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A NEW TABLE OF VALUES OF GENERAL PHYSICAL CONSTANTS¹
(as of August 1941)
R. T. Birge, Berkeley, California
In 1929 I published¹,² a set of mutually consistent values of the general physical constants which, with a few exceptions, appeared to be quite satisfactory. In a later publication³ a more logical treatment was given of the interrelated atomic constants: \(e\), \(h\), \(\frac{e}{m}\), and \(\alpha\), but without essential changes in the accepted values. The system of constants published in 1929 was satisfactory in the sense that, in almost all those cases where a check was possible, the experimental values that seemed to be the best were in satisfactory agreement with the values calculated indirectly. The only serious discrepancy that existed there—the discrepancy between the “spectroscopic” value of \(\frac{e}{m}\) and the value of this ratio obtained from experiments on the deflection of an electron beam—has by the present time been almost completely eliminated³.
However, already in 1929 signs began to appear of another discrepancy—the discrepancy in the values of the electron charge \(e\), obtained by the oil-drop method and by the new method based on measuring the wavelengths of X-rays with the aid of gratings⁴. The reality of this discrepancy was soon established with certainty, and the unsatisfactory state of affairs caused by this circumstance existed until 1935, when it was shown beyond doubt that the value of the viscosity of air used by Millikan in his work with oil drops differed substantially from the true value. Numerous determinations of the viscosity of air made in recent years, together with new determinations of \(e\) by Millikan’s method, have now brought the two values of the electron charge into satisfactory agreement with one another. But since the value of \(e\) obtained from measurements of X-ray wavelengths is now considered significantly more accurate, the value of the electron charge adopted in the present article is simply the value of \(e\) obtained by this method.
¹ Translation by N. S. Khlebnikov.
² Hereafter cited as G. C. (General Constants), 1929.
When computing \(e\) from data of measurements of X-ray wavelengths, it is not \(e\) itself that is initially determined, but the Avogadro number \(N_0\). The value of \(e\) is then found on the basis of the relation
\[ e=\frac{F}{N_0}, \]
where \(F\) is the Faraday number. Thus \(N_0\) now proves to be a fundamental constant and for this reason has been placed in table \(a\), whereas \(e\) is a derived quantity and, in essence, should be placed in table \(c\). Only for reasons of convenience have I nevertheless retained \(e\) in table \(a\).
In 1929 the value of \(e\), in round numbers, was expressed as \(4.77\cdot 10^{-10}\) CGSE. The new value of \(e\) is approximately \(4.80\cdot 10^{-10}\) CGSE. Such an exceptionally large change inevitably leads to the appearance of a whole series of discrepancies in the values of other constants. Indeed, as I have already indicated, precisely because all the values of the atomic constants agreed so well with one another when Millikan’s value of \(e\) was adopted, I for a number of years believed that this value had to be exact.
Among the new discrepancies, the largest in percentage terms are given by the values of the radiation constants, namely: the second radiation constant \(c_2\) and the constant of the Stefan—Boltzmann law \(\sigma\). It is quite possible, however, that the true errors in the experimentally found values of these constants are much greater than the estimated experimental errors, and therefore these discrepancies should not be taken too seriously.
On the other hand, there exists a discrepancy which until recently seemed very large and very serious. Its character can be explained most simply in the following way.
If in Bohr’s formula one substitutes, as the Rydberg constant, the very accurately known value \(R_\infty\) and the best values, found by direct methods, of \(e\) and \(\frac{e}{m}\), then the value of the ratio \(\frac{h}{e}\) calculated from these data proves to be substantially larger than any found directly. I pointed out\(^5\) the seriousness of the situation that had arisen immediately after the new “high” value of \(e\) had been established with certainty. Since then this problem has been examined in detail by many authors, among whom Dennison\(^6\) and Du Mond\(^7\) should be mentioned.
The most recent experimental data indicate that in the former direct determinations of \(\frac{h}{e}\) there may have been serious systematic errors and that, when these are eliminated, the indicated discrepancies may disappear. This applies, in particular, to the experimental determination of \(\frac{h}{e}\) from the limit of the continuous X-ray spectrum—a method which at present makes possible the most accurate computation of \(\frac{h}{e}\).
Even before the new experimental data concerning \(\frac{h}{e}\) were published, it seemed to me, on the basis of an assessment of all the available material, that the “indirect” values, rather than the “direct” values of \(\frac{h}{e}\), should be the correct ones. Proceeding from this assumption, in August 1939 I compiled a new table of general physical constants and reproduced it by mimeograph1. The purpose of this preliminary table was only to elicit corrections and suggestions from competent persons, so that this material might be used in a detailed article on this topic, which I intended to write soon. For a whole series of reasons such a detailed article has still not been written; but at the present time I have completed the study of the fundamental constants and of various auxiliary constants necessary for calculating the fundamental ones. The final impetus that prompted me to write the present article was the proposal of the London Physical Society to write, for the annual publication it issues, Reports on Progress in Physics, a short article on this question. The manuscript of that article has already been sent to London, but in view of the current international situation there may be a delay in its publication. It seems desirable, however, to make the new table of constants available as soon as possible. Therefore, in the present article I give the tables of constants taken from the manuscript just mentioned, accompanying them only with the explanations necessary to make them intelligible. I hope, however, in the not too distant future, to publish a more extensive article, similar to G. C. 1929, in which the origin of each of the quantities included in the table will be fully explained.
