Summary
The bremsstrahlung cross-section at the high-frequency limit of the spectrum has been evaluated to the first three nonvanishing orders inαZ for high-energy incident electrons. The power series inαZ which occurs in the cross-section apparently converges rapidly enough to yield accurate results over a wide range of values ofZ. The values obtained for the cross-section are somewhat smaller than those predicted by the lowest-order (Fano) theory and are in good agreement with recent experimental measurements.
Riassunto
Si è valutata, per elettroni incidenti di alta energia, la sezione efficace della bremsstrahlung al limite delle alte frequenze dello spettro sino ai primi tre ordini diαZ, che non si annullano. La serie di potenze diαZ, che compare nella sezione efficace, apparentemente converge abbastanza rapidamente da dare risultati accurati in un vasto campo di valori diZ. I valori della sezione efficace così ottenuti sono alquanto minori di quelli predetti dalla teoria del più basso ordine (Fano) e concordano abbastanza bene con le recenti misure sperimentali.
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References
U. Fano:Phys. Rev.,116, 1156 (1959). The bremsstrahlung cross-section at the high-frequency limit of the spectrum was previously obtained byNagasaka [F. Nagasaka: Thesis, University of Notre Dame (1955) (unpublished)], using Sommerfeld-Maue wave functions for both initial and final electron states. A similar calculation was made byMihailovic (M. V. Mihailovic:Nuovo Cimento,9, 331 (1958)].
U. Fano: ref. (1).
W. R. Johnson andC. J. Mullin:Phys. Rev.,119, 1270 (1960). See alsoR. H. Pratt:Phys. Rev.,120, 1717 (1960).
The validity of this argument for the use ofψ 1 as given by eq. (6) in the bremsstrahlung matrix element has been verified to the first two nonvanishing orders ofαZ by evaluating the matrix element with modified Sommerfeld-Maue wave functions (Johnsons andMullin, reference (4) used for bothψ 1 andψ 2. We have found that form/p 1≪1 andp 2=0 the only contributions to the matrix element arise from that part ofψ 1 which is given by eq. (6). A proof of the validity of the use of the-energy wave functions given by eq. (6) to describe the outgoing electron in the relativistic photoeffect has been given byPratt (R. H. Pratt:Phys. Rev.,117, 1017 (1960)]. The criterion for the validity of the use of eq. (6) in the bremsstrahlung matrix element (forp 2=0) is the same as the criterion for the validity of the use of eq. (6) in the matrix element for the relativistic photoeffect.
W. R. Johnson andR. T. Deck:Journ. Math. Phys.,3 319 (1962).
K. Mitchell:Phil. Mag.,40, 351 (1949); gives nine-place tables forL 2(x), −1 ≤x ≤1.
H. Bethe andW. Heitler:Proc. Roy. Soc., A146, 83 (1934);W. Heitler:The Quantum Theory of Radiation, 3rd edition (Oxford, 1954), p. 245.
H. E. Hall, A. O. Hanson andD. Jamnik:Phys. Rev.,129, 2207 (1963).
For a comparison between this work and the recent results ofPratt, see eq. (34) ofR. J. Jabbur andR. H. Pratt:Phys. Rev.,129, 184 (1963).
A. Erdelyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Higher Transcendental Functions, Bateman Manuscript Project, vol.I (New York, 1953), p. 13, Sect. 1.6, eq. (2).
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Supported in part by the U. S. Atomic Energy Commission.
The research reported in this article was initiated while two of the authors (C. L. H. and C. J. M.) were summer visitors at the Midwestern Universities Research Association. These authors wish to express their gratitude for the gracious hospitality extended to them by the M.U.R.A. group.
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Deck, R.T., Mullin, C.J. & Hammer, C.L. High-frequency limit of the bremsstrahlung spectrum. Nuovo Cim 32, 180–197 (1964). https://doi.org/10.1007/BF02732602
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DOI: https://doi.org/10.1007/BF02732602