THE COMPTON EFFECT.

L. Ya. Shtrum.

1. Introduction. — 2. Fundamentals of the Debye–Compton theory. — 3. Displacement of spectral lines. — 4. The Compton effect for bound electrons. — 5. The wave theory of the Compton effect. — 6. Motion of electrons. — 7. Intensity of the secondary rays.

1. According to the theory of light quanta in the form given to it by Einstein \([1]\), radiation propagates in space not in the form of waves, but in the form of limited discrete quantities of energy, quanta, enclosed within a small volume and possessing a definite amount of momentum. The energy of such a quantum is equal to \(h\nu\), and the momentum is equal to \(\frac{h\nu}{c}\), where \(h\) is Planck’s constant, \(\nu\) is the number of oscillations per second, and \(c\) is the velocity of light. The interaction of quanta with material particles takes place on the basis of the laws of conservation of energy and conservation of momentum.

This theory proved extremely useful in explaining various phenomena. For certain phenomena of interaction between matter and radiant energy it gives the only rational explanation. These include the phenomena of the photoelectric effect, in which the energy of a quantum, absorbed in its entirety, is expended in imparting energy to an electron, and—in a certain sense the reverse of the first—the phenomenon of the production of X-rays upon the stopping of moving electrons, when the kinetic energy of the stopped electron appears in the form of the energy of a quantum of X-radiation. Starting from this theory, Einstein obtained a simple and exceedingly elegant derivation of Planck’s formula for black-body radiation, and, taking into account Wien’s displacement law, also derived the quantum condition of frequency (Bohr’s second postulate) \([2]\). On the basis of the postulate concerning the momentum of a quantum, Schrödinger obtained an expression for the Doppler principle \([3]\).

Despite all the successes of the quantum theory, there remains an extensive domain of phenomena connected with the propagation of light rays and their interaction

THE COMPTON PHENOMENON

...among themselves, which can be explained only on the basis of the wave theory. Such are the phenomena of reflection, refraction, interference, and polarization. Until recently, the phenomenon of the scattering of light also belonged among the phenomena not explained by the theory of quanta. The classical theory gave a complete explanation not only of the scattering of rays with optical wavelengths: J. J. Thomson gave an electromagnetic theory of the scattering of X-rays [4], on the basis of which he succeeded in calculating the number of electrons in an atom. However, investigations carried out in recent years by Debye, Compton, and others have shown that the theory of quanta gives a more complete explanation, in agreement with experiment, of phenomena connected with the scattering of X-rays. Thus the domain of phenomena explained by the theory of quanta is expanding.

According to Thomson’s theory, an electromagnetic wave of X-rays, falling upon some body, imparts acceleration to the electrons entering into the composition of the atoms of that body; and the electrons which have received acceleration, in turn, radiate energy in the form of secondary scattered X-rays. Thomson’s classical theory leads to the following results:

1) In the scattering of X-rays, as in the scattering of light rays, the wavelength remains unchanged.

2) The energy of the secondary rays emitted by each electron does not depend on the wavelength of the primary rays and is equal to the quantity

\[ \frac{8}{3}\pi \frac{e^{4}}{m^{2}c^{4}}\,P, \qquad\ldots\ldots\ldots\ldots\ldots (1), \]

where \(e\) is the charge of the electron, \(m\) is the mass of the electron, \(c\) is the velocity of light, and \(P\) is the energy of the primary rays incident per unit time upon unit surface of the body emitting the secondary rays.

3) The intensity of the secondary rays depends on the angle of scattering and is expressed by the formula

\[ I=I_{0}\cdot \frac{1+\cos^{2}\theta}{2}, \qquad\ldots\ldots\ldots\ldots\ldots (2), \]

where \(I\) is the intensity of the secondary rays in a direction making an angle \(\theta\) with the direction of the primary rays, and \(I_{0}\) is the intensity of the rays scattered in the direction of the primary rays, i.e. at \(\theta=0\). From formula (2) it follows that at \(\theta=\pi\), \(I=I_{0}\). This means that when X-rays pass through a thin layer of some substance, the intensity of the scattered rays must be the same on both sides of the layer.

Thomson’s theory is in good agreement with experiment for X-rays of ordinary hardness. But in the investigation of scat-

scattering of very hard rays, i.e. rays with a short wavelength, and also of $\gamma$-rays, whose wavelength is still shorter, it turned out that quite noticeable deviations from the results predicted by Thomson’s theory are observed. Thus, for example, it was noted that in the scattering of hard X-rays the secondary rays prove to be softer, i.e. are distinguished by a greater wavelength [5, 6, 7]. Still earlier this phenomenon had been observed for $\gamma$-rays [8]. The intensity of the scattered hard rays assumes values that deviate from those obtained from formula (2) [9, 10]. The intensity of the secondary rays is greater at acute angles than at obtuse angles $\theta$. The total intensity of the secondary rays proves to be less than is obtained according to Thomson’s theory.

These deviations of the observed phenomena from the predictions of the classical theory led to an attempt to explain the scattering of X-rays by means of quantum theory. The quantum theory of the scattering of X-rays was proposed almost simultaneously and independently by P. Debye [11] and A. Compton [12, 13, 14, 15]. The essence of the Debye–Compton theory is as follows.

