Abstract

After a brief summary of the ideas underlying the quantum theory of dispersion it is shown that it can be applied to the refraction of x-rays, although the assumption that the number of atoms in a wave length cube is large is no longer satisfied. A general formula for the index of refraction in terms of the atomic absorption coefficient a and the critical frequencies is given. From the condition, experimentally verified, that the electrons in the atom for impressed frequencies, large compared to their natural frequencies, shall act like free electrons as far as the index of refraction is concerned, a relation is obtained for a. From the failure of this relation when applied to the groups of electrons separately, conclusions are drawn as to the coupling of the groups. Some considerations on the origin of the Compton shifted radiation are added, from which it appears that in the wave description this radiation must be regarded as coming from all the atoms and as being coherent with the incident waves; a result suited to stress the difficulty of harmonizing the wave picture with that of quantum processes in the atoms.

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  1. As Professor Kramers kindly informs me, he too has obtained most of the results derived in 3. (To be published shortly.)
  2. ai is then the integral of the absorption coefficient over all frequencies, considering the absorption curve as approaching zero on both sides of the natural frequency. In reality the absorption coefficient at large frequencies approaches a small positive value corresponding to the scattering of a free charge. This part of the absorption coefficient has to be subtracted.
  3. N. Bohr, Zs. f. Phys. 13, p. 161; 1923.
  4. H. A. Kramers, Nature, 113, p. 673; 114, p. 310, 1924; H. A. Kramers and W. Heisenberg, Zs. f. Phys., 31, p. 681; 1925. See also M. Born, Zs. f. Phys., 26, p. 379; 1924 and J. H. van Vleck, Phys. Rev., 24, p. 344; 1924.
  5. This is identical with a formula obtained previously by R. Ladenburg, Zs. f. Phys., 4, p. 451; 1921 in a different way; see also R. Ladenburg and F. Reiche, Naturwiss., 11, p. 584; 1923.
  6. For an elaborate mathematical discussion of the case of few atoms in a wavelength cube leading to the same conclusion the reader is referred to F. Reiche, Ann. d. Phys. 50, pp. 1 and 121, 1916.
  7. Disregarding the different possible orientations of the orbits of the valence electrons.
  8. These are the values customarily used in x-ray notation; their ratio is that of the statistical weights of the two states.
  9. For a detailed discussion of the experimental evidence and references to the literature see H. A. Kramers, Phil. Mag., 46, p. 836; 1923; see also F. K. Richtmyer, Phys. Rev., 27, p. 1; 1926.
  10. Bergen Davis and H. M. Terrill, Proc. Nat. Ac., 8, p. 537; 1922;A. H. Compton, Phil. Mag., 45, p. 1121; 1922;C. C. Hatley and Bergen Davis, Phys. Rev., 23, p. 290; 1924;Bergen Davis and R. von Nardroff, Phys. Rev., 23, p. 291; 1924; Proc. Nat. Ac., 10, p. 60, 384; 1924;C. C. Hatley, Phys. Rev., 24, p. 486; 1924;R. von Nardroff, Phys. Rev., 24, p. 143; 1924;E. Hjalmar and M. Siegbahn, Nature, 115, p. 85; 1925;A. Larsson, M. Siegbahn and T. Waller, Phys. Rev., 25, p. 235; 1925.Bergen Davis and C. M. Slack, Phys. Rev., 25, p. 881; 1925;M. Siegbahn, Journ. de Phys., 6, p. 228; 1925;Bergen Davis and C. M. Slack, Phys. Rev., 27, p. 18; 1926.A. Larson, Zs. f. Phys. 35, p. 401; 1926;I am indebted to Professor Davis and Mr. Slack for some additional unpublished material. This deals with the refraction of X-rays in silver on the long wavelength side of the K-limit. Below are given the values of δ.106. as observed and as computed from the Lorentz formula and from (9) respectively: .707 Å.U.; -5.3±.6, -5.7, -6.0; .515 Å.U.; -3.0±.4, -2.1, -3.1; .500 Å.U., -2.5±.3, -1.0, -2.9; .485 Å.U., -2.2±.3, 16.4, -2.5.
  11. W. Kuhn, Zs. f. Phys., 33, p. 408; 1925.
  12. W. Thomas, Naturwiss., 13, p. 627; 1925. See also F. Reiche and W. Thomas, Zs. f. Phys., 34, p. 510; 1925.
  13. W. Heisenberg, Zs. f. Phys., 31 p. 617; 1925.
  14. E. C. Stoner, Phil. Mag., 48, p. 719; 1924.
  15. The shifted radiation can be considered, as far as the variation of its frequency with the angle of observation is concerned, as coming from sources moving in the direction of the incident beam with a velocity depending on the frequency of this beam (see A. H. Compton, Phys. Rev., 21, p. 483; 1923). With coherence of the shifted radiation we mean that, if by a Lorentz transformation we go to a coordinate system in which the sources are at rest, then in this system the radiation coming from them has a definite phase relation to the incident waves. The fact that one does not observe selective reflection in crystals at angles corresponding to the shifted radiation does not contradict this, as the moving sources need not start moving all at the same time so that they no longer form a lattice.
  16. W. Bothe and H. Geiger, Zs. f. Phys., 32, p. 639; 1925.
  17. A. H. Compton and A. W. Simon, Phys. Rev., 26, p. 289; 1925.
  18. Bohr, Kramers and Slater, Phil. Mag., 47, p. 785, 1924; Zs. f. Phys., 24, p. 69; 1924.

