DOI: 10.1016/s0081-1947(08)60071-5. Pseudopotential Theory of Cohesion and Structure

Pseudopotential Theory of Cohesion and Structure

10.1016/s0081-1947(08)60071-5

Crossref book-chapter

Elsevier

Solid State Physics (78)

Bibliography

Heine, V., & Weaire, D. (1970). Pseudopotential Theory of Cohesion and Structure. Solid State Physics, 249â463.

Authors 2

Volker Heine (first)
D. Weaire (additional)

References 422 Referenced 297

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A word about notation: we shall talk about Brillouin zone planes labeled by the reciprocal lattice vectors g drawn in the usual way6 according to the geometry of the primitive unit cell irrespective of whether or not the structure factor3 vanishes due to the internal arrangement of the atoms inside the unit cell. We shall restrict the expression “the Brillouin zone” to denote the first or “reduced” zone. All larger zones constructed out of appropriate zone planes we shall refer to asJones zones. Usually the planes selected for this have reasonably large structure factor and lie close to the free electron Fermi sphere. Our k will always be in extended k space6 unless the contrary is stated
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We shall usez to denote the total number of s and p electrons of an atom and usually refer to it as the “valency” for short since this is the number of electrons contributed to the conduction band (or “valence band” in the case of a semiconductor). In alloys the same symbolz denotes the electrons-per-atom ratio
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We shall use atomic units (s.u.) throughout (h = m = e = 1). In the past numerical values of energy have often been quoted in rydbergs (1 Ry = 1/2 a.u. ≈ 13.6 eV).
See Heine,31,33 Harrison,32 Ziman6 (Chapter 3), and Cohen and Heine34 (Parts I and II).
V. Heine, this volume.
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Strictly speaking, the pseudopotential is a function of four variables and we should writev(q, k | k + q |, E), but the variation withq is the most important so that we shall omit explicit mention of the other variables for the present. See Sections 9–12 and Sections 2 and 16 of Cohen and Heine.34
We shall use q to denote a continuous variable and g when we wish to imply a reciprocal lattice vector.
In a complete NFE expression for the band gap, there would be higher order terms of perturbation theory which we shall ignore. Also if there is more than one atom per primitive unit cell, we should include a factor S(g) whereS is the X-ray structure factor (see Section 5).
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A. O. E. Animalu, Tech. Rept. No. 3 (unpublished, 1965) obtainable from the Secretary, Solid State Theory Section, Cavendish Laboratory, Cambridge, England. This report gives tables ofv(q) with and without damping factor, for 25 elements corresponding to the graphs of Animalu and Heine47 with the exception of Cd, Ca, Sr, and Ba.
A. O. E. Animalu, Tech. Rept. No. 4 (unpublished, 1965) obtainable from the Secretary, Solid State Theory Section, Cavendish Laboratory, Cambridge, England. This gives tables ofv(q) for Cd, Ca, Sr, Ba, B, C, and P as in Animalu.49
We shall use capital V to denote the pseudopotential in the whole crystal, and lower case v that of a single atom or ion.
The proof of (5.4) is straightforward and one always gets this form when Fourier transforming a superposition of N localized functions in a volume NΩ. The coefficient F(q) in a Fourier expression is which gives the form (5.4). The first term is the usual X-ray structure factor S. In the case of a regular crystal, it is nonzero only when q is a reciprocal lattice vector g, in which case all cells contribute identically to the sum and S can be simplified to (5.6). For a liquid or disordered solid, S(q) is a continuous function of q and the term in brackets is identical with (5.6) if we regard the whole volume with N atoms as the unit cell.
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For the reasons behind this choice the reader is referred to Heine.31
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We inciude the factor e2 explicitly so that the equation is valid not only in atomic units.
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The derivation of this equation for a free electron gas is given by Hedin and Lundqvist,120 see particularly Eqs. (13.10) and (33.1) and Section 22. See also Hedin.124 It also applies to atoms, molecules, and insulating crystals (Section 36 of Hedin and Lundqvist120).
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Figures 43 and 44 of Hedin and Lundqvist.120
Table IV of Hedin and Lundqvist.120
Table VII of Hedin and Lundqvist.120
As in Section 4,r3 is the radius of the sphere containing on average one electron.
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Section 16 of Cohen and Heine.84
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Note added since the manuscript was completed, Three further calculations (Singwi et al 137b Geldart and Taylor, 137 and Ferreira 137d) have appeared recently which essentially calculate Ep or Ee, although none of the results are expressed in terms off so that an immediate comparison is not possible. However, visual inspection of Geldart and Taylor137c suggests that theirf would rise to a peak greater than 1/2 at intermediate values ofq. For the most complete recent discussion, see Shaw.137e
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Equations (1.47) and (3.62) of Pine and Nozieres.122
Contrary to Shaw and Pynn,112 we do not believe that the high density expansions have any relevance at metallic densities. Also if we use the high density expansion forUc [Eq. (5.121) of Pine and Nozieres122] in (10.59), we obtain a different value for the correlation contribution toa from the one quoted by Shaw and Pynn.112
Calculation by Ma reported by Shaw and Pynn.112
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For a discussion of the practical details of handling such integrals see Animalu.148 Relevant material may also be found in Shaw23 and Sloan.149
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This topic is extensively discussed by Cohen and Heine,34 particularly Sections 2, 11, 12, 14, and 21.
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Note that the effective mass calculations reported by Animalu and Heine47 (sT6) were found to be in error.156 Note also the need to distinguish contributions to the total average band effective mass m* from nonlocality and energy dependence respectively.154,156 The latter is described in the next section. One cannot simply put asterisks on all them's in a given theory in order to incorporate effective mass corrections, because energy dependence and nonlocality have rather different effects.