1. According to the conceptions of the theory of light quanta, the scattering of X-rays is due to collisions of radiation quanta with electrons. These collisions are entirely analogous to the phenomenon of elastic impact. Suppose that a quantum of X-rays, to which the frequency $\nu_0$ is inherent, moving in the direction $AB$ (Fig. 1), collides at the point $B$ with a stationary electron. As a result of the collision the electron is set in motion with some velocity $v$ in a direction making an angle $\varphi$ with the direction $AB$, i.e. with the direction of the primary ray. The energy of the moving electron is equal (on the assumption that the mass of the electron depends on the velocity) to the quantity

$$ E = mc^2 \left(\frac{1}{\sqrt{1-\beta^2}} - 1\right), \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (3) $$

where $E$ is the energy of the electron, $m$ is the mass of the stationary electron, $c$ is the speed of light, $\beta = \frac{v}{c}$. Since the laws of conservation of energy and conservation of momentum apply also to the collision of a quantum with an electron, when energy is imparted to the electron the energy of the quantum will decrease. If the initial energy of the quantum is equal to $h\nu_0$, then the energy of the quantum after the collision is equal to $h\nu$, where the new frequency of oscillations $\nu$ is less than $\nu_0$. From the law of conservation of energy we have:

$$ h\nu_0 = h\nu + E \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (4) $$

Substituting the value of $E$ from (3), we obtain:

$$ h\nu_0 = h\nu + mc^2 \left(\frac{1}{\sqrt{1-\beta^2}} - 1\right) \ldots \ldots \ldots \ldots \ldots (5) $$

COMPTON EFFECT

After the collision with the electron, not only the frequency of the quantum changes, but also the direction of its motion. Let the new direction of the quantum make an angle \(\theta\) with its original direction. The absolute value of the momentum of the quantum before the collision is equal to \(\dfrac{h\nu_0}{c}\), and after the collision to \(\dfrac{h\nu}{c}\). The absolute value of the momentum of the electron is equal to \(\dfrac{m\beta c}{\sqrt{1-\beta^2}}\). Since the momentum of the primary quantum is equal to the vector sum of the momenta of the secondary quantum and the electron, we have

\[ \left(\frac{m\beta c}{\sqrt{1-\beta^2}}\right)^2 = \left(\frac{h\nu_0}{c}\right)^2 + \left(\frac{h\nu}{c}\right)^2 + 2\,\frac{h\nu_0}{c}\cdot\frac{h\nu}{c}\cdot\cos\theta \qquad\ldots\ldots (6) \]

Fig. 1.

Equations (5) and (6) are fundamental in the quantum theory of the scattering of X-rays. Taking the quantities \(\nu\) and \(\theta\) as independent variables and solving these two equations with respect to \(\nu\) and \(\beta\), we obtain:

\[ \nu=\frac{\nu_0}{1+a(1-\cos\theta)} \qquad\ldots\ldots\ldots\ldots (7) \]

\[ \beta = 2a\sin\frac{\theta}{2}\, \frac{ \sqrt{\,1+\left(2a+a^2\right)\sin^2\frac{\theta}{2}\,} }{ 1+2\left(a+a^2\right)\sin^2\frac{\theta}{2} } \qquad\ldots\ldots (8) \]

where

\[ a=\frac{h\nu_0}{mc^2} \qquad\ldots\ldots\ldots\ldots (9) \]

The constant \(a\) expresses the ratio of the mass of the quantum \(\dfrac{h\nu_0}{c^2}\) to the mass of the electron \(m\).

Equation (7) shows that the frequency of oscillation of X-rays does not remain constant upon scattering, as follows from the classical theory, but decreases as a function of the scattering angle \(\theta\).

Expressing the frequency of oscillation of the quantum in terms of the wavelength, we have:

\[ \nu_0=\frac{c}{\lambda_0} \]

\[ \nu=\frac{c}{\lambda} \]

and formula (7) can be rewritten in the following form:

\[ \lambda-\lambda_0=\frac{2h}{mc}\sin^2\frac{\theta}{2}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (10) \]

Formula (10) clearly shows that the spectral line of homogeneous X-rays is shifted under the influence of scattering toward longer wavelengths. It is noteworthy that this shift is the same for all wavelengths and depends only on the scattering angle. Substituting the numerical values of the quantities \(h, m, c\) into formula (10), we obtain:

\[ \lambda-\lambda_0=0.0486\sin^2\frac{\theta}{2}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (10a) \]

In the scattering of X-rays at a right angle, the magnitude of the shift, the same for all rays, is equal to \(0.0243\ \text{\AA}\).

3. As we have already mentioned, it had been observed quite some time ago that hard rays become softer upon scattering, i.e., their wavelength increases. Precise quantitative measurements confirming formula (10) were first carried out by A. Compton \([12,14,16]\). Since the expected change in wavelength was rather small, of the order of several hundredths of an angstrom, it was necessary for the observations to isolate one definite spectral line. As a source of rays Compton used a molybdenum anticathode and made measurements on the molybdenum \(K_a\) line. The radiator (i.e., the body scattering the X-rays) was a piece of graphite. The principal difficulty of the experiment lay in the fact that the intensity of homogeneous rays scattered in a definite direction is very small; it amounts to only one twenty-five-thousandth part of the total intensity of the primary rays. Therefore it was necessary to obtain the possi-

COMPTON PHENOMENON

a more intense beam of rays can be obtained. In Fig. 2, borrowed from Compton’s article, the arrangement of the experiment is shown. The source of rays is the molybdenum anticathode 1 of an X-ray tube of special design, which makes it possible to place the source of secondary rays as close as possible to the anticathode. The secondary rays from graphite \(R\), after passing through a series of slits, fall on the crystal of the spectrometer and then into the ionization chamber. In Fig. 3 are reproduced the curves obtained by Bragg’s method. Curve \(A\) shows the unchanged primary molybdenum \(K_\alpha\) line. Curves \(B\), \(C\), and \(D\) depict the spectrum of rays scattered at angles of \(45^\circ\), \(90^\circ\), and \(135^\circ\). The difference between the right-hand and left-hand curves is only that in the latter case the slit was narrower. In addition to the primary line, another line, shifted toward longer wavelengths, is also clearly visible on the curves, the shift being

Fig. 2.

Fig. 3.

greater the larger the scattering angle, as follows from formula (10). The accuracy of the observations corresponds to \(0.001\ \text{\AA}\). The wavelength of the molybdenum \(K_\alpha\) line is \(\lambda_0 = 0.708\ \text{\AA}\). The wavelength of the shifted line at a scattering angle of \(90^\circ\) is \(0.730\ \text{\AA}\). Thus the observed shift is \(0.022\ \text{\AA}\). The theoretically expected shift according to formula \((10a)\) is \(0.023\ \text{\AA}\). The agreement with experiment is complete.

