Quadrupolar contact fields: Theory and applications

American Journal of Physics

Volumn 77, Issue 9, 2009, Pages 807-817

  Gray, C G ^a   Karl, G ^a   Novikov, V A ^b,c  

^a UNIVERSITY OF GUELPH   (Canada)

^b ITEPct   (Russian Federation)

^c PERIMETER INSTITUTE FOR THEORETICAL PHYSICS   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 69149101944     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.3138700     Document Type: Article

References (77)

1. Cartesian tensor forms for the multipole expansion for the electric field are given without derivation in Ref. 2, p. 146, up to the quadrupole term and are derived in Refs. 3 and 4, p. 51, to all orders. Spherical tensor forms are derived to all orders in Ref. 2, p. 145, and in Ref. 4, p. 53. Cartesian tensor forms for the multipole expansion for the magnetic field are derived in Ref. 2, p. 184, up to the dipole term and in Ref. 5 to all orders. The spherical tensor form is derived to all orders in Ref. 6.
2. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
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8. For the definition of the magnetic scalar potential see Ref. 6. Various definitions of the Cartesian magnetic quadrupole moment are discussed in C. G. Gray, "Definition of the magnetic quadrupole moment," Am. J. Phys. 48, 984-985 (1980).
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11. E. Fermi, "Über die magnetischen momente der atomkerne," Z. Phys. 60, 320-333 (1930);
12. Casimir (unpublished) simultaneously derived the magnetic dipole contact term.
13. See L. Pauling and S. Goudsmit, Structure of Line Spectra (McGraw-Hill, New York, 1930), p. 208; See also Ref. 43.
14. Reference 2, p. 188.
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25. -5 term), 312, 112, and 110, the last three being contact terms with 312 the largest of the three for the cases studied. The 314 term affects all but s- and p-states, 312 and 112 affect only p-states, and 110 affects only s- and p-states.
26. H. B. G. Casimir and C. Karreman, "Magnetic octupole moment of a nucleus," Physica 9, 494-503 (1942). This paper is reproduced as Appendix B in Casimir (1963), Ref. 43.
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29. Reference 2, p. 149.
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32. Reference 2, p. 162.
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35. G. Karl and V. A. Novikov, "Quadrupolar contact terms and hyperfine structure," Phys. Rev. C 74, 024001-1-7 (2006);
36. 77, 039901(E) (2008);
37. See also G. Karl and V. Novikov, "Hyperfine structure in Ω-nucleus system," Fiz. B 14, 75-78 (2005) and M. I. Krivoruchenko and A. Faessler, "Decays, contact p-wave interactions and hyperfine structure in omega exotic atoms," Nucl. Phys. A 803, 173-209 (2008).
38. Reference 2, p. 148. In a private communication Dr. Jackson notes that his method gives an alternative simple derivation of our results, Eqs. (30) and (31).
39. μα. The general rule is contract nearest neighbors first, then next nearest neighbors, etc. (When the tensors are symmetric, as in this paper, we can be cavalier about the order of the indices.) Further discussion of this Milne Chapman notation for Cartesian tensors is given in Appendix B in Ref. 4. Note also that with flat-space Cartesian components, there is no need to distinguish covariant (indices down) and contravariant (indices up) components.
40. More rigorous arguments can be given. See, for example, C. P. Frahm, "Some novel delta-function identities," Am. J. Phys. 51, 826-829 (1983).
41. Nonsymmetric second-rank tensors contain also an antisymmetric l = 1 (vector) part. For a discussion, with examples, see Ref. 4, pp. 490 and 544.
42. Reference 4, p. 489.
43. (2) is used (see, for example, Ref. 2, pp. 146 and 151).
44. Reference 4, p. 63.
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48. Reference 4, p. 452.
49. i.
50. d, and the nuclear contribution can be calculated from the equilibrium bond lengths and angles.
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53. Similar theorems are proved for the magnetic field B in Ref. 2, p. 187. Alternative proofs for the electric and magnetic cases are given in Ref. 9.
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56. 0 at an assumed minimum, we get a contradiction. The same argument shows that maxima also cannot exist. Thus, if an equilibrium point r exists satisfying E(r)=-∇φ(r)= 0, it must be a saddle point of φ(r) or belong to a region of constant φ.
57. S. Earnshaw, "On the nature of molecular forces which regulate the constitution of the luminiferous ether," Trans. Cambridge Philos. Soc. 7, 97-112 (1842); W. T. Scott, "Who was Earnshaw?," Am. J. Phys. 27, 418-419 (1959).
58. 0, which is derived similarly.
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61. J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd ed. (Oxford U.P., Oxford, 1891), p. 31;
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63. H. B. G. Casimir, On the Interaction Between Atomic Nuclei and Electrons (Teyler's Tweede Genootschap, Haarlem, Netherlands, 1936);
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65. See, for example, Ref. 2, p. 150, or Ref. 4, p. 72.
66. See, for example, Ref. 4, p. 78.
67. See, for example, Ref. 4, pp 81-83.
68. N. F. Ramsey, Nuclear Moments (Wiley, New York, 1953), p. 23.
69. Note that our quadrupole moment Q is half that defined in Ref. 21, and our Coulomb law is q/r [see Eq. (1)] and that in Ref. 21 is q/πr.
70. E. Durand, Électrostatique (Masson, Paris, 1964), Tome I, p. 93. For the field line equation of a general axial multipole, see p. 97.
71. Also see D. M. Wills and L. R. Young, "Equations for the field lines of an axisymmetric magnetic multipole," Geophys. J. R. Astron. Soc. 89, 1011-1022 (1987);
72. B. Jeffreys, "Derivations of the field lines of an axisymmetric multipole," Geophys. J. 92, 355-356 (1988).
73. J. Jeans, The Mathematical Theory of Electricity and Magnetism, 5th ed. (Cambridge U.P., Cambridge, 1925), p. 50;
74. Kristjansson, "On the drawing of lines of force and equipotenciais," Phys. Teach. 23, 202-206 (1985).
75. N. Namsrai, ELECTRIC HELD 〈www.physics-software.com〉.
76. C. P. Slichter, Principles of Magnetic Resonance (Harper & Row, New York, 1963).
77. K. B. Pomeranz, "Two theorems concerning the Laplace operator," Am. J. Phys. 31, 622-623 (1963).