit. The number of droplets whose sizes exceed this limit is too small to form a visible fog.
§ 4. Condensation on Positive and Negative Ions
C. T. R. Wilson established (1896) that on negative ions condensation begins earlier than on positive ones. He found that in the air–water-vapor system condensation on negative ions begins with an expansion equal to 1.25 \((S = 4)\), whereas on positive ions it occurs starting with an expansion of 1.31 \((S = 6)\). Experimenting with water and a number of other organic liquids, Laby established that in air the condensation of vapors of all the liquids he investigated (with the exception of water) begins on positive ions earlier than on negative ones. The liquids studied by Laby were: acetic acid, amyl alcohol, chloroform, ethyl acetate, ethyl alcohol, ethyl iodide, heptyl alcohol, isoamyl alcohol, methyl butyrate, and propyl acetate.
Similar investigations were carried out by Sharrer (1939), who studied the condensation of supersaturated vapors on natural ions and on ions formed by X-rays. He investigated vapors of water, ethyl and methyl alcohols, benzene, carbon tetrachloride, chloroform, chlorobenzene, and a mixture of vapors of water and ethyl alcohol. It turned out that water and chlorobenzene condense earlier on negative ions, while vapors of ethyl and methyl alcohols, as well as vapors of chloroform, condense earlier on positive ions. For benzene and carbon tetrachloride, condensation on negative and positive ions begins at approximately the same time.
Such a difference can be explained by the fact that a double layer of charges is formed on the surface of the droplet: one on the surface of the droplet itself, the other (of opposite sign) in the gas surrounding the droplet at a very small distance from the first. The basis for this assumption is the well-known fact that the spraying of water or the blowing of air through it is accompanied by electrification.
From equation (I,12) we saw that the quantity
\[ \frac{M}{RT\rho}\left[\frac{2\sigma}{r}-\frac{e^{2}}{8\pi k r^{4}}\right] \]
determines the excess pressure at the surface of the droplet in comparison with the pressure of saturated vapors. The first term is due to surface tension, the second (equal to \(kE^{2}/8\pi\), where \(E=\frac{e}{kr^{2}}\) is the field strength) to the electric field. The presence of a double layer of charges must lead to a change in the influence of the second term.
If \(V\) is the potential difference between the layers, caused by their charges, and \(d\) is the distance between them, then the influence of the double layer will be characterized by the quantity \(\frac{k}{8\pi}\cdot\left(\frac{V}{d}\right)^{2}\), and equation (I,12) is transformed into the following:
\[ \ln \frac{P_r}{P_\infty} = \frac{M}{RT\rho} \left\{ \frac{2\sigma}{r} - \frac{k}{8\pi} \left( \frac{V}{d} + \frac{e}{kr^{2}} \right)^{2} \right\}, \tag{I,15} \]
and the corresponding equation:
\[ \ln \frac{P_r}{P_\infty} = \frac{M}{RT\rho} \left\{ \frac{2\sigma}{r} - \frac{kV^{2}}{8\pi d^{2}} \right\} \tag{I,16} \]
for the case of an uncharged droplet, when \(e=0\).
Comparing expressions (I,15) and (I,16), we see that for a charged droplet the term standing in brackets on the right-hand side of the equality is decreased by the quantity
\[ \frac{e^{2}}{8\pi k r^{4}}+\frac{eV}{4\pi r^{2}d}. \tag{I,17} \]
Since here the first term is always positive, we must consider the influence of only the second term.
If \(eV/4\pi r^{2}d\) is positive, the right-hand side of equation (I,15), and consequently, for a given \(r\), also the magnitude \(P_r/P_\infty\), decrease; i.e., condensation on the surface of the droplet will occur at a smaller supersaturation than before, which facilitates the conditions for condensation. On the other hand, when \(eV/4\pi r^{2}d<0\), these conditions will be hindered, and, in order for condensation to begin, a greater supersaturation will be required. The electric field caused by the double layer has a definite direction, depending on the nature of the gas. If the ion has a charge that creates a field of the same sign, the product will be positive and
action of such an ion as a condensation center will be more effective.
When a clean water surface is brought into contact with air, the latter becomes negatively electrified, while an equal positive charge moves to the water surface, forming the outer shell of the double layer. The negative charges of this double layer are located on the water surface, and the positive ones on the air surface in contact with it. The field is directed inward, and therefore, in the case of water vapor, negative ions will prove more effective than positive ones.
Some liquids, when air is blown through them, become electrified not negatively but positively. In this case the field is directed outward, and consequently positive ions, as condensation centers, will be more effective. The correctness of these conclusions was experimentally verified by Leby (1908), Scharrer (1939), and Hazen (1944) in experiments with various alcohols. By creating a strong electric field, it is possible to separate from one another the droplets formed on negative and positive ions. A photograph obtained in this way, due to Hazen, is shown in Fig. 3. In this photograph the difference in the action of positive and negative ions is clearly visible. The denser column was formed on positive ions.
Fig. 3. Condensation on positive and negative ions. The positive and negative ions separated into two columns before droplets formed on them. The denser column was formed on positive ions (after Hazen, Phys. Rev. 65, 259 (1944).)
§ 5. The limit of condensation on ions and the limit of fog formation
Wilson established that in the air–water-vapor system the centers of condensation of individual droplets, appearing upon expansion equal to 1.25, are ions, whereas condensation of fog, forming throughout the entire volume of the chamber upon expansions exceeding 1.37, apparently occurs on molecular complexes of vapors contained in the chamber. These two limiting expansions may be called the limit of condensation on ions and the limit of fog formation. Wilson’s experimental results were later confirmed and supplemented by Powell (1928), Vollmer and Flood (1934), Flood (1934), and Beke (1941).
In order to establish the limit of condensation on ions and the limit of fog formation, Powell used γ-rays as the ionizing source. He found that in a chamber filled with air and water vapor, in the presence of γ-rays, the formation of droplets occurs at expansion 1.25 (the supersaturation is equal to four). After removal of the γ-ray source, condensation is not observed until the expansion reaches 1.37 (the supersaturation is equal to 8).
To remove natural ions immediately after their formation, Flood applied a strong electric field. In the presence of such a field ions are absent, and condensation occurs only on complexes of vapor molecules, which corresponds to the limit of formation of a continuous fog. On the other hand, the minimum expansion required for the formation of droplets in the absence of a field, when ions are present,
Table VIII
Limit of condensation on ions and limit of fog formation at various alcohol concentrations in the water–alcohol mixture
| C₂H₅OH content (in %) | C₂H₅OH content (in %) | Expansion | Expansion | C₂H₅OH content (in %) | C₂H₅OH content (in %) | Expansion | Expansion |
|---|---|---|---|---|---|---|---|
| by weight | by volume | without field (limit of condensation on ions) | with field (limit of fog formation) | by weight | by volume | without field (limit of condensation on ions) | with field (limit of fog formation) |
| 00.0 | 00.0 | 1.251 | 1.276 | 58.3 | 63.0 | 1.101 | 1.113 |
| 9.3 | 11 | 1.155 | 1.174 | 67.8 | 73 | 1.105 | 1.114 |
| 24.9 | 30 | 1.115 | 1.130 | 73.4 | 77 | 1.100 | 1.112 |
| 44.2 | 49 | 1.110 | 1.112 | 83.9 | 87 | 1.114 | 1.128 |
| 50.4 | 57 | 1.098 | 1.107 | 90.0 | 92 | 1.119 | 1.132 |
| 52.8 | 59 | 1.103 | 1.114 | 96.0 | 96 | 1.142 | 1.158 |
| [[unclear: marks in cell]] | [[unclear: marks in cell]] | 100.0 | 100 | 1.152 | 1.172 |
corresponds to the boundary of condensation on ions. Flude also studied the change of these boundaries as a function of the composition of the liquid mixture used. Flude’s data, characterizing the change in the boundary of fog formation and the boundary of condensation on ions as a function of the percentage content of alcohol in the water—alcohol mixture, are given in Table VIII.
From the table presented it is seen that, when alcohol alone is used (concentration \(100\%\)), the boundary of condensation on ions and the boundary of fog formation lie below the corresponding boundaries for pure water. It is also seen from the table that, for a certain concentration of the mixture of water and alcohol, these boundaries have a minimum value. Evidently, such a concentration of the mixture is optimal for chamber operation, provided that it produces the minimum amount of fog forming the background.
§ 6. Critical supersaturation
We have already seen earlier (Table IV) that, for one and the same expansion, alcohol vapors give a smaller supersaturation than water vapors. On the other hand, Flude’s results allow one to conclude that, in comparison with water vapor, in alcohol vapor the boundary of condensation on ions and the boundary of fog formation lie at smaller supersaturations. By using a mixture consisting, by volume, of \(70\%\) alcohol and \(30\%\) water, these boundaries can be lowered still further. The facts indicated can be explained by the fact that the position of the boundaries of condensation on ions and of fog formation ultimately depends on the number of droplets formed (at one and the same supersaturation) in one \(\mathrm{cm}^3\), which may be different for different liquids.
The number of droplets formed in one \(\mathrm{cm}^3\) was calculated by Volmer and Weber (1926) and by Farkas (1927), who, following Thomson, assumed that in saturated vapors there is always, in equilibrium, a definite number of embryonic droplets (centers). These droplets are in a state of formation and evaporation, but they are capable of becoming centers of condensation.
The work expended in creating an embryonic droplet of radius \(r\) is determined by the formula \(W=4\pi r^2\sigma/3\), derived by Gibbs on the basis of thermodynamic considerations. Here, as before, \(\sigma\) denotes the surface tension.
Volmer and Weber also found thermodynamically that the number of embryonic centers is proportional to the expression
\[ Z=Ae^{-4\pi r^2\sigma/3kT}, \tag{1.18} \]
where \(k\) is Boltzmann’s constant. The quantity \(A\) entering this expression, as shown by Farkas (1927), is equal to
\[ A=\frac{2C}{F}\alpha P_{\infty}\left(\frac{\sigma}{kT}\right)^{1/2}. \]
where \(C\) is a constant, \(F=4\pi r^2\), \(a=N/(2\pi RTM)^{1/2}\), \(M\) is the molecular weight of the liquid, and \(P_\infty\) is the pressure of the saturated vapor at temperature \(T\). Thus, the number of droplets formed in \(1\ \mathrm{cm}^3\) is determined by the formula
\[ Z=\frac{2C}{F}\,aP_\infty\left(\frac{\sigma}{kT}\right)^{1/2} e^{-\frac{4\pi r^2\sigma}{3kT}} . \tag{1,19} \]
To obtain an idea of the supersaturation necessary for obtaining good tracks, let us introduce the concept of the critical supersaturation \(S_c\), which we shall define as that supersaturation at which the number of droplets formed in one \(\mathrm{cm}^3\) is of the order of unity. Then, on the basis of equation (1,19), we have
\[ \frac{4\pi r^2\sigma}{3kT}=\ln \frac{2C}{F}\,aP_\infty\left(\frac{\sigma}{kT}\right)^{1/2}, \tag{1,20} \]
and, according to Thomson’s formula (1,10), the critical supersaturation is equal to
\[ S_c=\frac{P_r}{P_\infty}=e^{\frac{2\sigma M}{rRT\rho}} \]
or
\[ \ln S_c=\frac{\sigma}{T}\cdot\frac{M}{\rho}\cdot\frac{2}{Rr}. \tag{1,21} \]
Substituting here the value of \(1/r\) from (1,20), we obtain
\[ \ln S_c= \left(\frac{\sigma}{T}\right)^{1/2}\cdot\frac{M}{\rho}\cdot\frac{2}{R} \left(\frac{4\pi}{3k}\right)^{1/2} \cdot \left\{1/\ln\left[\frac{2C}{F}aP_\infty\left(\frac{\sigma}{kT}\right)^{1/2}\right]\right\}^{1/2} \simeq \]
\[ \simeq D\left(\frac{\sigma}{T}\right)^{3/2}\cdot\frac{M}{\rho}, \tag{1,22} \]
where \(D\) is a constant, if the small variation of the logarithmic term is neglected.
For pure liquids the value \(S_c\) can be calculated by means of equation (1,21) and compared with experiment. Table IX, taken from the work of Volmer and Flood, shows that the theoretical and experimental data are in quite satisfactory agreement. The empirical constant \(D\) was determined so that the theoretical and experimental data for water at a temperature of \(264^\circ\) abs. coincided.
For use in a Wilson chamber, the ideal mixture would be one having the lowest possible critical supersaturation, i.e., the smallest possible molecular volume in combination with a small surface tension. As is seen from Table IX, water has a small molecular volume but a large surface tension, whereas for alcohols the reverse is the case. A mixture of water and alcohol in the proper proportion makes it possible to obtain a critical supersaturation
than for a pure liquid. Such a mixture is a mixture of water and ethyl alcohol.
If vapors of two liquids are used as the condensing vapors, the droplets formed contain both liquids. Flood showed that in this case as well the critical supersaturation \(S'_c\) can be determined by an expression analogous to (1,21):
Table IX
Experimental and theoretical values of supersaturations
(according to Volmer and Flood)
| Substance | \(T\) | \(M\) | \(\rho\) | \(M/\rho\) | \(\sigma\) | \(S_c\) (theor.) | \(S_c\) (exp.) |
|---|---|---|---|---|---|---|---|
| Water | 264 | 18 | 1.00 | 18.0 | 77.0 | 4.85 | 4.85 |
| Ethyl alcohol | 273 | 46 | 0.81 | 56.8 | 24.0 | 2.30 | 2.34 |
| \(n\)-propyl alcohol | 270 | 60 | 0.81 | 74.3 | 25.4 | 3.20 | 3.05 |
| Isopropyl alcohol | 265 | 60 | 0.82 | 73.4 | 23.1 | 2.90 | 2.80 |
| Methyl alcohol | 270 | 32 | 0.81 | 39.5 | 24.8 | 1.84 | 3.20 |
\[ \ln S'_c = k(\sigma'/T)(M/\rho)', \tag{1,23} \]
where \((M/\rho)'\) now refers to the molecular volume of the mixture, and \(\sigma'\) is the surface tension of the droplets. Knowing the molar content of both liquids in the droplet, one can, with the aid of the equation given above, calculate the value of \(S'_c\) and compare it with the value found experimentally. For a mixture of water with ethyl alcohol, Flood obtained good agreement between the calculated and experimentally obtained values of \(S'_c\). The critical supersaturation reaches a minimum, equal to 1.68, for a mixture containing 60% alcohol and 40% water (by volume).
A number of other organic liquids suitable for use in the Wilson chamber were investigated by Loufridge and Trublud (1934).
In 1941 Beck carried out a series of experiments for the purpose of experimentally determining the optimal composition of the mixture, i.e., the composition giving small expansion, an insignificant fog background, and good electron tracks in a chamber filled with air at a pressure of the order of atmospheric pressure. In these experiments a small quantity (about \(5\ \mathrm{cm}^3\)) of a mixture of water and alcohol in various concentrations was introduced into the chamber. The expansion necessary for obtaining tracks in general, tracks of the best quality, and the onset of the appearance of a continuous fog were determined. The best tracks were considered to be those in which the droplets were present in considerable quantity and were well developed. At the same time the fog background had to be practically imperceptible.
Beck established that a mixture of two pure alcohols is unsuitable. In comparison with a water–alcohol mixture, it requires a greater
expansion, but gives less sharp tracks and a stronger fog background. Beck’s results, obtained by him for various concentrations of the water–alcohol mixture, are reproduced in Fig. 4. His investigations show that, when air is used as the noncondensing gas, the conditions for forming the best tracks are obtained at a minimum expansion equal to 1.125, and with a mixture consisting of 65% ethyl alcohol and 35% water. Still better results are obtained if a mixture consisting of 50% ethyl or normal propyl alcohol, 25% acetone, and 25% water is used. In this case the expansion is equal to 1.112. The presence of acetone increases the contrast of the tracks relative to the fog background.
Fig. 4. Dependence of the expansion required for obtaining good tracks on the concentration of a mixture of water and alcohol.
§ 7. Growth of droplets
In this section we shall consider the process (beginning with molecules or ions) of the formation of fog or rain droplets and their subsequent growth to a size visible to the naked eye.
In Fig. 5, borrowed from Simpson’s report (1941), particle sizes of interest for the present question are indicated. This figure shows a logarithmic scale of lengths, on which, for comparison, the positions of such well-known points are marked as the point corresponding to the limit of visibility with the naked eye \((5 \cdot 10^{-3}\ \mathrm{cm})\), the limit of the resolving power of the microscope \((2 \cdot 10^{-5}\ \mathrm{cm})\), and also the range of wavelengths of visible light, extending from \(4 \cdot 10^{-5}\) to \(7 \cdot 10^{-5}\ \mathrm{cm}\).
The size of fog particles and raindrops, which varies within rather wide limits, is marked by a vertical line or by brackets embracing the entire region within which the diameter may vary. The arrows on the vertical line indicate that the diameter of fog particles may go beyond these limits. The limits indicated in Fig. 5 for the sizes of fog particles from \(2 \cdot 10^{-3}\) to \(4 \cdot 10^{-4}\ \mathrm{cm}\) refer to particles of the most frequently encountered sizes. However, alongside them there are fog particles of smaller and larger diameters, so that the region indicated above in fact extends to the sizes of an ion on one side and the sizes of raindrops on the other. On the right-hand side of the figure two tables are given. In the upper of them are indicated the supersaturations necessary for condensation to occur on centers having siz—
measures indicated in the same line. In the lower table is given the falling velocity of droplets of different sizes, calculated according to Stokes’ law.
Particles of greatest interest have the following sizes:
| Particle type | Size |
|---|---|
| Molecules | \(10^{-8}\) cm |
| Small ions | \(10^{-7}\) ” |
| Ions of medium size | \(10^{-6}\) ” |
| The largest ions | \(10^{-5}\) ” |
| Fog particles | \(10^{-4}\)—\(10^{-3}\) cm |
| Rain droplets | \(10^{-2}\)—\(10^{-1}\) ” |
According to the views of J. J. Thomson, set forth in the preceding paragraphs, a space saturated with vapor must always contain water droplets of radius \(5.2\cdot 10^{-8}\) cm or less, which arise and again rapidly evaporate. The initial physical process leading to the formation of these droplets is still not entirely clear. The radius of the water molecule, found from measurements of viscosity, has the value \(2\cdot 10^{-8}\) cm. A water droplet of the dimensions supposed by Thomson requires the combination (coalescence) of about \(10^4\) molecules. This number may be compared with the number of collisions experienced by one molecule, equal (at normal pressure and temperature) to approximately \(7.1\cdot 10^9\) per sec. Bearing in mind that the thermal velocity of a water molecule under normal conditions is \(6.15\cdot 10^4\) cm/sec, it is difficult to imagine a process by which so large a number of molecules would associate together as a result of inelastic collisions. Moreover, at such small radii the vapor pressure must considerably exceed the pressure of saturated vapor.
Fig. 5. Relative sizes of molecules, ions, fog particles, and rain droplets.
However, whatever the process of formation of embryonic droplets may be \((r \simeq 5.2\cdot 10^{-8}\ \text{cm})\), they can serve as centers for further condensation, provided only that the supersaturation is greater than four.
As has already been noted earlier, in this region the equilibrium vapor pressure near a droplet decreases as its size increases, as a result of which vapor from the surrounding space diffuses toward the surface of the droplet. The growth of droplets can be calculated with the aid of the diffusion equation
\[ \frac{dm}{dt}=4\pi r^{2}D\frac{d\rho}{dr}, \tag{1,24} \]
where \(D\) is the diffusion coefficient, and \(d\rho/dr\) is the density gradient at a distance \(r\) from the center of the droplet. Instead of \(d\rho/dr\) one may take the approximate value \((\rho_D-\rho_2)/r_0\), assuming that the density changes from the liquid density \(\rho_D\) at the center of the droplet to the density \(\rho_2\) of the saturated vapor at its surface. Under this assumption, taking into account that \(m=(4/3)\pi r^{3}\rho\), equation (1,24) can be rewritten in the following form:
\[ \frac{dr_0^{2}}{dt}=\frac{2D}{\rho}(\rho_D-\rho_2). \tag{1,25} \]
Fig. 6. Growth of droplets. Photograph of droplets falling in an atmosphere of helium and vapors of 95-percent alcohol, taken with intermittent (30 times per second) illumination. [Hazen, Rev. Sci. Instr. 13, 247 (1942).]
Closely connected with equation (1,24) is the equation of latent heat
\[ \lambda \frac{dm}{dt}=4\pi r^{2}K\frac{dT}{dr}, \tag{1,25} \]
where \(\lambda\) is the latent heat, \(K\) is the thermal conductivity, and \(dT/dr\) is the temperature gradient. As in the preceding case, here one may take,
that the temperature changes from \(T_D\) at the center to \(T_2\) at the surface of the drop. \(T_2\) is thus the temperature of the space. Then the equation corresponding to (1,25) will be
\[ \frac{dr_0^2}{dt}=\frac{2K}{\rho\lambda}(T_D-T_2). \tag{1,27} \]
With the aid of the equations given in the first paragraph of this chapter and equations (1,25) and (1,27), one can obtain an expression for the rate of growth of the droplets in terms of quantities already known. Having done this, one can show that \(r_0^2\) varies with time approximately linearly. This result is apparently confirmed by Hazen’s experiments, to the description of which we now turn.
The size of the droplets formed in the Wilson chamber can be determined either with the aid of Stokes’ law (Brode, 1939), or directly by a photographic method. The second of these methods is less reliable because the illumination, the quality of the emulsion, and the diffraction phenomena caused by the lenses affect the character of the image of the droplet.
Hazen determined the rate of growth of the droplets by photographing them during their fall under periodic illumination. A typical photograph is shown in Fig. 6. Taking into account that the magnification is linearly related to the distance measured on the film, he determined the true value of \(r_0^2\) for various time intervals and found that its change with time occurs linearly. For nitrogen at an expansion of 1.15 the value of \(dr_0^2/dt\) is \(5\cdot 10^{-6}\ \mathrm{cm^2/sec}\). This result agrees with the calculated data. However, in the case of hydrogen the calculated values of \(dr_0^2/dt\) and the values of this quantity obtained from experiment do not agree with one another.
Table X
Temperature of droplets in vapors of 95% alcohol.
Total pressure \(1.1\text{–}1.2\ \mathrm{atm}\)
| Gas | \(N_2\) | \(H_2\) | He |
|---|---|---|---|
| Expansion | 1.16 | 1.15 | 1.10 |
| \(T_D-T_2\) (1,27) | 10 | 4 | 4 |
| \(T_1-T_2\) (adiabatic) | 15.5 | 15 | 15 |
From the measured value of \(dr_0^2/dt\), using formula (1,27), one can calculate the temperature of the droplet. In Table X, compiled by Hazen, the values of \(T_D-T_2\) found in this way are compared with the adiabatic lowering of the temperature \(T_1-T_2\).
From the table given it is evident that the temperature of the droplet is always somewhat higher than the temperature of the surrounding gas. The greatest temperature difference is observed for nitrogen.
II. VARIOUS TYPES OF WILSON CHAMBER
§ 1. Early designs of the Wilson chamber
The design of C. T. R. Wilson’s first chamber (Fig. 7) is described in detail in many manuals. In this chapter, therefore, we shall consider only the various modifications of this instrument that appeared after
Fig. 7. Wilson chamber of the original design:
$AB$ — a closed cylindrical chamber.
$B$ — a movable piston sliding inside the cylinder.
$F$ — a rubber gasket fastened to a brass disk, stopping the motion of piston $B$ during expansion.
$D$ — an evacuated vessel connected with the volume under $B$ by means of tap $C$.
$WW$ — wooden liners used to reduce the volume under $B$.