The change in wavelength as a function of the scattering angle was also traced by Compton for \(\gamma\)-rays \([14,17]\). The wavelength

was determined in the latter case already indirectly, by the absorption coefficient. In the appended table the results of the observations (third column) are compared with the calculated values.

WAVELENGTH OF PRIMARY AND SECONDARY γ-RAYS.

Duane directed his criticism against the interpretation of Compton’s experiments set forth by us; in a series of works carried out together with several collaborators, he repeated Compton’s experiments and defended the point of view that the appearance of the new spectral line is explained by the occurrence of so-called “tertiary” rays [18, 19, 20, 21]. In Duane’s opinion, X-rays, falling on the body that scatters them, cause the emission of photoelectrons. The kinetic energy of each photoelectron is somewhat less than the energy of the quantum, since part of this energy is spent on the work of detaching the electron. Striking the atoms of the body, these photoelectrons in turn cause the appearance of a continuous X-ray spectrum, and since the energy of the electrons corresponds to a lower frequency than the frequency of the primary rays, the boundary of this spectrum is shifted toward longer waves. As secondary radiators Duane and his collaborators used various substances: graphite, ice, aluminum, common salt, sulfur. The magnitude of the shift, in accordance with Duane’s theory, increases with the atomic weight. And since for graphite Duane’s results coincide with Compton’s results, Duane comes to the conclusion that the tertiary rays do not represent merely a phenomenon accompanying the Compton phenomenon, but that the cause of the shift of the spectral line is contained precisely in them.

Despite the fact that the observed shift does not numerically contradict Duane’s theory, that theory is not able to explain certain features of the phenomenon. First of all, X-rays produced by impacts of electrons and making an angle of $90^\circ$ with the direction of the electron stream are polarized by no more than 25% [22]. In the present case the degree of polarization should be still smaller, since the photoelectrons move in different directions. In actual fact, however, the seco-

THE COMPTON EFFECT

secondary rays from light elements at an angle of \(90^\circ\) to the primary rays are polarized almost entirely, by no less than 98% \([23]\).

Moreover, the energy of the secondary rays produced in graphite and in paraffin by the molybdenum \(K_\alpha\) line amounts to 24% of the energy of the primary rays \([24]\), whereas for photoelectrons of the corresponding velocity only 1% of the energy is converted into the energy of X-rays.

According to Duane, photoelectrons emitted from the wooden box surrounding the apparatus play a prominent role in the formation of tertiary rays. Check experiments with a box lined with lead, and with no box at all, were carried out by Ross and Webster \([25]\), and also by Compton, Birt and Woo \([26]\). The results obtained were the same as before, in agreement with Compton’s theory.

The most convincing confirmation of the Debye—Compton theory is provided by precise spectroscopic observations made by a number of investigators \([27,28,29,30,31,32,33,34,35,36,37,38,39]\). The displacement of the spectral line was measured not only by means of an ionization chamber, but also photographically. As we have already mentioned, the intensity of the secondary rays in a given direction is very small; therefore, when photographing spectral lines, the exposure must last a rather long time, not less than one hundred hours with a powerful tube \([27]\). As primary rays there were usually used rays of the molybdenum \(K_\alpha\) line (wavelength \(0.708\ \text{Å}\)) or of the tungsten \(K_\alpha\) line (wavelength \(0.209\ \text{Å}\)). In the photographs obtained by Ross, Webster, Becker, Kalman and Mark, and Sharp, both the primary, undisplaced line and the displaced line are clearly visible, the magnitude of the displacement being in exact quantitative agreement with Compton’s theory. Whereas, according to the theory of tertiary rays, in place of the displaced line there should be a broad band, sharply bounded only on the side of short wavelengths, in most photographs the displaced line appears quite sharp (though still less sharp than the primary line). Thus, for example, in Ross’s photograph \([30]\), in which he observed the molybdenum \(K\) line scattered by paraffin at an angle of \(55^\circ\), it was possible to distinguish on the displaced line even the components \(a_1\) and \(a_2\), for which the difference in wavelengths is \(0.004\ \text{Å}\). A very distinct spectrogram, obtained by Becker together with four collaborators \([34]\), is shown in Fig. 4. Molybdenum rays scattered by aluminum at an angle of \(100^\circ\) were photographed. In the figure one clearly sees both the undisplaced lines \(a\) and \(\beta\), and the displaced lines, denoted \(a_c\) and \(\beta_c\). The magnitude of the displacement is \(0.027\ \text{Å}\). In the original spectrograms it is possible to resolve even the components \(a_1\) and \(a_2\).

The most precise measurements up to the present time have apparently been made by Sharp, in Compton’s laboratory \([35]\) (in December

1925). The \(K_{\alpha}\) line of molybdenum, scattered by paraffin at an angle of \(169.7^\circ\), was photographed. In Fig. 5, which reproduces Sharp’s spectrogram, the shifted and unshifted lines \(K_{\alpha}\) of molybdenum are visible. For comparison, the \(K_{\alpha}\) line of zirconium, of greater wavelength, is included. The experimentally observed magnitude of the shift is \((0.04825 \pm 0.00017)\ \text{\AA}\); that calculated theoretically from formula (10) is \((0.04798 \pm 0.00009)\ \text{\AA}\). It is noteworthy that Sharp, using his results and substituting them in formula (10), calculates the value of the electron mass \(m\), independently of the ratio \(\frac{e}{m}\). One obtains \(m = (8.99 \pm 0.034)\cdot 10^{-28}\ \text{g}\). From observations of the deflection of moving electrons one obtains \(m = 8.98\cdot 10^{-28}\ \text{g}\). (V. T. Bridge, Phys. Rev. 14, 369, 1919.)

Fig. 4.

