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

## Keywords

Physical Review Letter Weizmann Institute Pauli Principle Nuclear Mass Nonrelativistic Quantum
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## Reference

- 1.E. H. Lieb, I. Sigal, B. Simon, and W. Thirring, to be published.Google Scholar
- 2.We choose units of length and energy so that /i
^{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. 1.Google Scholar - 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.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.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.
*N(Z)*denotes the smallest number obeying this condition.Google Scholar - 7.I. 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).MathSciNetCrossRefGoogle Scholar
- 9.R. Benguria and E. Lieb, Phys. Rev. Lett. 50, 50 (1983).CrossRefGoogle Scholar
- 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.B. Baumgartner, “On the Thomas-Fermi-von Weizsäcker and Hartree energies as functions of the degree of ionization” (to be published).Google Scholar
- 12.He also needs a method to control quantum corrections. 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 x
_{a}in an N-dependent way.Google Scholar - 15.G. Choquet, C. R. Acad. Sci. 244, 1606 - 1609 (1957).Google Scholar
- 16.Since Ix
_{a}IXI„ for all*a*,we can replace the right-hand side of (6) by CN^{I}/^{2}R^{-}Ix_{x}I^{-}. 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’^{°2}RGoogle Scholar - 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.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.

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