$I$ — a clamp connecting the lower volume of the chamber with the atmosphere. When it is opened, piston $B$ rises to its initial position.
$J$ — a clamp by means of which the initial position of the piston is regulated and thus the magnitude of the expansion is selected.
$K$ — a battery from which a voltage is applied to the chamber in order to remove ions present in it before expansion.
Wilson’s first original work and which had the purpose of adapting the chamber for the study of special types of phenomena.
One of the shortcomings of Wilson’s chamber, in its original form, is the relatively large expenditure of time required to obtain a single photograph. The very small probability of nuclear processes leads to the fact that, in order to obtain a photograph of the phenomenon under investigation, it is often necessary to make a considerable number of expansions. Blackett, for example, indicates (1925) that among the $10^6$ photographs of $\alpha$-particle tracks in nitrogen obtained by him, the process of capture of this particle by a nitrogen nucleus and the emission of a proton was observed in only 20 cases. It is obvious that photographing such rare phenomena requires a device that would permit a considera-
but to reduce the interval of time between repeated expansions. The earliest construction of a chamber of this type belongs to Shimizu (1921). In this construction the motion of the chamber piston occurs continuously, producing successive compressions and expansions.
Shimizu also developed a method for simultaneously obtaining two images of tracks by photographing from different directions that form an angle of 90° with each other. Subsequent authors, including Blackett, Harkins and Ryan, and Auger and Perrin, applied Shimizu’s method to the study of the complex problem of collisions occurring during the passage of $\alpha$- and $\beta$-particles.
Blackett showed that although Shimizu’s chamber is very effective for continuous photography of $\alpha$-particle tracks, the tracks obtained do not have the same sharpness as in the Wilson chamber in its original form. To obtain sharp tracks it is necessary for the expansion to take place very rapidly, which is precisely what is characteristic of the first Wilson chamber.
In order to combine this advantage of the original chamber design with the possibility of obtaining photographs more frequently, Blackett (1927) somewhat modified Shimizu’s device. In this version of Shimizu’s chamber, the sharpness of the expansion is achieved by a jerky motion of the piston under the action of a spring. In 1927 and 1929 Blackett made other improvements to the chamber-control mechanism as well, which enabled him to obtain up to 1270 photographs daily, on each of which up to 20 tracks of $\alpha$-particles were recorded [Blackett and Lee (1931, 1932)].
An essential feature of all chambers used before 1933 is that the specified expansion is produced in them by a sudden motion of the piston forming the bottom of the chamber. The stop occurs when the piston comes into contact with the base of the chamber, so that after the expansion is completed the volume of the chamber remains constant, while the pressure in it increases somewhat owing to the rise in temperature (see III, § 5). In all these chambers, the necessity of using water or oil to create seals permits their use only in a horizontal position.
Wilson (1933) was the first to introduce an important change in the method of producing the expansion. In the construction he proposed, the chamber has a fixed bottom formed by a stretched wire mesh. Below the mesh there is a rubber diaphragm, under which compressed air is supplied. The expansion is produced by releasing the air from under the diaphragm into the atmosphere or into a large vessel, the final pressure in which can be regulated at will. As a result of the reduction of pressure beneath the diaphragm, the latter descends, thereby producing an expansion of the working volume of the chamber.
A chamber with a diaphragm is simpler in construction and can be used both in a horizontal and in a vertical position. In addition, in a chamber of this type the increase in temperature of the working volume,
occurring as a result of condensation and heating by the walls, does not proceed as rapidly as in chambers of the earlier type. The elasticity of the diaphragm hinders abrupt changes in pressure; therefore the state of the necessary supersaturation is preserved after expansion for a longer interval of time. Almost all chambers now being manufactured belong to this type.
We shall dwell very briefly on other modifications of the chamber proposed by various authors with the aim of eliminating certain specific difficulties. Owing to the presence of very small openings, after several days of operation a chamber usually requires adjustment of the magnitude of the expansion. To eliminate this shortcoming, Dahl, Hafstad, and Tuve (1933) constructed a hermetic chamber, using sylphons for this purpose. In their design the sylphons are part of the chamber and can be compressed or stretched mechanically or by means of compressed gas. Variants of a chamber of this type were developed by Dempster (1934), Brubaker and Bonner (1935), and Crane (1937), and they were used mainly for the investigation of nuclear disintegrations. Experience shows that in these chambers fewer vortices are formed than in the original piston chambers or in later chambers with a rubber diaphragm.
In 1935 C. T. R. Wilson and J. G. Wilson proposed another, extremely interesting, chamber design in which the expansion occurs in the radial direction. In this design the expansion is accomplished by sharply lowering the pressure in the annular space surrounding the chamber, which communicates with the volume of the chamber itself by means of slits of suitable size and shape located in its cylindrical part. The pressure in the annular space is lowered either by connecting it with the surrounding atmosphere, or with the aid of a device with a rubber diaphragm. A chamber with radial expansion permits illumination from the side of the bottom (which in this case must be made of flat glass), which considerably facilitates the problem of illumination (see III, § 1). It also makes it possible to place sources or targets near its axis without substantially disturbing the regime of its operation. The advantages of the chamber with radial expansion were also examined by Trew (1938).
At the same time C. T. R. Wilson and J. G. Wilson developed (1935) a method of working with the so-called “falling chamber,” consisting in the fact that the chamber and the camera rigidly connected with it begin free fall immediately after expansion, during which all subsequent operations are performed—illumination, photography, etc. The advantage of the falling-type chamber is that it eliminates the distorting traces of the action of gravity, since in it there is no displacement of droplets relative to the gas, and the influence of convection is minimal. The interval of time between the moment of expansion and photography can be
increased without risk that the tracks may be distorted as a result of the fall of droplets. In addition, the exposure can be increased by using light sources that make it possible to illuminate for the desired duration. With regard to this chamber it should be noted that, owing to mechanical difficulties, it has not found application.
Locher described (1933) a rectangular chamber with which he photographed showers of cosmic rays. This chamber found only limited use, because the distortions caused by eddies appear in it more strongly than in a chamber of cylindrical form.
Among other modifications of the Wilson chamber one should mention the chamber that continuously maintains the state of sensitivity. One attempt to develop such a chamber was made by Langsdorf (1939), who used the diffusion of heated saturated vapors through a non-condensing gas into a space maintained at a low temperature. The diffusion took place in the vertical direction between a heated cover and a cooled bottom, with the vapor coming from a liquid contained in a glass vessel heated from above. However, obtaining sharp tracks by means of such a chamber encounters great difficulties. Another method of producing a state of continuous supersaturation, proposed by Vollrath (1936), is based on the counter-diffusion of hydrochloric-acid vapor and water vapor. Trey (1940) attempted to solve the same problem by cooling vapors through thermal conduction. Finally, Brinkman described a chamber giving several expansions per second.
The designs of portable chambers of low weight were described by Locher (1933), Livingston (1936), Bauer (1936), Ratenau (1938), Hilsch (1939), Thomas and Ramsay (1939), Kuntze (1941), and Herzog (1941). Wilson chambers for demonstration purposes were described by Herzog (1937), Hilsch (1939), Livingston (1936), and Ratenau (1938).
§ 2. Chamber Controlled by Counters
In the study of nuclear phenomena, when strong sources can be used, obtaining many tracks of α- or β-particles on each photograph presents no difficulty. In the study of rare phenomena, however—for example, processes caused by cosmic rays—the use of a chamber whose expansions are produced at random is disadvantageous. The flux of cosmic particles falling on the chamber is negligible, and therefore many of the photographs will contain no tracks at all.
Blekett and Occhialini (1933), and almost simultaneously Anderson (1933), developed a method for controlling the expansion of a chamber by means of counters. In a chamber controlled by counters, the expansions are not produced at random, but occur only when a cosmic particle passes through two counters, one of which is placed above the chamber and the other below it. The passage of the particle through both coun—
Fig. 8. Mechanism for controlling the chamber, monitored by counters:
\(V_1\) — expansion valve.
\(M_3\) — rod, rigidly connected with plate \(V_1\) and resting on one arm of the T-shaped metal part \(X_3\), rotating about \(Q\). When \(X_3\) is moved to position \(X'_3\), \(M_3\) slips off, producing expansion.
\(Th\) — relay \(G_1T1C\), connected (through \(S_2\) and \(S_4\)) in parallel with magnet \(U\). When a pulse from the counters arrives at the thyrotron grid, the relay short-circuits (shunts) the electromagnet.
\(U\) — electromagnet holding the armature in position \(X_5\). When the current in the electromagnet ceases, the magnet armature moves from position \(X_5\) to position \(X'_5\). Simultaneously with this, arm \(X_3\) moves to position \(X'_3\), and expansion occurs.
\(H_3\) — spring pulling back \(X_5\) when the current through the magnet ceases.
\(X_4\) — brass part by means of which, when the armature is moved from position \(X_5\),
The chamber is ready for expansion. As soon as a pulse from the coincidence circuit reaches the grid of the thyratron, the thyratron fires, and the valve \(V_1\) opens. Expansion takes place and, at the same time, a flash of light occurs. Since the lens shutter is open the whole time, photographing takes place. As soon as expansion has occurred, \(S_3\) switches on the motor, and the cam mechanism \((K_1, K_2,\) and \(K_3)\) brings the chamber into readiness for the next expansion. The film is advanced automatically by means of a mechanical linkage with the roller \(J\). Starting and stopping of the compressor are also carried out automatically, as soon as the air pressure in the supply reservoir becomes higher or lower than prescribed limits, set as desired.
Fig. 9. Electrical control circuit of a chamber controlled by counters.
to position \(X_5'\), the rod \(X_6\) is given a longitudinal motion. When \(X_6\) moves, the contacts at \(S_3\) and \(S_5\) close, while at \(S_1\) and \(S_6\) they open.
\(S_5\) — key closing the circuit of \(C_1Z_1\). As a result of the discharge of capacitor \(C_1\), relay \(Z_1\) operates and in turn closes the circuit \(N_3N_4Z_2\), short-circuiting the resistance \(R_2\), owing to which the full voltage is applied to the lamp. The duration of illumination is determined by the discharge time of capacitor \(C_1\) through the inductive load \(Z_1\).
\(R_1\) and \(R_2\) — resistances regulating the current through the lamp \((R_1 \ll R_2)\).
\(S_3\) — contacts for switching on motor \(M\), which rotates the shaft \(J\) with the cams \(K_1, K_2,\) and \(K_3\) mounted on it.
\(K_2\) — cam which returns valve \(V_1\) and the armature of magnet \(U\) to their initial position. At the moment when, during rotation of the cam, point 2 comes opposite 1, the arm \(X_3\) transmits (through \(H_2, X_1\), and \(X_2\)) a push returning \(X_3\) to its former position relative to \(M_3\). The valve closes and is held in this position by magnet \(U\). At the same time, the lever \(X_4\) pulls the rod \(X_6\) to the left, closing contacts \(S_4\) (thyratron) and \(S_6\) (charging of the capacitor).
\(K_1\) — cam. As the shaft \(J\) rotates, \(K\) first comes to position 2 and begins to admit air through \(V_2\) and \(O_2\). When \(K_1\) is in position 3, air exits through \(O_2\). A supplementary expansion takes place, removing the rest of the ions in order to clean the chamber before the principal expansion. When \(K\) is in position 4, air is admitted again; in position 5, \(V_2\) is completely open to the compressor installation. Finally, when \(K\) reaches position 1, communication between the compressor and the chamber is interrupted.
\(K_3\) — cam with rod \(M_2\), which, at the moment when \(K_1\) and \(K_2\) are in positions 1, presses on \(X_7\) and opens the contacts at \(S_1\) (motor) and closes those at \(S_2\) (anode of the thyratron).
Fig. 10. Vertical section of the chamber through the axis of the cylinder:
\(G_1\) — glass cylinder.
\(G_2\) — thick glass plate covering the cylinder, through which photography is performed.
\(R_1, R_2, R_3\) — rubber seals.
\(R\) — rubber diaphragm, by the motion of which compression and expansion of the volume of the chamber occur.
\(W\) — wire mesh used to reduce turbulence.
\(T\) — black velvet moistened with alcohol. It creates a black background and reduces turbulence.
\(P\) — brass disk, fastened by means of a rubber ring and used as a piston.
\(M_1\) — rod screwed into plate \(P\). When the piston moves it slides in tube \(E\).
\(N\) — stop device, rigidly connected with tube \(E\), and specifying the maximum forward motion of \(P\) in compression.
\(E\) — brass tube with an external thread. By screwing it into or out of tube \(F\), its position and the position of the stop can be changed; this changes the magnitude of the compression.
\(O_2\) — opening for admitting air under the piston. By means of valve \(V_2\), shown in Fig. 10, compressed air is supplied through this opening.
\(O_1\) — opening for releasing air during expansion. In the uppermost position of the valve it is closed.
\(V_1\) — expansion valve. When this valve rebounds, air exits through openings \(O_1\) and \(O_2\).
\(O_4\) — opening for admitting the water–alcohol mixture. During operation of the chamber it is closed.
\(O_5\) — opening for filling the working volume of the chamber with gas. During operation it is also closed.
\(O_6\) — opening in brass part \(B\), connecting the space between the rubber diaphragm and the piston with the atmosphere. It facilitates the forward and reverse motion of the piston.
counter triggers a series of relays, as a result of which expansion and photographing take place. The complete process of obtaining a single photograph proceeds in the following sequence:
1) passage of the ionizing particle through the upper counter, the chamber, and the lower counter,
2) expansion of the chamber,
3) switching on the light and illuminating the droplets,
4) exposure, lasting until the droplet sizes become sufficiently large, and
5) bringing the chamber into readiness for the next expansion.
The operations listed take place each time a coincident discharge occurs in the counters. Blackett and Occhialini indicate that 80% of the photographs obtained by this method contain traces of cosmic particles, and in most of them showers consisting of many particles are recorded.
The operation of the various mechanisms of this chamber is clear from the diagrams shown in Figs. 8, 9, and 10.
Various types of valve circuits for monitoring the operation of the chamber, and also for establishing the desired sequence of operation of its individual mechanisms, were described by Richardson (1938), Berendett and Sizu (1939), Getting (1939), Street and Stevenson (1936), and Jones (1937).
§ 3. An Uncontrolled Chamber with Increased Sensitivity Time
A Wilson chamber controlled by counters considerably reduces the consumption of film and, especially, the time spent on carrying out an investigation. However, as was noted above, this chamber was developed chiefly for solving certain narrow problems connected with the study of cosmic rays, and therefore it has certain shortcomings.
First, the controlled chamber possesses a certain selectivity, as a result of which the results obtained with it do not reflect the true relative frequency of occurrence of particles of different kinds and different energies. In particular, compared with the true statistical distribution, it records an increased number of shower particles and, conversely, a reduced number of particles of low energy. The second shortcoming is connected with obtaining a magnetic field for measuring the energy of the recorded particles. A controlled chamber requires the use either of a permanent magnet, in which case it is impossible to obtain a sufficiently strong magnetic field, or of an electromagnet. In the latter case the magnet must be switched on all the time, which, besides a considerable increase in energy consumption, creates great difficulties connected with regulating the temperature of the chamber.
Obviously, the way out of the situation consists in developing such a chamber which, operating automatically, would ensure the poss—
possibility of obtaining several tracks at each forced expansion. Such a chamber would give a true picture of the statistical distribution, and the magnetic field could be switched on one or two seconds before the expansion and switched off after its completion.
In 1935 Berden showed that the number of tracks recorded in one expansion can be increased by increasing the sensitive time of the chamber (see III, § 5). Berden worked with a siphon-type chamber, operated by compressed air and having dimensions of 20 cm in diameter and 4 cm in depth. By artificially slowing the expansion of the chamber (by a corresponding change in the aperture of the valve), he increased the sensitive time of the chamber to two seconds, thanks to which he could obtain on average four tracks of cosmic particles in each photograph. A construction essentially similar in principle was also proposed by Frisch (1935). In Berden’s chamber the bottom is the surface of a liquid; therefore it can work only in a horizontal position and, consequently, is not adapted for the investigation of cosmic rays. It should also be noted that in a chamber with an increased sensitive time the tracks are obtained more blurred.
The control mechanisms of an uncontrolled chamber are analogous to the corresponding mechanisms of a chamber controlled by counters. The electromagnet controlling the chamber valve is switched on at definite time intervals by means of a clockwork mechanism. By this time the magnetic field used to deflect the particles, which is switched on several seconds earlier, has time to become established. After the expansion is completed it is switched off, and the chamber is brought back to its initial position.
Williams (1939a) showed that the sensitive time of a chamber can be increased by increasing its depth. The chamber constructed by him (with a rubber diaphragm) has a depth of 30 cm with a diameter of 30 cm and is connected to a Helmholtz-type coil producing a magnetic field of 2200 oersteds. This chamber gives 2–3 tracks in each photograph and can be used in both the horizontal and the vertical position. In comparison with Berden’s chamber it has the further advantage that, owing to its greater depth, it permits the use of the oblique illumination method (see III, § 1). Later Williams and Roberts (1940) constructed a chamber of still larger dimensions—60 cm in diameter and 50 cm deep—with the aid of which they succeeded in photographing the decay of a mesotron. The long sensitive time and considerable volume make this chamber a very sensitive detector of radioactive radiations. For these purposes it was used by Wollan, Williams, and Evans (1939).
Among other chambers of the indicated type one should mention the Meyer-Leibnitz chamber (1939), in which the sensitive time reached one second; the deep chamber of Hazen (1942), with a sensitive time of about 0.5 sec.; and the large chamber of Herzog (1935).
It is essential to note that the productivity of the chamber (the number of tracks per one expansion) is directly related to its geometrical dimensions and to the magnitude of the sensitive time. The average number of tracks of cosmic particles recorded on one photograph is equal to \(j d x t_s\), where \(j\) is the number of cosmic particles falling on \(\text{cm}^2\) per second, \(d\) is the diameter of the volume being photographed, \(x\) is the depth of focus of the objective used, and \(t_s\) is the sensitive time of the chamber. In Williams’s chamber \(d \simeq 24\ \text{cm}\), \(x \simeq 5\ \text{cm}\), and \(t_s \simeq 0.4\ \text{sec}\). Since \(j \simeq 0.03\), the average number of tracks recorded on one photograph is approximately 1.5.
§ 4. Low- and High-Pressure Chambers
In certain cases—for example, when photographing tracks of fission products—chambers with reduced pressure in the working volume are used. Chambers of this kind are subject to great requirements with respect to their airtightness. The lower volume of such a chamber is connected to a vessel whose pressure is somewhat higher than that in the working volume. Expansion is produced by connecting the lower volume with an evacuated vessel. Such a chamber, operating at various pressures, was described by Joliot (1934). Low-pressure chambers play an important role in the study of fission products and other particles having a short range—of the order of several millimeters of air under normal conditions.
Recently, another major step has been taken in the development of the Wilson chamber method: a chamber has been constructed that operates at ultrahigh pressures. The usual method for studying the interaction of high-energy particles (cosmic rays) with matter consists in observing the secondary radiation emerging from a layer of dense matter after the process under study has occurred somewhere in the depth of this layer. Since the probability of the processes under investigation is usually very small, in order to obtain any appreciable results one has to take considerable thicknesses of materials. The disadvantage of this method is that we observe secondary or tertiary particles formed in the thickness of the material, and not the primary process itself, about which this method does not give a complete picture. The phenomenon can be interpreted fully only in the case where it takes place in the gas of the chamber, when it is possible to measure angles and energies with the necessary accuracy.
Unfortunately, owing to the low absorbing power of gas, the probability of observing such events is extremely small. One of these rare events, of particular interest, is the decay of the mesotron at the end of its range. Weighty evidence for the existence of this process was obtained thanks only to two or three photographs taken with the aid of a Wilson chamber. Another, equally rare event is represented by explosive showers of the Heisenberg type.
For the investigation of these phenomena it is necessary to increase the probability of observing them in the gas of the chamber, which can be accomplished either by increasing the volume of the chamber or by increasing the pressure. In the latter case, the increase in the probability of observation occurs as a result of an increase in the number of acts of interaction per unit path of the particle. Since increasing the geometrical dimensions of the chamber creates difficulties connected with photography, a satisfactory solution of this problem can be obtained by increasing the pressure. A high-pressure chamber is also of special interest for purposes of studying nuclear processes caused by artificially accelerated particles, in view of the fact that the improvement of cyclotrons and betatrons makes it possible to obtain these particles with ever greater energy.
The first high-pressure chamber intended for measuring neutron energies was built by Mott-Smith (1934). It consists of two volumes separated by a rubber diaphragm. The upper of them is the working volume, while the lower is connected to the expansion device. The chamber body is made of a strong brass ring with a thick glass window provided in it for illumination. The upper cover is made of a quartz plate 2.5 cm thick, capable of withstanding a pressure of up to 50 atm. Work with this chamber was carried out at 15 atm. Brubaker and Bonner developed (1935) a chamber (of the sylphon type), operating at a pressure of 25 atm, which functioned fully automatically. Kipper (1935) constructed a two-centimeter chamber in which the pressure could be brought up to 100 atm. Williams and Evans (1940) also built a chamber operating at 80 atm. Their chamber had a depth of 16 cm with a diameter of 20 cm and was filled with argon.
Recently Johnson, Benedetti, and Shutt built a chamber (diameter 30 cm, depth 9 cm) of a new design, calculated for operation at 200, and possibly also at 300 atm (Fig. 11). The glass cylinder of this chamber has a thickness of 6 mm, and the front cover, of the same diameter as the cylinder, is 9 mm. In a chamber of this design the high pressure is created in a cast steel vessel filled with transparent oil. The chamber proper is placed inside this vessel, so that its walls do not experience a large pressure difference.
The shape of the steel vessel makes it possible to use it as the yoke of an electromagnet capable of creating a strong magnetic field inside the chamber. The volume of the available space permits the placement in it of up to one ton of copper wire, which makes it possible to obtain a field of the order of 10,000 oersteds without serious heating of the chamber.
The bottom of the chamber is formed by a movable disk, which is joined to the cylindrical walls of the chamber by means of a diaphragm of synthetic rubber. Expansion is produced by releasing from the vessel a certain amount of oil through a high-pressure valve controlled by compressed air of low pressure (~3 kg/cm²), acting on the large surface (diameter 13 dm) of an aluminum piston.
The chamber is filled with argon saturated with a mixture of water vapor and isopropyl alcohol. As the pressure is increased, the expansion required to obtain good tracks decreases; therefore, in a high-pressure chamber fewer vortices are formed. The expansion, which at atmospheric pressure is 1.07–1.08 (for the vapor mixture used), at 110 atm decreases to 1.04 (see I, § 16). The maximum increase in the volume of this chamber is about 20% of the initial volume. It is illuminated by a parallel beam from a capillary arc placed inside a steel vessel at the focus of a cylindrical parabolic reflector. Photography is carried out through a small-diameter window made of glass five centimeters thick, in order to withstand the full pressure. Owing to the greater ionization density, the illumination requirements for a high-pressure chamber are somewhat lower than for ordinary chambers operating at atmospheric pressure. However, the noted advantage of the high-pressure chamber is neutralized by the absorption of light in the oil.
Fig. 11. Schematic drawing of the high-pressure chamber of Johnson, Shutt, and Benedetti (Rev. Sci. Instz. 14, 265 (1943)).
The principal advantage of the chamber described, as compared with Blackett’s chamber, is that in a given time it makes it possible to observe a greater length of cosmic-ray tracks. At 200 atmospheres the effective length of a track passing along the diameter of the chamber is equivalent to 60 meters of path in argon under normal conditions. Thus, the high-pressure chamber can register the complete “history” of a particle over so long a path. Thanks to
for a longer period, during which the state of supersaturation is maintained in the chamber; its sensitive time (see III, § 5), estimated from good tracks, is almost 10 times greater than that of chambers of the ordinary type.