As has already been noted, the most reliable value of \(\frac{h}{e}\) at present appears to be the one calculated indirectly, with the aid of the Rydberg constant. This value of \(\frac{h}{e}\) has been adopted by us, as a result of which both quantities—\(e\) and \(h\) and \(\frac{h}{e}\)—become derived constants and are accordingly placed in table \(c\). Other quantities, such as, for example, the gas constant per mole, the Boltzmann constant, the Stefan–Boltzmann constant, and the radiation constants, which were placed in table \(b\) of G. C. 1929, now appear in table \(c\), since only indirect (i.e. derived) values are given for them. In general, however, tables \(a\), \(b\), and \(c\) of the present article correspond to the tables with the same designations in G. C. 1929. Because the number of atomic-weight values whose knowledge is necessary has increased considerably, the atomic weights are now set apart in a separate table, designated \(a'\). No probable errors are assigned to the values for \(O^{17}\) and \(O^{18}\), since the errors that do exist have an entirely negligible effect on the resultant value for \(O\) \((=16.004357)\).
It must also be noted that the table c given here is very incomplete. In the detailed article I am preparing, I intend to include everything that was placed in table c of G. C. 1929, and I would welcome suggestions concerning other derived constants that could be included there. With the exception of \(\dfrac{h}{e}\) and \(h\), the elements of table c are arranged in alphabetical order. In the table c of G. C. 1929 I tried to group the constants according to their natural relations, but the results obtained, since the question is one of practical use of the tables, proved to be wholly unsatisfactory.
The system of values of the constants published here differs very little from the mimeographed table of 1939. The only important change concerns Faraday’s number \(F\). I now take for \(F\) \(96501_2\) international coulombs on the chemical scale. This new value is a weighted mean of the values 96494 and 96511, obtained with silver and iodine voltameters respectively, whereas previously I simply took the value obtained with the silver voltameter.
On the other hand, the tables published here differ very greatly from G. C. 1929. Indeed, with the exception of the atomic weight of silver\(^1\), in tables a and b there is not a single value that has not undergone a greater or lesser change. The greatest change concerns the magnitude \(e\), and, as a consequence of this change alone, not only \(h\), but almost all the other derived constants prove to have changed appreciably in magnitude. It should be noted in this connection that the values of certain functions of \(e\), \(h\), and \(m\) obtained by direct observation in recent times, for example \(\left(\dfrac{e}{m}\right)\), \(\left(\dfrac{h}{e}\right)^2\), \(\dfrac{h}{m}\), \(\dfrac{e^2}{h}\), and \(\left(\dfrac{e}{m}\right)\left(\dfrac{e}{h}\right)\), agree quite satisfactorily with the values of \(e\), \(h\), and \(m\) adopted here. As already indicated, the most recent experimental results for \(\dfrac{h}{e}\) are in good agreement with the values obtained indirectly; they turn out to be somewhat higher than the indirect values.
As a result of recent important investigations\(^3\), the Rydberg constant \(R_{\infty}\) has changed by a rather substantial amount. All details concerning the value of \(R_{\infty}\) adopted here, just as also the value of \(\dfrac{e}{m}\), are contained in my recently published article \(^{3b}\).
Many changes in the 1929 tables were caused by the discovery of the isotopes of oxygen and hydrogen. The existence of oxygen isotopes makes possible two scales of atomic weights—the physical and the chemical—and it becomes very important to indicate on which scale a given published value is assigned. The existence of the hydrogen isotope \(\mathrm{H}^2\) leads to the conclusion that the symbol H, used in G. C. 1929, refers in some cases to the normal mixture of hydrogen isotopes, while in others—to the isotope \(\mathrm{H}^1\).
In order to follow as closely as possible the latest proposals of the Committee, I have made several changes in symbols and in names. One of the important changes in symbols I make here, however, on my own responsibility.
Since in table c, for each arbitrary constant its expression in terms of the others is given, it is essential that the corresponding symbol denote not only the constant itself, but also its connection with a definite unit. Meanwhile, in G. C. 1929 and in a number of papers published later, I followed the widespread but quite imprecise custom, using \(e\) to denote the electron charge in electrostatic units and \(\dfrac{e}{m}\) to denote the specific charge of the electron expressed in electromagnetic units.