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This type of statement has been recently challenged by Pendry,117 who gives a justification of pseudopotential theory independent of considerations of the smoothing of wave functions. See also J. B. Pendry (to be published).
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We are indebted to J. Shively for this calculation
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The zone drawn by Mott and Jones for graphite contains three electrons per atom, not four as stated, and relates only to the σ bonds. It can be extended to describe the π electrons.
We have already emphasized in Section 17 that weakening the potential by pseudiz-ing away the core states is always a mathematically valid procedure even in NaCl for example190 but what remains may or may not be small enough for perturbation theory to be a good approximation. The potentials encountered in covalent structures are “large” in this latter sense.
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For further details of this procedure see Shaw,23 Harrison,32 Tosi,207 and Johnson.208 The method is based on the work of Epstein.209,210
10.1016/S0081-1947(08)60515-9 / Solid State Phys. by Tosi (1964)
10.1088/0022-3719/1/6/309 / J. Phys. C by Johnson (1969)
10.1007/BF01444309 / Math. Ann. by Epstein (1903)
10.1007/BF01449900 / Math. Ann. by Epstein (1907)
For the details of this expansion see Harrison,32 Chapter 7.
For a review of attempts to fit phonon dispersion relations by the alternative procedure of adjustment of a parameterized form for πbe (q) see Cohen and Heine.34
10.1103/PhysRev.177.1139 / Phys. Rev. by Metzbower (1969)
10.1103/PhysRevLett.15.634 / Phys. Rev. Letters by Stedman (1965)
10.1088/0022-3719/2/3/306 / J. Phys. C by Sharp (1969)
See Huntington,203 Section 4.
{'key': '10.1016/S0081-1947(08)60071-5_bib224', 'series-title': '“Dynamical Theory of Crystal Lattices,”', 'author': 'Pindor', 'year': '1954'} / “Dynamical Theory of Crystal Lattices,” by Pindor (1954)
10.1088/0022-3719/2/6/315 / J. Phys. C by Pindor (1969)
{'key': '10.1016/S0081-1947(08)60071-5_bib225_2', 'first-page': '481', 'volume': '3', 'year': '1970', 'journal-title': 'J. Phys. C'} / J. Phys. C (1970)
10.1016/0001-6160(67)90042-9 / Acta Met. by Hodges (1967)
10.1016/0022-3697(65)90121-6 / J. Phys. Chem. Solids by Freeman (1965)
Strictly speaking, such structures are close-packed only if the interplanar spacingh, which we assume constant although this is not dictated by symmetry in all cases, is related to the interatomic spacinga of the close-packed layers byh = (f)1/2a as in fee. The theory as presented here is not restricted to this case, although some of the qualitative remarks made about it are not valid if the structure is not reasonably near to close-packing.
{'key': '10.1016/S0081-1947(08)60071-5_bib229', 'volume': '5', 'author': 'Harrison', 'year': '1970'} by Harrison (1970)
One pseudopotential calculation reported by Krasko et al 223a in stands rather puzzling contradiction of this conclusion since a large positive stacking fault energy is obtained for fee copper.
{'key': '10.1016/S0081-1947(08)60071-5_bib231', 'first-page': '2414', 'volume': '9', 'author': 'Krasko', 'year': '1968', 'journal-title': 'Soviet Phys.-Solid State'} / Soviet Phys.-Solid State by Krasko (1968)
{'key': '10.1016/S0081-1947(08)60071-5_bib232', 'author': 'Gallagher', 'year': '1970', 'journal-title': 'Trans. AIME'} / Trans. AIME by Gallagher (1970)
The unusual form of the energy-wavenumber characteristic in this calculation derives from the fact that the various structure-dependent terms were grouped in a way rather different from what is now usual, leading to a different definition of effective charges and a form for the energy-wavenumber characteristic which was not negative-definite as it usually is.
Sham also included an estimate of Born-Mayer repulsion between ion cores in his calculation of phonon dispersion in Na. Effects due to Born-Mayer repulsion appear to be quite small and of uncertain magnitude for the simple metals due to the small sizes of the core, and are ignored in this review.
10.1051/jphys:01967002807053900 / J. Phys. (France) by Pick (1967)
10.1103/PhysRev.156.769 / Phys. Rev. by Roy (1967)
The equations used by Roy and Venkataraman228 for the calculation of phonon frequencies are incorrect. An unpublished erratum gives corrected results in qualitative agreement with the observed dispersion curves, with quantitative discrepancies of up to 20%.
10.1016/0375-9601(68)90290-9 / Phys. Letters by King (1968)
10.1016/0038-1098(69)90403-7 / Solid State Commun. by King (1969)
See Section 47.
10.1103/PhysRev.175.699 / Phys. Rev. by Hayes (1969)
10.1103/PhysRev.163.667 / Phys. Rev. by Shyu (1967)
{'key': '10.1016/S0081-1947(08)60071-5_bib243', 'series-title': '“Neutron Inelastic Scattering.”', 'author': 'Brovman', 'year': '1968'} / “Neutron Inelastic Scattering.” by Brovman (1968)
10.1080/14786436908228637 / Phil. Mag. by Weaire (1969)
10.1088/0022-3719/2/4/126 / J. Phys. C by Cousins (1969)
10.1088/0022-3719/2/7/320 / J. Phys. C by Inglesfield (1969)
10.1088/0022-3719/2/7/321 / J. Phys. C by Inglesfield (1969)
10.1016/0001-6160(69)90001-7 / Acta Met. by Inglesfield (1969)
10.1088/0022-3719/3/4/010 / J. Phys. C by Coulthard (1970)
{'key': '10.1016/S0081-1947(08)60071-5_bib250', 'author': 'Cousins', 'year': '1970', 'journal-title': 'J. Phys. C'} / J. Phys. C by Cousins (1970)
10.1103/PhysRev.155.682 / Phys. Rev. by Ashcroft (1967)
Apart from this the formalism is identical with that described in Part I
10.1103/PhysRev.159.500 / Phys. Rev. by Ashcroft (1967)
10.1103/PhysRev.170.687 / Phys. Rev. by Shyu (1968)
10.1088/0022-3719/1/1/326 / J. Phys. C by Ashcroft (1968)
10.1103/PhysRev.150.487 / Phys. Rev. by Cowley (1966)
10.1016/0031-9163(66)90196-X / Phys, Letters by Sahni (1966)
10.1016/0038-1098(67)90509-1 / Solid State Commun. by Krebs (1967)
10.1002/pssb.19670230208 / Phys. Status Solidi by Ho (1968)
10.1103/PhysRev.169.523 / Phys. Rev. by Ho (1968)
10.1103/PhysRev.178.713 / Phys. Rev. by Dynes (1969)
10.1103/PhysRev.185.971 / Phys. Rev. by Gilat (1969)
D. Taylor (1970) (to be published).
{'key': '10.1016/S0081-1947(08)60071-5_bib264', 'first-page': '330', 'volume': '5', 'author': 'Schneider', 'year': '1966', 'journal-title': 'Phys. Kondens. Materie'} / Phys. Kondens. Materie by Schneider (1966)
10.1016/0375-9601(67)90423-9 / Phys. Letters by Schneider (1967)
{'key': '10.1016/S0081-1947(08)60071-5_bib266', 'series-title': '“Neutron Inelastic Scattering.”', 'author': 'Schneider', 'year': '1968'} / “Neutron Inelastic Scattering.” by Schneider (1968)
{'key': '10.1016/S0081-1947(08)60071-5_bib267', 'first-page': '2414', 'volume': '9', 'author': 'Krasko', 'year': '1968', 'journal-title': 'Soviet Phys—Solid State'} / Soviet Phys—Solid State by Krasko (1968)