A disadvantage of the high-pressure chamber is the necessity of a comparatively long pause (of the order of 15 min.) between expansions. This interval is needed for the dissipation of the heat liberated during compression of the gas in the working volume of the chamber.
III. FACTORS AFFECTING THE QUALITY OF TRACKS
§ 1. Illumination
In practice it often happens that tracks which are clearly visible to the eye nevertheless do not appear in the photograph. The reasons for this may be:
a) insufficient illumination, b) an incorrect choice of magnification, or c) a small depth of focus of the camera lens.
Let us consider an object having area \(S\), situated normally to the axis of the lens at a distance \(u\) from its optical center. If \(a\) is the diameter of the diaphragm used, then the total luminous flux of the source passing through the lens is approximately equal to \((IS/u^2)\cdot(\pi a^2/4)\), where \(I\) is a quantity proportional to the brightness of the object under consideration. Denoting by \(S'\) the area of the object in the image plane and neglecting losses due to reflection and scattering in the lens, one may write the following expression for the illumination in the image plane:
\[ I_{S'}=\frac{I}{u^2}\cdot\frac{S}{S'}\frac{\pi a^2}{4} \tag{III, 1} \]
or, introducing the factor \(m=v/u=(S'/S)^{1/2}\), characterizing the linear magnification,
\[ I_{S'}=\frac{\pi I}{4}\frac{A^2}{(1+m)^2}, \tag{III, 2} \]
where \(A=a/f\) is the relative aperture of the lens, and \(v=f(1+m)\). Consequently, in order to increase \(I_{S'}\), it is necessary to increase \(I\) and \(A\) and to decrease \(m\). The method of increasing \(I\) with a minimum expenditure of power will be considered later. We shall now consider the question of the maximum permissible value of \(A\), or \(a/f\), which, as the subsequent discussion will show, is limited by the requirement of having a definite depth of focus.
If \(u\) and \(v\), respectively, are the distances of the object and the image, and \(f\) is the focal length of the lens, then a change of \(u\) by \(\delta u\) will cause a change of \(v\) by \(\delta v\), determined by the expression:
\[ \frac{\delta v}{\delta u}=-\frac{v^2}{u^2}=-m^2 . \tag{III, 3} \]
Since the change \(\delta v\) is accompanied by a corresponding change in the path difference of the marginal and central rays, equal to \((a^2/\delta)\cdot(\delta v/v^2)\)*, the condition for obtaining a sufficiently perfect image, requiring that the indicated difference not exceed \(\lambda/4\), gives
\[ \frac{\lambda}{4}=\frac{a^2\delta v}{8v^2} \tag{III, 4} \]
and
\[ \delta u=\frac{\delta v}{m^2}=\frac{2\lambda}{A^2}\left(\frac{1+m}{m}\right)^2 . \tag{III, 5} \]
Equations (III, 5) and (III, 2) show that, as \(A\) increases, the depth of focus \(\delta u\) decreases, whereas the illumination \(I_{SI}\) increases. Consequently, the choice of the magnitude \(A\) is determined by a compromise. Usually the relative aperture of the objective is chosen equal to some critical value \(A_c\), consistent with the required depth of focus, the magnitude \(A_c\) being given by the expression:
\[ A_c=\left(1+\frac{1}{m}\right)(2\lambda/\delta u)^{1/2}. \tag{III, 6} \]
Thus, if \(m=1/12\), \(\lambda=4.4\cdot10^{-5}\ \text{cm}\) and \(\delta u=5\ \text{cm}\), then \(A_c=1/18.3\), and, consequently, the aperture of the diaphragm must not exceed \(f:18.3\). At the critical value of the relative aperture \(A_c\) we have:
\[ I_{SI}=\frac{\pi I}{4}\cdot\frac{(1+m)^2}{m^2}\cdot\frac{2\lambda}{\delta u}\cdot\frac{1}{(1+m)^2} =\frac{\pi I\lambda}{2m^2\delta u}, \tag{III, 7} \]
from which it is seen that, for a given value of \(\delta u\), the quantity \(I_{SI}\) increases as \(m\) decreases; moreover, the smaller \(m\) is, the easier it is to obtain not only better image sharpness, but also a larger number of tracks per unit magnification. The smallest value of \(m\) that can be used is limited by the resolving power (the maximum number of strokes per one mm, reproduced separately) of the photographic emulsion employed. The practically usable minimum value of \(m\) is such a value at which the width of the image of the track is equal to the width of the finest line reproducible on the film. For very sensitive emulsions this quantity is of the order of \(20\ \mu\).
The width of the image of a track may be taken equal to the sum of the width \(\varepsilon\) of the diffraction pattern obtained from a linear source and the width \(D\) of the geometrical image of the track.
For the first of these quantities we have:
\[ \varepsilon=\frac{2v\lambda}{a}=2\lambda(1+m)/A, \tag{III, 8} \]
) Rayleigh, Collected Papers* (Cambr. Univer. Press, England, 1920), vol. I, p. 415.
and for the second, \(mD_0\), where \(D_0\) is the track width. Thus the total width of the image of the track \(\Delta\) is equal to:
\[ \Delta = mD_0 + 2\frac{\lambda}{A}(1+m) = m\left[D_0 + (2\lambda\delta u)^{1/2}\right], \tag{III, 9} \]
where for \(A\) its critical value (III, 6) has been used. Taking \(D_0 = 0.1\ \mathrm{mm}\), \(\lambda = 4.4 \cdot 10^{-5}\ \mathrm{cm}\), and \(\delta u = 5\ \mathrm{cm}\), we obtain \(\Delta = m \cdot 310\ \mu\). For Kodak Super-XX film \(\Delta\) is approximately equal to 20, and therefore the smallest usable value of \(m\) will be \(m \geq 20/310 = 1/15.5\). If, instead of \(m = 1/12\), this value is taken, then as a result we obtain either an increase in the depth of focus (III, 5) or an increase in the illumination of the image (III, 2). As a rule, it is recommended to use film of the greatest sensitivity, which makes it possible, by using small \(A\), to increase the depth of focus and thereby increase the number of tracks observed in a single expansion of the chamber. When working with reduced-pressure chambers, in which, owing to the smaller specific ionization, the tracks are weaker, the advantage of small magnifications is preferable to use not for increasing the depth of focus, but for increasing the illumination of the image. When the value of the relative aperture deviates from the critical value, both in the direction of increase and in the direction of decrease, the width of the image increases [Blackett (1929, 6)]. In the first case this occurs because of the imperfection of the focusing action of the objective, in the second because of the increase in the diffraction circle (Airy disk). It is therefore advantageous to use the critical value of the relative aperture.
We shall now consider precisely those methods of increasing \(I_{SI}\) which can be carried out only by increasing \(I\), while leaving all other conditions unchanged. Since increasing \(I\) does not affect the depth of focus, it is desirable to obtain the greatest possible \(I\) with the minimum expenditure of energy. By applying strong illumination of the tracks, one can make \(I_{SI}\) greater than is required for normal exposure, and then, by stopping down the objective, increase the depth of focus. This will increase the productivity of the chamber and at the same time improve the quality of the image.
For illuminating the chamber Blackett (1934) used the spark discharge of condensers in a quartz mercury lamp. Later he switched to illumination by means of a capillary mercury tube, connecting it directly into the secondary winding of an 8000-volt transformer. Through the primary winding, for a short interval of time (from 0.02 to 0.05 sec.), a current of the order of 100–200 amperes from a 220-volt mains supply was passed. The resulting flash gave illumination quite sufficient for photographing individual droplets at right angles to the direction of illumination.
Kren (1937) used 1000-watt (110-volt) “Mazda” projection lamps equipped with parabolic mirrors for illumination. At the moment of photographing, the voltage on the lamps was raised to 220 volts. Such illumination proved sufficient for obtaining
images on Super-X film with an exposure of 0.2 sec. and a lens aperture \(f:1.9\).
The tracks of charged particles become visible owing to the scattering by droplets of the light with which they are illuminated. Webb experimentally established (1935) that the intensity of light scattered by droplets increases rapidly as the scattering angle decreases (see Fig. 12). At an angle of \(20^\circ\) to the direction of the illuminating beam, water droplets scatter light approximately 100 times more strongly than at an angle of \(90^\circ\); droplets formed in a mixture of water and alcohol (taken in equal quantities) give, for the same angles, a ratio of scattered-light intensities of more than fiftyfold. From the figures quoted it is clear that tilting the axis of the camera lens toward the direction of illumination can considerably increase the brightness of the image. When working with chambers of small depth this possibility is absent; therefore photography has to be carried out at right angles to the direction of illumination, which requires strong light sources. Deep chambers, however, permit oblique illumination and, consequently, in this case it is possible to photograph tracks from directions forming angles of less than \(90^\circ\) with the direction of illumination.
Fig. 12. Intensity of light scattered by droplets at different angles to the direction of the illuminating beam.
Williams and Terroux, as early as 1930, worked with a deep chamber. They were therefore able, while keeping the axis of the camera lens parallel to the magnetic field, to reduce the angle between it and the direction of illumination. In their apparatus only 8–10 lamps of 100 watts, equipped with condensers and switched on for a brief moment into a 220-volt circuit, gave sufficient illumination of a chamber 30 cm deep and 30 cm in diameter.
§ 2. Photography
Investigations of nuclear processes and cosmic rays by means of the Wilson chamber require the development of a method of photography that would make it possible to determine the range, curvature, and angles between the directions of emission of various particles from the positions of the tracks of these particles in any planes. For this it is necessary that one and the same
the track was photographed from two different directions. One of the methods used for this purpose, which has become widespread, was described by Shimizu (1921). In this method the photography is carried out from two mutually perpendicular directions with the aid of mirrors \(B_1\) and \(B_2\) (see Fig. 13), placed at right angles to one another and at an angle of \(45^\circ\) to the plane of the chamber \(A\) and of the mirrors \(C_1\) and \(C_2\). The photograph is made with one objective and on the same film. The two images obtained in this way make it possible to calculate the angles of emission of particles and the true length of their path in the chamber [Blackett (1922) and (1923)]. At first sight this method appears advantageous, since it makes it possible to reduce considerably the consumption of film. However, owing to the fact that the planes of the object form a right angle, only a small part of the chamber can be well focused, which substantially reduces the number of tracks recorded in a single expansion.
Fig. 13. Stereoscopic photography with one objective.
To eliminate this drawback, Blackett (1929, b) proposed a more perfect method, based on the use of two objectives. In order to simplify the calculations and to increase the accuracy of determining the angles, photography is carried out on two mutually perpendicular planes \(P\) and \(P'\), as shown in Fig. 14. The objectives \(L\) and \(L'\) are then set at such an angle to the plane of the Wilson chamber that this plane is conjugate with the planes \(P\) and \(P'\). This is the case when the principal plane of each of the objectives passes through the line of intersection of the plane of the Wilson chamber with the corresponding image plane*).
Fig. 14. Stereoscopic photography with two objectives.
Photography by this method makes it possible to obtain fairly sharp photographs of the entire chamber. Since the optical axes of the objectives, passing through the center of the Wilson chamber, form a small angle with the normal to the corresponding image plane, the magnification in different parts of the image is not the same. However, this drawback can be circumvented by using a special method of track analysis, to the description of which we now turn.
*) K. Glazebrooke, Dictionary of Applied Physics (Mac-Millan and Company, London, 1927), vol. IV, p. 400.
The idea of the method is very simple and consists in the fact that the developed negative of the photograph is reprojected back into the plane of the object by means of the same camera with which it was taken [Williams and Terroux (1930), Curtiss (1930)]. It is essential here that the negative be placed in the camera in exactly the same position as it occupied during photographing. The image is then projected onto a thin, translucent white screen; by translational and rotational motion of the screen the two images of the track are brought into coincidence. In this way an exact geometrical reproduction of the track is obtained in the plane in which it was formed, and linear distortions are automatically corrected. Since the same camera is used, and since the position of the negative in it is the same as during photography, this method also eliminates other errors that may arise from poor assembly of the various parts of the camera. By replacing the white screen with photographic paper, one can obtain a photograph of the track in its natural form. A method for processing stereoscopic photographs of tracks was also proposed by Groshev et al. (1936).
In practice, both objectives are usually mounted on a single frame so that the camera can be moved as a whole without fear of changing the relative positions of its parts. Apparatus of this type, adapted for viewing stereoscopic photographs, has been described by Curtiss (1930) and by Jones and Ruark (1940). In order to facilitate the accurate placement of the negative in the position it occupied during exposure, the film is pierced during photography by a special pin, rigidly connected with the camera and controlled by two electromagnets. The placement of the negative in the camera is then reduced to smoothly pulling the film through until the pin drops into the hole previously punched by it. Experience shows that although stereoscopic photography carried out with two objectives requires twice as much film, it is nevertheless more advantageous than photography (also stereoscopic) with one objective, since it gives 4–5 times as many tracks, and all tracks are well focused.
Of course, in those cases where it is necessary only to establish whether some rare process occurs inside the chamber (for example, the formation of delta rays), one may confine oneself to photographing with a single objective (not stereoscopic), placing its principal plane parallel to the plane of the chamber.
§ 3. Sharpness of tracks
When passing through a gas, a fast ionizing particle creates equal numbers of positive and negative ions, forming a narrow column surrounding its path. If recombination and the influence of the electric field are not taken into account, the spreading of the ions will occur only as a result of diffusion, and their distribution
after a time \(\tau\) will be determined by the formula
\[ n(r)=\frac{N_0}{4\pi D\tau}\,e^{-\frac{r^2}{4D\tau}}, \tag{III,10} \]
where \(r\) is the distance from the track, \(N_0\) is the total number of ions per centimeter of track length, and \(D\) is the coefficient of diffusion of the ions.
Suppose that supersaturation is created after a time \(\tau\) following the passage of the charged particle. As soon as this occurs, the ions become covered with water, forming droplets, and their mobility falls to zero. A photograph taken at this moment will depict the projections of the ions onto the plane of the film at the time \(\tau\). Introduce a rectangular coordinate system in which the axis \(OZ\) coincides with the direction of the track, the axis \(OX\) lies in the plane of the photographic plate, and the direction of observation is taken as the axis \(OY\). Then the distribution \(\rho(x)\) of the density of the images of ions (droplets) along the \(x\)-axis (i.e. across the width of the track) is obtained by integrating the quantity \(n(r)\) with respect to \(y\) over the limits from \(+\infty\) to \(-\infty\):
\[ \rho(x)=\frac{N_0}{4\pi D\tau}\int_{-\infty}^{+\infty} e^{-[(x^2+y^2)/4D\tau]}\,dy = \frac{N_0}{(4\pi D\tau)^{1/2}}\,e^{-x^2/4D\tau}. \tag{III, 11} \]
From this expression it is evident that \(\rho(x)\) decreases from the center of the track to its periphery; it is therefore convenient to define the width of the track as the width \(x_1\) of the band within which 90% of the droplet images are contained. The quantity \(x_1\) can be determined from the equality
\[ \frac{ \dfrac{N_0}{(4\pi D\tau)^{1/2}}\displaystyle\int_{0}^{x_1} e^{-x^2/4D\tau}\,dx }{ \dfrac{N_0}{(4\pi D\tau)^{1/2}}\displaystyle\int_{0}^{\infty} e^{-x^2/4D\tau}\,dx } =0.9, \]
which gives
\[ x_1=4.68\,(D\tau)^{1/2}. \tag{III, 12} \]
In air under normal conditions the mean value of \(D\) for positive and negative ions is \(0.034\ \mathrm{cm^2\,sec^{-1}}\)*), so that \(x_1=0.86\,\tau^{1/2}\). Consequently, if the track has a thickness of \(1\ \mathrm{mm}\), the corresponding \(\tau\) is approximately \(1/70\ \mathrm{sec}\).
In a controlled chamber, after the simultaneous discharge in the counters has caused the triggering mechanism to operate, some time is spent on the motion of the piston. Let this time, constituting part of \(\tau\), be equal to \(a\tau\) \((a<1)\). The distance \(d\) traversed by the pis—
) J. J. Thomson, Conduction of Electricity through Gases* (Cambridge University Press, England, 1928), vol. I, p. 77.
in this time, is equal to
\[ d=\frac{1}{2}(SP/m)\cdot(a\tau)^2, \tag{III, 13} \]
where \(S\) is the area of the piston, \(P\) is the pressure difference on its sides, \(m\) is the mass, and \(SP/m\) is the acceleration of the piston. The diffusion coefficient \(D\) is inversely proportional to the pressure, so that one may set \(D=\frac{k}{\rho}\), where \(\rho\) is the density of the gas in the chamber. Eliminating \(D\) and \(\tau\) from equation (III, 12), we obtain:
\[ x_1=5.5\left(\frac{k}{\rho}\right)^{1/2}\left(\frac{d\cdot m}{SPa^2}\right)^{1/4}. \tag{III, 14} \]
This formula shows that, in order to obtain sharp tracks, it is necessary:
1) to use, for filling the chamber, a gas and a mixture of vapors that permit operation at minimum expansions (small \(d\)),
2) to make the chamber piston as light as possible,
3) to operate at the highest possible pressures.
In Table XI the calculated values of \(x_1\) are compared with the values found experimentally (Blackett (1934)).
Table XI
Calculated and experimentally found values of \(x_1\)
| Gas | \(D\) (under normal conditions) | \(D\) (at a pressure of \(1.48\ atm\)) | \(x_1\) calculated (in mm) | \(x_1\) experimental (in mm) |
|---|---|---|---|---|
| O\(_2\) | 0.032 | 0.022 | 0.71 | 0.85 |
| H\(_2\) | 0.135 | 0.092 | 1.42 | 1.78 |
In the calculations the quantity \(a\) was taken equal to unity. Taking into account the difficulties associated with the experimental determination of the quantity \(x_1\), it must be acknowledged that the data in the fourth and fifth columns of the table given agree quite well.
§ 4. Distortions of tracks
Distortions of track images may be caused by the following reasons: a) imperfection of the optical system, b) refraction in the upper glass of the chamber, and c) vortices inside the chamber.
a) Distortions produced by the optical system. These distortions are connected with the necessity of using objectives with a large relative aperture. The use of such objectives is caused, first, by the desire to increase the illumination of the image-
...and, secondly, in those cases where the photographing is carried out through the aperture of the electromagnet pole—the desire to obtain tracks of greater length. Even the highest-quality lenses give distortions, which may be positive in some cases and negative in others. Consequently, if these distortions are not taken into account, the track of a negative particle of very high energy which has undergone some displacement may be attributed to a positive particle of approximately the same energy.
Fig. 15. Distortion of a linear object produced by a lens.
Certain details of the question of distortions caused by lenses were considered by Blackett and Brode (1936). In Fig. 15, \(O\) is the optical center of the image, i.e. the point of intersection of the lens axis with the photographic plate or film. If the object is a straight line, then in the absence of distortions its image will also be a straight line. In the presence of distortion the image of the object will be a curve. Let these images be \(MN\) (undistorted) and \(M'N'\) (actual image). Let, further, \(P(x_0, y_0)\) be the position of the undistorted image of some point of the object space, and \(P'(x_0+\delta x, y_0+\delta y)\) the actually obtained one. Then
\[ \delta x=x_0\frac{\delta r}{r}. \tag{III, 15} \]
The quantity \(\delta r\), representing the displacement of the true image \(P\) occurring as a result of distortion, may be represented in the form of a series of odd powers of the distance \(r\) from point \(P\) to point \(O\)*, i.e.,
\[ \delta r=a_1r^3+a_2r^5+\cdots, \tag{III, 16} \]
and therefore,
\[ \delta x=x_0(a_1r^2+a_2r^4)=A+By^2+Cy^4, \tag{III, 17} \]
where \(A=a_1x_0^3+a_2x_0^5,\ B=ax_0+2a_2x_0^3,\ C=a_2x_0\). Setting \(y=0\), we obtain \(DD'=\delta x=A\).
* R. Glazebrooke, Dictionary of Applied Physics (Mac Millan and Company, London, 1927), vol. 4, p. 403.
To determine the coordinate \(x\) of the point \(P\) relative to the straight line drawn through \(D'\) parallel to \(MN\), we may write the equation of the curve \(M'D'N'\):
\[ x = By^{2} + Cy^{4} = (a_{1}x_{0} + 2a_{2}x_{0}^{3})y^{2} + a_{2}x_{0}y^{4}, \tag{III,18} \]
whence, for \(a_{2}=0\):
\[ x = a_{1}x_{0}y^{2}. \tag{III,19} \]
The curvature is parabolic, increasing linearly with increasing distance \(x_{0}\) from the axis. It will be positive or negative depending on whether the sign of \(a_{1}\) is positive or negative. Approximating the curve by an arc of a circle, we obtain \(\sigma = 2x/y^{2} = 2a_{1}x_{0}\), where \(\sigma\) is the curvature. For \(a_{1}\ne 0\) the distorted curve can be approximated by an arc of a circle only for small values of \(y\) near the point \(D'\). In this case we may write:
\[ x \simeq (a_{1}x_{0} + 2a_{2}x_{0}^{3})y^{2} \tag{III,20} \]
and
\[ \sigma_{\text{center}} = 2a_{1}x_{0} + 4a_{2}x_{0}^{3}. \tag{III,21} \]
Consequently, when the signs of \(a_{1}\) and \(a_{2}\) are the same, the curvature at the point \(D'\) has a minimum and increases with distance from this point along the track. If \(a_{1}\) and \(a_{2}\) are of opposite signs, the curvature at \(D'\) is maximal. Toward the end of the track it decreases and may even change sign.
As can be seen from Fig. 16, the curvature calculated by this method and produced by the distortion agrees well with its values found experimentally [Blekett and Brode (1936)].
Fig. 16. Distortions calculated and found experimentally.
Experimental determination of the distortion introduced by the objective is carried out by photographing parallel wires and subsequently measuring the curvature of their image. Since the axis of the objective is parallel to the magnetic field, the total curvature of the track on the photographic plate is composed of its true curvature and the curvature arising from distortion by the objective. Subtracting the latter from the total curvature, one can obtain the true curvature of the track.
The distortion considered above can be reduced to a minimum if a specially made objective is used, designed to operate in combination with the upper glass of the chamber that is being used.
b) Distortions caused by the upper glass (cover) of the chamber. A parallel plate of thickness \(t\), with refractive index \(n\), placed at right angles to the axis of the objective between the object and the objective, displaces a ray making an angle \(\theta\) with the axis by an amount \(\delta\theta\), equal to
\[ \delta\theta=\frac{t}{u}\sin\theta\{1-\cos\theta\,(n^{2}-\sin^{2}\theta)^{-1/2}\}, \tag{III,22} \]
where \(u\) is the object distance. As a result, the image is displaced by an amount \(\delta r\), which, as can be shown, is given by
\[ \delta r=A_{0}r+A_{1}r^{3}, \tag{III,23} \]
where
\[ A_{0}=\frac{t}{u}\left(1-\frac{1}{n}\right),\qquad A_{1}=\frac{t}{2uv^{2}}\left(1-\frac{1}{n^{3}}\right), \]
and \(v\) is the image distance. It can also be shown that the curvature \(\sigma\) in the image of a straight line, due to the action of the glass plate, will be equal to \(\sigma=2A_{1}x_{0}\), where \(x_{0}\) is the distance from the center of the plate to the point at which the image would have been obtained in the absence of distortion. The plate introduces a positive distortion; therefore an objective specially made to obtain photographs free from distortions must be matched to the upper glass of the chamber being used. Consequently, there is the possibility of compensating for a small negative distortion of the objective by selecting the upper glass of the chamber.