In now establishing a single system of units for \(e\) and \(\dfrac{e}{m}\), I have relied on the circumstance that in a real experiment all electrical quantities are measured in practical units (volts, ohms, etc.) and that, in order to obtain the corresponding quantities expressed in the electromagnetic system, it is sufficient to carry out the proper transfer of the decimal point. Therefore, in the present article all quantities are, as a rule, expressed in electromagnetic units and are denoted by symbols without primes, whereas symbols with primes show that the quantity is expressed in the electrostatic system. Thus \(F\) and \(\dfrac{e}{m}\) have the same meaning as in G. C. 1929, but \(e\) denotes \(4.80\cdot 10^{10}\) CGSE, while \(e\left(\dfrac{e'}{c}\right)\) denotes \(1.602\cdot 10^{-20}\) CGSM, with the corresponding changes for \(\dfrac{h}{e}\). The mass of the electron \(m\) is now expressed, as it should be, as \(e/\left(\dfrac{e}{m}\right)\), whereas in G. C. 1929 it had to be calculated from the relation \(e/\left(\dfrac{e}{m}\right)c\).
One of the essential questions in compiling tables such as those published here is the proper designation of certain quantities. A whole article could be devoted to this question, but I wish here to note only that some changes have been made here in comparison with the designations in G. C. 1929, as well as a number of other changes required from the standpoint of logic. What is necessary is a consistent system of designations such that, if we write, for example, \(R_0T_0=V_0A_0\) (see table c), then both sides must be identical. For this reason, precisely in the designations of certain units there appear such quantities as mole\(^{-1}\), gram-equivalent\(^{-1}\), atm\(^{-1}\), etc. Further, in order to make such a check possible in all cases (in particular, in cases where electron-volts occur), it is necessary to write \(e'=4.80\cdot 10^{-10}\) CGSE electron\(^{-1}\), instead of simply CGSE, and to make the corresponding changes in the expressions for other units connected with \(e\) or \(e'\). In the same way, the mass of the electron \((m)\) must be expressed as gram-electron\(^{-1}\).
In tables a and b the designations of the sections (from A to M) have been kept the same as those adopted in G. C. 1929. These designations are useful in that they show which fundamental constants of table a are connected with each of the auxiliary constants of table b.
Table a
Principal constants and relations¹)
| Section | Constant | Designation and value |
|---|---|---|
| A | Speed of light | $$c=(2.99776\pm0.00004)\cdot10^{10}\ \mathrm{cm}\cdot\mathrm{sec}^{-1}$$ |
| B | Gravitational constant | $$G=(6.670\pm0.005)\cdot10^{-8}\ \mathrm{dyn}\cdot\mathrm{cm}^{2}\cdot\mathrm{g}^{-2}$$ |
| C | Liter (= 1000 ml-liter) | $$L=1000.028\pm0.002\ \mathrm{cm}^{3}$$ |
| D | Volume of an ideal gas (0°C, \(A_0\)) | $$V_0=(22.4146\pm0.0006)\cdot10^{3}\ \mathrm{cm}^{3}\cdot\mathrm{atm}\cdot\mathrm{mole}^{-1}$$ |
| D | The same | $$V'_0=22.4140\pm0.0006\ \mathrm{l}\cdot\mathrm{atm}\cdot\mathrm{mole}^{-1}$$ |
| D | Volume of an ideal gas (0°C, \(A_{45}\)) | $$V_{45}=(22.4157\pm0.0006)\cdot10^{3}\ \mathrm{cm}^{3}\cdot\mathrm{atm}\cdot\mathrm{mole}^{-1}$$ |
| D | The same | $$V'_{45}=22.4151\pm0.0006\ \mathrm{l}\cdot\mathrm{atm}\cdot\mathrm{mole}^{-1}$$ |
| E | International ohm (\(=p\) abs. ohm) | $$p=1.00048\pm0.00002$$ |
| E | International ampere (\(=q\) abs. amp.) | $$q=0.99986\pm0.00002$$ |
| F | Atomic weights (see Table \(a'\)) | — |
| G | Standard atmosphere | $$A_0=(1.013246\pm0.000004)\cdot10^{6}\ \mathrm{dyn}\cdot\mathrm{cm}^{-2}\cdot\mathrm{atm}^{-1}$$ |