Of decisive significance for Compton’s theory are measurements of the displacement of a spectral line as a function of the scattering angle, since according to the theory of tertiary rays the magnitude of the displacement does not depend on the angle. Ross \([^{31}]\) observed the displacement at angles of \(110^\circ\) and \(160^\circ\). The magnitude of the displacement at \(110^\circ\) is \((0.035 \pm 0.002)\ \text{\AA}\) (theoretical \(0.0345\ \text{\AA}\)); at \(160^\circ\) it is \((0.047 \pm 0.002\ \text{\AA}\) (theoretical \(0.0469\)). Callihan and Mark \([^{32}]\) obtained photographs of the molybdenum \(K_{\alpha}\) line scattered by graphite at angles of \(90^\circ\) and \(72^\circ\) (Fig. 6). The observed magnitudes of the shifts amount—

Fig. 5.

are \(0.0241\ \text{Å}\) and \(0.0170\ \text{Å}\), while those calculated by formula (10) are \(0.0243\ \text{Å}\) and \(0.0168\ \text{Å}\).

It is interesting that Duane, who for a long time was an opponent of Compton’s theory, published at the end of 1925 the results of systematic experiments (together with Allison) on the scattering of X-rays by aluminum, lithium, and other elements, the results of the experiments, in Duane’s opinion, confirming Compton’s theory \([36]\). With this paper of Duane’s, apparently, the discussion of the “theory of tertiary rays” comes to an end.

\( \vartheta = 72^\circ \)

\( \vartheta = 40^\circ \)

Unshifted
\(K_{\beta}\)-line.

Unshifted
\(K_{\alpha}\)-line.

Fig. 6.

1. As we have seen, the Debye—Compton theory is confirmed by experimental data with great accuracy. The greatest agreement of the experimental data with the predictions of the theory is obtained in those cases (of which we have chiefly spoken so far) when X-rays are scattered by elements of small atomic weight—carbon, lithium, boron, etc.

But if an element with comparatively large atomic weight is exposed to X-rays, then the magnitude of the displacement of the spectral line does not always correspond to formula (10) \([40]\). On the other hand, in all spectrograms there is obtained, in addition to the shifted line, also the principal, unshifted one, and the intensity of the latter is in some cases greater, in others less, than the intensity of the shifted line. The original Debye—Compton theory gives no indication of the existence of an unshifted line, let alone of the relative intensity of both lines. Moreover, if one compares the structure of these lines, the shifted line usually proves to be somewhat broader and more diffuse. The theory we have set forth likewise gives no indication of this circumstance. From all this it follows that the theory, while in general correctly representing the essence of the phenomenon, but not illuminating certain details, requires certain supplements or generalization.

Indeed, when we discussed the distribution of energy between the quantum and the electron [equation (5)], it was assumed that all the energy communicated by the quantum to the electron has the form of kinetic energy. In other words, we assume that before the collision with the quantum the electron was at rest and was not bound by appreciable forces to the atoms of the substance. The original Compton theory is a theory

scattering of X-rays by free electrons. But, as is known, electrons are located inside the atom in stationary orbits with different energy levels, and in order merely to remove an electron from the sphere of attraction of the remaining part of the atom, it is necessary to expend a certain amount of energy, equal to the ionization energy of the atom for the corresponding level. If this ionization energy is small, it may be neglected; but for elements with a large charge of the atomic nucleus it is necessary to take it into account. Moreover, when a quantum collides with an electron that is bound to an atom, part of the energy and momentum is imparted also to the remaining part of the atom. All these circumstances must be taken into account in constructing a generalized theory of the Compton phenomenon.

A generalized quantum theory of the scattering of X-rays was proposed by Compton himself \[41, 40\] and, somewhat later, by Jauncey \[42, 43, 44, 45\].

According to Compton’s generalized theory, the energy of the quantum in a collision with a bound electron is distributed as follows. Part of the energy is expended in removing the electron from the atom. This part is equal to \(h\nu_s\), where \(\nu_s\) is the critical ionization frequency, i.e., the lowest frequency of those rays capable of tearing the electron from the atom. Part of the energy goes into imparting kinetic energy to the electron, another part into imparting energy to the atom, and, finally, the last part remains in the form of the energy of the changed quantum. The momentum of the primary quantum is also equal to the (vector) sum of the momenta of the secondary quantum, the electron, and the atom. Hence the energy equation takes the form:

\[ \frac{hc}{\lambda_0} = \frac{hc}{\lambda} + \frac{hc}{\lambda_s} + mc^2\left(\frac{1}{\sqrt{1-\beta^2}}-1\right) + \frac{1}{2}MV^2, \ldots\ldots (11) \]

where \(\lambda_s\) is the wavelength of the ionizing rays, \(M\) and \(V\) are the mass and velocity of the atom; the remaining notation is as before.

Let the primary quantum move along the coordinate axis \(X\). We choose the direction of the \(Y\)-axis in such a way that the path of the deflected quantum lies in the plane \(XY\). The motion of the electron occurs, in the general case, outside this plane. Let the cosines of the angles formed by the deflected quantum with the coordinate axes be denoted by \(l_1, m_1, 0\). The corresponding cosines for the path of the electron are \(l_2, m_2, n_2\); for the path of the atom, \(l_3, m_3, n_3\). The momentum of the electron is

\[ p=\frac{m\beta c}{\sqrt{1-\beta^2}}\ldots\ldots\ldots\ldots\ldots\ldots (11a) \]

The momentum of the atom is: \(P=MV\ldots\ldots\ldots\ldots\ldots\ldots (12)\)

The equations of momentum (in projections on the coordinate axes):

\[ \frac{h}{\lambda_o}=\frac{h l_1}{\lambda}+p l_2+P l_3 \ . . . . . . . . . . . (13) \]

\[ 0=\frac{h}{\lambda}m_1+p m_2+P m_3 \ . . . . . . . . . . . (14) \]

\[ 0=0+p n_2+P n_3 \ . . . . . . . . . . . (15) \]