The distortion introduced by the glass cover is determined by measuring the curvature of the image of a straight wire photographed once without the glass and a second time with the glass between it and the camera.
c) Distortions due to vortices inside the chamber. In a Wilson chamber controlled by counters, the time interval between the passage of the particle and the moment at which it is photographed is about \(10^{-2}\) sec. During this interval the gas in the chamber expands, and any irregularities in this process may lead to distortion of the tracks. Experience shows that placing, above the rubber diaphragm, a copper mesh or a perforated plate covered with velvet moistened with a mixture of alcohol and water considerably reduces the turbulent motion of the gas during and after expansion. The tracks may also be distorted if there is a convective gas current in the chamber before expansion. After each expansion the gas in the chamber is heated, and this heating occurs first of all near the walls. As a result, con-
convective currents, causing the rapid descent of masses of colder air in the central part of the chamber. In vertical chambers these currents are considerably stronger than in horizontal ones. The forces producing the motion of the gas increase with time as \(t^3\); therefore photography must be carried out immediately after the expansion. The falling of droplets also leads to distortion of the tracks, so that \(0.25\) sec after the end of the expansion it is practically impossible to photograph tracks without distortion. Another type of distortion is connected with the appearance inside the chamber of vortical motions, the cause of which has not yet been precisely established.
Distortions in the chamber increase not only with increasing distance from the axis of the objective, but also with increasing inclination of the tracks to the vertical. Therefore, in order to obtain good results, the direction of the tracks should be as close as possible to the vertical. Distortions increase with increasing temperature. In order to get rid of distortions associated with changes in room temperature, it is useful to cool the metal frame of the chamber with running water. [Blackett and Wilson (1937), (1938).] The best results can be obtained by placing the chamber in a heat-insulating box made of fiber board, providing in it openings for the poles of the magnet. In this case illumination is carried out through a thick glass window made on one side of the box. Experience shows that the temperature inside the box should not differ by more than one degree from the temperature of the magnet. These conditions are created by maintaining the temperature of the room close to the temperature of the magnet. A small amount of water cooling the bottom of the chamber maintains the humidity of the walls, helping to preserve the stability of the thermal conditions inside the chamber. One should avoid spending large amounts of power on feeding the magnet, since in this case, although the curvature of the tracks increases, the distortions simultaneously increase as well because of the less favorable conditions arising as a result of heating of the magnet.
The measures described can considerably reduce distortions of tracks in the chamber even in the case when metal plates are placed inside the chamber. The total curvature caused by all types of distortions occurring inside the chamber may be determined by measuring the curvature of tracks taken in the absence of a magnetic field (see IV, § 4).
§ 5. Sensitivity time of the Wilson chamber
After the expansion has occurred, the gas in the chamber gradually warms up, the supersaturation decreases, and after a certain time condensation on the ions ceases. The sensitivity time of the chamber can be defined as the interval of time during which the supersaturation remains sufficient for condensation to occur
condensation on the ions along the track of the ionizing particle. [Williams (1939,a).] An experimental estimate of this time can be made by observing the formation of tracks by radioactive sources exposed at various intervals of time after expansion. The magnitude of the expansion must in this case be maximal, but such that fog formation does not occur.
The sensitive time of the chamber was determined by Hazen (1942), who used various methods for this purpose. In one of these methods a radioactive source containing Cl was rapidly moved inside the chamber during the expansion and after it. The successive positions of the source (and consequently also of the tracks of the particles which it emitted) were then photographed, by means of intermittent illumination, on one and the same photographic plate. Photographs obtained in this way make it possible to find the sensitive time of the chamber as the interval of time after the expansion during which tracks of a definite density are still formed.
Let us consider the physical processes taking place in the chamber after expansion. Immediately after expansion the gas in the chamber has a temperature \(T_2\), lower than the temperature \(T_1\) of the surrounding walls. Owing to the fact that the heat capacity and thermal conductivity of the walls are greater than those of the gas, their temperature remains practically unchanged at \(T_1\) all the time. Therefore the gas adjoining the walls is heated. The greatest heating takes place directly at the walls, decreasing with distance from them. As a result of this heating, the layer of gas adjoining the walls expands and, compressing the inner mass of gas, raises its temperature. The problem with which we are dealing here is analogous to the classical problem of heat transfer in the inner layers of the earth considered by Lord Kelvin, the solution of which can be found in all standard treatises on heat*). The temperature \(T_x\), established after a time \(t\) at a distance \(x\) from the boundary layer, is given by the following expression:
\[ T_x = T_1 - (T_1 - T_2)\frac{2}{\pi^{1/2}}\int_0^\xi e^{-\xi^2}\,d\xi, \tag{III, 24} \]
where \(\xi = x/2ht^{1/2}\), \(h^2 = K/c\rho\), \(K\) is the thermal conductivity, \(c\) the specific heat, \(\rho\) the density, and \(h^2\) the diffusion coefficient.
The increase in volume of a layer of given thickness \(dx\) is equal to \(S\,dx\,\Delta T/T_2\), where \(\Delta T = T_x - T_2\). Consequently,
\[ d\Delta V = \frac{S\Delta T}{T_2}\,dx = \frac{S(T_1-T_2)}{T_2} \left\{ 1-\frac{2}{\pi^{1/2}}\int_0^\xi e^{-\xi^2}\,d\xi \right\}dx. \]
\[ \text{*) M. N. Saha and B. N. Svjvastava, Treatise on Heat (Indian Press, Allahabad, India, 1935).} \]
The total change in volume is obtained by integrating this expression with respect to \(x\) over the limits from 0 to \(\infty\), i.e.
\[ \Delta V=\frac{S(T_1-T_2)}{T_2}\,2ht^{1/2} \int_0^\infty\left\{1-\frac{2}{\pi^{1/2}}\int_0^y e^{-\xi^2}\,d\xi\right\}dx =1.14\,\frac{S(T_1-T_2)}{T_2}\,ht^{1/2}. \tag{III, 25} \]
The expansion of the boundary layer tends to compress the main volume of the gas \(V\). The resulting temperature increase \(\Delta T\) is given by the following formula:
\[ \Delta T=(k-1)\frac{\Delta V}{V}T_2 =1.14\,\frac{S}{V}(T_1-T_2)\,ht^{1/2}(k-1). \tag{III, 26} \]
Denoting by \(\Delta T_s\) the temperature increase after which the chamber becomes insensitive, we obtain:
\[ t_s=\left(\frac{\Delta T_s}{T_1-T_2}\right)^2 \cdot\left(\frac{V}{S}\right)^2 \cdot\left(\frac{\rho c}{K}\right) \cdot\left(\frac{1}{k-1}\right)^2 \cdot\left(\frac{1}{1.14}\right)^2. \tag{III, 27} \]
It remains for us to find the quantity \(\Delta T_s\). Suppose that \(1+\varepsilon\) is the expansion at which tracks only just begin to appear, and \(1+\varepsilon+\Delta\varepsilon\) is the maximum permissible expansion at which fog has not yet appeared. Since further
\[ \frac{T_2}{T_1}=(1+\varepsilon)^{-(k-1)}, \]
then, taking logarithms, we obtain approximately:
\[ (T_2-T_1)/T_1=-(k-1)\varepsilon=-T/T_1, \tag{III, 28} \]
where \(T\) is the temperature difference. Consequently, we have:
\[ \Delta T/T=\Delta\varepsilon/\varepsilon, \tag{III, 29} \]
and, substituting this value in (III, 27):
\[ t_s=\left(\frac{\Delta\varepsilon}{\varepsilon}\right)^2 \cdot\left(\frac{V}{S}\right)^2 \cdot\left(\frac{\rho c}{K}\right) \cdot\left(\frac{1}{k-1}\right)^2 \cdot 0.77. \tag{III, 30} \]
The expression obtained shows that the time of sensitivity of the chamber increases proportionally to the following quantities:
1) the density of the gas \(\rho\),
2) the square of the ratio of the volume of the chamber to its surface \((V/S)^2\),
3) the square of the quantity \(\dfrac{\Delta\varepsilon}{\varepsilon}\).
The first of these conclusions is in agreement with the observations of Joliot (1934), who found that when the pressure is changed from \(P=40\) to \(P=76\ \mathrm{cm}\ Hg\), the value of \(t_s\) changes together with \(P\). However, at very low pressures \(t_s\) decreases more slowly, which, apparently—
therefore, is explained, on the one hand, by an increase in \(\Delta\varepsilon/\varepsilon\), occurring as a result of an increase in the ratio of the vapor pressure to the gas pressure and, on the other hand, by the rapid decrease of \(k\) associated with the lowering of the total pressure (see I, § 1b).
The second conclusion is confirmed by comparing the values of \(t_s\) obtained for different chambers. Williams (1939, b) measured the sensitivity time of two chambers, one of which had a diameter of \(16\) cm and a depth of \(4\) cm \((V/S \simeq 1.3\ \text{cm})\), and the other a diameter of \(30\) cm and a depth of \(30\) cm \((V/S \simeq 4\ \text{cm})\). The sensitivity time of the second of these chambers should therefore be 10 times greater than that of the first. The experimental results showed that for the first chamber \(t_s \simeq 0.05\) sec and for the second \(t_s \simeq 0.4\) sec. Hazen’s measurements, performed with two chambers for which the value \((V/S)^2\) differed by a factor of two, gave values of \(t_s\) in the ratio \(1.7:1.0\).
Experimental confirmation of the third conclusion, that \(t_s\) varies as \((\Delta\varepsilon/\varepsilon)^2\), was given by Hazen (1942). In this case, however, a preliminary determination of \(\Delta\varepsilon\) is required. For this one may approximately put:
and then
\[
\left.
\begin{aligned}
T &= T_1 - T_2 = -T_1(k-1)\varepsilon \simeq 100\varepsilon,\\
\Delta\varepsilon &= -(\varepsilon/T)\Delta T = 0.01\Delta T.
\end{aligned}
\right\}
\tag{III,31}
\]
If it is assumed that \(\Delta\varepsilon\) has the constant value \(\simeq 0.03\), then \(t_s \sim 1/\varepsilon^2\). The graph obtained by Hazen (see Fig. 4 of his paper) shows that up to \(\varepsilon = 0.26\) formula (III, 30) agrees well with experiment. However, for \(\varepsilon\) greater than this value, \(t_s\), having reached a maximum, begins to decrease. The reason for this may be the appearance of fog at \(\varepsilon \geq 0.26\). The appearance of a large number of fog droplets lowers the humidity and, in addition, causes heating, as a result of which Williams’ formula becomes inapplicable. However, so long as condensation occurs only on ions, formula (III, 30) correctly determines the sensitivity time of the chamber.
Pressure changes after adiabatic expansion. Let us now consider how the pressure inside the chamber changes after expansion. If the expansion occurs sufficiently rapidly, then the pressure decrease caused by it is determined by the adiabatic equation. Upon reaching the minimum, which occurs immediately after expansion, the pressure, owing to the inflow of heat from the chamber walls, begins to rise. If \(P\) is the gas pressure, then
\[ \Delta P/P = -k(\Delta V/V). \]
Substituting here the value of \(\Delta V\) from (III, 25), we obtain:
\[ \Delta P = -(kP/V)\Delta V = -1.14k(PS/V)[(T_1 - T_2)/T_2]ht^{1/2}. \tag{III,32} \]
Consequently, \(\Delta P\) increases as \(t^{1/2}\).
The indicated conclusion was verified by Hazen’s experiments. In these experiments the pressure inside the chamber was measured by an aneroid, whose needle readings were recorded photographically.
Hazen’s results show that, for a short interval of time immediately following the expansion, formula (III, 32) is valid.
At a low rate of expansion it is necessary to take into account the influx of heat during the finite time of expansion. This influx of heat can be compensated by an additional expansion. Equation (III, 31) shows that each degree of temperature rise requires an increase of the expansion by one percent.
§ 6. Magnets for Work with the Wilson Chamber
Observation of cosmic rays in a Wilson chamber placed between the poles of an electromagnet led Anderson (1932) to the discovery of the positron. This fact alone emphasizes the importance of the use of a magnetic field in investigations with the Wilson chamber. Besides the possibility of determining the sign of charged particles, the magnetic field also makes it possible to measure their momentum (see IV, § 2).
For determining the momentum of particles whose energies lie in the range \(10^5\)—\(10^7\) eV, a field of 1000 oersteds is sufficient. The curvature of tracks in such a field is quite measurable. For deflecting particles of very high energy—of the order of \(10^{10}\) eV—even a field of 20,000 oersteds is insufficient. It is obvious that the magnitude of the magnetic field must be chosen so that the bending of the tracks produced by the field is not so strong that the particle makes several turns, and at the same time not so weak that the particle passes almost undeflected. Here we shall dwell chiefly on methods of obtaining magnetic fields used in the study of cosmic rays. Similar methods of obtaining a magnetic field can also be used in investigations with alpha and beta particles. The remark about the correct choice of the field strength remains valid in this case as well.
Determinations of the energy of cosmic rays from the curvature of their tracks in a strong magnetic field were made by Kunze (1933, a), (1933, b), Anderson (1933), Blackett (1936), (1937), Blackett and Brode (1936), Jones and Hughes (1940), and Jones (1939). Measurements in weaker fields were carried out by Williams (1939, b). Data on the types of magnets used by various authors and on the magnitudes of the fields they obtained are given in Table XII.
It is seen from the table that, before Blackett, improperly designed magnets were used. This led not only to an excessively large expenditure of power without a corresponding increase in the magnetic field, but also increased the difficulties of cooling. In the case of an uncontrolled chamber, where the magnetic field is switched on for several ...
seconds before expansion; such an expenditure of energy can still be tolerated. However, for a chamber controlled by counters, when the magnetic field must be switched on all the time, a proper design of the magnet is necessary.
Table XII
Data on magnets for investigations with a Wilson chamber
| Author | Type of magnet | Coil weight | Power used | Field strength | Chamber size |
|---|---|---|---|---|---|
| Kunze (1933) | Solenoid, water-cooled | Copper 1100 kg | 500 kW | 18,400 oersteds | 16.4 cm in diameter |
| Anderson (1933) | Heavy water-cooled coil with a relatively light yoke | Copper 896 kg, iron 500 kg | 440 kW | 15,000 oersteds | 16 cm in diameter |
| Blackett (1936) | Water-cooled coil with a heavy yoke | Copper 3000 kg, iron 8000 kg | 25 kW | 14,000 oersteds | 17 cm in diameter |
| Jones and Hughes (1940) | Oil-cooled coil with a core | Copper 1000 kg, steel 9000 kg | From 35 to 125 kW | From 12,400 to 16,000 oersteds | 30×4.2 cm |
For a very deep chamber (\(\sim 30\) cm), it is practically impossible to construct a magnet that would produce a large field without a considerable expenditure of power. Therefore, when working with such chambers, two coils of the Helmholtz type are usually used to obtain the magnetic field. In these cases the region of field homogeneity is determined by the depth of focus of the objective used for photography. For measuring very large energies only small chambers are used.
It should be noted that the field of a solenoid is less homogeneous than the field formed between the flat poles of an electromagnet; therefore the accuracy in determining the energy will be correspondingly lower.
Since the deflection occurs at right angles to the direction of the magnetic field, and since the axis of the objective must be perpendicular to the path of the particle, an aperture is usually drilled in one pole of the magnet through which the photography is carried out. When stereoscopic photography is required, the photography is carried out with the aid of plane mirrors fixed to the walls of this aperture.
IV. PHYSICAL MEASUREMENTS PERFORMED WITH THE AID OF THE WILSON CHAMBER
§ 1. Specific ionization
The term “specific ionization” denotes the number of ion pairs formed per unit length of the track of a charged particle as it passes through a medium. According to Bloch (1933a), (1933b), the specific ionization is directly proportional to the square of the charge and, over a wide range, inversely proportional to the square of the velocity of the moving particle [see (V, 1), Appendix 3]. As regards the medium, the specific ionization is directly proportional to the number of electrons in a cubic centimeter of the substance. Consequently, in a medium consisting of heavy molecules, for example xenon molecules, the ionization will be higher than in a medium formed by light molecules. For the same charge and in one and the same medium, the specific ionization increases as the velocity of the particle decreases.
The energy transferred by the incident particle to the electrons may considerably exceed the ionization potential of the electrons of the various shells of the atoms of the medium. Therefore the electrons knocked out by the incident particle, the so-called primary electrons, may in turn produce ionization. This process of secondary ionization continues as long as the energy of the electrons remains greater than the ionization potential of the outer shell of the atoms. The fastest of the knocked-out electrons are observed on some tracks in the form of branches from the main path of the particle.
The specific ionization of a charged particle may be determined by various methods, in particular with the aid of an ionization chamber, Geiger–Müller counters, etc. However, the Wilson-chamber method is the only one that permits the primary and the total ionization to be measured separately. When a charged particle passes through the chamber immediately after expansion, condensation on the ions occurs before they have time to diffuse from the place of their formation. Under these conditions the width of the track is determined by the expression \(x_1 = 4.68(Dt)^{1/2}\). The primary ionization, in which the knocked-out electrons possess energy sufficient to form, say, 20 secondary electrons, appears in the photograph as a bold point (cluster). The measurement of the specific primary ionization is therefore reduced to counting the number of clusters and branches formed per unit length of the main track; for such measurements clear and unblurred tracks must be selected. In those cases where a magnetic field is used to deflect the particles, the specific primary ionization may be obtained as a function of \(H\rho\) or of the particle momentum.
If the ionizing particle passes through the chamber not long before the expansion, the ions, owing to diffusion, have time to spread out, and instead of
of a fat droplet on the photograph, a group of separate ions will be obtained. The scattering of droplets will then depend on the delay, i.e., on the magnitude of the time interval between the passage of the particle and the moment of expansion.
For a given delay time the scattering of the clusters will depend on the nature of the gas, varying as \(D^{1/2}\), where \(D\) is the diffusion coefficient of the ions in the medium under consideration. If the delay is too small, the primary ions cannot be resolved into their secondary components.
On the other hand, with a very large delay the ions that have formed have time to move far away from the place of their origin and, as a result, become mixed with the droplets forming the chamber background, which is always present. In practice the magnitude of the delay is chosen depending on the nature of the gas used. When a magnetic field is applied simultaneously, the delay must be chosen so that the ions separate completely, while the track is not too greatly distorted as a result of the motion of the gas, since otherwise the errors in measuring the curvature may prove considerable. From equation (III, 12) we see that for \(\mathrm{O}_2\) \((D \sim 0.022\ \mathrm{cm}^2\ \mathrm{sec}^{-1})\), \(x \simeq 3.1\ \mathrm{mm}\) at \(\tau \simeq 0.2\ \mathrm{sec}\), whereas for air \((D \simeq 0.034\ \mathrm{cm}^2\ \mathrm{sec}^{-1})\), \(x \simeq 3.9\ \mathrm{mm}\) for the same value of \(\tau\).
Thus, in determining the total specific ionization, when the use of the delay method is essential, only a chamber controlled by counters can be used. The desired duration of the delay is set by the time interval between the instants of coincidence in the counters and the operation of the magnet \(U\) (see Fig. 10).
The magnitude of this interval is set either by means of a thyratron delay circuit or by means of a cam mechanism triggered by the counters. The successive stages of the process, for an expansion occurring with a delay of about \(0.2\ \mathrm{sec}\), are as follows: passage of the cosmic particle \(t = 0.0\), switching off the field \(t = 0.01\ \mathrm{sec}\), expansion \(t = 0.2\ \mathrm{sec}\), switching on the illumination \(t = 0.245\ \mathrm{sec}\), and, finally, switching off \(t = 0.250\ \mathrm{sec}\). With such a sequence of events, Brode (1939) measured “nests” containing up to 250 ions.
The illumination of chambers with delayed expansion must be stronger than that of chambers operating without delay. In the latter case the ions are grouped more densely, owing to which it is possible to obtain a photograph of the track even when the scattering action of an individual droplet is small. Increasing the aperture used raises the illumination of the droplet images; however, the distortion produced by the optical system also increases. To reduce distortions caused by the fall of the droplets, the exposure time should be limited to the minimum necessary duration.
After the total number of ions has been counted, the image of the track is projected back into object space, where its actual length is measured. The mean specific ionization of the particle is obtained by dividing the total number of ions by the length of the track. The use of a focus of small depth (increase of aperture) makes it possible easily to isolate ions belonging to the chamber background, which lie in one projection with the track but at other depths. The error due to the presence of background can be determined by counting the density of droplets in regions adjacent to the track. The absolute magnitude of the chamber background decreases on passing to smaller expansions.
When photographing tracks in a magnetic field, the probability of recombination of positive and negative ions is considerably reduced, since ions of opposite charge are completely separated from one another.
§ 2. Momentum and curvature of tracks in a magnetic field
A particle having mass \(M\), charge \(Ze\), and momentum \(p\), moving at right angles to the magnetic field \(H\), describes a trajectory with radius of curvature \(\rho\), satisfying the equality
\[ \begin{gathered} pc=\frac{M\beta c^2}{(1-\beta^2)^{1/2}}=ZeH\rho,\\ \text{or}\\ (pc)_{\mathrm{ev}}=300ZH\rho, \end{gathered} \tag{IV, 1} \]
if the quantity \(pc\) is expressed in electron-volts, and \(H\rho\) in oersted-cm. Thus, if the charge \(Z\) is known, the momentum of the particle can be obtained directly from measurement of \(H\rho\).
If \(E\), \(T\), and \(\beta c\) are, respectively, the total energy, kinetic energy, and velocity of the particle, then the equations relating \(H\rho\) to these quantities will have the following form:
\[ E=Mc^2\left\{1+\left(\frac{ZeH\rho}{Mc^2}\right)^2\right\}^{1/2}, \tag{IV, 2} \]
\[ T=Mc^2\left[\left\{1+\left(\frac{ZeH\rho}{Mc^2}\right)^2\right\}^{1/2}-1\right], \tag{IV, 3} \]
\[ \beta=\frac{ZeH\rho}{Mc^2\left\{1+\left(\frac{ZeH\rho}{Mc^2}\right)^2\right\}^{1/2}}. \tag{IV, 4} \]
If \(T \ll Mc^2\), we have:
\[ T\simeq \frac{(ZeH\rho)^2}{2Mc^2} \quad \text{and} \quad T_{\mathrm{ev}}=0.088\,\frac{m}{M}(ZH\rho)^2; \tag{IV, 5} \]
\[ \beta\simeq \frac{ZeH\rho}{Mc^2} =5.86\cdot 10^{-4}\,\frac{m}{M}ZH\rho. \tag{IV, 6} \]
For \(T \gg Mc^2\):
\[ \begin{gathered} T = ZeH\rho - Mc^2 \\ \text{or} \\ T_{\mathrm{eV}} \simeq 300\,ZeH\rho - (Mc^2)_{\mathrm{eV}}, \end{gathered} \tag{IV,7} \]
where \(T_{\mathrm{eV}}\) is the kinetic energy expressed in electron-volts, and \(H\rho\) is expressed in oersted·cm.
To determine the velocity or energy of a particle from a measurement of the quantity \(H\rho\), it is, generally speaking, necessary to know its mass [(IV, 1) and (IV, 6)]. However, if the kinetic energy is so large that the rest energy may be neglected, equation (IV, 7) makes it possible to obtain the kinetic energy from the measured value of \(H\rho\).
Tables of numerical values of the quantities \(\beta\), \(H\rho\), \(p/Mc^2\), and \(T\) are given in Appendix II [see, in this connection, Bhattacharyya (1941)]*).