10.1103/PhysRev.166.723 / Phys. Rev. by Floyd (1968)
10.1103/PhysRev.176.832 / Phys. Rev. by Wallace (1968)
10.1103/PhysRev.178.900 / Phys. Rev. (1969)
P. S. Ho (to be published).
{'key': '10.1016/S0081-1947(08)60071-5_bib271', 'volume': 'II', 'author': 'Pearson', 'year': '1958'} by Pearson (1958)
We will generally refer to Pearson,263 for well-known structures, and will give the appropriate detailed references only in unfamiliar cases.
10.1002/pssb.19670230135 / Phys. Status Solidi by Teichler (1967)
10.1103/PhysRev.153.673 / Phys. Rev. by Heine (1967)
10.1088/0370-1328/92/4/313 / Proc, Phys. Soc. by Hubbard (1967)
10.1088/0022-3719/1/2/324 / J. Phys. C by Jacobs (1968)
10.1088/0022-3719/2/6/316 / J. Phys. C by Pettifor (1969)
10.1103/PhysRev.152.505 / Phys. Rev. by Hodges (1966)
10.1103/PhysRev.153.659 / Phys. Rev. by Mueller (1967)
10.1103/PhysRev.181.1036 / Phys. Rev. by Harrison (1969)
10.1103/PhysRev.186.619 / Phys. Rev. by Deegan (1969)
10.1088/0022-3719/1/5/318 / J. Phys. C by Jacobs (1968)
10.1080/14786436608219015 / Phil. Mag. by Harris (1966)
R. Jacobs, thesis, University of Cambridge (1969) and to be published 1970.
{'key': '10.1016/S0081-1947(08)60071-5_bib282', 'series-title': '“Magnesium and Its Alloys.”', 'author': 'Roberts', 'year': '1960'} / “Magnesium and Its Alloys.” by Roberts (1960)
10.1103/PhysRev.89.282 / Phys. Rev. by Goodenough (1953)
10.1103/PhysRev.98.449 / Phys. Rev. by McClure (1955)
See the preface to Pearson, 263 also Roberts.271
10.1098/rsta.1962.0010 / Phil. Trans. Roy. Soc. by Falicov (1962)
10.1080/14786435908238237 / Phil. Mag. by Hardie (1959)
10.1016/0001-6160(65)90079-9 / Acta Met. by Novotny (1965)
10.1080/14786436908228076 / Phil. Mag. by Weaire (1969)
Because the inclusion of effective mass corrections in the crude form employed here destroys the cancellation of the Ewald and band structure terms at lowq, which is irrelevant to the comparison of simple crystal structures but would lead to pair interactions of an unusual form, an extra factor of (mKmE)-1 was arbitrarily included in E in the calculations of Fig. 24 to rectify this while making little difference at highq.
The idea that the distorted structures of Zn and Cd could be explained in terms of pairwise central forces of reasonable form, provided that there were additional large volume-dependent energy terms, appears to have been first suggested by Nabarro and Varley.281 However, the idea did not gain general acceptance (see e.g. Wallace282).
10.1017/S0305004100027663 / Proc. Cambridge Phil. Soc. by Nabarro (1952)
10.1063/1.1740739 / J. Chem. Phys. by Wallace (1955)
This discrepancy, which casts some doubt on the discussion of axial ratios given in this section, may be at least partially attributable to the fact that Shaw did not use effective mass corrections as did Weaire.64 Such corrections are rather small for Na, Mg, and Al, for which Shaw's work gives such excellent results, but are large enough to be significant for most other simple metals (see Shaw154 and Weaire156).
10.1098/rspa.1966.0132 / Proc. Roy. Soc. by Harris (1966)
10.1080/14786436608219004 / Phil. Mag. by Howie (1966)
10.1103/PhysRev.142.392 / Phys. Rev. by Perez-Albuene (1966)
10.1016/S0081-1947(08)60410-5 / Solid State Phys. by Drickamer (1965)
10.1063/1.1714059 / J. Appl. Phys. by McWhan (1965)
R. Pynns, private communication (1969). See footnote 283.
10.1107/S0365110X57000134 / Acta Crystall. by Barrett (1957)
10.1063/1.1730663 / J. Chem. Phys. by Atoji (1959)
10.1080/14786436808227467 / Phil. Mag. by Doidge (1968)
J. S. Abell, thesis, Univ. of Surrey (1969); private communication.
10.1016/0036-9748(68)90210-X / Scripta Met. by Abell (1968)
See e.g. Atoji.291
According to Animalu, and Heine,47Ra = 2.1 a.u.,Ra = 3.35 a.u. The nearest neighbor distance in α-Hg is 5.7 a.u. ~ 1.4 (2Re).
10.1103/PhysRev.154.535 / Phys. Rev. by Vasvari (1967)
10.1016/0079-6786(67)90037-4 / Progr. Solid State Chem. by Klement (1966)
10.1103/PhysRev.145.465 / Phys. Rev. by Cotterill (1966)
M. Appapillai (to be published).
{'key': '10.1016/S0081-1947(08)60071-5_bib312', 'series-title': '“Theory of Alloy Phases.”', 'author': 'Raynor', 'year': '1956'} / “Theory of Alloy Phases.” by Raynor (1956)
{'key': '10.1016/S0081-1947(08)60071-5_bib313', 'first-page': '643', 'volume': '20', 'author': 'Svechkarev', 'year': '1965', 'journal-title': 'Soviet Phys. JETP'} / Soviet Phys. JETP by Svechkarev (1965)
10.1016/0001-6160(69)90159-X / Acta Met. by Preece (1969)
10.1103/PhysRev.98.391 / Phys. Rev. by Goodenough (1951)
10.1038/185029b0 / Nature by Graham (1960)
{'key': '10.1016/S0081-1947(08)60071-5_bib316', 'first-page': '894', 'volume': '9', 'author': 'Vereshchagin', 'year': '1965', 'journal-title': 'Soviet Phys.-Doklady'} / Soviet Phys.-Doklady by Vereshchagin (1965)
D. B. McWhan, private communication (1968).
{'key': '10.1016/S0081-1947(08)60071-5_bib318', 'first-page': '260', 'volume': '84', 'author': 'Curien', 'year': '1961', 'journal-title': 'Bull. Soc. Fr. Miner, Cristallogr.'} / Bull. Soc. Fr. Miner, Cristallogr. by Curien (1961)
{'key': '10.1016/S0081-1947(08)60071-5_bib319', 'first-page': '378', 'volume': '14', 'author': 'Reichardt', 'year': '1969', 'journal-title': 'Bull. Am. Phys. Soc.'} / Bull. Am. Phys. Soc. by Reichardt (1969)
10.1126/science.141.3585.1041 / Science by Barnett (1963)
Not to be confused with the hexagonal close-packed structure.
{'key': '10.1016/S0081-1947(08)60071-5_bib322', 'first-page': '833', 'volume': '239', 'author': 'Giessen', 'year': '1966', 'journal-title': 'Trans. Metall. Soc. A.I.M.E.'} / Trans. Metall. Soc. A.I.M.E. by Giessen (1966)
10.1016/0001-6160(66)90068-X / Acta Met. by Kane (1966)
{'key': '10.1016/S0081-1947(08)60071-5_bib324', 'first-page': '111', 'volume': '12', 'author': 'Giessen', 'year': '1969', 'journal-title': 'Advan. X-Ray Anal.'} / Advan. X-Ray Anal. by Giessen (1969)
10.1016/0001-6160(54)90197-2 / Acta Met. by Raynor (1954)
{'key': '10.1016/S0081-1947(08)60071-5_bib326', 'series-title': '“The Mechanism of Phase Transformations in Crystalline Solids,”', 'author': 'Weaire', 'year': '1969'} / “The Mechanism of Phase Transformations in Crystalline Solids,” by Weaire (1969)
{'key': '10.1016/S0081-1947(08)60071-5_bib327', 'first-page': '365', 'volume': '25', 'author': 'Brovman', 'year': '1967', 'journal-title': 'Soviet Phys. JETP'} / Soviet Phys. JETP by Brovman (1967)
10.1007/BF02710327 / Nuovo Cimento by Bonsignori (1968)
10.1103/PhysRev.128.1099 / Phys. Rev. by Brockhouse (1962)
Added in proof. Coulthard241 has recently published such a calculation
10.1016/0001-6160(62)90115-3 / Acta Met. by Bolling (1962)
10.1147/rd.83.0215 / IBM J. Res. Develop. by Cohen (1964)
See Mott and Jones,3 p. 167.