Bohr, N.

N. Bohr, Zs. f. Phys. 13, p. 161; 1923.

Bothe, W.

W. Bothe and H. Geiger, Zs. f. Phys., 32, p. 639; 1925.

Compton, A. H.

A. H. Compton and A. W. Simon, Phys. Rev., 26, p. 289; 1925.

Davis, Bergen

Bergen Davis and H. M. Terrill, Proc. Nat. Ac., 8, p. 537; 1922;A. H. Compton, Phil. Mag., 45, p. 1121; 1922;C. C. Hatley and Bergen Davis, Phys. Rev., 23, p. 290; 1924;Bergen Davis and R. von Nardroff, Phys. Rev., 23, p. 291; 1924; Proc. Nat. Ac., 10, p. 60, 384; 1924;C. C. Hatley, Phys. Rev., 24, p. 486; 1924;R. von Nardroff, Phys. Rev., 24, p. 143; 1924;E. Hjalmar and M. Siegbahn, Nature, 115, p. 85; 1925;A. Larsson, M. Siegbahn and T. Waller, Phys. Rev., 25, p. 235; 1925.Bergen Davis and C. M. Slack, Phys. Rev., 25, p. 881; 1925;M. Siegbahn, Journ. de Phys., 6, p. 228; 1925;Bergen Davis and C. M. Slack, Phys. Rev., 27, p. 18; 1926.A. Larson, Zs. f. Phys. 35, p. 401; 1926;I am indebted to Professor Davis and Mr. Slack for some additional unpublished material. This deals with the refraction of X-rays in silver on the long wavelength side of the K-limit. Below are given the values of δ.106. as observed and as computed from the Lorentz formula and from (9) respectively: .707 Å.U.; -5.3±.6, -5.7, -6.0; .515 Å.U.; -3.0±.4, -2.1, -3.1; .500 Å.U., -2.5±.3, -1.0, -2.9; .485 Å.U., -2.2±.3, 16.4, -2.5.

Geiger, H.

W. Bothe and H. Geiger, Zs. f. Phys., 32, p. 639; 1925.

Heisenberg, W.

W. Heisenberg, Zs. f. Phys., 31 p. 617; 1925.

Kramers, H. A.

H. A. Kramers, Nature, 113, p. 673; 114, p. 310, 1924; H. A. Kramers and W. Heisenberg, Zs. f. Phys., 31, p. 681; 1925. See also M. Born, Zs. f. Phys., 26, p. 379; 1924 and J. H. van Vleck, Phys. Rev., 24, p. 344; 1924.

For a detailed discussion of the experimental evidence and references to the literature see H. A. Kramers, Phil. Mag., 46, p. 836; 1923; see also F. K. Richtmyer, Phys. Rev., 27, p. 1; 1926.

Kuhn, W.

W. Kuhn, Zs. f. Phys., 33, p. 408; 1925.

Ladenburg, R.

This is identical with a formula obtained previously by R. Ladenburg, Zs. f. Phys., 4, p. 451; 1921 in a different way; see also R. Ladenburg and F. Reiche, Naturwiss., 11, p. 584; 1923.

Simon, A. W.

A. H. Compton and A. W. Simon, Phys. Rev., 26, p. 289; 1925.

Stoner, E. C.

E. C. Stoner, Phil. Mag., 48, p. 719; 1924.

Terrill, H. M.

Bergen Davis and H. M. Terrill, Proc. Nat. Ac., 8, p. 537; 1922;A. H. Compton, Phil. Mag., 45, p. 1121; 1922;C. C. Hatley and Bergen Davis, Phys. Rev., 23, p. 290; 1924;Bergen Davis and R. von Nardroff, Phys. Rev., 23, p. 291; 1924; Proc. Nat. Ac., 10, p. 60, 384; 1924;C. C. Hatley, Phys. Rev., 24, p. 486; 1924;R. von Nardroff, Phys. Rev., 24, p. 143; 1924;E. Hjalmar and M. Siegbahn, Nature, 115, p. 85; 1925;A. Larsson, M. Siegbahn and T. Waller, Phys. Rev., 25, p. 235; 1925.Bergen Davis and C. M. Slack, Phys. Rev., 25, p. 881; 1925;M. Siegbahn, Journ. de Phys., 6, p. 228; 1925;Bergen Davis and C. M. Slack, Phys. Rev., 27, p. 18; 1926.A. Larson, Zs. f. Phys. 35, p. 401; 1926;I am indebted to Professor Davis and Mr. Slack for some additional unpublished material. This deals with the refraction of X-rays in silver on the long wavelength side of the K-limit. Below are given the values of δ.106. as observed and as computed from the Lorentz formula and from (9) respectively: .707 Å.U.; -5.3±.6, -5.7, -6.0; .515 Å.U.; -3.0±.4, -2.1, -3.1; .500 Å.U., -2.5±.3, -1.0, -2.9; .485 Å.U., -2.2±.3, 16.4, -2.5.