From (IV, 1) it is easy to see that a particle having energy \(pc = 10^8 \, \mathrm{eV}\) and charge \(Z = 1\), deflected by a field of \(1000\) Oe, will give a track with radius of curvature \(3.3\) m. To obtain the same curvature for a particle possessing energy \(10^9 \, \mathrm{eV}\), a field of \(10^4\) Oe is required. In investigations in the field of cosmic rays, where one has to deal with particles of very high energies, the use of very large fields is necessary. Therefore, serious attention must be paid to the design of the magnet and to the development of measures against heating of the chamber. Some data on magnets used in the study of cosmic rays have already been given in III, § 6. Here we shall dwell on methods of measuring tracks with very small curvature and on the errors in determining the momentum and energy of particles arising from false curvatures in the chamber.
As already noted above, measurements of tracks are carried out by reprojecting two images back into object space. If \(P\) and \(P'\) are two stereoscopic images of one and the same point \(O\), then upon reprojection the position of this point in object space will lie at the intersection of the lines \(PL\) and \(P'L'\), where \(L\) and \(L'\) are the optical centers of the objectives. Using this, one can construct in object space a wire model of the track in its natural size. In practice, arrows \(A\), \(B\), \(C\), etc., mounted on hinges in front of the camera, are usually used. By moving one of these arrows, one can bring its end into coincidence with the point of intersection of the lines \(PL\) and \(P'L'\) and thus obtain the position of the point \(O\).
Having performed similar operations for several other points, one can then project the ends of all the arrows onto a screen and carry out all the necessary measurements on it.
) J. B. Hoag, Electron and Nuclear Physics* (D. Van Nostrand Company, Inc., New York, 1938), pp. 464–466.
When working with a chamber controlled by counters, when the tracks lie in a plane perpendicular to the magnetic field and, consequently, to the axis of symmetry of the optical system, the process of measurement is simplified by the fact that the reprojection of the images is made onto a screen perpendicular to the direction of the magnetic-field lines. By rotating the screen about its vertical and horizontal axes, both images can be brought into exact coincidence. The coordinates of the points of the track obtained in the indicated manner are then measured with a microscope with a micrometer eyepiece, after which they are plotted on paper, where \(\rho\) is determined. The method described was used by Blackett (1936), Anderson (1933), and other authors.
Instead of a microscope, Jones and Hughes (1940) used a special device, shown in Fig. 17. The projected track intersects the straight line \(DE\) present on this instrument at two points \(A\) and \(C\). The sagitta of the middle part of the track is found by moving a disk mounted on a movable platform until its edge coincides with the track. The reading is taken from the micrometer screw by which the platform is moved, and can be made with an accuracy of up to \(0.005\) cm. This value also determines the smallest deflection (the sagitta) that can be measured on this instrument. For a track length of 20 cm and a magnetic field of 16,000 oersteds, the indicated displacement corresponds to a particle energy of \(5 \cdot 10^{10}\) eV. The value of \(\rho\) determined by this method agrees well with those results obtained in measurements by the method described above.
Fig. 17. Instrument for rapid measurement of the radius of curvature.
For processing tracks having smaller curvature, another method has been developed, with certain advantages as regards the speed of measurement and its accuracy. It is based on the remarkable ability of the eye to judge straightness when looking along a line along its length. In working by this method, the curvature, by means of some optical device, is compensated to such an extent that the reprojection of the track is transformed into a straight line, the measurement itself being reduced to the correct assessment of straightness. In practice this is accomplished by, as before, obtaining the reprojection of the track on a white screen, after which a device producing the required change in curvature is placed in front of the camera lenses. For tracks of very small curvature, the use of an inclined plane-parallel glass plate is quite sufficient as such a device [Blackett and Brode (1936)]. For measuring tracks of medium curvature one may use an achro-
matical prism. For small curvatures the additional curvature \(1/\rho\) introduced by such a prism into the image of a straight line is equal to
\[ \rho=\rho_0 \operatorname{cosec}\theta, \tag{IV, 8} \]
where \(\theta\) is the angle between the straight line and the principal plane of the prism. To determine the curvature of the track, it is necessary to measure the angle \(\theta\) at which the image of the track on the screen appears as a straight line. The accuracy of the estimate of rectilinearity increases as the angle at which the observer looks at the screen decreases. By using screens coated with MgO and strong illumination, this angle can be reduced to 2–3 degrees. The apparatus is first calibrated by measuring lines of known curvature, from which \(\rho_0\) is determined. After this has been done, the value of \(\rho\) corresponding to a definite value of \(\theta\) can be found directly from (IV, 8), or from a calibration curve of the dependence of \(\rho\) on \(\theta\).
The theory of the distortion produced by a combination of lenses and a prism is very complicated. When the prism is in the position corresponding to the angle of minimum deviation, the radius of curvature \(\rho\) due to the action of the prism, according to Popu (1920), is expressed by the following formula:
\[ \rho_0=\frac{\mu f(1-\mu^2\sin^2\alpha)^{1/2}}{2(\mu^2-1)\sin\alpha}, \tag{IV, 9} \]
where \(\mu\) is the refractive index of the prism, \(2\alpha\) is its refracting angle, and \(f\) is the focal length of the objective used.
Working with an objective \(f=35\ \mathrm{mm}\) (relative aperture \(1:2\)) at a magnification of \(1/5.6\), and placing an achromatic prism (angles: crown—\(29^\circ\), flint—\(16^\circ\)) in front of the objectives, Blackett (1937) measured radii of curvature down to \(0.33\ \mathrm{m}\), the probable error being \(0.0016\ \mathrm{m}\).
The maximum energy that can be measured depends on the accuracy with which the curvature is determined; for small curvature this, in turn, depends on how accurately the sagitta \(d\) of a track of definite length \(l\) is measured. If \(\rho\) is the radius of curvature, then the mathematical expression for the relation between the quantities of interest to us is given by the following formulae:
\[ \rho=\frac{l^2}{8d}\quad \text{and}\quad pc=300Hl^2/8d. \tag{IV, 10} \]
As a result of the distortion of tracks caused by the motion of the gas and by other factors discussed in the preceding sections, tracks even in the absence of a magnetic field always have some curvature. The influence of these distortions is shown in Fig. 18, borrowed from the work of Yusa (1940). This figure shows a graph of the distribution of 110 tracks, taken in the absence of a magnetic field, as a function of the magnitude of the sagitta \(d\). Similar measurements
curvatures of tracks taken in the absence of a magnetic field make it possible to estimate the probable error in determining \(d\). In the present case it is \(0.012\ \text{cm}\). The sagitta corresponding to particles of different energy can be calculated from (IV, 10). Knowing the probable error in measuring \(d\), one can calculate the error in determining the energy. In the case under consideration the field was \(12400\) oersteds and \(l=20\ \text{cm}\); consequently, the value of \(d\) for particles with energies \(10^9\) and \(10^{10}\ \mathrm{eV}\) is, respectively, \(0.186\) and \(0.0186\ \text{cm}\). With the above-mentioned probable error in determining \(d\) (\(0.012\ \text{cm}\)), this means that the error in determining the energy of the particles is \(6\%\) for \(E=10^9\ \mathrm{eV}\), \(66\%\) for \(E=10^{10}\ \mathrm{eV}\), and \(100\%\) for particles with energy \(2\cdot10^{10}\ \mathrm{eV}\). These figures represent the maximum energy that can be measured by the indicated method.
Fig. 18. Curvatures of tracks caused by distortions inside the chamber (in the absence of a magnetic field). Distribution of tracks according to the magnitude of the sagitta.
Measurements of changes in momentum \(p\). Another important physical quantity that can be determined by the Wilson chamber method is the change in the particle momentum. Moving in a material medium, a particle, as a result of ionization, radiation losses, etc., continuously expends its energy, in connection with which the value \(H\rho\) gradually changes along the track. In a medium of low density all kinds of losses, referred to one centimeter of path, are very small, and the change of \(\rho\) from one point of the track to another is barely perceptible, so that the particle retains a constant (average) curvature over a fairly large portion of the track. Therefore, in a chamber with a magnetic field, where just these conditions occur, it is possible with sufficient justification to regard the curvature of the track as constant.
However, in materials of high density (for example, Pb, Au) the specific losses (which, moreover, depend on the nature of the particle) are considerably greater. Therefore, if inside the chamber, in the path of the particles, one places a lead plate \(1\text{--}2\ \text{cm}\) thick, the curvature of the track on the two sides of the plate will, generally speaking, be different. In particular, the part of the track situated on the side where the particle enters the lead will have a smaller
greater curvature than the part on the side where it exits the lead. From the measurement of the curvature of the particle track before and after its passage through the lead, one can, using (IV, 1), determine the loss of momentum
\[ c(dp/dx)=300ZH(d\rho/dx). \tag{IV, 11} \]
In the case when the energy of the particles is small, it is necessary to pay special attention to the proper choice of the thickness of the plate, since with a very large difference in the curvature of the track on the two sides of the plate the accuracy of the measurement of \(\rho\) decreases, and with it the accuracy in determining the loss of momentum also decreases.
Placing a plate inside the chamber considerably increases distortions. Since in this case the nature of the distortion in the two halves of the chamber is different, the distortion must be determined separately for each of them. Usually for this purpose, in each series of experiments, after a certain number of photographs the magnetic field is switched off and a photograph is taken without the field. From measurements of these photographs, for each of the halves of the chamber one finds distortion curves, giving, for the given series of measurements, the error in curvature as a function of position in the chamber. Owing to the difference in the behavior of the two halves of the chamber, an apparent increase of momentum after the passage of the particle through the lead, and even a change in the sign of its charge, may sometimes be observed. Therefore the elucidation of possible errors in determining the curvature is a task of primary importance.
In experiments with a Wilson chamber placed in a magnetic field, the difference in the curvature of the track on the two sides of the plate at once indicates the direction of motion of the particle and, consequently, the sign of its charge. Anderson discovered the positron (1933) precisely by this method.
§ 3. Range
The range is a very important characteristic of a particle, since in investigations in the field of nuclear physics the nature and energy of the particle are in many cases determined from observations of its range inside the Wilson chamber. It is customary to refer the range of a particle to dry air at \(15^\circ\) C and a pressure of \(76\ \text{cm Hg}\), measured under standard barometric conditions (sea level at latitude \(45^\circ\) at \(0^\circ\) C). Under identical conditions the stopping power of water vapor relative to air is taken to be equal to 0.74. Corrections for temperature, pressure, humidity, and the latitude of the location may in some cases reach a considerable magnitude and therefore must be carefully taken into account.
For a rapid determination of the range in air one proceeds as follows. On the film one measures the length \(L_0\) of the image of the track of a full-range alpha particle, which in air at \(15^\circ\) C and at \(760\ \text{mm Hg}\) has range \(R_0\). Then if \(L_\alpha\) is the apparent length of an un-
of whose track, then its range \(R_a\), reduced to air, will be equal to
\[ R_a = L_a (R_0/L_0), \tag{IV, 12} \]
on the assumption that the stopping power of the gas mixture relative to air is the same along the entire range.
Unfortunately, the exact determination of the magnitude of the range is associated with difficulties arising from the absorption of particles in the source itself, and also because of the spread of ranges. Rutherford, Ward, and Lewis (1931) established that the appearance on the source of even the slightest deposit reduces the value of the mean range of \(9\ \mathrm{cm}\) alpha particles by approximately \(0.3\ \mathrm{mm}\).
As a result of fluctuations in the number of collisions, an initially homogeneous beam, after passing through some thickness of absorber, becomes inhomogeneous. If, in passing through a layer of substance, the particles have on average \(n\) collisions, the fluctuation in the number of collisions will be of order \(n^{1/2}\), and the fluctuation in the energy of individual particles after passing through the absorber will be \(n^{1/2} I\), where \(I\) is the mean energy spent on the formation of an ion pair. In reality, along with fluctuations in the number of collisions, there are also fluctuations in \(I\). As a result, the ranges of individual particles of a homogeneous beam are not the same, but are distributed about the mean value according to the Gaussian law.
Fig. 19. For the determination of the mean and extrapolated ranges: \(a)\) differential range-distribution curve \([p(R)\,dR]\), \(b)\) integral curve \([p(R)]\).
The spread of alpha-particle ranges by the Wilson-chamber method was studied by Curie (1923), Meitner (1926), Frisch and Nimmo (1929), and Freytag. For this purpose a narrow, horizontal beam of homogeneous alpha particles was introduced into the Wilson chamber. The tracks of these particles were photographed, after which the ranges were measured and their spread determined. The results of the measurements show that the spread of ranges follows the Gaussian distribution well:
\[ p(R)\,dR = \frac{\alpha}{\pi^{1/2}} e^{-\alpha^2 (R - R_0)^2}\,dR, \tag{IV, 13} \]
where \(R_0\) is the mean range, \(\alpha\) is the linear parameter of the spread, depending on the properties of the particles and the medium, and \(p(R)\,dR\) is the probability of the appearance of particles with a range lying between \(R\) and \(R + dR\).
The differential range distribution curve is given in Fig. 19 (curve a). The distance reached by half of all the particles is called the “mean range.” The concept of mean range is meaningful in the case of complete homogeneity of the initial beam (as is the case for some radioactive substances) and with an infinitely thin source (so that there is no absorption in the source itself). Artificial radioactive sources, for reasons of intensity, must have a sufficient thickness. Since the energy of the emitted particles, moreover, depends on the angle of emission, in this case the beam is nonhomogeneous from the very beginning. For determining the range, the fast particles are of greater importance; in this connection the concept of the so-called extrapolated range is used.
The extrapolated range is readily obtained from the integral curve of the distribution of particles by ranges, i.e., from the curve giving the number of particles \(P(R)\) (ordinate) having a range greater than a given one (abscissa). This curve is expressed by the formula
\[ P(R)=\int_R^\infty p(R)\,dR=\frac{1}{2}\left[1-\Phi\{a(R-R_0)\}\right], \tag{IV, 14} \]
where \(\Phi\) is the probability integral. It has the form shown in the same Fig. 19 (curve b). The extrapolated range is determined by the point of intersection of the tangent to the steepest part of curve b with the abscissa axis.
Since the steepest section of the integral curve \(P(R)\) is at \(R=R_0\), it is easy to show that
\[ R_{\text{extr}}=R_0+\frac{1}{2}\frac{\pi^{1/2}}{a}=R_0+S. \tag{IV, 15} \]
The difference between the mean and extrapolated ranges has been measured for alpha particles of ThC′ and RaC′. The good agreement of the results of these experiments with the results of calculations, especially in the region of high energies, makes it possible to use the calculated values of \(S\) to determine the mean range \(R_0\) from the measured extrapolated range \(R_{\text{extr}}\). The spread of alpha-particle ranges varies from \(1.2\%\) of their full range at an energy of 4 MeV to \(0.85\%\) at an energy of 50 MeV. The values of \(S\) for alpha particles and protons of various energies are given by Livingston and Bethe (1937). (See also, in this connection, Appendix 3b, where a derivation is given of the approximate range–velocity relation and where it is shown that the range of particles is approximately proportional to \(M/Z^2\) and to the velocity.)
It should be noted, however, that the range–energy relation given by theory is not completely exact. The reasons for this are, first, multiple changes of the particle charge over the last several centimeters of its path (as a result of alternating loss and capture of electrons) and, second, the uncertainty in the value of the mean excitation potential \(I\).
As a result of the difficulties indicated, Livingston and Bethe proposed the following method for establishing a conditional range—energy scale. The mean excitation energy \(I\) and the constant entering into (V, 1) are chosen so that the theoretically calculated range on the average coincides with the observed range of individual groups of alpha particles, whose energies can be determined with great accuracy from their curvature in a magnetic field (Rutherford, Lewis, and Bowden (1933)). In this way the range—energy relation can be determined in the interval from 5.3 to 11.5 MeV. For alpha particles and protons of lower energies, where the theory is much less satisfactory, the experimental data of Lee are used (1932, a), (1932, b). At higher energies, on the contrary, one can make use of the theoretical relation, since in this region the theory is more reliable. The range—energy curves obtained by this method for alpha particles and protons with maximum energy up to 15 MeV are given by Livingston and Bethe. To determine the energy corresponding to some range, these data are usually used.
In the tables given in the Appendix, values of the ranges and the corresponding velocities and energies are given for alpha particles, protons, and electrons. Using these tables, from the measured range in a Wilson chamber one can determine the velocity and energy of the particle. Table XIV gives the values of the quantity \(RZ^2/m/M\), as functions of \(\beta/(1-\beta^2)^{1/2}\). These results are of a general character and may be applied to any heavy particle.
§ 4. Determination of Particle Mass from Wilson-Chamber Data
The kinetic energy of heavy charged particles rarely exceeds several million electron-volts and, consequently, in comparison with the rest energy is a small quantity. The velocity of a particle is small compared with the speed of light \(c\) and varies inversely proportional to the particle mass:
\[ \beta=\frac{pc}{(p^2c^2+M^2c^4)^{1/2}}\simeq\frac{pc}{Mc^2}. \tag{IV, 16} \]
The ionization density of such comparatively slow particles varies as the reciprocal of the square of the velocity; as a result, the ionizing powers of particles of different mass differ greatly, even if their kinetic energies are comparable with one another. Therefore the nature of a particle can be determined from consideration of the ionization density along its track in a Wilson chamber.
The energy of cosmic particles is comparable with, or even exceeds, the rest energy. In this case the velocities of particles of all kinds—electrons, mesotrons, or protons—are close to the speed of light, and the density of the ionization produced by them is approximately the same. In connection with this
determination of the mass of particles possessing such high energies becomes a very difficult problem. Other commonly used methods of determining mass, for example from deflection in an electrostatic or magnetic field, also become unreliable.
Below we describe comparatively recently developed methods for determining the mass of fast cosmic particles from Wilson-chamber data. These methods have been applied chiefly to determine the mass of a new particle—the meson; moreover, it has turned out that the mass values obtained by different ones of these methods do not agree with one another. The reason for these discrepancies may lie in imperfections of the experimental methods, as will be discussed below; however, the possibility is not excluded that in cosmic rays there are mesons having different rest masses.
The mass of a particle can be determined by the Wilson-chamber method from measurements of the following quantities:
a) the specific ionization \(I\) and the quantity \(H\rho\),
b) the specific ionization \(I\) and the range \(R\),
c) the range \(R\) and the quantity \(H\rho\),
d) from the change in the particle momentum \(d(H\rho)/dx\) on traversing an absorbing plate inside the Wilson chamber, and, finally,
e) from observations of elastic collisions of the incident particle with an electron.
a) Determination of mass from the specific ionization and the curvature in a magnetic field. Under certain assumptions (see Appendix 3a), the specific energy loss \((-dE/dx)\), due to ionization, does not depend on the mass and is a function only of the velocity of the particle. The form of this function is given by equation (V, 9) in Appendix 3.
In general we may put:
\[ \left(-\frac{dE}{dx}\right)_{\text{ion}} = f\left(\frac{\beta}{(1-\beta^2)^{1/2}}\right). \tag{IV,17} \]
The values of \((-dE/dx)\), computed as a function of the argument \(\beta/(1-\beta^2)^{1/2}\), are given in Table XXXIII. In Fig. 20 the data of this table are represented graphically (curve 1).
The method for determining the number of ions formed in one centimeter of track was discussed in detail in § 1 of this chapter (the method of delayed expansion). For dense (unresolved) tracks the number of primary ions can be determined by the indirect method proposed by Williams (1939, b). If \(I_0\) is the mean specific ionization, then the mean free path between two successive collisions is equal to \(1/I_0\). Along the length of track \(L\) observed in the chamber there will be \(I_0 L\) such mean free paths. The probability \(P\) that over the length of track \(L\) there will occur a gap of magnitude \(l\) cm is given by the expression:
\[ p = I_0 L e^{-I_0 l}. \tag{IV,18} \]
Consequently, by observing the number, along the length of the track, of gaps greater than \(l\), one can, with the aid of equation (IV,18), determine the value \(I_0\). From the value of \(I_0\) found in this way (the number of ion pairs formed in one centimeter of the particle’s path), the specific energy loss \((-dE/dx)\) is determined, assuming that 32 eV is expended in the formation of one ion pair.
Fig. 20. Curves of the dependence of the quantities
\(dE/dx,\ H\rho \left/ \dfrac{M}{m} \right.,\ R \left/ \dfrac{M}{m} \right.\) and \(H\rho/R\) on \(\beta/(1-\beta^2)^{1/2}\).
To determine the mass, in addition to the specific ionization, it is also necessary to measure the curvature of the track. It should be noted, however, that for an accurate measurement of the curvature the track must be sharp, whereas the determination of \((-dE/dx)\), carried out by counting droplets, on the contrary requires a certain blurring of it. Therefore a simultaneous determination of each of these two quantities cannot be performed with sufficiently high accuracy.
Putting \(Z=1\), from equation (IV,1) we have:
\[ \frac{M}{m} = \frac{eH\rho}{mc^2}\cdot \frac{\beta}{(1-\beta^2)^{1/2}} = \frac{H\rho}{1704}\cdot \frac{\beta}{(1-\beta^2)^{1/2}}, \tag{IV,19} \]
where \(H\rho\) is expressed in oersted·cm, and \(M/m\) is the unknown mass of the particle expressed in terms of the electron mass. Consequently, if from the measurement of \(dE/dx\) the quantity \(\beta/(1-\beta^2)^{1/2}\) has been found, then the above formula makes it possible at once to determine the particle mass from the obtained value of \(H\rho\).
The graph of the dependence of the quantity \(H\rho\sqrt{M/m}\) on \(\beta/(1-\beta^2)^{1/2}\) is given in Fig. 20 (curve (2)). From the measured values of \(dE/dx\) and \(H\rho\), this curve makes it possible to determine the quantity \(M/m\) directly, without any calculations.
In the following Table XIII are given the results of several determinations of the meson mass, carried out by this method.
Table XIII
Values of the meson mass found from the measurement of specific ionization and \(H\rho\)
| Author | \(-dE/dx\) per cm of air, in \(10^3\) eV | \(H\rho\) in \(10^5\) oersted·cm | \(M/m\) from curves 1 and 2 |
|---|---|---|---|
| Williams and Pickup (1938) | 12.5 | 1.1 | 230 |
| Street and Stevenson (1937) | 7.5 | 1.47 | 210 |
| Yuz (1941) | 15.0 | 0.96 | 230 |
| Brode, MacPherson, and Starr (1936) | 12.5 | 1.00 | 214 |
| Brode, MacPherson, and Starr (1936) | 25.0 | 0.55 | 184 |
| Anderson and Neddermeyer (1936) | 25.0 | 0.60 | 200 |
| Anderson and Neddermeyer (1936) | 7.5 | 2.5 | 370 |
b) Determination of the mass from range and specific ionization. For particles whose mass is much greater than the electron mass, radiation losses are relatively small even at very high energies (\(\beta=0.9\)) and in substances with a large atomic number (\(Z=82\)). Thus, for example, the radiative losses of a mesotron having an energy of \(1.5\cdot 10^8\) eV over 1 cm of path in lead amount to
\[ (\delta E)_{\mathrm{rad}}=\frac{\ln 2E}{200^2\cdot 0.5}\simeq 0.01\ \mathrm{MeV}, \]
whereas the ionization losses in the same thickness of lead are equal to 10 MeV. Thus, the total energy loss of heavy particles is determined chiefly by their ionization losses. Under this condition the range of a particle \(R\) may be expressed, as shown in Appendices 3a and b, by a formula of the following form:
\[ R=\frac{M}{m}\,g\left(\frac{\beta}{(1-\beta^2)^{1/2}}\right), \tag{IV,20} \]
where \(Z\) has been put equal to unity. From this formula it is seen that the quantity \(R/(M/m)\) is a function only of \(\beta/(1-\beta^2)^{1/2}\). The quantities \(R/(M/m)\) corresponding to different values of \(\beta/(1-\beta^2)^{1/2}\) are given in Table XXIV. The dependence of these quantities is shown graphically by curve (3) in Fig. 20, which is applicable to heavy particles of any mass. Польз-
using it, one can, from the range found and the value \(\beta/(1-\beta^2)^{1/2}\), determine the mass of the particle in terms of the electron mass.
