Understanding alpha decay

American Journal of Physics

Volumn 64, Issue 8, 1996, Pages 1061-1071

  Holstein, Barry R ^a,b  

^a UNIVERSITY OF MASSACHUSETTS   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030505821     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18308     Document Type: Article

References (33)

1. M. Von Laue, "Notiz zur Quantentheorie des Atomkernes," Z. Phys. 52, 726-734 (1928); R. W. Gurney and E. U. Condon, "Quantum Mechanics and Radioactive Disintegration," Phys. Rev. 33, 127-140 (1929).
2. M. Von Laue, "Notiz zur Quantentheorie des Atomkernes," Z. Phys. 52, 726-734 (1928); R. W. Gurney and E. U. Condon, "Quantum Mechanics and Radioactive Disintegration," Phys. Rev. 33, 127-140 (1929).
3. See, e.g., D. Park, Introduction to the Quantum Theory (McGraw-Hill, New York, (1974), Chap. 12.
4. Hereafter, unless explicitly noted, we shall take ℏ = c = 1.
5. I. Perlman and J. O. Rasmussen, "Alpha Radioactivity," in Handbuch der Physik, edited by S. Flugge (Springer-Verlag, Berlin, 1957), Vol. XLLII, pp. 109-204.
6. G. Gamow, "Zur quantentheorie des Atomkernes," Z. Phys. 51, 204-212 (1928); an equivalent approach is that of M. G. Fuda, "Time-dependent theory of alpha decay," Am. J. Phys. 52, 838-842 (1984).
7. G. Gamow, "Zur quantentheorie des Atomkernes," Z. Phys. 51, 204-212 (1928); an equivalent approach is that of M. G. Fuda, "Time-dependent theory of alpha decay," Am. J. Phys. 52, 838-842 (1984).
8. See, e.g., D. Park, Ref. 2, Sec. 4.5.
9. For the case of uranium, with a radius of about 7 fm, K∼π/R∼90 MeV, while κ ∼ √2MZα/R ∼ 500 MeV.
10. We neglect terms involving (dA/dκ)(dκ/dE) as they are suppressed by at least K/κ≪1 with respect to the leading terms.
11. An exception is the article by A. Bohm, M. Gadella, and G. B. Mainland, "Gamow vectors and decaying states," Am. J. Phys. 57, 1103-08 (1989).
12. E. Fermi, Nuclear Physics (Univ. Chicago Press, Chicago, 1950).
13. E. C. Kemble, The Fundamental Principles of Quantum Mechanics (McGraw-Hill, New York, 1937).
14. F. Rasetti, Elements of Nuclear Physics (Prentice-Hall, New York, 1936).
15. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), #3.354.5.
16. P. Ramond, Field Theory: a Modern Primer (Addison Wesley, Reading, MA, 1989); R. J. Rivers, Path Integral Methods in Modern Field Theory (Cambridge U.P., New York, 1987).
17. P. Ramond, Field Theory: a Modern Primer (Addison Wesley, Reading, MA, 1989); R. J. Rivers, Path Integral Methods in Modern Field Theory (Cambridge U.P., New York, 1987).
18. For this equation only, in order to emphasize the semiclassical limit discussed below, we include the explicit factor of ℏ.
19. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
20. B. R. Holstein and A. R. Swift, "Barrier penetration via path integrals," Am. J. Phys. 50, 833-39 (1982).
21. R. D. Carlitz and D. A. Nicole, "Classical Paths and Quantum Mechanics," Ann. Phys. (NY) 164, 411-462 (1985).
22. B. R. Holstein and A. R. Swift, "Path integrals and the WKB approximation," Am. J. Phys. 50, 829-833 (1982).
23. C. P. Burgess, "Quantum tunneling and imaginary-time paths," Am. J. Phys. 59, 994-998 (1991).
24. See, e.g., F. Rasetti, Ref. 8, Ch. VI, sec. 31.
25. A. I. Vainshtein et al., "ABC of Instantons," Sov. Phys. Usp. 25, 195-215 (1982).
26. B. R. Holstein, "Semiclassical treatment of the double well," Am. J. Phys. 56, 338-345 (1988).
27. Note here that the usual negative sign which accompanies the right-hand side of this equation in classical mechanics is not present because of our use of imaginary time.
28. 0)1n(r-α), i.e., the particle spends a logarithmically large time in the vicinity of r≃α.
29. I. S. Gradshteyn and I. M. Ryzhik, Ref. 10, #1.431.2.
30. S. Coleman, "The Uses of Instantons," in Aspects of Symmetry (Cambridge U.P., New York, 1985).
31. Here the "extra" factor of 2 on the right-hand side is associated with the analytic continuation of Eq. (91), as described by S. Coleman in the preceding reference.
32. See, e.g., A. Das, Field Theory: a Path Integral Approach (World Scientific, Singapore, 1993).
33. 1 has not been performed.