Thomas, W.

W. Thomas, Naturwiss., 13, p. 627; 1925. See also F. Reiche and W. Thomas, Zs. f. Phys., 34, p. 510; 1925.

Other (18)

As Professor Kramers kindly informs me, he too has obtained most of the results derived in 3. (To be published shortly.)

ai is then the integral of the absorption coefficient over all frequencies, considering the absorption curve as approaching zero on both sides of the natural frequency. In reality the absorption coefficient at large frequencies approaches a small positive value corresponding to the scattering of a free charge. This part of the absorption coefficient has to be subtracted.

N. Bohr, Zs. f. Phys. 13, p. 161; 1923.

H. A. Kramers, Nature, 113, p. 673; 114, p. 310, 1924; H. A. Kramers and W. Heisenberg, Zs. f. Phys., 31, p. 681; 1925. See also M. Born, Zs. f. Phys., 26, p. 379; 1924 and J. H. van Vleck, Phys. Rev., 24, p. 344; 1924.

This is identical with a formula obtained previously by R. Ladenburg, Zs. f. Phys., 4, p. 451; 1921 in a different way; see also R. Ladenburg and F. Reiche, Naturwiss., 11, p. 584; 1923.

For an elaborate mathematical discussion of the case of few atoms in a wavelength cube leading to the same conclusion the reader is referred to F. Reiche, Ann. d. Phys. 50, pp. 1 and 121, 1916.

Disregarding the different possible orientations of the orbits of the valence electrons.

These are the values customarily used in x-ray notation; their ratio is that of the statistical weights of the two states.

For a detailed discussion of the experimental evidence and references to the literature see H. A. Kramers, Phil. Mag., 46, p. 836; 1923; see also F. K. Richtmyer, Phys. Rev., 27, p. 1; 1926.

Bergen Davis and H. M. Terrill, Proc. Nat. Ac., 8, p. 537; 1922;A. H. Compton, Phil. Mag., 45, p. 1121; 1922;C. C. Hatley and Bergen Davis, Phys. Rev., 23, p. 290; 1924;Bergen Davis and R. von Nardroff, Phys. Rev., 23, p. 291; 1924; Proc. Nat. Ac., 10, p. 60, 384; 1924;C. C. Hatley, Phys. Rev., 24, p. 486; 1924;R. von Nardroff, Phys. Rev., 24, p. 143; 1924;E. Hjalmar and M. Siegbahn, Nature, 115, p. 85; 1925;A. Larsson, M. Siegbahn and T. Waller, Phys. Rev., 25, p. 235; 1925.Bergen Davis and C. M. Slack, Phys. Rev., 25, p. 881; 1925;M. Siegbahn, Journ. de Phys., 6, p. 228; 1925;Bergen Davis and C. M. Slack, Phys. Rev., 27, p. 18; 1926.A. Larson, Zs. f. Phys. 35, p. 401; 1926;I am indebted to Professor Davis and Mr. Slack for some additional unpublished material. This deals with the refraction of X-rays in silver on the long wavelength side of the K-limit. Below are given the values of δ.106. as observed and as computed from the Lorentz formula and from (9) respectively: .707 Å.U.; -5.3±.6, -5.7, -6.0; .515 Å.U.; -3.0±.4, -2.1, -3.1; .500 Å.U., -2.5±.3, -1.0, -2.9; .485 Å.U., -2.2±.3, 16.4, -2.5.

W. Kuhn, Zs. f. Phys., 33, p. 408; 1925.

W. Thomas, Naturwiss., 13, p. 627; 1925. See also F. Reiche and W. Thomas, Zs. f. Phys., 34, p. 510; 1925.

W. Heisenberg, Zs. f. Phys., 31 p. 617; 1925.

E. C. Stoner, Phil. Mag., 48, p. 719; 1924.

The shifted radiation can be considered, as far as the variation of its frequency with the angle of observation is concerned, as coming from sources moving in the direction of the incident beam with a velocity depending on the frequency of this beam (see A. H. Compton, Phys. Rev., 21, p. 483; 1923). With coherence of the shifted radiation we mean that, if by a Lorentz transformation we go to a coordinate system in which the sources are at rest, then in this system the radiation coming from them has a definite phase relation to the incident waves. The fact that one does not observe selective reflection in crystals at angles corresponding to the shifted radiation does not contradict this, as the moving sources need not start moving all at the same time so that they no longer form a lattice.

W. Bothe and H. Geiger, Zs. f. Phys., 32, p. 639; 1925.

A. H. Compton and A. W. Simon, Phys. Rev., 26, p. 289; 1925.

Bohr, Kramers and Slater, Phil. Mag., 47, p. 785, 1924; Zs. f. Phys., 24, p. 69; 1924.

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