As was already indicated, the specific ionization \(-dE/dx\) uniquely determines the quantity \(\beta/(1-\beta^2)^{1/2}\). Consequently, simultaneous measurement in a Wilson chamber of the range and of the specific ionization makes it possible to determine the mass of the particle. The method described is not, however, especially accurate, because measurements of both the ionization and the range usually give underestimated values of these quantities. Both errors act in the same direction, and therefore this method gives a lower limit for the particle mass.
In Table XIV, by way of example, data of various authors are given (see the references in the work of Wheeler and Ladenburg (1942) on the meson mass found by this method).
Table XIV
Meson mass found from measurements of range and specific ionization
| Author | \(-dE/dx\) per 1 cm of air, in \(10^3\) eV | \(\left\lvert R/\dfrac{M}{m}\right\rvert\), obtained from curves 1 and 2 of Fig. 20 | \(R\) in cm of air (observed) | \(\dfrac{M}{m}\) |
|---|---|---|---|---|
| Brode, MacPherson and Starr (1936) | 25.0 | 0.11 | \(>18\) | \(>167\) |
| Corson and Brode (1938) | 13.7 | 0.29 | \(>15\) | \(>50\) |
| Street and Stevenson (1936) | 15.0 | 0.25 | \(>7\) | \(>28\) |
| Anderson and Neddermeyer (1936) | 25.0 | 0.11 | \(<8\) | \(<73\) |
c) Determination of the mass from the curvature of the track and the range. The quantity \(H\rho\) and the range \(R\), taken separately, cannot determine the velocity of the particle. But the ratio of these quantities \(H\rho/R\) is a function of the velocity and can be calculated theoretically.
From (IV,19) we have:
\[ H\rho = 1704\,\frac{M}{m}\frac{\beta}{(1-\beta^2)^{1/2}} \]
and from (IV,20):
\[ R = \frac{M}{m}\, g\left(\frac{\beta}{(1-\beta^2)^{1/2}}\right). \]
Consequently,
\[ \frac{H\rho}{R} = 1704\, \frac{\beta}{(1-\beta^2)^{1/2}} \bigg/ g\left(\frac{\beta}{(1-\beta^2)^{1/2}}\right) = h[\beta/(1-\beta^2)^{1/2}] \tag{IV,21} \]
where \(H\rho\) is expressed in oersted-cm, and \(R\) in cm of air. The function \(\mu\{\beta/[(1-\beta^2)^{1/2}]\}\) is easily calculated. Its computed values are represented graphically by curve 4 in Fig. 20. Thus, from the measured values of \(H\rho\) and \(R\), one can, from curve 4, find the corresponding quantity \(\beta/(1-\beta^2)^{1/2}\), and then determine \(M/m\) from (IV,19) or (IV,20).
Of all the methods described here for determining the mass, the latter is the most accurate, since the curvature and range can be determined rather reliably from measurements of sharp tracks. However, often the measurements make it possible to establish only the lower limit of the range. In these cases the indicated method gives only an upper limit for the particle mass. Table XV gives values of the mesotron mass \((M/m)\), obtained by various authors from measurements of curvature and range. The rather good agreement of the results obtained confirms the accuracy of this method.
Table XV
Mass of the mesotron found from measurements of curvature and range
| Author | \(H\rho\) in \(10^4\) oersted-cm | \(R\) in cm of air | \(\dfrac{H\rho}{R}\cdot 10^{-3}\) | \(\dfrac{H\rho}{M/m}\cdot 10^{-2}\) | \(M/m\) |
|---|---|---|---|---|---|
| Brode, Macpherson and Starr (1936) | 5.5 | \(>18\) | \(<3.06\) | \(>2.7\) | \(<204\) |
| Corson and Brode (1938) | 5.5 | \(>4\) | \(<13.5\) | \(>1.55\) | \(<350\) |
| Nishina and Takeuchi (1937) | 3.87 | 6.5 | 5.95 | 2.1 | 184 |
| Anderson and Neddermeyer (1936) | 1.74 | 770 | 0.226 | 220 |
g) Determination of the mass from the change in curvature.
In Chapter IV, § 2, a method was described for measuring the loss of momentum of a particle, based on the use of metal partitions inside a Wilson chamber. The magnitude of the change in momentum is an important parameter that makes it possible to determine the particle mass. This method was used by Corson and Brode (1938) and was considered in detail by Wheeler and Ladenburg (1941).
From equations (IV,11) and (V,11), for \(Z=1\) we have:
\[ \frac{d(H\rho)}{dx} = \frac{1}{300}\,c\,\frac{dp}{dx} = \frac{1}{300}\,\frac{1}{\beta}\,\frac{dE}{dx} = i\left[\beta(1-\beta^2)^{1/2}\right]. \tag{IV,22} \]
Since the dependence of \((dE/dx)_{\mathrm{ion}}\) on \(\beta/(1-\beta^2)^{1/2}\) is known (V,9), the equation given makes it possible to calculate the value \(d(H\rho)/dx\) corresponding to some specified value \(\beta(1-\beta^2)^{1/2}\). The values of \(d(H\rho)/dx\) obtained in this way are given in Table XXV.
and are graphically represented in Fig. 21. Using this curve, from the measured value of \(d(H\rho)/dx\) one can find the corresponding value of \(\beta/(1-\beta^2)^{1/2}\) and then, from equation (IV,19), determine the mass of the particle \(M/m\). The values of the mesotron mass obtained by this method by Anderson and Neddermeyer (1936) are given in Table XVI.
Fig. 21. Curve of the dependence of \(d(H\rho)/dx\) on \(\beta(1-\beta^2)^{1/2}\).
d) Determination of mass from observations of elastic collisions. The method of determining the mass from data obtained by observing the elastic collision of a particle with an electron is the most direct. In Appendix 3a it is shown that if \(T\) is the kinetic energy transferred to the electron in collision with a primary particle of mass \(M\) and momentum \(p\), and \(\theta\) is the angle between the directions of motion of the primary particle and the knocked-on electron, then the following relation holds:
Table XVI
Mesotron mass obtained from measurements of the change of curvature
| \((H\rho)_{\mathrm{av}}\) in \(10^5\) oerst.·cm | \(\dfrac{d(H\rho)}{dx}\cdot 10^{-5}\) | \(M/m\) |
|---|---|---|
| 3.27 | 0.67 | 161 |
| 2.20 | 2.67 | 250 |
| 6.15 | 0.522 | 226 |
\[ T = 2mc^2 \frac{p^2c^2\cos^2\theta}{\left\{mc^2 + (p^2c^2 + M^2c^4)^{1/2}\right\}^2 - p^2c^2\cos^2\theta}. \tag{IV,23} \]
The tracks of the primary particle and of the secondary electron are observed with the aid of a Wilson chamber placed in a strong magnetic field. The quantities \(p\) and \(T\), found from measurements of the curvature of the primary particle and the secondary electron, and the angle \(\theta\), make it possible to determine the mass \(M\) of the primary particle.
Compared with those described earlier, the advantage of this method consists in the fact that it is based on the fundamental laws of physics—the laws of conservation of energy and momentum—whereas in the preceding methods formulas of limited significance are used. However, the applicability of this method is limited by the fact that elastic collisions of particles with electrons occur extremely rarely. Recently Juz (1941) and Lépore-Ringe with collaborators (1941) obtained photographs of elastic collisions of a mesotron with an electron. The mass of the primary particles, found from these photographs by means of the method described, is respectively \(180\,m\) and \(240\,m\).
V. APPENDICES
§ 1. Vapor pressure of water and ethyl alcohol at different temperatures and different compositions of mixtures
Table XVII
Pressure of water vapor and ethyl-alcohol vapor at different temperatures (Landolt und Börnstein, Phys. Chem. Tabellen)
| Temperature in degrees C | Pressure in mm Hg: water | Pressure in mm Hg: ethyl alcohol | Temperature in degrees C | Pressure in mm Hg: water | Pressure in mm Hg: ethyl alcohol |
|---|---|---|---|---|---|
| 35 | 42,188 | 5 | 6,54 | 17,70 | |
| 30 | 31,834 | 78,41 | 4 | 6,01 | 16,62 |
| 28 | 28,35 | 70,09 | 3 | 5,68 | 15,69 |
| 25 | 23,76 | 59,03 | 2 | 5,29 | 14,60 |
| 24 | 22,38 | 55,70 | 1 | 4,92 | 13,65 |
| 23 | 21,07 | 52,54 | 0 | 4,58 | 12,73 |
| 22 | 19,83 | 49,54 | −1 | 4,25 | |
| 21 | 18,65 | 46,69 | −2 | 3,95 | |
| 20 | 17,54 | 44,00 | −2,8 | 9,49 | |
| 19 | 16,48 | 41,45 | −3,0 | 3,67 | |
| 18 | 15,48 | 39,05 | −4,0 | 3,40 | |
| 17 | 14,53 | 36,77 | −5,0 | 3,16 | |
| 16 | 13,64 | 34,62 | −6,0 | 2,93 | |
| 15 | 12,79 | 32,60 | −7,0 | 2,71 | |
| 14 | 11,99 | 30,69 | −8,0 | 2,51 | |
| 13 | 11,23 | 28,89 | −9,0 | 2,32 | |
| 12 | 10,52 | 27,19 | −10,0 | 2,14 | 6,47 |
| 11 | 9,84 | 25,59 | −10,6 | 5,20 | |
| 10 | 9,21 | 24,08 | −11,0 | 1,98 | |
| 9 | 8,61 | 22,66 | −12,0 | 1,83 | |
| 8 | 8,05 | 21,31 | −13,0 | 1,68 | |
| 7 | 7,51 | 20,04 | −16,5 | 3,23 | |
| 6 | 7,01 | 18,84 | −24,6 | 1,72 |
Table XVIII
Vapor pressure of alcohol and of water vapor at different concentrations of the water–ethyl alcohol mixture
| Percentage content of ethyl alcohol (by weight) | 20°C: Water | 20°C: Ethyl alcohol | 40°C: Water | 40°C: Ethyl alcohol |
|---|---|---|---|---|
| 0 | 17.5 | 0.0 | 54.3 | 0.0 |
| 10 | 16.8 | 6.7 | 51.6 | 26.9 |
| 20 | 15.9 | 12.6 | 47.6 | 43.5 |
| 30 | 15.1 | 17.1 | 46.2 | 54.7 |
| 40 | 14.7 | 20.7 | 45.5 | 62.5 |
| 50 | 14.5 | 23.5 | 44.6 | 68.2 |
| 60 | 14.1 | 25.6 | 42.9 | 74.8 |
| 70 | 13.1 | 28.0 | 40.5 | 82.8 |
| 80 | 11.3 | 31.2 | 35.9 | 91.8 |
| 90 | 7.5 | 35.8 | 24.7 | 106.4 |
| 98 | 1.9 | 42.4 | 6.5 | 123.0 |
| 100 | 0.0 | 43.6 | 0.0 | 134.0 |
Table XIX
Values of the quantity \(C_p/C_v\)
| Substance | \(k = C_p/C_v\) | Substance | \(k = C_p/C_v\) |
|---|---|---|---|
| Air | 1.401 | Methyl alcohol | 1.256 |
| Argon | 1.667 | Ethyl alcohol | 1.133 |
| CO\(_2\) | 1.300 | Water vapor | 1.305 |
§ 2. Energy, range, velocity, and the value of \(H\rho\) for electrons, protons, and alpha particles
Table XX
Energy, range, velocity, and magnetic curvature for electrons
| \(\beta\) | \(p/Mc\) or \(\beta(1-\beta^2)^{-1/2}\) | Energy in MeV | \(H\rho\) in \(10^5\) oersted·cm | Range in cm of air at 76 cm Hg and 15° C | Range in g/cm² Al |
|---|---|---|---|---|---|
| 0,10 | 0,1005 | 0,00257 | 0,00171 | 0,04 | 0,00005 |
| 0,15 | 0,1517 | 0,00584 | 0,00258 | ||
| 0,20 | 0,2042 | 0,01053 | 0,00348 | 0,23 | 0,00027 |
| 0,25 | 0,2582 | 0,01675 | 0,00440 | ||
| 0,30 | 0,3145 | 0,02466 | 0,00536 | 1,00 | 0,0012 |
| 0,35 | 0,3736 | 0,03448 | 0,00636 | ||
| 0,40 | 0,4364 | 0,04652 | 0,00743 | 3,40 | 0,0041 |
| 0,45 | 0,5039 | 0,06117 | 0,00858 | ||
| 0,50 | 0,5773 | 0,07900 | 0,00893 | 7,3 | 0,0087 |
| 0,525 | 0,6168 | 0,08934 | 0,01051 | ||
| 0,550 | 0,6585 | 0,1008 | 0,01122 | ||
| 0,575 | 0,7276 | 0,1135 | 0,01197 | ||
| 0,600 | 0,750 | 0,1277 | 0,01278 | 17,9 | 0,0215 |
| 0,625 | 0,800 | 0,1435 | 0,01365 | ||
| 0,650 | 0,8554 | 0,1613 | 0,01458 | ||
| 0,675 | 0,915 | 0,1815 | 0,01559 | ||
| 0,700 | 0,980 | 0,2044 | 0,01670 | 37,5 | 0,045 |
| 0,725 | 1,052 | 0,2308 | 0,01793 | ||
| 0,750 | 1,134 | 0,2614 | 0,01931 | ||
| 0,775 | 1,23 | 0,2984 | 0,02089 | ||
| 0,800 | 1,33 | 0,3404 | 0,02271 | 83,0 | 0,10 |
| 0,810 | 1,38 | 0,3602 | 0,02353 | ||
| 0,820 | 1,43 | 0,3816 | 0,02440 | ||
| 0,830 | 1,49 | 0,4049 | 0,02535 | ||
| 0,840 | 1,55 | 0,4305 | 0,02637 | ||
| 0,850 | 1,61 | 0,4587 | 0,02749 | ||
| 0,860 | 1,69 | 0,4901 | 0,02871 | ||
| 0,870 | 1,77 | 0,5251 | 0,03006 | ||
| 0,880 | 1,85 | 0,5645 | 0,03156 | ||
| 0,890 | 1,95 | 0,6093 | 0,03325 | ||
| 0,900 | 2,06 | 0,6609 | 0,03517 | 217,0 | 0,26 |
| 0,910 | 2,195 | 0,7230 | 0,03739 | ||
| 0,920 | 2,34 | 0,7923 | 0,03999 | ||
| 0,930 | 2,53 | 0,8787 | 0,04310 | ||
| 0,940 | 2,76 | 0,9861 | 0,04693 | ||
| 0,950 | 3,04 | 1,125 | 0,05182 | 437,0 | 0,525 |
| 0,960 | 3,43 | 1,313 | 0,05845 | ||
| 0,970 | 3,98 | 1,590 | 0,06797 | ||
| 0,980 | 4,93 | 2,056 | 0,08389 | 860,0 | 1,03 |
| 0,990 | 7,018 | 3,109 | 0,11950 | 1300,0 | 1,57 |
| 0,995 | 9,962 | 4,602 | 0,16970 | ||
| 0,996 | 11,147 | 5,204 | 0,19060 | ||
| 0,997 | 12,881 | 6,087 | 0,21940 | ||
| 0,998 | 15,788 | 7,568 | 0,26990 | ||
| 0,999 | 22,344 | 10,911 | 0,38060 |
Table XXI
Energy, range, and velocity for alpha particles
| Velocity in \(10^9\) cm/sec | \(\beta^2 \cdot 10^3\) | \(P/Mc\) or \(\beta(1-\beta^2)^{-1/2}\) | Energy in MeV | \(H\rho\) in \(10^5\) oersted·cm | Range in cm of air at 76 cm Hg and 15° C |
|---|---|---|---|---|---|
| 0,75 | 0,626 | 0,0249 | 1,167 | 1,555 | 0,55 |
| 0,80 | 0,712 | 0,0266 | 1,328 | 1,659 | 0,62 |
| 0,85 | 0,804 | 0,0282 | 1,499 | 1,763 | 0,70 |
| 0,90 | 0,901 | 0,030 | 1,681 | 1,867 | 0,90 |
| 0,95 | 1,004 | 0,0316 | 1,873 | 1,971 | 0,91 |
| 1,00 | 1,113 | 0,0332 | 2,075 | 2,074 | 1,04 |
| 1,05 | 1,227 | 0,0348 | 2,288 | 2,178 | 1,18 |
| 1,10 | 1,346 | 0,0366 | 2,511 | 2,282 | 1,32 |
| 1,15 | 1,471 | 0,0382 | 2,745 | 2,386 | 1,48 |
| 1,20 | 1,602 | 0,0398 | 2,989 | 2,490 | 1,67 |
| 1,25 | 1,739 | 0,0415 | 3,244 | 2,594 | 1,87 |
| 1,30 | 1,881 | 0,0432 | 3,509 | 2,698 | 2,09 |
| 1,35 | 2,028 | 0,0447 | 3,785 | 2,802 | 2,33 |
| 1,40 | 2,181 | 0,0464 | 4,071 | 2,906 | 2,58 |
| 1,45 | 2,340 | 0,0480 | 4,368 | 3,010 | 2,86 |
| 1,50 | 2,504 | 0,0497 | 4,674 | 3,114 | 3,17 |
| 1,55 | 2,673 | 0,0515 | 4,991 | 3,218 | 3,50 |
| 1,60 | 2,849 | 0,0531 | 5,319 | 3,322 | 3,85 |
| 1,65 | 3,029 | 0,0548 | 5,658 | 3,426 | 4,24 |
| 1,70 | 3,216 | 0,0565 | 6,008 | 3,530 | 4,65 |
| 1,75 | 3,408 | 0,0581 | 6,367 | 3,635 | 5,09 |
| 1,80 | 3,605 | 0,0598 | 6,737 | 3,739 | 5,57 |
| 1,85 | 3,808 | 0,0614 | 7,117 | 3,843 | 6,08 |
| 1,90 | 4,017 | 0,0632 | 7,508 | 3,947 | 6,62 |
| 1,95 | 4,231 | 0,0648 | 7,910 | 4,052 | 7,20 |
| 2,00 | 4,451 | 0,0666 | 8,322 | 4,156 | 7,82 |
| 2,05 | 4,677 | 0,0681 | 8,745 | 4,260 | 8,48 |
| 2,10 | 4,907 | 0,0700 | 9,178 | 4,369 | 9,18 |
| 2,15 | 5,144 | 0,0714 | 9,622 | 4,469 | 9,92 |
| 2,20 | 5,386 | 0,0730 | 10,077 | 4,574 | 10,71 |
| 2,25 | 5,633 | 0,0749 | 10,543 | 4,678 | 11,54 |
| 2,30 | 5,887 | 0,0765 | 11,018 | 4,783 | 12,42 |
| 2,35 | 6,145 | 0,0780 | 11,504 | 4,888 | 13,43 |
| 2,40 | 6,410 | 0,0800 | 12,001 | 4,992 | 14,32 |
| 2,45 | 6,680 | 0,0821 | 12,508 | 5,137 | 15,35 |
| 2,50 | 6,955 | 0,0831 | 13,027 | 5,202 | 16,44 |
Table XXII
Energy, range, and velocity for protons
| \(V_0\) in \(10^9\) cm/sec | \(\beta^2 \cdot 10^3\) | \(P/Mc\) or \(\beta(1-\beta^2)^{1/2}\) | Energy in MeV | \(H\rho\) in \(10^5\) oersted·cm | Range in cm of air at 76 cm Hg and \(15^\circ\) C |
|---|---|---|---|---|---|
| 1.0 | 1.113 | 0.03334 | 0.522 | 1.044 | 0.8 |
| 1.2 | 1.602 | 0.04005 | 0.753 | 1.253 | 1.4 |
| 1.4 | 2.181 | 0.04675 | 1.025 | 1.463 | 2.3 |
| 1.6 | 2.849 | 0.05343 | 1.340 | 1.673 | 3.6 |
| 1.8 | 3.605 | 0.06014 | 1.697 | 1.883 | 5.3 |
| 2.0 | 4.451 | 0.0668 | 2.095 | 2.093 | 7.5 |
| 2.2 | 5.386 | 0.0737 | 2.536 | 2.304 | 10.4 |
| 2.4 | 6.410 | 0.0804 | 3.021 | 2.514 | 14.0 |
| 2.6 | 7.523 | 0.0872 | 3.649 | 2.725 | 18.4 |
| 2.8 | 8.724 | 0.0938 | 4.120 | 2.936 | 23.9 |
| 3.0 | 10.015 | 0.1008 | 4.734 | 3.148 | 30.4 |
| 3.2 | 11.395 | 0.1077 | 5.391 | 3.360 | 38.2 |
| 3.4 | 12.863 | 0.1142 | 6.092 | 3.573 | 47.4 |
| 3.6 | 14.442 | 0.1212 | 6.838 | 3.786 | 58.2 |
| 3.8 | 16.063 | 0.1280 | 7.628 | 4.000 | 70.5 |
| 4.0 | 17.804 | 0.1350 | 8.463 | 4.214 | 84.8 |
| 4.2 | 19.629 | 0.1420 | 9.343 | 4.428 | 101.3 |
| 4.4 | 21.543 | 0.1482 | 10.270 | 4.643 | 119.9 |
| 4.6 | 23.547 | 0.1558 | 11.240 | 4.858 | 140.8 |
| 4.8 | 25.638 | 0.1622 | 12.250 | 5.075 | 164.6 |
| 5.0 | 27.819 | 0.1690 | 13.310 | 5.293 | 191.1 |
§ 3. Dependence of specific ionization, range, and change of momentum on the velocity of the particle
a) Specific ionization. For fast particles the mean loss of energy to ionization and excitation is expressed by the following formula [Rossi and Greisen (1941)]:
\[ \left(-\frac{dE}{dx}\right)_{\mathrm{ion}} = \frac{2\pi n e^4 Z^2}{mV^2} \left\{ \ln \frac{2mv^2 T_m}{I^2(1-v^2/c^2)} - 2\frac{v^2}{c^2} \right\}, \tag{V,1} \]
where \(M\) is the mass of the particle, \(v=\beta c\) is its velocity, \(Ze\) is its charge, \(n\) is the number of electrons in \(1\ \mathrm{cm}^3\) of the retarding medium, \(I\) is the mean excitation energy, equal for air to \(11.5\,Z\) or \(82.6\ \mathrm{eV}\) (Wilson (1941)), and \(T_m\) is the maximum kinetic energy that can be transferred to an electron in an elastic collision. In the formula given, the mass of the primary particle enters only through \(T_m\); in other words, the energy loss is practically independent of \(M\).
The maximum energy \(T_m\) can be determined by using the law of conservation of energy and momentum. If \(E\) is the total energy of the primary particle, and \(T\) is the kinetic energy which it transfers to the electron in the collision, then the law of conservation of energy gives:
\[ (E-T)^2 = E_r^2 = (p_r c)^2 + M^2 c^4, \tag{V,2} \]
where \(E_r\) and \(p_r\) are the energy and momentum of the primary particle after the collision.
Next, from the law of conservation of momentum it follows:
\[ p_r^{\,2}c^2=p_e^{\,2}c^2+p^2c^2-2pcp_ec\cos\theta, \tag{V,3} \]
where \(p_e\) is the momentum of the electron after the collision, and \(\theta\) is the angle between the direction of its emission and the trajectory of the primary particle.
