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 nonrelativistic quantum theory. It is proved that N(Z)/Z → 1 as Z → ∞. The Pauli principle plays a critical role.

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Reference

  1. 1.
    E. 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 /i2/ 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. 1.Google Scholar
  3. 3.
    We 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.
    The minimum without any symmetry restriction occurs on a totally symmetric state, so that we could just as well view Eh(N,Z) as a Bose energy.Google Scholar
  5. 5.
    The result for Ea 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
  6. 6.
    N(Z) denotes the smallest number obeying this condition.Google Scholar
  7. 7.
    I. Sigal, to be published.Google Scholar
  8. 8.E. Lieb, Phys. Rev. A (to be published). A summary appears in E. H. Lieb, Phys. Rev. Lett. 52, 315 (1984).MathSciNetCrossRefGoogle Scholar
  9. 9.R. Benguria and E. Lieb, Phys. Rev. Lett. 50, 50 (1983).CrossRefGoogle Scholar
  10. 10.See R. Benguria, H. Brezis, and E. Lieb, Commun. Math. Phys. 79, 167 (1981); E. Lieb, Rev. Mod. Phys. 53, 603 (1981), and 54, 311 (E) (1982).Google Scholar
  11. 11.
    B. Baumgartner, “On the Thomas-Fermi-von Weizsäcker and Hartree energies as functions of the degree of ionization” (to be published).Google Scholar
  12. 12.
    He also needs a method to control quantum corrections. 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
  14. 14.
    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
  15. 15.G. Choquet, C. R. Acad. Sci. 244, 1606 - 1609 (1957).Google Scholar
  16. 16.
    Since Ixa IXI„ for all a,we can replace the right-hand side of (6) by CNI/2R-IxxI-. Since the gradients are all zero if 1,1..1 (1- e)R,we can replace the right-hand side of (6) also by C (1-e ‘N’°2RGoogle Scholar
  17. 17.This formula is easy to prove by expanding if„ [j„H]]. 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 who realized its significance for bound-state questions.Google Scholar
  18. 18.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).CrossRefGoogle Scholar
  19. 19.
    For bosons, the “electron” density collapses as Z-, not Z1i3; see Ref. 9.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

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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