| G | 45° atmosphere | $$A_{45}=(1.013195\pm0.000004)\cdot10^{6}\ \mathrm{dyn}\cdot\mathrm{cm}^{-2}\cdot\mathrm{atm}^{-1}$$ |
| H | Melting point of ice (abs. scale) | $$T_0=273.16\pm0.01^\circ\mathrm{K}$$ |
| I | Mechanical equivalent of heat | $$J_{15}=4.1855\pm0.0004\ \mathrm{abs.}\ \mathrm{J}\cdot\mathrm{cal}_{15}^{-1}$$ |
| I | The same, according to electrical-measurement data | $$J'_{15}=4.1847\pm0.0003\ \mathrm{int.}\ \mathrm{J}\cdot\mathrm{cal}_{15}^{-1}$$ |
| J | Faraday constant (1) on the chemical scale | $$F=96501_2\pm10\ \mathrm{int.}\ \mathrm{coul}\cdot\mathrm{g\text{-}eqv}^{-1}=96587_7\pm10\ \mathrm{abs.}\ \mathrm{coul}\cdot\mathrm{g\text{-}eqv}^{-1}=9648.7_7\pm1.0\ \mathrm{CGSM}\cdot\mathrm{g\text{-}eqv}^{-1}$$ $$F'=Fc=(2.89247\pm0.00030)\cdot10^{14}\ \mathrm{CGSE}\cdot\mathrm{g\text{-}eqv}^{-1}$$ |
| J | (2) on the physical scale | $$F=96514_0\pm10\ \mathrm{abs.}\ \mathrm{coul}\cdot\mathrm{g\text{-}eqv}^{-1}=9651.4_0\pm1.0\ \mathrm{CGSM}\cdot\mathrm{g\text{-}eqv}^{-1}$$ $$F'=Fc=(2.89326\pm0.00030)\cdot10^{14}\ \mathrm{CGSE}\cdot\mathrm{g\text{-}eqv}^{-1}$$ |
| K | Avogadro number (on the chemical scale) | $$N_0=(6.0228_3\pm0.0011)\cdot10^{23}\ \mathrm{mole}^{-1}$$ |
| K | Charge of the electron | $$e=\frac{F}{N_0}=(1.60203_3\pm0.00034)\cdot10^{-20}\ \mathrm{CGSM}$$ $$e'=ec=(4.8025_1\pm0.0010)\cdot10^{-10}\ \mathrm{CGSE}$$ |
| L | Specific charge of the electron | $$\frac{e}{m}=(1.7592\pm0.0005)\cdot10^{7}\ \mathrm{CGSM}\cdot\mathrm{g}^{-1}$$ $$\frac{e'}{m}=\frac{ec}{m}=(5.2736_6\pm0.0015)\cdot10^{17}\ \mathrm{CGSE}\cdot\mathrm{g}^{-1}$$ |
| M | Planck constant | \(h\) (see Table \(c\)) |
¹) Unless specially stated, all quantities in these tables that are connected with moles or gram-equivalents are given on the chemical scale.
In conclusion I should like to note that, as a result of the very extensive and carefully planned work on the general physical constants that has been carried out since 1929, the general situation here has improved considerably. Further, at the present time it is recognized much more widely than in 1929 that even broadly conceived and costly investigations may be associated with serious and quite unexpected sources of systematic errors. For this reason it is necessary that each of the important constants be determined by as large a number as possible of different methods, and it is also desirable that these methods differ from one another as much as possible. Only on the condition that radically different methods give results in full agreement with one another can one really rely on the final weighted mean value of the quantity being determined.
Table a′
Atomic weights
| (1) | By the physical scale \((\mathrm{O}^{16}=16.000)\) \(\mathrm{H}^{1}=1.00813 \pm 0.00001_{7}\) \(\mathrm{H}=1.00827_{6}\pm 0.00001^{?}\) \(\mathrm{He}^{4}=4.00389\pm 0.00007\) \(\mathrm{C}^{12}=12.00386\pm 0.00004\) \(\mathrm{C}=12.01465\pm 0.00023\) \(\mathrm{N}^{14}=14.00753\pm 0.00005\) \(\mathrm{N}=14.01121\pm 0.00009_{5}\) \(\mathrm{O}^{16}=16.0000\) \(\mathrm{O}=16.00435_{7}\pm 0.00008_{6}\) |
\(\mathrm{H}^{2}=2.01473\pm 0.00001_{9}\) (with the abundance ratio of \(\mathrm{H}^{1}\) to \(\mathrm{H}^{2}\) equal to \(6900\pm 100\)) \(\mathrm{C}^{13}=13.00761\pm 0.00015\) (with the abundance ratio of \(\mathrm{C}^{12}\) to \(\mathrm{C}^{13}\) equal to \(92\pm 2\)) \(\mathrm{N}^{15}=15.0049\pm 0.0002\) (with the abundance ratio of \(\mathrm{N}^{14}\) to \(\mathrm{N}^{15}\) equal to \(270\pm 6\)) \(\mathrm{O}^{17}=17.0045\) \(\mathrm{O}^{18}=18.0049\) (with the isotope abundance ratio \(\mathrm{O}^{16}:\mathrm{O}^{17}:\mathrm{O}^{18}=(506\pm 10):1:(0.204\pm 0.008)\)) |