Eliminating, with the aid of this equation, \(p_r\) from (V,2) and expressing \(p_e\) in terms of the kinetic energy \(T\) of the electron, we obtain:
\[ T\left(E+mc^2\right)=pc\cos\theta\left[T\left(T+mc^2\right)\right]^{1/2}. \tag{V,4} \]
Or, after simplification:
\[ T=2mc^2\frac{p^2c^2\cos^2\theta}{\left(E+mc^2\right)^2-p^2c^2\cos^2\theta}, \tag{V,5} \]
whence for the maximum energy we have:
\[ T_m=2mc^2\frac{p^2c^2}{m^2c^2+M^2c^4+2Emc^2}. \tag{V,6} \]
For \(M\gg m\) and \(E\ll \left(\frac12 M/m\right)Mc^2\)
\[ T_m=2mc^2\frac{p^2}{M^2c^2}=2mc^2\frac{\beta^2}{1-\beta^2}. \tag{V,7} \]
The critical energy \(E_c\), below which equation (V,7) remains valid, is \(10^{12}\) eV for protons and \(10^{10}\) eV for mesotrons with mass 200. Consequently, for all practical purposes it may be assumed that the quantity \(T_m\) is given by equation (V,7) and does not depend on the mass of the primary particle, being a function only of its velocity.
Substituting the value of \(T_m\) from (V,7) into (V,1), we obtain:
\[ -\frac{dE}{dx}=\frac{4\pi ne^4Z^2}{m\beta^2c^2} \left\{\ln\frac{2m\beta^2c^2}{I(1-\beta^2)}-\beta^2\right\}. \tag{V,8} \]
Substituting here the numerical values for air \(n=3.9\cdot10^{20}\), \(I=82.6\) eV [Wilson (1941)] and putting \(Z=1\), we obtain the following expression for \(dE/dx\) in eV per 1 cm of air:
\[ -\frac{dE}{dx}=\frac{2\cdot10^2}{\beta^2} \left\{9.43+2\ln\frac{\beta}{(1-\beta^2)^{1/2}}-\beta^2\right\}. \tag{V,9} \]
Table XXIII
Values of the quantity \(\left(-dE/dx\right)\) as a function of \(p/Mc\)
| \(p/Mc\) or \(\beta(1-\beta^2)^{-1/2}\) | \(-\left(dE/dx\right)\) in \(10^3\) eV per cm of air | \(p/Mc\) or \(\beta(1-\beta^2)^{-1/2}\) | \(-\left(dE/dx\right)\) in \(10^3\) eV per cm of air |
|---|---|---|---|
| 0.126 | 45.2 | 0.894 | 3.28 |
| 0.18 | 25.9 | 1.15 | 2.83 |
| 0.255 | 14.8 | 1.39 | 2.69 |
| 0.315 | 10.8 | 1.61 | 2.52 |
| 0.366 | 8.7 | 2.68 | 2.41 |
| 0.411 | 7.4 | 4.74 | 2.49 |
| 0.60 | 4.68 |
Table XXIV
Values of the quantity \(RZ^2(M/m)\) as a function of \(p/Mc\)
| \(p/Mc\) or \(\beta/(1-\beta^2)^{1/2}\) | \(RZ^2(M/m)\) \(R\) expressed in \(10^{-2}\) cm of air, under normal conditions |
Notes |
|---|---|---|
| 0.02 | 0.05 | According to the data of Livingston and Bethe (1937) |
| 0.04 | 0.10 | According to the data of Livingston and Bethe (1937) |
| 0.06 | 0.275 | According to the data of Livingston and Bethe (1937) |
| 0.08 | 0.70 | According to the data of Livingston and Bethe (1937) |
| 0.10 | 1.60 | According to the data of Livingston and Bethe (1937) |
| 0.12 | 3.0 | According to the data of Livingston and Bethe (1937) |
| 0.14 | 5.25 | According to the data of Livingston and Bethe (1937) |
| 0.16 | 8.40 | According to the data of Livingston and Bethe (1937) |
| 0.18 | 12.35 | According to the data of Livingston and Bethe (1937) |
| 0.25 | 21.00 | From the approximate formula of Euler and Heisenberg |
| 0.30 | 41 | From the approximate formula of Euler and Heisenberg |
| 0.35 | 61 | From the approximate formula of Euler and Heisenberg |
| 0.40 | 102 | From the approximate formula of Euler and Heisenberg |
| 0.45 | 183 | From the approximate formula of Euler and Heisenberg |
| 0.50 | 265 | From the approximate formula of Euler and Heisenberg |
Table XXV
Values of \(d(H\rho)/dx\) in lead as a function of \(p/Mc\)
| \(p/Mc\) or \(\beta/(1-\beta^2)^{1/2}\) | \(\beta\) | \(d(H\rho)/dx \times 10^5\) oersted·cm per 1 cm of lead |
|---|---|---|
| 0.3 | 0.288 | 8.33 |
| 0.4 | 0.372 | 4.33 |
| 0.5 | 0.449 | 2.71 |
| 0.6 | 0.515 | 1.91 |
| 0.7 | 0.574 | 1.45 |
| 0.8 | 0.625 | 1.15 |
| 0.9 | 0.667 | 0.94 |
| 1.0 | 0.706 | 0.80 |
| 1.1 | 0.739 | 0.72 |
| 1.2 | 0.768 | 0.67 |
| 1.3 | 0.793 | 0.62 |
| 1.4 | 0.814 | 0.56 |
| 1.5 | 0.832 | 0.545 |
| 1.6 | 0.848 | 0.543 |
| 1.8 | 0.874 | 0.542 |
The quantities \((-dE/dx)\), calculated for different values of \(p/Mc\) or \(\beta/(1-\beta^2)^{1/2}\), are given in Table XXIII.
The described method for calculating the energy loss is valid for all particles for which losses due to radiation may be neglected. In Fig. 20
the magnitude of the losses is represented graphically (curve 1). This curve can be used to determine the mass of a particle from the ionization it produces (see IV, § 4).
b) Calculation of the range as a function of \(p/Mc\). Table XXIV gives the values of \(RZ^2/(M/m)\) as a function of \(p/Mc\), or \(\beta/(1-\beta^2)^{1/2}\). The first nine values are taken from the curve of Livingston and Bethe (1937). They were obtained from experimental determinations of the range of alpha particles and protons. To obtain the dependence of the range of fast cosmic particles on their energy, we use the approximate calculation of Euler and Heisenberg (1938). Neglecting in equation (V,8) the weak dependence of \(dE/dx\) on \(\ln[\beta/(1-\beta^2)^{1/2}]\), one may write approximately
\[ -dE/dx=\frac{aZ^2}{\beta^2}, \tag{V,10} \]
where \(a=2.5\cdot 10^3\) eV per 1 cm of air. Then for the range we have:
\[ \begin{gathered} R=\int \frac{dE}{-dE/dx}=\frac{1}{aZ^2}\int \beta^2\,dE,\\ \text{and since } E^2=p^2c^2+M^2c^4,\text{ then:}\\ dE=\frac{pc^2}{E}\,dp=\beta c\,dp. \end{gathered} \tag{V,11} \]
Consequently,
\[ \begin{aligned} R&=\frac{c}{aZ^2}\int_0^p \beta^3\,dp =\frac{c}{aZ^2}\int_0^p \frac{p^3\,dp}{(p^2+M^2c^2)^{3/2}}\\ &=\frac{c}{aZ^2}\left[\frac{2M^2c^2+p^2}{(M^2c^2+p^2)^{1/2}}\right]_0^p =\frac{Mc^2}{aZ^2}\left\{\frac{2+(p/Mc)^2}{[1+(p/Mc)^2]^{1/2}}-2\right\}. \end{aligned} \tag{V,12} \]
A graph of the dependence of the quantity \(RZ^2/(M/m)\) on \(\beta/(1-\beta^2)^{1/2}\) is shown in Fig. 20. These data can be used to determine the mass of a particle (see IV, § 4).
c) Calculation of \(d(H\rho)/dx\) as a function of \(\beta/(1-\beta^2)^{1/2}\). Using the calculated values of \(dE/dx\), with the aid of equation (IV,22) the quantity \(d(H\rho)/dx\) can easily be expressed as a function of \(\beta(1-\beta^2)^{1/2}\). The data obtained in this way are given in Table XXV.
Thus, measurement of the loss of momentum occurring when a particle passes through lead placed in a Wilson chamber makes it possible to determine the mass of the particle. Examples of the use of this method were given in IV, § 4.
§ 4. Basic relations for Compton scattering of \(\gamma\)-rays. The elementary theory of the Compton effect is well known. Let \(h\nu\) and \(h\nu'\) be, respectively, the energies of the incident and scattered quantum, and let \(\theta\) and \(\varphi\) be the angles which the scattered quantum and the recoil electron make with the direction of motion of the primary quantum,
Then, applying the conservation laws, we can write:
\[ h\nu=h\nu' + mc^2\left(\frac{1}{(1-\beta^2)^{1/2}}-1\right), \tag{V,13} \]
\[ \frac{h\nu}{c}=\frac{h\nu'}{c}\cos\theta+\frac{m\beta c}{(1-\beta^2)^{1/2}}\cos\varphi, \tag{V,14} \]
\[ 0=\frac{h\nu'}{c}\sin\theta-\frac{m\beta c}{(1-\beta^2)^{1/2}}\sin\varphi . \tag{V,15} \]
Introducing the notation
\[ h\nu/mc^2=\gamma;\quad h\nu'/mc^2=\gamma_1;\quad \frac{1}{(1-\beta^2)^{1/2}}-1=T/mc^2=\gamma_2; \tag{V,16} \]
equations (V,13), (V,14), and (V,15) can be rewritten in the following form:
\[ \gamma=\gamma_1+\gamma_2;\quad \gamma=\gamma_1\cos\theta+(\gamma_2^2+2\gamma_2)^{1/2}\cos\varphi; \]
\[ 0=\gamma_1\sin\theta-(\gamma_2^2+2\gamma_2)^{1/2}\sin\varphi . \tag{V,17} \]
From this we obtain:
\[ \gamma_1=\frac{\gamma}{1+\gamma(1-\cos\theta)}, \tag{V,18} \]
\[ \gamma_2=\frac{\gamma^2(1-\cos\theta)}{1+\gamma(1-\cos\theta)} =\frac{2\gamma^2}{1+2\gamma+(1+\gamma)^2\operatorname{tg}^2\varphi}, \tag{V,19} \]
\[ \operatorname{ctg}\varphi=(1+\gamma)\operatorname{tg}\frac{\theta}{2}. \tag{V,20} \]
When \(\theta\) changes from \(0^\circ\) to \(180^\circ\), the angle \(\varphi\) changes from \(90^\circ\) to \(0^\circ\). As \(\varphi\) decreases from \(90^\circ\) to \(0^\circ\), the energy transferred to the electron,
Fig. 22. Differential transverse emission cross section of Compton electrons as a function of \(E_\varphi\) for \(h\nu=2.5\cdot 10\,\mathrm{eV}\).
increases from zero to the maximum value
\[ \frac{2\gamma^2}{1+2\gamma}, \]
i.e., to \(2\gamma(1+2\gamma)\) of the energy of the primary quantum. For small \(\gamma\), an insignificant amount of energy is transferred to the electron.
The differential cross section for the deflection of a quantum through an angle lying between \(\theta\) and \(\theta+d\theta\), calculated per unit solid angle and per electron, is expressed by the formula
\[ \sigma(\gamma,\theta)= \frac{r_0^2}{2}\, \frac{(1+\cos^2\theta)}{\{1+\gamma(1-\cos\theta)\}^2} \left\{ 1+ \frac{\gamma^2(1-\cos\theta)^2}{(1+\cos^2\theta)} \cdot \frac{1}{1+\gamma(1-\cos\theta)} \right\}, \tag{V,21} \]
where \(r_0=e^2/mc^2\).
The differential cross section for the emission of a Compton electron in the direction \(\varphi\), referred to unit solid angle and to one electron, is equal to
\[ \sigma(\varepsilon,\varphi)= \frac{[2(1+x)]^{1/2}}{\pi}\, \frac{32r_0^2(1+\gamma)^2}{\{\gamma^2+4\gamma+2-\gamma^2\cos2\varphi\}^2} \times \]
\[ \times \left[ 1+ \frac{2(1+\cos2\varphi)} {(\gamma^2+2\gamma+2)-(\gamma^2+2\gamma)\cos2\varphi} \left\{ \frac{\gamma^2(1+\cos2\varphi)} {\gamma^2+4\gamma+2-\gamma^2\cos2\varphi} - \frac{(\gamma+1)^2(1-\cos2\varphi)} {\gamma^2+2\gamma+2-(\gamma^2+2\gamma)\cos2\varphi} \right\} \right]. \tag{V,22} \]
The curve in Fig. 22 gives the dependence of the quantity \(\sigma(\varepsilon,\varphi)\) on the energy \(E_\varepsilon\) transferred to the electron. This curve corresponds to \(\gamma=5\), i.e. \(h\nu=2.5\cdot10^6\ \mathrm{eV}\). For \(\varphi=0\), \(\sigma(\varepsilon\varphi)\) and \(E_\varepsilon\) have their maximum value. The energy transferred to the electron is in this case equal to
\[ (\gamma_\varepsilon)_{\max}=\frac{2\gamma^2}{1+2\gamma}. \tag{V,23} \]
Table XXVI
Values of \(H\rho\) of Compton electrons and the corresponding electron and \(\gamma\)-ray energies
| \(H\rho\) in \(10^3\) oersted·cm |
\(T\) in MeV | \(\gamma\)-ray energy in MeV |
|---|---|---|
| 1.00 | 0.08 | 0.19 |
| 1.50 | 0.17 | 0.309 |
| 2.00 | 0.28 | 0.436 |
| 2.50 | 0.40 | 0.570 |
| 3.00 | 0.52 | 0.710 |
| 3.50 | 0.65 | 0.850 |
| 4.00 | 0.80 | 0.99 |
| 4.50 | 0.93 | 1.14 |
| 5.00 | 1.08 | 1.28 |
| 5.50 | 1.22 | 1.43 |
| 6.00 | 1.37 | 1.57 |
| 6.50 | 1.51 | 1.72 |
| 7.00 | 1.65 | 1.86 |
| 7.50 | 1.80 | 2.01 |
| 8.00 | 1.95 | 2.16 |
| 8.50 | 2.09 | 2.31 |
| 9.00 | 2.24 | 2.46 |
| 9.50 | 2.31 | 2.60 |
| 10.00 | 2.53 | 2.76 |
From the measured value \(H\rho\), corresponding to this maximum energy, one can determine the energy of the quantum by which the Compton electron was produced. From (V,23) we have:
\[ \gamma=\frac{\gamma_2+(\gamma_2^2+2\gamma_2)^{1/2}}{2} \tag{V,24} \]
and since
\[ T=E-mc^2=[(eH\rho)^2+m^2c^4]^{1/2}-mc^2, \]
then
\[ \gamma_2+1=[(eH\rho/mc^2)^2+1]^{1/2} \]
and
\[ \gamma_2(\gamma_2+2)=(eH\rho/mc^2)^2. \]
Using these equalities in (V,24), we finally obtain:
\[ \gamma=\frac{(y^2+1)^{1/2}+y-1}{2}, \tag{V,25} \]
where \(y=\dfrac{eH\rho}{mc^2}\).
In Table XXVI the values of \(H\rho\) for Compton electrons are compared with the values of their energies and the energies of the primary quanta. By measuring the magnetic curvature of Compton electrons and using this table, one can determine the energy of the \(\gamma\)-rays by which these electrons are knocked out.
LITERATURE
Below is given a list of works and general handbooks that were mentioned in the present article. Although this list is not exhaustive, nevertheless it includes publications illuminating all aspects of work with the Wilson chamber. Experimental works whose data were used, but to which no references were made in the text, are also included in this list. The Wilson chamber method has become so widely used in laboratories throughout the world, and the material obtained with its aid is so great, that the authors, limited by the size of the article and by time, were unfortunately unable to do justice to all the investigators working in this field.
In writing this review the authors made extensive use of materials from the general handbooks and individual papers mentioned in the text, published in various scientific journals. Many of the diagrams have been reproduced by us from originals placed in these journals. To the authors of these works we express our gratitude.
J. Aitken, Collected Scientific Papers 1880—1916 (Cambridge University Press, Cambridge, England, 1923).
T. Alper, Zschr. f. Physik 76, 172 (1932).
On the \(\delta\)-rays and the range-velocity relation of slow electrons.
C. D. Anderson, Phys. Rev. 41, 405 (1932).
Energies of cosmic-ray particles.
C. D. Anderson, Phys. Rev. 44, 406 (1933).
Cosmic-ray negative and positive electrons.
C. D. Anderson and S. H. Neddermeyer, Phys. Rev. 50, 263 (1936).
Cloud-chamber observations of cosmic rays at 4300 meters elevation and at sea level.
P. Auger, J. de phys. et rad. 6, 208 (1925).
On composite photoelectric effect.
P. Auger, and P. Ehrenfest, J. de phys. et rad. 6, 285 (1935).
Counter controlled cloud-chamber results at different altitudes.
P. Auger and P. Ehrenfest, J. de phys. et rad. 8, 204 (1937).
Showers produced by cosmic rays.
R. H. Bacon, E. N. Grisewood and C. W. Merwe, Phys. Rev. 59, 531 (1941).
The Radioactivity of Mn^56^ and I^128^.
F. Barendregt and G. F. Sizoo, Physica 6, 1077 (1939).
Cloud-chamber with electrical automatic control.
H. Bauer, Physik. Zschr. 37, 627 (1936).
Portable Wilson cloud-chamber.
J. A. Bearden, Rev. Sci. Inst. 6, 256 (1935).
Wilson chamber with an increased time of sensitivity.
C. Beck, Rev. Sci. Inst. 12, 602 (1941).
Optimum liquid combination for cloud-chamber.
R. A. Becker, W. A. Fowler and C. C. Lauritsen, Phys. Rev. 62, 186 (1942).
Short range α-particles from flourine bombarded with protons.
H. J. Bhabha, Proc. Roy. Soc. 166, 501 (1938).
On the theory of heavy electrons and nuclear forces.
P. C. Bhattacharya, Proc. Nat. Inst. Sci. Ind. 7, 275 (1941).
A search for the double proton.
P. M. S. Blackett, Proc. Roy. Soc. 102, 294 (1922).
On the analysis of α-ray photographs.
P. M. S. Blackett, Proc. Roy. Soc. 103, 62 (1923).
The study of forked α-ray tracks.
P. M. S. Blackett, Proc. Roy. Soc. 107, 349 (1925).
Ejection of protons from nitrogen nuclei.
P. M. S. Blackett, J. Sci. Inst. 4, 433 (1927).
An automatic cloud-chamber for the rapid production of α-ray photographs.
P. M. S. Blackett, J. Sci. Inst. 6, 184 (1929).
a) On the automatic use of Wilson cloud-chamber.
P. M. S. Blackett, Proc. Roy. Soc. 123, 619 (1929).
b) Double camera for artificial disintegrations.
P. M. S. Blackett, Proc. Roy. Soc. 135, 132 (1932).
On the collision loss of α-particles and H-particles.
P. M. S. Blackett, Proc. Roy. Soc. 146, 281 (1934).
Technique of counter controlled Wilson chamber.
P. M. S. Blackett, Proc. Roy. Soc. 154, 564 (1936).
The measurement of the energy of cosmic rays.
P. M. S. Blackett, Proc. Roy. Soc. 159, 1 (1937).
Measurement of cosmic-ray energy spectrum.
P. M. S. Blackett and R. Brode, Proc. Roy. Soc. 154, 573 (1936).
Curvature measurement and energy spectrum of cosmic rays.
P. M. S. Blackett and D. S. Lees, Proc. Roy. Soc. 134, 658 (1931).
Range velocity relation of α-rays.
P. M. S. Blackett and D. S. Lees, Proc. Roy. Soc. 134, 658 (1932).
a) The range velocity of recoil atoms.
P. M. S. Blackett and D. S. Lees, Proc. Roy. Soc. 136, 325 (1932).
b) On the photography of artificial disintegration collision by α-rays.
P. M. S. Blackett and G. P. S. Occhialini, Proc. Roy. Soc. 139, 699 (1933).
Some photographs of the tracks of penetrating radiations,
P. M. S. Blackett and J. G. Wilson, Proc. Roy. Soc. 160, 304 (1937).
Energy loss of cosmic-ray particles in metal plates.
P. M. S. Blackett and J. G. Wilson, Proc. Roy. Soc. 165, 290 (1938).
Scattering of cosmic-ray particles in matter.
F. Bloch, Ann. d. Physik 16, 285 (1933).
a) Stopping power of matter for swiftly moving charged particles.
F. Bloch, Zschr. f. Physik 81, 363 (1933).
b) Stopping power of atoms with several electrons.
N. Bohr, Phys. Rev. 58, 864 (1940).
Successive transformations in nuclear fission.
N. Bohr, J. K. Boggild, K. J. Brostrøm, and T. Lauritsen, Phys. Rev. 58, 839 (1940).
Velocity range relation for fission fragments.
N. Bohr, J. K. Boggild, K. J. Brostrøm, and T. Lauritsen, Phys. Rev. 59, 270 (1941). Velocity range relation for fission fragments
T. W. Bonner, Phys. Rev. 59, 237 (1941).
Determination of the energy of neutrons.
T. W. Bonner and W. M. Brubaker, Phys. Rev. 49, 778 (1936).
a) The disintegration of nitrogen by slow neutrons.
T. W. Bonner and W. M. Brubaker, Phys. Rev. 50, 308 (1936).
b) Disintegration of Be, B, and carbon by deuterons.
T. W. Bonner and L. M. Mott-Smith, Phys. Rev. 46, 258 (1934).
Energy spectrum of neutrons from the disintegration of F, B, and Be by α-particles.
D. M. Bose, B. Chaudhuri, and M. Sinha, Phys. Rev. 65, 341 (1944).
Cosmic-ray meson spectra.
W. H. Bostick, Phys. Rev. 61, 557 (1942).
Cloud-chamber photographs at 4300 meters altitude.
H. Brinkman, K. Akad. Amsterdam. Proc. 39, 1185 (1939).
a) Continuously active cloud-chamber.
H. Brinkman, Physica 6, 519 (1939).
b) Wilson chamber with several expansions per second.
R. B. Brode, Rev. Mod. Phys. 11, 222 (1939).
Specific ionization of high speed particles.
R. B. Brode, H. G. Macpherson, and M. A. Starr, Phys. Rev. 50, 587 (1936).
The heavy particle component of cosmic radiation.
K. J. Brostrøm, J. K. Boggild, and T. Lauritsen, Phys. Rev. 58, 651 (1940).
Cloud-chamber studies of fission fragment track.
K. J. Brostrøm, J. K. Boggild, and T. Lauritsen, Phys. Rev. 59, 275 (1941).
Range and straggling of fission fragments.
W. M. Brubaker and T. W. Bonner, Rev. Sci. Inst. 6, 143 (1935).
Automatic high pressure Wilson cloud-chamber.
J. Chadwick, N. Feather, and Davies, Proc. Camb. Phil. Soc. 30, 357 (1934).
J. Chadwick, N. Feather, and E. Bretscher, Proc. Roy. Soc. 163, 366 (1937).
Range, angle, projection of protons from photo-disintegration of deuterium.
F. C. Champion, Proc. Roy. Soc. 134, 672 (1932).
a) The distribution of energy in the β-ray spectrum of RaE.
F. C. Champion, Proc. Roy. Soc. 136, 631 (1932).
b) On some close collisions of fast β-particles with electrons photographed by expansion method.
F. C. Champion, Proc. Roy. Soc. 153, 353 (1936).
Scattering of fast β-particles by nitrogen nuclei.
F. L. Code, Phys. Rev. 59, 229 (1941).
Scattering of mesotrons in tungsten.
D. R. Corson and R. B. Brode, Phys. Rev. 53, 773 (1938).
The specific ionization and mass of cosmic-ray particles.