Since \(M\) is large in comparison with \(m\), the velocity of the atom \(V\) is small and the quantity \(\frac{1}{2}MV^2\) in equation (11) may be neglected. Solving equations (11), (13), (14), and (15), we obtain that the change of wavelength

\[ \delta\lambda=\lambda-\lambda_o= \]

\[ =\frac{\lambda_o}{1-A}\left[ a(1-l_1)+s\left(1-\frac{1}{2}as\right) +B\left(l_1l_3+m_1m_3-l_3+\frac{B}{2a}\right) \right], \quad (16) \]

where the following abbreviated notations have been introduced:

\[ a=\frac{h}{mc\lambda_o};\quad s=\frac{\lambda_o}{\lambda_s};\quad B=\frac{P}{mc};\quad A=s\left(1+a-\frac{1}{2}ds\right)-B\left(l_3-\frac{B}{2a}\right). \ . . (17) \]

For the kinetic energy of the electron one obtains the expression:

\[ E=h\nu_o\left[ 1- \frac{ 1-a\left(l_1s+\frac{1}{2}s^2\right) +B\left(l_3+l_1l_3s+m_1m_3s-\frac{B}{2a}\right) }{ 1+a(1-l_1-s)+B(l_1l_3+m_1m_3) } \right] \ . . (18) \]

If the electrons are free, then \(\nu_s=0\); \(\lambda_s=\infty\); \(P=0\) (the momentum is imparted only to the electron). In this case expressions (16) and (18) are simplified and become the Debye–Compton equations for free electrons:

\[ \delta\lambda=\frac{h}{mc}(1-l_1)\ . . . . . . . . . . (16\,a) \]

\[ E=h\nu_o\frac{a(1-l_1)}{1+a(1-l_1)}\ . . . . . . . . . . (18\,a) \]

Equations (16 a) and (18 a) are identical with equations (10) and (8). As is seen from formula (16), the displacement of the spectral line depends on the magnitude \(s\), i.e. on the ratio of the wavelength of the primary rays to the wavelength of the ionizing rays. For a certain value of \(s\) the displacement of the spectral line will be smallest. This will occur in the case when the ionization energy is so great that all the energy imparted to the electron is expended in separating it from the atom. The kinetic energy of the electron is equal to zero, and all the momentum is imparted to the ionized atom.

The magnitude of the displacement (we shall denote it by the letter \(D\)) is obtained from equation (11), by setting equal to zero in this equation the third and fourth terms. We have:

\[ D=\frac{\lambda_0^2}{\lambda_s-\lambda_0}\ . . . . . . . . . . . . . . (19) \]

Compton represents the very process of imparting momentum to the electron and to the ionized atom in the following way. In order to remove an electron from the atom, the ionization energy \(\frac{hc}{\lambda_s}\) is expended. When this energy is imparted, the electron remains at rest outside the atom, and the corresponding quantity of motion is imparted to the remainder of the atom. As soon as the electron has been freed, the further action of the radiation is no longer imparted to the atom, but only to the electron, just as in the case when the electron is free from the very beginning. Thus, by the end of this process the atom receives the quantity of motion \(P=\frac{h}{\lambda_s}\) in the direction of the primary ray. In equation (16) we have:

\[ B=\frac{P}{mc}=\frac{h}{mc\lambda_s}=as;\quad l_3=1;\quad m_3=0. \]

Substituting these values in equation (16), we obtain:

\[ \delta\lambda=\frac{\lambda_0^2}{\lambda_s-\lambda_0}+\frac{h}{mc}(1-l_1)\ . . . . . . . . . . . . . (20) \]

In the right-hand side of equation (20), the first term is equal to the quantity \(D\) in equation (19), i.e. to the magnitude of the smallest displacement, while the second term is equal to the displacement in the case of a free electron, according to equations (16a) and (10). Denoting the latter quantity by \(F\), we have:

\[ \delta\lambda=D+F\ . . . . . . . . . . . . . (20a) \]

where

\[ F=\frac{h}{mc}(1-l_1)\ . . . . . . . . . . . . . (21) \]

\[ D=\frac{\lambda_0^2}{\lambda_s-\lambda_0}\ . . . . . . . . . . . . . (19) \]

Formulas (20) and (20a) are confirmed by Compton’s experiments [40]. In the appended table the results of several experiments are listed.

In the first column of the table are indicated the spectral lines of the primary rays (the $K_{\alpha}$ and $K_{\beta}$ lines of molybdenum and tungsten), in the second column—the radiator (the body scattering the rays), in the third—the scattering angle, in the fourth—the observed magnitude of the shifts, in the fifth—the value calculated by formula (20), and in the sixth—the limiting value of the shift $D$, calculated by formula (19). From the table it is seen that in those cases where the rays are scattered by light substances (the first three rows), for which the ionization energy may be neglected, the observed shift corresponds to the case of free electrons, and $D = 0$. For heavier substances the value $D$ already constitutes a significant part of the observed shift. Finally, in the case when the atomic weight of the radiator is considerable (tungsten rays are scattered by molybdenum), the ionization energy is so large that almost all the energy imparted to the quantum is spent on individual electrons, and the whole shift is exhausted by the value $D$.

Let us suppose that the electron occupies such a position in the orbit at the moment of collision that all the energy imparted to it by the quantum proves insufficient to remove it from the atom. In such a case the electron remains inside the atom; to the mass of the electron is added the mass of the entire atom; in formula (10) one must replace $m$ by the quantity $M$, and the magnitude of the shift proves negligibly small. This explains the existence of the unshifted line.

Duane’s theory is essentially similar to Compton’s theory, but goes further, giving a calculation for the width of the shifted line and for the ratio of the intensities of the shifted and unshifted lines. Unlike Compton, Duane introduces into the equations of energy and momentum chiefly not the ionization energy, but the velocity of the electron (which, in turn, can be expressed as a function of the ionization energy). As a result of the calculations, the wavelength of the shifted line is expressed as a function of the angle between the direction of the pri-

...of the quantum and the direction of the electron’s velocity at the moment of collision. Having calculated the maximum and minimum of this function, one can find the greatest and smallest values of the wavelength of the shifted line. The difference between these quantities determines the width of the shifted line. If they are denoted by \(\lambda_{max}\) and \(\lambda_{min}\), then, according to Duane:

\[ \lambda_{max}-\lambda_{min}=4\lambda_0 \sin \frac{\theta}{2}\sqrt{\frac{2h}{mc\lambda_s}}\ . \ . \ . \ . \ . \ . \ (22) \]

(the remaining notation is as before).

