The Stability of Matter: From Atoms to Stars pp 100-102 | Cite as

# Asymptotic Neutrality of Large-*Z* Ions

Chapter

## 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}E. H. Lieb, I. Sigal, B. Simon, and W. Thirring, to be published.Google Scholar - 2.w
_{e}choose units of length and energy so that fi^{2}/ 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}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 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.The result for E
_{b}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.N(Z denotes the smallest number obeying this condition.Google Scholar
- 7.Sigal, to be published.Google Scholar
- 8.E. Lieb, Phys. Rev. A (to be published). A summary appears in E. H. Lieb, Phys. Rev. Lett. 52, 315 (1984).Google Scholar
- 9.R. Benguria and E. Lieb, Phys. Rev. Lett. 50, 50 (1983). loSee R. Benguria, H. Brezis, and E. Lieb, Commun. Math. Phys. 79, 167 (1981); Google Scholar
- 10.E. Lieb, Rev. Mod. Phys. 53,603 (1981), and 54, 311(E) (1982).Google Scholar
- 11.B. Baumgartner, “On the Thomas-Fermi--von Weizsacker and Hartree energies as functions of the degree of ionization” (to be published).Google Scholar
- 12.He also needs a method to control quantum correc-tions. This method is discussed later.Google Scholar
- 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.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.G. Choquet, C. R. Acad. Sci. 244, 1606–1609 (1957).Google Scholar
- 16.Since lxai< XI&o for all
*a*we can replace the righthand side of (6) by CA/^{1/2}R^{-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.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
- 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).Google Scholar
- 19.

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