| (2) | By the chemical scale \((\mathrm{O}=16.0000)\) Ratio of atomic weights on the physical scale to atomic weights on the chemical scale \[r=(16.004357\pm 0.000086):16=1.000272\pm 0.000005\] \(\mathrm{H}^{1}=1.00785_{6}\pm 0.00001_{8}\) (recalculated from the physical scale) \(\mathrm{H}^{2}=2.01418_{2}\pm 0.00002_{1}\) » » » » \(\mathrm{H}=1.00800_{2}\pm 0.00001_{8}\) » » » » \(\mathrm{He}^{4}=4.00280\pm 0.00007\) » » » » \(\mathrm{C}=12.01139\pm 0.00024\) » » » » \(\mathrm{N}=14.00740\pm 0.00012\) » » » » \(\mathrm{N}=14.0086\pm 0.0007\) (direct observation) \(\mathrm{Na}=22.994\pm 0.003\) \(\mathrm{Cl}=35.457\pm 0.001\) \(\mathrm{Ca}=40.080\pm 0.005\) \(\mathrm{Ag}=107.880\pm 0.002\) \(\mathrm{I}=126.915\pm 0.004\) |
Table b
Additional quantities calculated or used in connection with table a
| Section | Constant | Designation and value |
|---|---|---|
| A | Ratio of electrostatic units to electromagnetic units (observed directly) | $c'=(2.9971_{0}\pm0.0001\cdot10^{10})$ $\mathrm{cm}^{1/2}\cdot\mathrm{sec}^{-1/2}\cdot\mathrm{ohm}^{1/2}=(2.9978_{1}\pm0.0001_{0})\cdot10^{10}\,\mathrm{cm}\cdot\mathrm{sec}^{-1}$ |
| A | The same (by an indirect method) | $c'=c=(2.99776\pm0.00004)\cdot10^{10}\,\mathrm{cm}\cdot\mathrm{sec}^{-1}$ |
| B | Mean density of the Earth | $\delta=5.517\pm0.004\,\mathrm{g}\cdot\mathrm{cm}^{-3}$ |
| C | Maximum density of water | $\delta_m(\mathrm{H_2O})=0.999972\pm0.000002\,\mathrm{g}\cdot\mathrm{cm}^{-3}$ |
| D | Acceleration of gravity (standard value) | $g_{0}=980.655\,\mathrm{cm}\cdot\mathrm{sec}^{-2}$ |
| D | The same, at latitude $45^\circ$ | $g_{45}=980.616\,\mathrm{cm}\cdot\mathrm{sec}^{-2}$ |
| D | Density of gaseous oxygen $(0^\circ\mathrm{C},\, A_{45})$ | $L_{1}=1.42897\pm0.00003\,\mathrm{g}\cdot\mathrm{l}^{-1}$ |
| D | Limiting density of gaseous oxygen $(0^\circ\mathrm{C},\, A_{45})$ | $L_{\mathrm{lim}}=1.427609\pm0.000037\,\mathrm{g}\cdot\mathrm{l}^{-1}$ |
| D | Factor for converting oxygen $(0^\circ\mathrm{C},\, A_{45})$ to an ideal gas | $1-x=1.000953_{3}\pm0.000009_{4}$ |
| E | International coulomb $(=q$ abs. coulombs$)$ | $q=0.99986\pm0.00002$ |
| E | International gauss $(=q$ abs. gausses$)$ | — |
| E | International henry $(=p$ abs. henries$)$ | $p=1.00048\pm0.00002$ |
| E | International volt $(=pq$ abs. volts$)$ | $pq=1.00034\pm0.00003$ |
| E | International joule $(=pq^{2}$ abs. joules$)$ | $pq^{2}=1.00020\pm0.00004_{5}$ |
| G | Specific weight of Hg $(0^\circ\mathrm{C},\, A_{0})$, relative to air-free $\mathrm{H_2O}$ at maximum density | $\rho_{0}=13.59542_{0}\pm0.00005$ |
| G | Density of Hg $(0^\circ\mathrm{C},\, A_{0})$ | $D_{0}=13.59504_{0}\pm0.00005_{7}\,\mathrm{g}\cdot\mathrm{cm}^{-3}$ |
Continuation of Table 6
| Section | Constant | Symbol and value |
|---|---|---|
| J | Electrochemical equivalents (on the chemical scale) Silver (apparent) |
\(E^*_{\mathrm{Ag}} = 1.11800 \cdot 10^{-3}\) \(g \cdot \text{international coul}^{-1}\) |
| J | Silver (apparent) | \(E^*_{\mathrm{Ag}} = (1.11807 \pm 0.00012) \cdot 10^{-3}\) \(g \cdot \text{abs. coul}^{-1}\) |
| J | ” (corrected) | \(E_{\mathrm{Ag}} = (1.315026 \pm 0.000025) \cdot 10^{-3}\) \(g \cdot \text{abs. coul}^{-1}\) |
| J | Iodine (apparent) | \(E_{\mathrm{I}} = (1.315026 \pm 0.000025) \cdot 10^{-3}\) \(g \cdot \text{abs. coul}^{-1}\) |
| J | Iodine (corrected) | \(E_{\mathrm{I}} = (1.31535 \pm 0.00014) \cdot 10^{-3}\) \(g \cdot \text{abs. coul}^{-1}\) |
| K | Effective lattice constant of calcite (18°C) according to Siegbahn’s system | \(d''_{18} = 3.02904 \cdot 10^{-8}\ \mathrm{cm}\) |