{'key': '10.1016/S0081-1947(08)60071-5_bib334', 'author': 'Williams', 'year': '1970', 'journal-title': 'J. Phys. Chem. Sol.'} / J. Phys. Chem. Sol. by Williams (1970)
10.1103/PhysRev.142.441 / Phys. Rev. by Lin (1966)
10.1103/PhysRevLett.22.1251 / Phys. Rev. Letters by Gillis (1969)
See below Eq. (4.2), or Eq. (4.4) of Cohen and Heine.34 In the present context we identifyre of the empty core model with the radiusRe of the ion core.
10.1103/PhysRev.122.1821 / Phys. Rev. by Cohen (1961)
The (R) for 29 elements has been calculated by R. Bowley from the model potential (Section 9) using the ‘local’ form of the theory as in Section 5. The results are not being published in full because of the criticisms of such an approximation (Section 11), and because full nonlocal calculations have been started by M. Appapillai using Shaw's optimized form of the model potential (Section 9). We are indebted to Mr. Bowley for making his results available to us.
Readers unfamiliar with this item should note that it is a hard boiled egg, shelled and surrounded by a layer of sausage meat approximately one inch thick.
{'key': '10.1016/S0081-1947(08)60071-5_bib341', 'series-title': '“Phase Stability in Metals and Alloys”', 'first-page': '3', 'author': 'Hume-Rothery', 'year': '1966'} / “Phase Stability in Metals and Alloys” by Hume-Rothery (1966)
10.1051/jphysrad:019620023010060700 / Phys. Radium by Massalski (1962)
{'key': '10.1016/S0081-1947(08)60071-5_bib342_2', 'series-title': '“Metallic Solid Solutions”', 'author': 'Friedel', 'year': '1963'} / “Metallic Solid Solutions” by Friedel (1963)
{'key': '10.1016/S0081-1947(08)60071-5_bib343', 'first-page': '1', 'volume': '10', 'author': 'Massalski', 'year': '1961', 'journal-title': 'Progr. Mater. Sci.'} / Progr. Mater. Sci. by Massalski (1961)
{'key': '10.1016/S0081-1947(08)60071-5_bib344', 'series-title': '“Phase Stability in Metals and Alloys”', 'first-page': '281', 'author': 'Nevitt', 'year': '1966'} / “Phase Stability in Metals and Alloys” by Nevitt (1966)
See Section 41.
See e.g. Mott and Jones,3 Chapter IV.
{'key': '10.1016/S0081-1947(08)60071-5_bib347', 'volume': '2', 'author': 'Slater', 'year': '1965'} by Slater (1965)
10.1016/0022-3697(69)90344-8 / J. Phys. Chem. Solids by Hafemeister (1969)
10.1103/PhysRev.141.553 / Phys. Rev. by Higgins (1966)
10.1143/JPSJ.24.467 / J. Phys. Soc. Japan by Fujiwara (1968)
{'key': '10.1016/S0081-1947(08)60071-5_bib351', 'series-title': '“Quantum Theory of Solids,”', 'author': 'Kittel', 'year': '1963'} / “Quantum Theory of Solids,” by Kittel (1963)
10.1080/00018735400101233 / Advan. Phys. by Friedel (1954)
{'key': '10.1016/S0081-1947(08)60071-5_bib353_1', 'series-title': '“Soft X-Ray Band Spectra”', 'author': 'Rooke', 'year': '1968'} / “Soft X-Ray Band Spectra” by Rooke (1968)
10.1080/14786436508218867 / Phil. Mag. by Appleton (1965)
10.1016/0375-9601(69)90284-9 / Phys. Letters by Jacobs (1969)
W. Kohn and C. Majumdar, Phys. Rev. 138, A1617 (10.1103/PhysRev.138.A1617)
See Cohen,25 p. 11.
{'key': '10.1016/S0081-1947(08)60071-5_bib356', 'series-title': '“Structure of Metals,”', 'author': 'Barrett', 'year': '1966'} / “Structure of Metals,” by Barrett (1966)
Rather similar results have been obtained in recent calculations by Shaw,23,222 who performs lattice sums using the asymptotic form of the interatomic potential for the hep, fee, and bee structures.
10.1051/jphysrad:019620023010063700 / J. Phys. Radium by Jones (1962)
{'key': '10.1016/S0081-1947(08)60071-5_bib358_2', 'series-title': '“Metallic Solid Solutions”', 'author': 'Friedel', 'year': '1963'} / “Metallic Solid Solutions” by Friedel (1963)
10.1080/00018735800101307 / Advan. Phys. by Cohen (1958)
10.1080/14786436108238353 / Phil. Mag. by Hume-Rothery (1961)
10.1080/14786436508230088 / Phil. Mag. by Lettington (1965)
We are grateful to T. B. Massalski for pointing this out to us.
See e.g. Jacobs.181,182
For instance, in the case of Ga alloys unusual structures are found351,352 which may be interpreted in terms of the tendency of Ga atoms to favor an interatomic distance rather shorter than that which would be normally dictated by considerations of atomic volumes, exactly as in the case of the pure metal (Section 5 and Part VIII).
10.1107/S0567740869002536 / Acta Crystall. by Smith (1969)
10.1107/S0567740869002548 / Acta Crystall. by Smith (1969)
The definition ofequivalence for these purposes is that every site has the sameradial distribution of other sites.
10.1021/ja01529a005 / J. Am. Chem. Soc. by Sterrett (1959)
{'key': '10.1016/S0081-1947(08)60071-5_bib369', 'series-title': '“Constitution of Binary Alloys,”', 'author': 'Hansen', 'year': '1958'} / “Constitution of Binary Alloys,” by Hansen (1958)
{'key': '10.1016/S0081-1947(08)60071-5_bib370', 'series-title': '“Thermochemical Data of Alloys,”', 'author': 'Kubaschewski', 'year': '1956'} / “Thermochemical Data of Alloys,” by Kubaschewski (1956)
10.1088/0022-3719/2/5/305 / Proc. Phys. Soc. (London) (Solid State Phys.) by K-Berggren (1969)
10.1051/jphys:01969003005-6045100 / J. Phys. (France) by Conte (1969)
10.1016/0031-9163(66)90399-4 / Phys. Letters by Tamaki (1966)
10.1080/00018736700101285 / Advan. Phys. (1967)
10.1103/PhysRev.110.1 / Phys. Rev. by Percus (1958)
10.1103/PhysRevLett.8.462 / Phys. Rev. Letters by Percus (1962)
10.1103/PhysRev.145.83 / Phys. Rev. by Ashcroft (1966)
10.1103/PhysRev.143.36 / Phys. Rev. by Ascarelli (1966)
10.1016/0375-9601(69)90231-X / Phys. Lett. by North (1969)
10.1063/1.1698864 / J. Chem. Phys. by Sharrah (1953)
10.1515/zna-1957-0203 / Z. Naturforsch by Muller (1957)
10.1515/zna-1947-0906 / Z. Naturforsch by Hendus (1947)
10.1515/zna-1948-0708 / Z. Naturforsch (1948)
10.1515/zna-1965-0302 / Z. Naturforsch by Wagner (1965)
10.1103/PhysRev.139.A83 / Phys. Rev. by Christman (1965)
10.1103/PhysRev.153.217 / Phys. Rev. by Christman (1967)
10.1103/PhysRev.153.225 / Phys. Rev. by Christman (1967)
10.1103/PhysRev.159.108 / Phys. Rev. by Christman (1967)
By fee we mean a structure of fee-type stacking throughout this section (i.e. rhombo-hedral, unlessc/a is ideal).
10.1103/PhysRev.187.994 / Phys. Rev. by Hodges (1969)
10.1103/PhysRev.185.983 / Phys. Rev. by Wannier (1969)
{'key': '10.1016/S0081-1947(08)60071-5_bib391', 'author': 'Kleinman', 'year': '1970', 'journal-title': 'Phys. Rev.'} / Phys. Rev. by Kleinman (1970)
{'key': '10.1016/S0081-1947(08)60071-5_bib392', 'author': 'Shaw', 'year': '1970', 'journal-title': 'Phys. Rev.'} / Phys. Rev. by Shaw (1970)
10.1103/PhysRev.182.778 / Phys. Rev. by Wallace (1969)
10.1103/PhysRev.187.991 / Phys. Rev. (1969)
10.1103/PhysRev.187.808 / Phys. Rev. by Prakash (1969)
10.1016/0375-9601(70)90201-X / Phys. Lett. by King (1970)
W. M. Hartmann, to be published (1970).
10.1016/0375-9601(70)90576-1 / Phys. Letters by Ducharme (1970)
10.1016/0375-9601(69)91015-9 / Phys. Lett. by Evans (1969)
D. Stroud and N. W. Ashcroft (to be published) (1970).