D. R. Corson and R. L. Thornton, Phys. Rev. 55, 509 (1939).
Disintegration of uranium.
Coulier, J. de Pharm. et de Chimie 22, 165 (1875).
H. R. Crane, Rev. Sci. Inst. 8, 440 (1937).
Cloud-chamber for nuclear disintegration studies.
H. R. Crane, Phys. Rev. 53, 789 (1938).
New experimental evidence for a neutrino.
H. R. Crane, E. R. Gaerttner, and J. J. Turin, Phys. Rev. 50, 302 (1936).
A cloud-chamber study of the Compton effect.
H. R. Crane and O. H. Halpern, Phys. Rev. 56, 232 (1939).
Experiments on the recoil of nucleus in β-decay.
M. Curie, J. de phys. et rad. 4, 170 (1923).
Distribution of α-ray ranges.
I. Curie and F. Joliot, J. de phys. et rad. 4, 494 (1933).
Penetrating radiation.
L. F. Curtiss, Bur. Stand. J. Research 4, 663 (1930).
A new method of analyzing α-ray photographs.
L. F. Curtiss, Bur. Stand. J. Research 8, 579 (1932).
Simplified automatic cloud-chamber.
B. R. Curtiss and J. R. Richardson, Phys. Rev. 57, 1121 (1940).
Radiations from radioactive In (116).
O. Dahl, L. R. Hafstad, and M. A. Tuve, Rev. Sci. Inst. 4, 373 (1933).
Permanently sealed chamber with metallic sylphon bellows.
P. I. Dee, Nature 133, 564 (1934).
Disintegration of the diplon.
P. I. Dee, Proc. Roy. Soc. 148, 623 (1935).
Cloud track method for artificial transmutations.
P. I. Dee and C. W. Gilbert, Proc. Roy. Soc. 149, 200 (1935).
The transmutation of heavy hydrogen investigated by the cloud track method.
P. I. Dee and C. W. Gilbert, Proc. Roy. Soc. 154, 294 (1936).
The disintegration of boron into three α-particles.
P. I. Dee and E. T. S. Walton, Proc. Roy. Soc. 141, 733 (1933).
Transmutations of Li and B.
L. A. Delsasso, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 48, 848 (1935).
Protons from the disintegration of Li by deuterons.
L. A. Delsasso, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 51, 391 (1937).
a) Energy and absorption of the γ-radiation from Li¹ + H¹.
L. A. Delsasso, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 51, 527 (1937).
b) Gamma-radiations from F bombarded with protons.
A. J. Dempster, Rev. Sci. Inst. 5, 158 (1934).
Automatic Wilson cloud-chamber of simple design.
P. Duhem and M. Margules, Zschr. f. Physik. Chemie 35, 483 (1900).
Observations on the vapor pressure of binary liquid mixtures.
J. R. Dunning, Phys. Rev. 45, 586 (1934).
Emission and scattering of neutrons.
H. Euler and W. Heisenberg, Ergeb. d. exakt. Naturwiss. 17, 1 (1938).
Theoretical considerations for the interpretations of cosmic radiation.
L. Farkas, Zschr. f. physik. Chemie 125, 236 (1927).
Rate of formation of drops in supersaturated vapor.
N. Feather, Proc. Roy. Soc. 136, 709 (1932).
Collisions of neutrons with nitrogen nuclei.
N. Feather, Proc. Roy. Soc. 141, 194 (1933).
a) Collisions of \(\alpha\)-particles with F nuclei.
N. Feather, Proc. Roy. Soc. 142, 689 (1933).
b) Collision of neutrons with light nuclei.
N. Feather and F. Nimmo, Proc. Camb. Phil. Soc. 25, 198 (1929).
Distribution of ranges of \(\alpha\)-particles.
H. Flood, Zschr. f. physik. Chemie 170, 294 (1934).
Formation of drops in supersaturated ethyl alcohol water vapor mixture.
W. A. Fowler, L. A. Delsasso, and C. C. Lauritsen, Phys. Rev. 49, 561 (1936).
Radioactive elements of low atomic numbers.
W. A. Fowler and C. C. Lauritsen, Phys. Rev. 51, 1103 (1937).
Radioactive \(\alpha\)-particles from \(\mathrm{Li}^{1} + \mathrm{H}^{2}\).
W. A. Fowler, E. R. Gaerttner, and C. C. Lauritsen, Phys. Rev. 53, 628 (1938).
\(\gamma\)-radiation from B bombarded with protons.
J. A. Froemke, C. R. Bloomquist, and E. X. Anderson, Zschr. f. physik. Chemie 166, 305 (1933).
Formation of drops in methyl alcohol water vapor mixture.
D. K. Froman and J. C. Stearns, Rev. Mod. Phys. 10, 133 (1938).
Cosmic-ray showers and bursts.
L. Fussel, Rev. Sci. Inst. 10, 321 (1939). Exhaust valve for pneumatic cloud-chamber.
E. R. Gaerttner and L. A. Pardue, Phys. Rev. 57, 386 (1940).
\(\gamma\)-radiations from N bombarded with deuterons.
T. N. Gautier and A. E. Ruark, Phys. Rev. 57, 1040 (1940).
Composition of mixed vapor in cloud-chamber.
I. A. Getting, Rev. Sci. Inst. 10, 332 (1939).
A cloud-chamber control circuit.
L. Grosev, N. Dobrotin, and J. Frank, Comptes rendus, U. S. S. R. 3—6, 289 (1936). Stereocomparator for work with cloud-chamber.
O. Hahn and F. Strassman, Naturwiss. 27, 11 (1939).
Fission tracks in Wilson chamber.
J. Halpern and H. R. Crane, Phys. Rev. 55, 260 (1939).
The internal conversion coefficient in the \(\mathrm{F}^{19} + \mathrm{H}^{1}\) reaction and measurements on the \(\gamma\)-ray spectrum.
J. Hamilton, W. Heitler, and H. W. Peng, Phys. Rev. 64, 78 (1943).
Cosmic-ray mesons.
W. D. Harkins, D. M. Gans, and H. W. Newson, Phys. Rev. 47, 52 (1935). The disintegration of the nuclei of light atoms by nitrogen.
W. E. Hazen, Rev. Sci. Inst. 13, 247 (1932). Some operating characteristics of the Wilson cloud-chamber.
W. E. Hazen, Phys. Rev. 64, 7 (1943). Electrons in equilibrium with the penetrating component of cosmic-rays in lead at 10 000 ft. and at sea level.
W. E. Hazen, Phys. Rev. 65, 67 (1944) a) Cascade showers and nuclear disintegrations.
W. E. Hazen, Phys. Rev. 65, 259 (1944) b) Average, energy loss of mesotrons in air.
G. Herzog, J. Sci. Inst. 12, 153 (1935). A large cloud-chamber.
G. Herzog, Helv. Phys. Acta 10, 68 (1937).
Wilson chamber for projection purpose.
G. Herzog, Phys. Rev. 59, 117 (1941).
Cloud track of cosmic-rays in the substratosphere.
G. Herzog and W. H. Bostick, Phys. Rev. 59, 122 (1941).
Cloud-chamber picture of cosmic-rays at 29,000 ft. altitude.
H. Hilsch, Physik. Zschr. 40, 594 (1939).
Cloud-chamber for lecture experiments.
R. Holm, Zschr. f. Physik. 101, 138 (1936).
Cloud-chamber investigation of electric discharges through gases.
D. J. Hughes, Phys. Rev. 57, 592 (1940). Positive excess and electron component in the cosmic-ray spectrum.
D. J. Hughes, Phys. Rev. 60, 414 (1941). Cloud-chamber photographs of slow mesotron pair.
L. Janossy, Phys. Rev. 64, 345 (1943). Note on the production of cosmic-ray mesons.
W. Jenstchke and F. Prankal, Physik. Zschr. 40, 706 (1939).
Nuclear disintegration products of U.
T. H. Johnson, J. G. Barry, and R. P. Shutt, Phys. Rev. 57, 1047 (1940).
Direct evidence of the proton component of cosmic radiation.
T. H. Johnson, S. D. Benedetti, and R. P. Shutt, Rev. Sci. Inst. 14, 265 (1943). A hydrostatically supported cloud-chamber of new design at high pressures.
T. H. Johnson and R. P. Shutt, Phys. Rev. 61, 380 (1942).
Track of a decaying mesotron in cloud-chamber.
F. Jolliot, J. de phys. et rad. 5, 216 (1934).
Wilson apparatus for variable pressures.
F. Jolliot, Comptes rendus 208, 647 (1939).
Trajectories of products of uranium fission.
C. C. Jones, Rev. Sci. Inst. 8, 319 (1937).
Time delay circuit for Wilson cloud-chamber.
C. C. Jones and A. E. Ruark, Am. Phil. Soc. Proc. 82, 353 (1940).
Apparatus for viewing and measurements on stereoscopic cloud-chamber photographs.
H. Jones, Rev. Mod. Phys. 11, 235 (1939). Energy distribution and positive excess of mesotrons.
H. Jones and D. J. Hughes, Rev. Sci. Inst. 11, 79 (1940).
Magnet and cloud-chamber for cosmic-ray studies.
Kiessling, Naturwiss. Verein d. Hamburg—Altona 8 (1884).
P. Kipfer, Nature 135, 431 (1935).
A high pressure Wilson cloud-chamber.
F. Kirchner and H. Neuert, Physik. Zschr. 36, 54 (1935).
On the transformation of Be by slow protons.
P. Kunze, Zschr. f. Physik. 80, 559 (1933). a) Magnetic deflections of the cosmic radiations in the Wilson chamber.
P. Kunze, Zschr. f. Physik. 83, 18 (1933). b) Investigations of cosmic-rays in the Wilson chamber.
P. Kunze, Physik. Zschr. 42, 405 (1941). A portable cloud-chamber for demonstration purposes.
F. N. D. Kurie, Rev. Sci. Inst. 3, 655 (1932). Use of Wilson chamber for measuring the range of α-particles from weak sources.
F. N. D. Kurie, Phys. Rev. 45, 904 (1934). New mode of disintegrations induced by neutrons.
T. H. Laby, Phil. Trans. Roy. Soc. 208, 445 (1908).
The supersaturation and nuclear condensation of certain organic vapors.
W. E. Lamb, Phys. Rev. 58, 696 (1940). Passage of fission fragments through matter.
W. E. Lamb, Phys. Rev. 59, 687 (1941).
Range of fission fragments.
R. M. Langer, Phys. Rev. 56, 851 (1938).
Growth of droplets in Wilson chamber.
A. Langsdorf, Rev. Sci. Inst. 10, 91 (1939). A continuously sensitive diffusion cloud-chamber.
P. Leprince-Ringuet, and J. Crussard, J. de Phys. et rad. 8, 207 (1937). Study of high energy cosmic particles in the Bellevue electromagnet.
W. K. Lewis and E. Y. Murphee, J. Am. Chem. Soc. 46, 1 (1924).
Relation between vapor pressure and vapor composition in binary mixtures of volatile liquids.
W. B. Lewis and C. E. Wynn-Williams, Proc. Roy. Soc. 136, 349 (1932).
The range of α-particles from radioactive emanations and A products.
M. S. Livingston, Am. Phys. Teach. 4, 33 (1936).
Projection cloud-chambers.
M. S. Livingston and H. Bethe, Rev. Mod. Phys. 9, 285 (1937).
Nuclear dynamics.
J. J. Livingood and G. T. Seaborg, Rev. Mod. Phys. 12, 30 (1940).
A table of induced radioactivities.
G. L. Locher, J. Frank. Inst. 216, 673 (1933). a) Cloud-chamber photographs of cosmic-ray stosse.
G. L. Locher, Rev. Sci. Inst. 7, 471 (1933).
b) Wilson cloud-chamber for portable use.
D. H. Loughridge and H. C. Trueblood, Phys. Rev. 46, 323 (1934).
Organic liquids suitable for cloud expansion works.
A. C. B. Lovell, Proc. Roy. Soc. 172, 568 (1939). Showers produced by penetrating cosmic-rays.
H. Maier-Leibnitz, Zschr. f. Physik. 112, 569 (1939).
Investigations with slow Wilson chambers.
L. Meitner, Zschr. f. Physik. 37, 481 (1926).
Long range α’s from Th C.
L. Meitner and K. Philipp, Zschr. f. Physik. 87, 484 (1934).
Further measurements with neutrons.
J. M. W. Milatz and G. A. W. Rutgers, Physica. 7, 13 (1940).
Total and specific ionization of Po α-particles.
L. Mott-Smith, Rev. Sci. Inst. 5, 346 (1934).
A high pressure Wilson chamber.
E. B. M. Murrell and C. L. Smith, Proc. Roy. Soc. 173, 410 (1939).
Transmutations of Na by deuterons.
U. Nakaya and F. Yamasiki, Proc. Roy. Soc. 148, 446 (1939).
Applications of Wilson chamber to the study of spark discharge.
S. H. Neddermeyer and C. D. Anderson, Phys. Rev. 51, 884 (1937).
Note on the nature of cosmic-rays.
H. Neurt, Physik. Zschr. 36, 629 (1935). Range measurements of fragments of a light element bombarded by fast protons.
H. Neurt, Physik. Zschr. 37, 629 (1936).
Simple Wilson chamber.
C. E. Nielson and W. M. Powell, Phys. Rev. 63, 384 (1943).
Mesotron mass and heavy tracks on Mt. Evans.
Y. Nishina, M. Takeuchi, and T. Ichimiya, Phys. Rev. 52, 1198 (1937).
On the nature of the cosmic-ray particles.
R. Peierls, Report Prog. Phys. 6, 78 (1939).
The meson.
C. F. Powell, Proc. Roy. Soc. 119, 553 (1928). Condensation phenomenon at different temperatures.
W. Powell, Phys. Rev. 58, 474 (1940).
Photon production of mesotrons.
W. Powell, Phys. Rev. 61, 670 (1942).
Stars and protons at 14,125 ft.
H. Raether, Zschr. f. Physik. 94, 567 (1935).
Gas discharge in a cloud-chamber.
H. Raether, Physik. Zschr. 37, 560 (1936).
Electrical discharge in cloud-chamber.
H. Raether, Physik. Zschr. 38, 990 (1937). Examination of electron surge in the expansion chamber.
H. Raether, Zschr. f. Physik. 110, 611 (1938). Ionizing radiation accompanying a spark discharge.
G. Rathenau, Physica 5, 427 (1938). Simple Wilson chamber for demonstration purpose.
Lord J. W. S. Rayleigh, Collected Scientific Papers 1899—1920 (Cambridge University Press, Cambridge, England), vol. I, p. 415.
W. M. Rayton and T. R. Wilkins, Phys. Rev. 51, 818 (1937). A Wilson cloud-chamber investigation of the alpha-particles from uranium.
J. R. Richardson, Phys. Rev. 53, 124 (1938). a) Radiations produced from artificially produced ratio elements.
J. R. Richardson, Rev. Sci. Inst. 9, 152 (1938). b) Valve control circuits for Wilson chamber.
J. R. Richardson, Phys. Rev. 55, 609 (1939). Radiations from radioactive substances, Au\(^{198}\), Eu\(^{152}\), Ag\(^{106}\), Cu\(^{64}\), N\(^{13}\).
J. R. Richardson and L. Emo, Phys. Rev. 53, 234 (1938). Photo-disintegration of H\(^2\) by \(\gamma\)-rays from Na\(^{24}\).
J. R. Richardson and F. N. D. Kurie, Phys. Rev. 50, 999 (1936). The radiations emitted from artificially produced radioactive substances.
H. O. W. Richardson and A. Leigh-Smith, Proc. Roy. Soc. 162, 391 (1937). \(\beta\)-rays of Ra D.
F. Richarz, Ann. d. Physik. 19, 639 (1906). The value of the ratio of specific heats for a mixture of two gases.
D. Roaf, Proc. Roy. Soc. 153, 568 (1936). Disintegration of B by \(\alpha\)-particles.
M. Rohr, The formation of images in optical instruments (Dept. of Scientific and Industrial Research, H. M. Stationary Office, London, 1920).
B. Rossi and K. Greisen, Rev. Mod. Phys. 13, 240 (1941). Cosmic-ray theory.
B. Rossi, L. Janossy, R. Rochester, and M. Bound, Phys. Rev. 58, 762 (1940). Production of secondary ionizing particles by non-ionizing agents.
Lord Rutherford, W. B. Lewis, and B. V. Bowden, Proc. Roy. Soc. 142, 347 (1933). Analysis of long range-particles from radium C′ by the magnetic focusing method.
L. Schaffer, Ann. d. Physik. 35, 619 (1939). Condensation of supersaturated vapor on ions.
G. T. Seaborg, Rev. Mod. Phys. 16, 1 (1944). Table of isotopes.
R. L. Sen Gupta, Proc. Nat. Inst. Sci. Ind. 9, 295 (1943). Specific ionization of cosmic-ray particles.
L. Seren, Phys. Rev. 62, 204 (1942). Cloud-chamber study of collision electrons in equilibrium with mesons.
T. Shimizu, Proc. Roy. Soc. 99, 425 (1921). A reciprocating expansion apparatus for detecting ionizing rays.
K. Shinohara and M. Hatoyama, Phys. Rev. 59, 461 (1941). Pair production in the field of electrons.
R. P. Shutt, S. D. Benedetti, and T. H. Johnson, Phys. Rev. 62, 552 (1942). Cloud-chamber track of a decaying mesotron.
G. C. Simpson, Quat. J. Roy. Met. Soc. 67, 99 (1941). On the formation of cloud and rain.
M. Sinha, Trans. Bose Res. Inst. 15, 191 (1943). Cloud-chamber study of shower production in lead.
G. J. Sizoo and F. Barendregt, Physica 6, 1085 (1939). Production of positrons by \(\beta\)-particles.
D. Skobelzyn, Zschr. f. Physik. 43, 354 (1927). Intensity distribution in the spectrum of γ-rays from Ra C.
D. Skobelzyn, Zschr. f. Physik. 54, 686 (1929).
On a new type of fast β-rays.
L. B. Snoddy and C. D. Bradley, Phys. Rev. 45, 432 (1934). A method for investigating electrical breakdown process.
J. C. Street and E. C. Stevenson, Rev. Sci. Inst. 7, 347 (1936).
Design and operation of counter-controlled Wilson chamber.
J. C. Street and E. C. Stevenson, Phys. Rev. 52, 1003 (1937).
New evidence for the existence of a particle of mass intermediate between the proton and electron.
J. E. Thomas and W. E. Ramsay, J. Frank. Inst. 227, 789 (1939).
Small cloud-chamber for electron showers.
G. Tohmfor and M. Volmer, Ann. d. Physik. 33, 109 (1938).
Production of condensation nuclei in the presence of electrical charges.
J. J. Thomson, Phil. Mag. 46, 528 (1898). Charge carried by Röntgen ions.
F. Trey, Physik. Zschr. 39, 343 (1938). A new radially expanding cloud-chamber.
F. Trey, Physik. Zschr. 41, 415 (1940). Production of clouds in gases saturated with water vapor by removal of heat from the vapor by conduction.
B. Trumpy, Zschr. f. Physik. 111, 338 (1939). Secondary processes of the soft and penetrating components of cosmic rays.
J. J. Turin and H. R. Crane, Phys. Rev. 52, 63 (1937).
a) The absorption of high energy electrons, part I.
J. J. Turin and H. R. Crane, Phys. Rev. 52, 610 (1937).
b) The absorption of electrons, part II.
R. E. Vollrath, Rev. Sci. Inst. 7, 409 (1936).
Continuously active cloud-chamber.
M. Volmer and A. Weber, Zschr. f. physik. Chemie 119, 277 (1926). Number of drops formed per second in supersaturated space.
M. Volmer and H. Flood, Zschr. f. physik. Chemie 170, 273 (1934). Drop formation in saturated ethyl alcohol water vapor.
H. Walke, E. J. Williams, and G. R. Evans, Proc. Roy. Soc. 171, 360 (1939).
K electron capture, nuclear isomerism and the long period activities of titanium and scandium.
C. G. Webb, Phil. Mag. 19, 927 (1935).
On the scattering of light by water drops.
J. A. Wheeler and R. Ladenburg, Phys. Rev. 60, 754 (1941).
Mass of meson by the method of momentum loss.
E. J. Williams, Proc. Camb. Phil. Soc. 35, 512 (1939).
a) Sensitive time of a Wilson expansion chamber.
E. J. Williams, Proc. Roy. Soc. 172, 194 (1939).
b) Some observations on cosmic-ray using a large randomly operated cloud chamber.
E. J. Williams and G. R. Evans, Nature 145, 818 (1940).
Transformation of mesons into electrons.
E. J. Williams and E. Pickup, Nature 141, 684 (1938).
Heavy electrons in cosmic-rays.
E. J. Williams and G. E. Roberts, Nature 145, 102 (1940).
Track of a decay electron.
E. J. Williams and F. R. Terroux, Proc. Roy. Soc. 126, 289 (1930).
Investigations on the passage of fast β particles through gas.
C. T. R. Wilson, Phil. Trans. Roy. Soc. 189, 265 (1897). Condensation of water vapor in the presence of dust free air and other gases.
C. T. R. Wilson, Phil. Trans. Roy. Soc. 192, 403 (1899). a) On the condensation nuclei produced in gases by the action of Röntgen rays, uranium rays, ultraviolet light and other agents.
C. T. R. Wilson, Phil. Trans. Roy. Soc. 193, 289 (1899).
b) On the comparative efficiency as condensation nuclei of positively and negatively charged ions.
C. T. R. Wilson, Phil. Mag. 7, 681 (1904). Condensation method of demonstrating the ionization of air.
C. T. R. Wilson, Proc. Roy. Soc. 85, 285 (1911).
Cloud-chamber technique.
C. T. R. Wilson, Proc. Roy. Soc. 87, 277 (1912).
Cloud-chamber technique.
C. T. R. Wilson, Proc. Roy. Soc. 104, 1, 192 (1923). Investigation on x-rays and β-rays by the cloud method.
C. T. R. Wilson, Proc. Roy. Soc. 142, 88 (1933).
New type of expansion cloud-chamber.
C. T. R. Wilson and J. G. Wilson, Proc. Roy. Soc. 148, 523 (1935). Falling cloud-chamber and radially expanding cloud-chamber.
J. G. Wilson, Nature 142, 73 (1938). Production of secondary electrons by cosmic-ray particles.
J. G. Wilson, Proc. Roy. Soc. 174, 73 (1940). Scattering of mesotrons in metal plates.
K. Zuber, Helv. Phys. Acta 11, 366 (1938).
Automatic Wilson cloud-chamber.
General Surveys
- K. K. Darrow, Introduction to Contemporary Physics (D. Van Nostrand Company, Inc., New York, 1939).
- H. Geiger, Handbuch der Physik (Verlagsbuchhandlung Julius Springer, Berlin, 1927), vol. 22.
- R. Glazebrooke, Dictionary of Applied Physics (Macmillan and Company, London, 1927), vol. 4.
- W. Heitler, Quantum Theory of Radiation (The Clarendon Press, Oxford, 1935).
- Lord Rutherford, J. Chadwick, and C. D. Ellis, Radiation, from Radioactive Substances (Cambridge University Press, Cambridge, England, 1935).
- M. N. Saha and B. N. Srivastava, A Treatise on Heat (Indian Press, Allahabad, India, 1935).
- M. N. Saha and N. K. Saha, A Treatise on Modern Physics (Indian Press, Allahabad, India, 1934).
- J. J. Thomson, Application of Dynamics to Physics and Chemistry (Macmillan and Company, London, 1888).
- J. J. Thomson and G. P. Thomson, Conduction of Electricity through Gases (Cambridge University Press, Cambridge, England, 1928), vol. 1.
- F. A. B. Ward, Atom Tracks (London, 1937).