| K | True lattice constant of calcite (20°C) according to Siegbahn’s system | \(d'_{20} = 3.02951_{2} \cdot 10^{-8}\ \mathrm{cm}\) |
| K | True lattice constant of calcite (20°C) according to the CGS system | \(d_{20} = (3.03567_{4} \pm 0.00018) \cdot 10^{-8}\ \mathrm{cm}\) |
| K | Ratio of wavelength scales according to the CGS and Siegbahn systems | \(\dfrac{\lambda_g}{\lambda_s} = 1.002034 \pm 0.000060\) |
| K | Density of calcite (20°C) | \(\rho = 2.71029 \pm 0.00003\ g \cdot \mathrm{cm}^{-3}\) |
| K | Structure constant of calcite (20°C) | \(\varphi = 1.03594 \pm 0.00001\) |
| K | Molecular weight of calcite (on the chemical scale) | \(M = 100.091_{4} \pm 0.005\) |
| L | Rydberg constant for hydrogen (H¹) | \(R_{\mathrm{H}} = 109677.581_{2} \pm 0.007_{5}\ \mathrm{cm}^{-1}\) (on the international angstrom scale) |
| L | Rydberg constant for deuterium (H²) | \(R_{\mathrm{D}} = 109707.419_{3} \pm 0.007_{5}\ \mathrm{cm}^{-1}\) (on the same scale) |
| L | Rydberg constant for helium | \(R_{\mathrm{He}} = 109722.263 \pm 0.012\ \mathrm{cm}^{-1}\) (on the same scale) |
| L | Rydberg constant for infinite nuclear mass | \(R_{\infty} = 109737.303 \pm 0.017\ \mathrm{cm}^{-1}\) (on the same scale) or \(\pm 0.05\ \mathrm{cm}^{-1}\) (in the CGS system) |
NEW TABLE OF VALUES OF GENERAL PHYSICAL CONSTANTS
Table c
Some derived quantities¹)
Planck constant
\[ h=\left\{\frac{2\pi^{2}c^{3}F^{3}}{R_{\infty}N_{0}^{5}\left(\frac{e}{m}\right)}\right\}^{\frac13} =(6.624_{4}\pm0.002_{4})\cdot10^{-27}\ \mathrm{erg\cdot sec} \]
\[ \frac{h}{e}= \left\{\frac{2\pi^{2}c^{3}F^{2}}{R_{\infty}N_{0}^{2}\left(\frac{e}{m}\right)}\right\}^{\frac13} =(4.1349\pm0.0007_{1})\cdot10^{-8}\ \mathrm{erg\cdot sec\cdot CGSM^{-1}} \]
\[ \frac{h}{e'}=\frac{h}{ec}= \left\{\frac{2\pi^{2}F^{2}}{R_{\infty}N_{0}^{2}\left(\frac{e}{m}\right)}\right\}^{\frac13} =(1.3793_{3}\pm0.0002_{4})\cdot10^{-17}\ \mathrm{erg\cdot sec\cdot CGSE^{-1}} \]
Atomic weight of the electron
\[ E=\frac{F}{\left(\frac{e}{m}\right)} \]
by the physical scale
\[ E=(5.4862_{4}\pm0.0017)\cdot10^{-4} \]
by the chemical scale
\[ E=(5.4847_{6}\pm0.007)\cdot10^{-4} \]
Constant of band spectra, relating wave number and moment of inertia
\[ \frac{h}{8\pi^{2}c}= \left\{\frac{F^{5}}{256\pi^{4}R_{\infty}N_{0}^{5}\left(\frac{e}{m}\right)}\right\}^{\frac13} =(27.98_{66}\pm0.01_{0})\cdot10^{-40}\ \mathrm{g\cdot cm} \]
Boltzmann constant
\[ k=\frac{R_{0}}{N_{0}}= \frac{V_{0}A_{0}}{T_{0}N_{0}} =(1.38047_{4}\pm0.00026)\cdot10^{-16}\ \mathrm{erg\cdot degree^{-1}} \]
Charge of 1 g of H in electrolysis
\[ \frac{F}{H}=9572.1_{78}\pm1.0\ \mathrm{CGSM\cdot g^{-1}} \]
Charge of 1 g of H¹ in electrolysis
\[ \frac{e}{M_{H^{1}}}=\frac{F}{H^{1}}=9573.5_{60}\pm1.0\ \mathrm{CGSM\cdot g^{-1}} \]
Compton shift at \(90^\circ\)
\[ \frac{h}{mc}= \left\{\frac{2\pi^{2}F^{2}\left(\frac{e}{m}\right)^{2}}{R_{\infty}N_{0}^{2}}\right\}^{\frac13} =(0.024265_{14}\pm0.000005_{7})\cdot10^{-8}\ \mathrm{cm} \]
¹) In order to be able to calculate the probable error in a derived quantity, it is necessary to express it in terms of various fundamental constants of the tables, \(a\) or \(b\); this has been done in all cases. Since here \(e\) and \(h\) are treated as derived quantities, they do not appear in such defining expressions. However, when calculating the numerical values of derived quantities, the work is often greatly simplified by using previously calculated numerical values of other derived quantities, including, in particular, \(e\) and \(h\). In order to show how some derived quantities depend on \(e\) and \(h\), in many cases expressions of various forms are given for them below.