Thus, it follows from the theory that the shifted line must have a finite width. This conclusion is in agreement with experiments \([16, 46]\).

Applying his calculations to the scattering of molybdenum \(K_{\alpha}\) rays by the electrons of the carbon atom at the \(L\) level, Duane obtains for the width of the shifted line \(0.021\ \mathring{A}\). Compton’s experiments give \(0.027\ \mathring{A}\) \([16]\).

As was already indicated, the unshifted line owes its existence to those quanta which did not cause, in collision, the ejection of an electron. Let us suppose that the probability of a collision of this kind is \(P_0\), and that the probability of a collision in which the energy imparted by the quantum to the electron proves sufficient for ionization of the atom is \(P_1\). Then the ratio of the number of quanta that have produced the shifted line to the number of quanta that have produced the unshifted line is equal to the ratio \(\frac{P_1}{P_0}\). The same is the ratio of the intensity of the shifted line to the intensity of the unshifted line (more precisely, one should also take into account the different energy of the quanta themselves). We shall not dwell on the course of Duane’s not entirely rigorous calculations, but shall indicate only the results obtained: 1) The relative intensity increases with increasing angle of scattering. This may be explained as follows: the greater the angle of scattering, the greater the energy imparted by the quantum to the electron [see formula (27) below], and the greater the probability that this energy will prove sufficient for the separation of the electron, i.e., that the quantum will change its frequency. This result is confirmed by experiments \([16, 47]\) (cf., for example, Fig. 3). 2) The relative intensity of the shifted line at a given angle of scattering decreases with increasing atomic number of the scattering element: the greater the atomic number, the more strongly, in general, the electrons are bound, and the greater the probability that the energy imparted by the quantum will prove insufficient for ionization of the atom. This result is confirmed by the experiments of Davis \([48]\), Ross \([28, 31, 49]\), Webster \([50]\), Duane and Allison \([36]\). 3) The relative intensity of the shifted line increases with decreasing wavelength of the X-rays: the smaller the wavelength, the greater the energy of the quantum, and the greater the...

probability of detachment of the electron. This dependence has also been observed experimentally [49]. It is interesting to note, in this connection, Ross’s attempt [49] to obtain the Compton phenomenon for light rays, namely, for the green line of mercury. The negative result of this attempt may be explained by the insufficient energy of the quantum of the light rays of the visible spectrum.

1. Up to now we have spoken about the interpretation of the Compton phenomenon only from the point of view of the theory of light quanta. However, there exist a number of attempts to reconcile the Compton phenomenon with the existence of radiation which propagates in space in the form of waves, but is emitted and absorbed in whole quanta of energy \(h\nu\). Such are the “semiclassical” theories of Fürth [51], Halpern [52], Ya. I. Frenkel [53], Eckart [54], Bauer [55], and Mi [56, 57]. An essential feature of these theories is that they explain the displacement of the spectral line by the Doppler phenomenon, which is due to the motion of the scattering electrons. Compton, in one of his first works, pointed out that the same value of the displacement as is obtained from formula (10) can be obtained on the basis of the Doppler principle, if one assumes that the radiating electrons move in the direction of the primary rays with an “effective” velocity \(\bar{\beta}c\), where

\[ \bar{\beta}=\frac{\alpha}{1+\alpha}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (23) \]

According to the “semiclassical” theories, the electromagnetic wave of the X-rays, falling upon an electron, imparts to it accelerated motion. If the amount of energy absorbed by the electron from the primary wave is equal to \(h\nu_{0}\), then the mean velocity of the electron corresponds to formula (23). Eckart, basing himself on the general theory of relativity, derives from the wave equation and from the differential equation of a light ray, as a geodesic line, equations analogous to the Compton—Debye equation. Mi advances an original hypothesis, according to which not only whole atoms, but also electrons may pass into various stationary states, so that the scattering of X-rays represents, in some sense, fluorescent radiation of electrons. Mi combines his theory with the theory of virtual vibrators of Bohr, Kramers, and Slater [58], upon which Bohr [59], apparently, no longer insists at the present time.

We do not dwell in detail on these theories, not only for lack of space, but also in view of their insufficiency for a complete explanation of the Compton phenomenon: deriving, with the help of certain assumptions, the displacement formula for free electrons, they are not in a position to explain those phenomena which are connected with the motion

“recoil electrons” (i.e., electrons that have collided with radiation quanta). We shall now turn to a consideration of these phenomena.

1. When a quantum collides with an electron, the latter is imparted some part of the energy and momentum of the primary quantum. If the quantum is deflected through an angle \(\theta\) from its initial direction, then the velocity of the electron, as a function of the angle \(\theta\), is expressed by formula (8), where \(\beta=\frac{v}{c}\). Let us suppose that the direction of the electron’s velocity makes an angle \(\varphi\) with the direction of the primary quantum. It is not difficult to find the relation between the angles \(\varphi\) and \(\theta\). Let us write the momentum equations in projections on the direction \(AB\) (Fig. 1) and on the direction perpendicular to the latter:

\[ \frac{h\nu_0}{c} = \frac{m\beta c}{\sqrt{1-\beta^2}}\cos\varphi + \frac{h\nu}{c}\cos\theta \qquad \ldots \ldots \ldots \ldots (24) \]

\[ 0 = \frac{m\beta c}{\sqrt{1-\beta^2}}\sin\varphi + \frac{h\nu}{c}\sin\theta \qquad \ldots \ldots \ldots \ldots (25) \]

From the last two equations we find:

\[ \tg\varphi = \frac{1}{(1+a)\tg\frac{\theta}{2}} \qquad \ldots \ldots \ldots \ldots (26) \]

The kinetic energy of the electron, on the basis of formulas (3) and (8), is

\[ E = h\nu_0\, \frac{2a\sin^2\frac{\theta}{2}} {1+2a\sin^2\frac{\theta}{2}} \qquad \ldots \ldots \ldots \ldots (27) \]

With the aid of formula (26) one can express the energy \(E\) as a function of the angle \(\varphi\):

\[ E = \frac{2a\,h\nu_0\cos\varphi} {(1+a)^2-a^2\cos^2\varphi} \qquad \ldots \ldots \ldots \ldots (28) \]

In Fig. 7, borrowed from Debye’s article, the quantities \(h\nu\) are shown graphically by arrows in comparison with \(h\nu_0\), for various angles \(\theta\). The arrows drawn downward show the direction of motion and the magnitude of the electron energy for the same values of \(\theta\) (marked by the same numerals).

