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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1E. H. Lieb, I. Sigal, B. Simon, and W. Thirring, to be published.
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.
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).
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.
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.
N(Z denotes the smallest number obeying this condition.
Sigal, to be published.
E. Lieb, Phys. Rev. A (to be published). A summary appears in E. H. Lieb, Phys. Rev. Lett. 52, 315 (1984).
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);
E. Lieb, Rev. Mod. Phys. 53,603 (1981), and 54, 311(E) (1982).
B. Baumgartner, “On the Thomas-Fermi--von Weizsacker and Hartree energies as functions of the degree of ionization” (to be published).
He also needs a method to control quantum correc-tions. This method is discussed later.
The support of p denoted by suppp, is just those points x where an arbitrarily small ball about x has some charge.
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.
G. Choquet, C. R. Acad. Sci. 244, 1606–1609 (1957).
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 .
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.
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).
For bosons, the “electron” density collapses as not Z-1/3; see Ref. 9.
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Lieb, E.H., Sigal, I.M., Simon, B., Thirring, W. (1991). Asymptotic Neutrality of Large-Z Ions. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02725-7_9
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