Continuation of Table c
Energy corresponding to 1 abs. electron-volt (in ergs)
\[ E_0=10^8 e=10^8\frac{F}{N_0}=(1.60203_8\pm 0.00034)\cdot 10^{-12}\ \text{erg} \]
Energy in calories per mole for 1 abs. electron-volt per molecule
\[ \frac{F\ \text{(abs. coul. per g-equiv)}}{J_{15}\ \text{(abs. joule per cal)}}=23052.8_5\pm 3.2\ \text{cal}_{15}\cdot \text{mole}^{-1} \]
Fine-structure constant
\[ \alpha=\frac{2\pi(e')^2}{hc} = \left\{ \frac{4\pi R_{\infty}F\left(\dfrac{e}{m}\right)}{N_0} \right\}^{1/3} = \]
\[ =(7.2976_6\pm 0.0008_6)\cdot 10^{-3} \]
\[ \frac{1}{\alpha}=137.030_2\pm 0.016 \]
\[ \alpha^2=(5.3256\pm 0.0013)\,10^{-5} \]
Gas constant per mole
\[ R_0=\frac{V_0A_0}{T_0}=(8.31436\pm 0.00038)\cdot 10^7\ \text{erg}\cdot \text{degree}^{-1}\cdot \text{mole}^{-1} \]
\[ R'_0=R_0\cdot \frac{10^{-7}}{J_{15}}=1.98646_7\pm 0.00021\ \text{cal}_{15}\cdot \text{degree}^{-1}\cdot \text{mole}^{-1} \]
\[ R''_0=\frac{V'_0}{T_0}=(8.20544_7\pm 0.00037)\cdot 10^{-2}\ \text{l}\cdot \text{atm}\ \text{degree}^{-1}\ \text{mole} \]
\[ R'''_0=\frac{R_0}{A_0}=\frac{V_0}{T_0}=82.0566_7\pm 0.0037\ \text{cm}^3\cdot \text{atm}\cdot \text{degree}^{-1}\cdot \text{mole}^{-1}, \]
so that
\[ R_0T_0=V_0A_0=(2.27115_0\pm 0.00006)\cdot 10^{10}\ \text{erg}\cdot \text{mole}^{-1} \]
Loschmidt number \((0^\circ\mathrm{C}, A_0)\)
\[ n_0=\frac{N_0}{V_0}=(2.6370_{13}\pm 0.0005_0)\cdot 10^{19}\ \text{atm}^{-1}\cdot \text{cm}^{-3} \]
Magnetic moment of one Bohr magneton
\[ \mu_1=\left(\frac{h}{4\pi}\right)\left(\frac{e}{m}\right)=\frac{1}{4\pi} \]
\[ \left\{ \frac{2\pi^2c^3F^5\left(\dfrac{e}{m}\right)^2}{R_{\infty}N_0^5} \right\}^{1/3} =(0.9273_{45}\pm 0.0003_7)\cdot 10^{-20}\ \text{erg}\cdot \text{gauss}^{-1} \]
Magnetic moment per mole for one Bohr magneton per molecule
\[ \mu_1N_0=\frac{1}{4\pi} \left\{ \frac{2\pi^2c^3F^5\left(\dfrac{e}{m}\right)^2}{R_{\infty}N_0^2} \right\}^{1/3} =5585.2_4\pm 1.6\ \text{erg}\cdot \text{gauss}^{-1}\cdot \text{mole}^{-1} \]
Mass of the \(\alpha\)-particle
\[ M_{\alpha}=\frac{\mathrm{He}-2E}{N_0}=(6.6442\pm 0.0012)\cdot 10^{-24}\ \text{g} \]
Mass of an atom of unit atomic weight
\[ M_0=\frac{1}{N_0}=(1.66035\pm 0.00031)\cdot 10^{-24}\ \text{g} \]
Continuation of Table c
| Quantity | Formula and value |
|---|---|
| Electron mass | $$m=\frac{e}{\left(\dfrac{e}{m}\right)}=\frac{\left(\dfrac{F}{N_0}\right)}{\left(\dfrac{e}{m}\right)}=(9.1066_6\pm0.0032)\cdot10^{-28}$$ |
| Mass of the atom H¹ | $$M_{\mathrm{H}^1}=\frac{\mathrm{H}^1}{N_0}=(1.67339_3\pm0.00031)\cdot10^{-24}\ \mathrm{g}$$ |
| Proton mass | $$M_p=\frac{(\mathrm{H}^1-E)}{N_0}=(1.67248_2\pm0.00031)\cdot10^{-24}\ \mathrm{g}$$ |
| Radiation density constant | $$a=\frac{8\pi^5 k^4}{15c^3h^3}=\left(\frac{V_0A_0}{T_0}\right)^4\frac{4\pi^3N_0R_\infty\left(\dfrac{e}{m}\right)}{15c^6F^5}$$ $$=(7.569_{42}\pm0.004_9)\cdot10^{-15}\ \mathrm{erg}\cdot\mathrm{cm}^{-3}\cdot\mathrm{deg}^{-4}$$ |