Dates

Type	When
Created	15 years, 4 months ago (April 26, 2010, 8:32 a.m.)
Deposited	3 years, 10 months ago (Oct. 26, 2021, 3:09 p.m.)
Indexed	1 month ago (Aug. 2, 2025, 1:07 a.m.)
Issued	55 years, 8 months ago (Jan. 1, 1970)
Published	55 years, 8 months ago (Jan. 1, 1970)
Published Print	55 years, 8 months ago (Jan. 1, 1970)

Funders 0

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BibTeX

@inbook{Heine_1970, title={Pseudopotential Theory of Cohesion and Structure}, ISBN={9780126077247}, ISSN={0081-1947}, url={http://dx.doi.org/10.1016/s0081-1947(08)60071-5}, DOI={10.1016/s0081-1947(08)60071-5}, booktitle={Solid State Physics}, publisher={Elsevier}, author={Heine, Volker and Weaire, D.}, year={1970}, pages={249–463} }

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  "reference": [
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib1",
      "first-page": "295",
      "volume": "35",
      "author": "Hume-Rothery",
      "year": "1926",
      "journal-title": "J. Inst. Metals"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib2",
      "series-title": "\u201cThe Nature of the Chemical Bond.\u201d",
      "author": "Pauling",
      "year": "1960"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib3",
      "series-title": "\u201cThe Theory and Properties of Metals and Alloys.\u201d",
      "author": "Mott",
      "year": "1936"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib4",
      "doi-asserted-by": "crossref",
      "unstructured": "Mott and Jones, 3 pp. 125\u2013128.",
      "DOI": "10.1016/S0165-0114(01)00193-2"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib5",
      "unstructured": "A word about notation: we shall talk about Brillouin zone planes labeled by the reciprocal lattice vectors g drawn in the usual way6 according to the geometry of the primitive unit cell irrespective of whether or not the structure factor3 vanishes due to the internal arrangement of the atoms inside the unit cell. We shall restrict the expression \u201cthe Brillouin zone\u201d to denote the first or \u201creduced\u201d zone. All larger zones constructed out of appropriate zone planes we shall refer to asJones zones. Usually the planes selected for this have reasonably large structure factor and lie close to the free electron Fermi sphere. Our k will always be in extended k space6 unless the contrary is stated"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib6",
      "series-title": "\u201cPrinciples of the Theory of Solids.\u201d",
      "author": "Ziman",
      "year": "1964"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib7",
      "unstructured": "We shall usez to denote the total number of s and p electrons of an atom and usually refer to it as the \u201cvalency\u201d for short since this is the number of electrons contributed to the conduction band (or \u201cvalence band\u201d in the case of a semiconductor). In alloys the same symbolz denotes the electrons-per-atom ratio"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib8",
      "doi-asserted-by": "crossref",
      "first-page": "250",
      "DOI": "10.1088/0959-5309/49/3/307",
      "volume": "49",
      "author": "Jones",
      "year": "1937",
      "journal-title": "Proc. Phys. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib9",
      "doi-asserted-by": "crossref",
      "first-page": "1301",
      "DOI": "10.1103/PhysRev.130.1301",
      "volume": "130",
      "author": "Cohen",
      "year": "1963",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib10_1",
      "doi-asserted-by": "crossref",
      "first-page": "643",
      "DOI": "10.1051/jphysrad:019620023010064301",
      "volume": "23",
      "author": "Cohen",
      "year": "1962",
      "journal-title": "J. Phys. Radium"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib10_2",
      "series-title": "\u201cMetallic Solid Solutions\u201d",
      "author": "Friedel",
      "year": "1963"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib11",
      "doi-asserted-by": "crossref",
      "first-page": "2503",
      "DOI": "10.1103/PhysRev.129.2503",
      "volume": "129",
      "author": "Harrison",
      "year": "1963",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib12",
      "doi-asserted-by": "crossref",
      "first-page": "2512",
      "DOI": "10.1103/PhysRev.129.2512",
      "volume": "129",
      "author": "Harrison",
      "year": "1963",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib13",
      "doi-asserted-by": "crossref",
      "first-page": "1107",
      "DOI": "10.1103/PhysRev.136.A1107",
      "volume": "136",
      "author": "Harrison",
      "year": "1964",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib14",
      "unstructured": "L. J. Sham, Thesis, Univ. of Cambridge (1963). Available on microfilm or photocopy from Micromethods Ltd., East Ardsley, Wakefield, Yorkshire."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib15",
      "doi-asserted-by": "crossref",
      "first-page": "33",
      "DOI": "10.1098/rspa.1965.0005",
      "volume": "A283",
      "author": "Sham",
      "year": "1965",
      "journal-title": "Proc. Roy. Soc. (London)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib16",
      "series-title": "\u201cAlloying Behavior and Concentrated Solid Solutions\u201d",
      "first-page": "50",
      "author": "Blandin",
      "year": "1963"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib17",
      "series-title": "\u201cPhase Stability in Metals and Alloys\u201d",
      "first-page": "115",
      "author": "Blandin",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib18",
      "first-page": "128",
      "volume": "C3",
      "author": "Blandin",
      "year": "1966",
      "journal-title": "J. Phys. (Suppl.)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib19",
      "unstructured": "R. Pick, Thesis, Univ. of Paris (1965)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib20",
      "doi-asserted-by": "crossref",
      "first-page": "603",
      "DOI": "10.1103/PhysRev.152.603",
      "volume": "152",
      "author": "Heine",
      "year": "1966",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib21",
      "unstructured": "C. H. Hodges, Thesis, Univ. of Cambridge (1967)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib22",
      "unstructured": "D. Weaire, Thesis, Univ. of Cambridge (1968)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib23",
      "unstructured": "R. W. Shaw, Jr., Thesis, Stanford Univ. (1968)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib24",
      "doi-asserted-by": "crossref",
      "first-page": "786",
      "DOI": "10.1103/PhysRev.115.786",
      "volume": "115",
      "author": "Ehrenreich",
      "year": "1959",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib25",
      "series-title": "\u201cAlloying Behavior and Effects in Concentrated Solid Solutions\u201d",
      "first-page": "1",
      "author": "Cohen",
      "year": "1965"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib26",
      "doi-asserted-by": "crossref",
      "first-page": "221",
      "DOI": "10.1016/S0081-1947(08)60593-7",
      "volume": "15",
      "author": "Sham",
      "year": "1963",
      "journal-title": "Solid State Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib27",
      "doi-asserted-by": "crossref",
      "first-page": "336",
      "DOI": "10.1098/rspa.1958.0003",
      "volume": "A243",
      "author": "Hubbard",
      "year": "1958",
      "journal-title": "Proc. Roy. Soc. (London)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib28",
      "unstructured": "See e.g. Mott and Jones,3 p. 59, or Ziman,6 p. 70."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib29",