From formulas (26) and (28), and from the drawing, it is evident that the recoil electrons are always thrown forward, at an acute angle to the direction of the primary quantum. The smaller the angle formed by the electron velocity with the direction of the primary quantum, ...

direction of the primary ray, the greater is the energy of the electron. The electron energy attains its greatest value at \(\theta=0\). In this case:

\[ E_{\max}=h\nu_0\cdot\frac{2\alpha}{1+2\alpha}\ .\ .\ .\ (29) \]

As is known, when X-rays are absorbed, photoelectrons are emitted. Between these electrons and the electrons accompanying the Compton phenomenon there is an essential difference. Whereas in the emission of a photoelectron the complete X-ray quantum is absorbed, and the energy of the photoelectron is close to the energy of the quantum, for the Compton electron the amount of energy obtained is considerably less than the value \(h\nu_0\) (see formulas (28) and (29)).

Fig. 7.

Fig. 8.

The existence of such “recoil electrons” (recoil electrons, Rückstosselektronen), produced by the Compton effect, was confirmed by the experiments of Wilson \([60]\), who made use of his method of the condensation chamber and photographed the trajectories of electrons emit-

...appearing in a gas under the influence of a beam of X-rays. Wilson established that, besides photoelectrons with a large range and a high velocity corresponding to the absorption of a quantum, there are also electrons with a small range (“fish tracks”), moving at acute angles to the direction of the X-rays. Similar experiments with positive results were carried out by Ikeyti [61] and Bothe [62, 63]. Ikeyti showed that the magnitude of the range of the electrons, and hence also their velocity, is the greater the smaller the angle with the direction of the X-rays, in agreement with Compton’s formula (28). In Fig. 8 one of the photographs obtained by Bothe is reproduced.

Fig. 9.

D. V. Skobeltsyn [64, 65] investigated, by means of Wilson’s method, the ejection of electrons by $\gamma$-rays, whose wavelength is still shorter than that of X-rays. In order to establish the dependence between the range (velocity) of the electrons and the angle $\varphi$, D. V. Skobeltsyn subjected them to the action of a magnetic field, under whose influence the trajectories bent into a circle. The tangent at the beginning of the trajectory gives an approximate value of $\varphi$. Agreement with Compton’s theory was obtained. In Fig. 9 one of D. V. Skobeltsyn’s photographs is reproduced.

A detailed theoretical and experimental investigation of recoil electrons was carried out by Compton (together with collaborators), who compared the conclusions of his theory with the conclusions of the “semiclassical” theory [66, 67, 68].

That the electrons with a small range are actually caused by the scattering of X-rays, Compton confirms by comparing the number of electrons. If \(N_R\) is the number of electrons with a small range (recoil electrons), \(\sigma\) is the scattering coefficient (the fraction of the energy of the primary rays converted into the energy of the scattered rays), \(N_P\) is the number of electrons with a large range (photoelectrons), and \(\tau\) is the absorption coefficient, then the proportionality must hold:

\[ \frac{N_R}{N_P}=\frac{\sigma}{\tau}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (30) \]

The last formula is not quite exact, since not all scattered quanta cause the detachment of electrons (the existence of an unshifted line), and not all electrons can give a noticeable trajectory. Therefore, a correction coefficient must be introduced into the right-hand side of equality (30), and equality (30) takes the form:

\[ \frac{N_R}{N_P}=k\frac{\sigma}{\tau},\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (30a) \]

where \(k<1\). The quantity \(k\) was calculated by Jauncey and de Foe \([69, 70]\).

In the appended table, the results of Compton’s observations are compared with the results of Compton’s and Jauncey’s calculations:

According to the wave theory of the Compton phenomenon, all electrons move in the direction of the primary ray with the same velocity, which is determined by formula (23). In such a case the energy of the electrons would be expressed by the quantity:

\[ E=h\nu_0\cdot\frac{1}{a}\left[\frac{1+a}{\sqrt{1+2a}}-1\right]\ldots\ldots\ldots\ldots\ldots (31) \]

Meanwhile, according to the quantum theory, the energy of an electron depends on its direction [(see (28)]. For small values of \(a\), the quantity \(E\) calculated by formula (31) is four times smaller than the maximum value of \(E\) according to formula (29). Since the range of an electron is proportional to the square of its kinetic energy, the difference becomes still greater—

…larger, by a factor of 16. In the appended table the results of Compton’s observations are compared with the magnitude of the greatest range of the electrons, calculated by the “quantum” formula (29) and by the “wave” formula (31).

As the table shows, the quantities calculated on the basis of quantum theory are considerably closer to the observed ones.

The most direct confirmation of the quantum theory of the Compton phenomenon is provided by observations of the trajectories of individual electrons and secondary quanta produced by the impact of one and the same primary quantum. These remarkable observations were carried out by Compton and Simon \[71, 72\] by Wilson’s method. In the photographs they obtained, the trajectories of the recoil electrons are directly visible; as for the secondary quantum, the latter produces along its path the appearance of fast β-rays (photoelectrons), and the straight line connecting the beginning of the electron trajectory with the beginning of the β-ray gives the direction of the secondary quantum. The angles made by these two directions with the direction of the primary ray confirm formula (26) with rather great accuracy (if the difficulty of the observations is taken into account). The probability of a chance coincidence is, according to Compton’s calculations, about $\dfrac{1}{250}$. Compton’s experiments directly confirm the proposition that energy and momentum remain constant in the interaction between radiation and electrons, and that the scattering of each quantum corresponds to the recoil of one electron.