| Ratio of the mass of the atom H¹ to the electron mass | $$\frac{M_{\mathrm{H}^1}}{m}=\left(\frac{e}{m}\right)\left(\frac{\mathrm{H}^1}{F}\right)=1837.5_{61}\pm0.5_6$$ |
| Ratio of the proton mass to the electron mass | $$\frac{M_p}{m}=\left(\frac{e}{m}\right)\left(\frac{\mathrm{H}^1-E}{F}\right)=1836.5_{61}\pm0.5_6$$ |
| Second radiation constant | $$c_2=\frac{hc}{k}=\frac{c^2T_0}{V_0A_0}\left\{\frac{2\pi^2F^5}{R_\infty N_0^2\left(e/m\right)}\right\}^{1/3}=1.4384_8\pm0.0003_4\ \mathrm{cm}\cdot\mathrm{deg}$$ |
| Specific charge of the α-particle | $$\frac{2e}{M_\alpha}=\frac{2F}{\mathrm{He}-2E}=4822.3_3\pm0.5_1\ \mathrm{CGSM}\cdot\mathrm{g}^{-1}$$ |
| Specific charge of the proton | $$\frac{e}{M_p}=\frac{F}{\mathrm{H}^1-E}=9578.7_7\pm1.0\ \mathrm{CGSM}\cdot\mathrm{g}^{-1}$$ |
| Stefan–Boltzmann law constant | $$\sigma=\frac{ac}{4}=\frac{2\pi^5k^4}{15c^2h^3}=$$ $$=\left(\frac{V_0A_0}{T_0}\right)^4\frac{\pi^3N_0R_\infty\left(\dfrac{e}{m}\right)}{15(Fc)^5}=(5.6728_3\pm0.003_7)\cdot10^{-5}\ \mathrm{erg}\cdot\mathrm{cm}^{-2}\cdot\mathrm{deg}^{-4}\ \mathrm{sec}^{-1}$$ |
Continuation of Table c
Wavelength corresponding to 1 abs. volt\(^1\)
\[ \lambda_0 = 10^{-8} c^2 \left(\frac{h}{e}\right) = \frac{c^2}{10^8} \left\{ \frac{2\pi^2/F^2}{R_\infty N_0^2\left(\frac{e}{m}\right)} \right\}^{1/3} = (12395.4 \pm 2.1)\cdot 10^{-8}\ \mathrm{cm}\cdot \mathrm{abs.}\,V^{-1} \]
Wave number corresponding to 1 abs. volt
\[ s_0=\frac{1}{\lambda_0} = \frac{10^8}{c^2} \left\{ \frac{R_\infty N_0^2\left(\frac{e}{m}\right)}{2\pi^2 F^2} \right\}^{1/3} =8067.4_9 \pm 1.4\ \mathrm{cm}^{-1}\cdot \mathrm{abs.}\,V^{-1} \]
Constant of Wien’s displacement law\(^2\)
\[ A=\frac{c_2}{4.965114}=0.28971_8 \pm 0.00007\ \mathrm{cm}\cdot \mathrm{deg} \]
Zeeman displacement per gauss
\[ \frac{\frac{e}{m}}{4\pi c} = (4.6699_1 \pm 0.0013)\cdot 10^{-5}\ \mathrm{cm}^{-1}\cdot \mathrm{gauss}^{-1} \]
\(^1\) In G.C. 1929, in the equality for \(\lambda_0\), the factor \(10^{-8}\) was accidentally omitted.
\(^2\) The factor 4.965114 is the root of the equation
\[ e^{-\beta}+\left(\frac{\beta}{5}\right)-1=0. \]
LITERATURE
- R. T. Birge, Rev. Mod. Phys., 1, 1, 1929.
- R. T. Birge, Phys. Rev., 40, 288, 1932.
- a) R. T. Birge, Phys. Rev., 54, 972, 1938; b) Phys. Rev., 60, 766, 1941.
- See G.C. 1929, pp. 41–43.
- R. T. Birge, Phys. Rev., 48, 918, 1935; Nature, 137, 187, 1936.
- F. G. Dunnington, Rev. Mod. Phys., 11, 65, 1939.
- Y. W. M. Du Mond, Phys. Rev., 56, 153, 1939; 58, 457, 1940.
- Y. W. Drinkwater, O. Richardson and W. E. Williams, Proc. Roy. Soc., A 174, 164, 1940.
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Copies of this table were sent to many persons, and figures taken from it have repeatedly been published in various works. ↩