      "unstructured": "We shall use atomic units (s.u.) throughout (h = m = e = 1). In the past numerical values of energy have often been quoted in rydbergs (1 Ry = 1/2 a.u. \u2248 13.6 eV)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib30",
      "unstructured": "See Heine,31,33 Harrison,32 Ziman6 (Chapter 3), and Cohen and Heine34 (Parts I and II)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib31",
      "unstructured": "V. Heine, this volume."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib32",
      "series-title": "\u201cPseudopotentials in the Theory of Metals,\u201d",
      "author": "Harrison",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib33",
      "volume": "1",
      "author": "Heine",
      "year": "1968"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib34",
      "unstructured": "M. L. Cohen and V. Heine, this volume."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib35",
      "unstructured": "Strictly speaking, the pseudopotential is a function of four variables and we should writev(q, k | k + q |, E), but the variation withq is the most important so that we shall omit explicit mention of the other variables for the present. See Sections 9\u201312 and Sections 2 and 16 of Cohen and Heine.34"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib36",
      "unstructured": "We shall use q to denote a continuous variable and g when we wish to imply a reciprocal lattice vector."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib37",
      "unstructured": "In a complete NFE expression for the band gap, there would be higher order terms of perturbation theory which we shall ignore. Also if there is more than one atom per primitive unit cell, we should include a factor S(g) whereS is the X-ray structure factor (see Section 5)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib38",
      "doi-asserted-by": "crossref",
      "first-page": "23",
      "DOI": "10.1016/0031-9163(65)91031-0",
      "volume": "14",
      "author": "Ashcroft",
      "year": "1965",
      "journal-title": "Phys. Letters"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib39",
      "doi-asserted-by": "crossref",
      "first-page": "2055",
      "DOI": "10.1080/14786436308209098",
      "volume": "8",
      "author": "Ashcroft",
      "year": "1963",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib40",
      "unstructured": "Sections 2\u20134 of Cohen and Heine.34"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib41",
      "unstructured": "Sections 4\u20136 of Cohen and Heine.34"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib42",
      "doi-asserted-by": "crossref",
      "first-page": "928",
      "DOI": "10.1063/1.1727705",
      "volume": "45",
      "author": "Austin",
      "year": "1966",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib43",
      "series-title": "\u201cAlloying Behavior and Effects in Concentrated Solid Solutions\u201d",
      "first-page": "85",
      "author": "King",
      "year": "1965"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib44",
      "doi-asserted-by": "crossref",
      "first-page": "529",
      "DOI": "10.1080/14786436508218898",
      "volume": "12",
      "author": "Abarenkov",
      "year": "1965",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib45",
      "doi-asserted-by": "crossref",
      "first-page": "48",
      "DOI": "10.1016/0031-9163(66)90251-4",
      "volume": "23",
      "author": "Ashcroft",
      "year": "1966",
      "journal-title": "Phys. Letters"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib46",
      "unstructured": "See Sections 1 and 4 of Cohen and Heine34, also Heine.31"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib47_1",
      "doi-asserted-by": "crossref",
      "first-page": "1249",
      "DOI": "10.1080/14786436508228674",
      "volume": "12",
      "author": "Animalu",
      "year": "1965",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib47_2",
      "first-page": "376",
      "volume": "294",
      "author": "Animalu",
      "year": "1966",
      "journal-title": "Proc. Roy. Soc. (London)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib47_3",
      "doi-asserted-by": "crossref",
      "first-page": "956",
      "DOI": "10.1088/0370-1328/92/4/317",
      "volume": "92",
      "author": "Weaire",
      "year": "1967",
      "journal-title": "Proc. Phys. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib48",
      "doi-asserted-by": "crossref",
      "first-page": "379",
      "DOI": "10.1080/14786436508221864",
      "volume": "11",
      "author": "Animalu",
      "year": "1965",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib49",
      "unstructured": "A. O. E. Animalu, Tech. Rept. No. 3 (unpublished, 1965) obtainable from the Secretary, Solid State Theory Section, Cavendish Laboratory, Cambridge, England. This report gives tables ofv(q) with and without damping factor, for 25 elements corresponding to the graphs of Animalu and Heine47 with the exception of Cd, Ca, Sr, and Ba."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib50",
      "unstructured": "A. O. E. Animalu, Tech. Rept. No. 4 (unpublished, 1965) obtainable from the Secretary, Solid State Theory Section, Cavendish Laboratory, Cambridge, England. This gives tables ofv(q) for Cd, Ca, Sr, Ba, B, C, and P as in Animalu.49"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib51",
      "unstructured": "We shall use capital V to denote the pseudopotential in the whole crystal, and lower case v that of a single atom or ion."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib52",
      "unstructured": "The proof of (5.4) is straightforward and one always gets this form when Fourier transforming a superposition of N localized functions in a volume N\u03a9. The coefficient F(q) in a Fourier expression is which gives the form (5.4). The first term is the usual X-ray structure factor S. In the case of a regular crystal, it is nonzero only when q is a reciprocal lattice vector g, in which case all cells contribute identically to the sum and S can be simplified to (5.6). For a liquid or disordered solid, S(q) is a continuous function of q and the term in brackets is identical with (5.6) if we regard the whole volume with N atoms as the unit cell."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib53",
      "unstructured": "See Ziman,6 p. 133."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib54",
      "doi-asserted-by": "crossref",
      "first-page": "89",
      "DOI": "10.1080/00018736400101011",
      "volume": "13",
      "author": "Ziman",
      "year": "1964",
      "journal-title": "Advan. Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib55",
      "unstructured": "Chapter 5 of Ziman6; Section 3 of Cohen and Heine34; Heine and Abarenkov56 and Shaw and Harrison.57"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib56",
      "doi-asserted-by": "crossref",
      "first-page": "451",
      "DOI": "10.1080/14786436408222957",
      "volume": "9",
      "author": "Heine",
      "year": "1964",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib57",
      "doi-asserted-by": "crossref",
      "first-page": "604",
      "DOI": "10.1103/PhysRev.163.604",
      "volume": "163",
      "author": "Shaw",
      "year": "1967",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib58",
      "unstructured": "Harrison,32 p. 50."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib59",
      "doi-asserted-by": "crossref",
      "first-page": "1337",
      "DOI": "10.1088/0022-3719/1/5/324",
      "volume": "1",
      "author": "Inglesfield",
      "year": "1968",
      "journal-title": "J. Phys. C"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib60",
      "doi-asserted-by": "crossref",