If Compton’s experiments prove the spatial relation between the direction of the deflected quantum and the direction of the recoiling electron, then the experiments of Bothe and Geiger \[73, 74, 75\] indicate coincidence in time between the processes of electron recoil and scattering of X-rays. In these experiments an X-ray passed by two Geiger counters, one of which recorded the moment of appearance of the electron, the other—the moment of appearance of the secondary quantum (with an accuracy of one ten-thousandth of a second).

The number of coincidences considerably exceeds the mathematical expectation for the number of random coincidences. The probability that this difference was accidental is, according to the calculations of Bothe and Geiger, 0.0000025. (For more detail see the abstract by G. S. Landsberg, “Advances in the Physical Sciences,” p. 252, 1925, No. 3).

The experiments of Compton and Simon, Bothe and Geiger show, as Bohr admitted [59], the unsatisfactoriness of the theory of virtual oscillators of Bohr, Kramers, and Slater, according to which the laws of conservation of energy and momentum have only statistical significance.

1. At the beginning of the article we mentioned that, in the case of hard primary rays, the distribution of the intensity of the secondary rays as a function of the scattering angle does not agree with Thomson’s classical formula (2). Especially noticeable is the asymmetric distribution of intensity with respect to the plane perpendicular to the primary ray. The task of quantum theory—to derive, independently, a formula agreeing with experiment—turns out in this case to be difficult. According to quantum theory, the intensity of the rays scattered in a given direction depends on the number of quanta deflected in that direction, and this number, in turn, is proportional to the probability of deflection of a quantum through the given angle. But concerning such a probability we cannot yet express a definite quantitative judgment, since we know little about the very mechanism of the interaction between the quantum and the electron. In order to obtain at least an approximate solution of the question, contemporary theories make use of the method proposed by Bohr (the “principle”) of correspondence, according to which conclusions obtained on the basis of quantum theory asymptotically approach the conclusions of classical theory for oscillations of decreasing frequency. In using this method, some assumption of classical theory, confirmed by experiment for slow oscillations, is taken and introduced into the reasoning of quantum theory. The choice of this assumption is what characterizes the difference between the theories of different authors.

Thus, for example, Debye [11] concludes on the basis of formula (2) that the probability of deflection of a quantum through the given angle \(\theta\) is proportional to the quantity \(\dfrac{1+\cos^2\theta}{2}\). Since the intensity of radiation is proportional, moreover, to the energy of each quantum \(h\nu\), Debye obtains from this the expression:

\[ \frac{J}{J_o}=\frac{h\nu}{h\nu_o}\cdot\frac{1+\cos^2\theta}{2}. \]

Substituting here the values of \(\nu\) from formula (7), we obtain:

\[ J=J_o\,\frac{\nu_o}{1+\alpha(1-\cos\theta)}\cdot\frac{1+\cos^2\theta}{2}\ldots\ldots\ldots(32) \]

Compton [14] applies the following argument. As we have seen, the displacement of the spectral line is the same as if the scattering were produced, on the basis of the classical laws, by electrons moving with an “effective” velocity \(\beta \nu\) [(see (23)]. Compton assumes that the distribution of intensities also corresponds to the classical distribution, but for electrons moving with the same effective velocity. From this assumption, rather loosely motivated (by Compton’s own admission), he derives the following expression for the intensity of the scattered radiation:

\[ J = J_0 \cdot \frac{1+\cos^2\theta+2a(1+a)(1-\cos\theta)^2}{[1+a(1-\cos\theta)]^5}\ .\ .\ .\ .\ .\ (33) \]

Compton’s formula (33) proves to be in better agreement with the results of observations of \(\gamma\)-ray scattering at different angles [10] than Debye’s formula (32).

If the expression for the intensity of the secondary rays is integrated over all values of the angle \(\theta\), and the resulting value of the scattered energy is divided by the total intensity of the primary rays, we obtain the value of the scattering coefficient. If we denote the intensity of the primary rays by \(P\), then we obtain the following expression for the scattering coefficient:

\[ \sigma_s = \frac{2\pi R^2}{P}\int_0^\pi J \cdot \sin\theta\, d\theta\ .\ .\ .\ .\ .\ .\ .\ .\ (34) \]

We shall denote by \(\sigma_0\) the value of the scattering coefficient obtained from the classical theory. As is seen from formula (1),

\[ \sigma_0 = \frac{8\pi}{3}\frac{Ne^4}{m^2c^4},\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (35) \]

where \(N\) is the number of electrons per unit volume.

Substituting the value of \(J\) from (32) into (34), Debye obtains:

\[ \sigma_s = \sigma_0 \cdot \frac{3}{4}\cdot \frac{1}{a^3} \left[(1+2a+2a^2)\ln(1+a)-a\left(1+\frac{3}{2}a\right)\right]\ .\ .\ .\ (36) \]

An error (in the integration) slipped into Debye’s calculation. The corrected formula has the form:

\[ \sigma_s = \sigma_0 \cdot \frac{3}{8}\cdot \frac{1}{a^3} \left[(1+2a+2a^2)\ln(1+2a)-2a(1+a)\right]\ .\ .\ (36a) \]

Compton obtains from (32) and (34) the following expression:

\[ \sigma_s = \sigma_0 \cdot \frac{1+a}{(1+2a)^2}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (37) \]

The last formula, while agreeing rather closely with the experimental data for small wavelengths, gives considerable deviations in the case of long waves.

Proceeding from arguments somewhat different from Compton’s conclusions, Woo [76] obtains another expression for the scattering coefficient:

\[ \sigma_s=\sigma_0\cdot \frac{1+a}{1+2a}\ldots\ldots\ldots\ldots (37a) \]

J. J. Thomson [77, 78] attempted to give a purely quantum theory of the intensity of scattered X-rays. But the formula he obtained, still more complicated than Compton’s formula, was also derived on the basis of insufficiently justified assumptions.

As we see, on the question of the intensity of scattered rays quantum theory proves to be far from complete. Further development of this question is apparently directly connected with a deeper study of the mechanism of interaction between radiation quanta and material particles.

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