Asymptotic Neutrality of Large-Z Ions

  • Elliott H. Lieb
  • Israel M. Sigal
  • Barry Simon
  • Walter Thirring

Abstract

Let N ( Z) denote the number of electrons that a nucleus of charge Z binds in nonrelativis-tic quantum theory. It is proved that N(Z)/Z → 1 as Z → ∞. The Pauli principle plays a critical role.

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References

  1. 1.
    1E. H. Lieb, I. Sigal, B. Simon, and W. Thirring, to be published.Google Scholar
  2. 2.
    we choose units of length and energy so that fi2/ 2m = e 2 = 1. In (1), we have taken infinite nuclear mass; our proof of Eq. (3) below extends to finite nuclear mass and to the allowance of arbitrary magnetic fields. See Ref. I.Google Scholar
  3. 3.
    3We have in mind the Pauli principle with two spin states. The number of spin states (so long as it is a fixed finite number) does not affect the truth of Eq. (3).Google Scholar
  4. 4.
    4The minimum without any symmetry restriction oc-curs on a totally symmetric state, so that we could just as well view Eb( N, Z) as a Bose energy.Google Scholar
  5. 5.
    The result for Eb is due to M. B. Ruskai, Commun. Math. Phys. 82, 457 (1982). The fermion result was ob- tained by I. Sigal, Commun. Math. Phys. 85, 309 (1982). M. B. Ruskai, Commun. Math. Phys. 85, 325 (1982), then used her methods to obtain the fermion result.Google Scholar
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    N(Z denotes the smallest number obeying this condition.Google Scholar
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    E. Lieb, Phys. Rev. A (to be published). A summary appears in E. H. Lieb, Phys. Rev. Lett. 52, 315 (1984).Google Scholar
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    B. Baumgartner, “On the Thomas-Fermi--von Weizsacker and Hartree energies as functions of the degree of ionization” (to be published).Google Scholar
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    He also needs a method to control quantum correc-tions. This method is discussed later.Google Scholar
  13. 13.
    The support of p denoted by suppp, is just those points x where an arbitrarily small ball about x has some charge.Google Scholar
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    To be sure the limit exists and is not a delta function or zero, one may have to scale the xa in an N-dependent way.Google Scholar
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    Since lxai< XI&o for all a we can replace the righthand side of (6) by CA/1/2R -1. Since the gradients are all zero if I X„„: < (1 - €) R we can replace the right-hand side of (6) also by C(1- E) N 112 R .Google Scholar
  17. 17.
    This formula is easy to prove by expanding LIja, Lia,H]1. Versions of it were found in successively more general situations by R. Ismagilov, Soy. Math. Dokl. 2, 1137 (1961); J. Morgan, J. Operator Theory 1, 109 (1979), and J. Morgan and B. Simon, Int. J. Quantum Chem. 17, 1143 (1980). It was I. Sigal in Ref. 5 whorealized its significance for bound-state questions.Google Scholar
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    This is precisely the scaling for Thomas-Fermi and for the real atomic system; see E. Lieb and B. Simon, Adv. Math. 23, 22 (1977).Google Scholar
  19. 19.
    For bosons, the “electron” density collapses as not Z-1/3; see Ref. 9.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Israel M. Sigal
    • 2
  • Barry Simon
    • 3
  • Walter Thirring
    • 4
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Department of Pure MathematicsWeizmann InstituteRehovotIsrael
  3. 3.Division of Physics, Mathematics and AstronomyCalifornia Institute of TechnologyPasadenaUSA
  4. 4.Institute for Theoretical PhysicsUniversity of ViennaViennaAustria

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