      "first-page": "222",
      "DOI": "10.1088/0022-3719/1/1/325",
      "volume": "1",
      "author": "Heine",
      "year": "1968",
      "journal-title": "J. Phys. C"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib61",
      "doi-asserted-by": "crossref",
      "unstructured": "See Blandin,16,17 and p. 137 of Ziman.6",
      "DOI": "10.1086/102480"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib62",
      "unstructured": "Kittel,63 p. 95."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib63",
      "series-title": "\u201cIntroduction to Solid State Physics,\u201d",
      "author": "Kittel",
      "year": "1956"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib64",
      "doi-asserted-by": "crossref",
      "first-page": "210",
      "DOI": "10.1088/0022-3719/1/1/324",
      "volume": "1",
      "author": "Weaire",
      "year": "1968",
      "journal-title": "J. Phys. C"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib65",
      "doi-asserted-by": "crossref",
      "first-page": "213",
      "DOI": "10.1080/14786436808227326",
      "volume": "18",
      "author": "Weaire",
      "year": "1968",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib66",
      "doi-asserted-by": "crossref",
      "first-page": "471",
      "DOI": "10.1098/rspa.1940.0033",
      "volume": "A174",
      "author": "Hume-Rothery",
      "year": "1940",
      "journal-title": "Proc. Roy. Soc. (London)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib67_1",
      "doi-asserted-by": "crossref",
      "first-page": "647",
      "DOI": "10.1051/jphysrad:019620023010064701",
      "volume": "23",
      "author": "Massalski",
      "year": "1962",
      "journal-title": "J. Phys. Radium"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib67_2",
      "series-title": "\u201cMetallic Solid Solutions\u201d",
      "author": "Friedel",
      "year": "1963"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib68",
      "doi-asserted-by": "crossref",
      "first-page": "1075",
      "DOI": "10.1088/0022-3719/1/4/326",
      "volume": "1",
      "author": "North",
      "year": "1968",
      "journal-title": "Proc. Phys. Soc. (London) (Solid State Phys.)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib69",
      "doi-asserted-by": "crossref",
      "first-page": "343",
      "DOI": "10.1098/rspa.1949.0032",
      "volume": "196",
      "author": "Pauling",
      "year": "1949",
      "journal-title": "Proc. Roy. Soc. (London)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib70",
      "series-title": "\u201cTheory of Alloy Phases.\u201d",
      "author": "Pauling",
      "year": "1957"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib71",
      "unstructured": "Ziman,6 p. 149."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib72_1",
      "doi-asserted-by": "crossref",
      "first-page": "336",
      "DOI": "10.1098/rspa.1950.0104",
      "volume": "A202",
      "author": "Hall",
      "year": "1950",
      "journal-title": "Proc. Roy. Soc. (London)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib72_2",
      "doi-asserted-by": "crossref",
      "first-page": "429",
      "DOI": "10.1080/14786435808244564",
      "volume": "3",
      "year": "1958",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib73",
      "first-page": "386",
      "volume": "3",
      "author": "Williams",
      "year": "1970",
      "journal-title": "J. Phys. C"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib74",
      "unstructured": "W. M. Lomer,in Proc. Intern. Conf. Magnetism p. 127. Inst. Phys. and Phys. Soc., London, 1964."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib75",
      "doi-asserted-by": "crossref",
      "first-page": "846",
      "DOI": "10.1103/PhysRevLett.19.846",
      "volume": "19",
      "author": "McWhan",
      "year": "1967",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib76",
      "series-title": "\u201cPhase Stability in Metals and Alloys\u201d",
      "first-page": "273",
      "author": "Pearson",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib77",
      "series-title": "\u201cPhase Stability in Metals and Alloys\u201d",
      "first-page": "280",
      "author": "Lomer",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib78",
      "unstructured": "G. Yuval, private communication (1968)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib79",
      "doi-asserted-by": "crossref",
      "first-page": "1833",
      "DOI": "10.1103/PhysRev.124.1833",
      "volume": "124",
      "author": "Sato",
      "year": "1961",
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      "first-page": "295",
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      "unstructured": "See Pearson,76 p. 385."
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      "doi-asserted-by": "crossref",
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      "unstructured": "R. M. Bowley, V. Heine, and D. Weaire (to be published)."
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      "unstructured": "Sections 5 and 20 of Cohen and Heine.34"
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      "unstructured": "Harrison, 32 p. 223."
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      "author": "Phillips",
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      "unstructured": "J. C. Phillips, private communication (1969)."
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      "series-title": "\u201cLattice Dynamics\u201d",
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      "series-title": "\u201cPhonons\u201d",
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      "series-title": "\u201cPhonons\u201d",
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      "first-page": "371",
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      "volume": "I",
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    {
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    {
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib108",
      "unstructured": "For the reasons behind this choice the reader is referred to Heine.31"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib109",
      "unstructured": "For an elementary exposition of nonlocal operators, the reader is referred to Appendix A of Cohen and Heine.34"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib110",
      "unstructured": "For a theoretical discussion of why and when this is so, see Heine31 and Cohen and Heine.34"
    },
    {
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      "doi-asserted-by": "crossref",
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    {
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      "unstructured": "Also called \u201cdepletion hole\u201d (Shaw and Harrison67)."
    },
    {
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      "doi-asserted-by": "crossref",
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      "key": "10.1016/S0081-1947(08)60071-5_bib118",
      "unstructured": "We inciude the factor e2 explicitly so that the equation is valid not only in atomic units."
    },
    {
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      "doi-asserted-by": "crossref",
      "first-page": "A1133",
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    {
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      "volume": "23",
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      "journal-title": "Solid State Phys."
    },
    {
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      "series-title": "\u201cTheory of Interacting Fermi Systems.\u201d",
      "author": "Nozieres",
      "year": "1964"
    },
    {
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      "series-title": "\u201cThe Theory of Quantum Liquids, Vol. I, Normal Fermi Liquids.\u201d",
      "author": "Pines",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib123",
      "unstructured": "The derivation of this equation for a free electron gas is given by Hedin and Lundqvist,120 see particularly Eqs. (13.10) and (33.1) and Section 22. See also Hedin.124 It also applies to atoms, molecules, and insulating crystals (Section 36 of Hedin and Lundqvist120)."
    },
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      "first-page": "A796",
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      "journal-title": "Solid State Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib126",
      "unstructured": "Figures 43 and 44 of Hedin and Lundqvist.120"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib127",
      "unstructured": "Table IV of Hedin and Lundqvist.120"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib128",
      "unstructured": "Table VII of Hedin and Lundqvist.120"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib129",
      "unstructured": "As in Section 4,r3 is the radius of the sphere containing on average one electron."
    },
    {
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      "first-page": "631",
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib132",
      "unstructured": "Figure 13 of Hedin and Lundqvist.120"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib133",
      "doi-asserted-by": "crossref",
      "first-page": "442",
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      "first-page": "670",
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib135",
      "doi-asserted-by": "crossref",
      "unstructured": "See Cohen, 10 Sham, 14,15 and D. Pines, private communication (1966).",
      "DOI": "10.1525/sp.1966.14.1.03a00110"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib136",
      "unstructured": "Section 16 of Cohen and Heine.84"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib137",
      "doi-asserted-by": "crossref",
      "first-page": "74",
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      "volume": "13",
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    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib138",
      "unstructured": "Note added since the manuscript was completed, Three further calculations (Singwi et al 137b Geldart and Taylor, 137 and Ferreira 137d) have appeared recently which essentially calculate Ep or Ee, although none of the results are expressed in terms off so that an immediate comparison is not possible. However, visual inspection of Geldart and Taylor137c suggests that theirf would rise to a peak greater than 1/2 at intermediate values ofq. For the most complete recent discussion, see Shaw.137e"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib139",
      "doi-asserted-by": "crossref",
      "first-page": "176",
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      "year": "1970",
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    {
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      "first-page": "183",
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      "volume": "48",
      "author": "Geldart",
      "year": "1970",
      "journal-title": "Canad. J. Phys."
    },
    {
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      "first-page": "1113",
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      "volume": "30",
      "author": "Ferreira",
      "year": "1969",
      "journal-title": "J. Phys. Chem. Solids"
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    {
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      "first-page": "1140",
      "DOI": "10.1088/0022-3719/3/5/027",
      "volume": "3",
      "author": "Shaw",
      "year": "1970",
      "journal-title": "J. Phys. C"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib143",
      "unstructured": "L. Ballentine, private communication (1968)"
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib144",
      "unstructured": "Equations (1.47) and (3.62) of Pine and Nozieres.122"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib145",
      "unstructured": "Contrary to Shaw and Pynn,112 we do not believe that the high density expansions have any relevance at metallic densities. Also if we use the high density expansion forUc [Eq. (5.121) of Pine and Nozieres122] in (10.59), we obtain a different value for the correlation contribution toa from the one quoted by Shaw and Pynn.112"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib146",
      "unstructured": "Calculation by Ma reported by Shaw and Pynn.112"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib147",
      "unstructured": "See Harrison, 32 p. 320."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib148",
      "doi-asserted-by": "crossref",
      "first-page": "585",
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      "journal-title": "Phys. Letters"
    },
    {
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      "first-page": "673",
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      "series-title": "\u201cQuantum Mechanics,\u201d",
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      "unstructured": "For a discussion of the practical details of handling such integrals see Animalu.148 Relevant material may also be found in Shaw23 and Sloan.149"
    },
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      "doi-asserted-by": "crossref",
      "first-page": "376",
      "DOI": "10.1098/rspa.1966.0213",
      "volume": "294",
      "author": "Animalu",
      "year": "1966",
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      "first-page": "332",
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      "volume": "3",
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      "journal-title": "J. Comp. Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib156",
      "unstructured": "This topic is extensively discussed by Cohen and Heine,34 particularly Sections 2, 11, 12, 14, and 21."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib157",
      "doi-asserted-by": "crossref",
      "first-page": "241",
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      "first-page": "1319",
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib161",
      "unstructured": "Note that the effective mass calculations reported by Animalu and Heine47 (sT6) were found to be in error.156 Note also the need to distinguish contributions to the total average band effective mass m* from nonlocality and energy dependence respectively.154,156 The latter is described in the next section. One cannot simply put asterisks on all them's in a given theory in order to incorporate effective mass corrections, because energy dependence and nonlocality have rather different effects."
    },
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      "unstructured": "This type of statement has been recently challenged by Pendry,117 who gives a justification of pseudopotential theory independent of considerations of the smoothing of wave functions. See also J. B. Pendry (to be published)."
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      "unstructured": "A. R. Williams (1970) (to be published)."
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      "unstructured": "G. Yuval, private communication (1969)."
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      "unstructured": "We are indebted to J. Shively for this calculation"
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      "unstructured": "The zone drawn by Mott and Jones for graphite contains three electrons per atom, not four as stated, and relates only to the \u03c3 bonds. It can be extended to describe the \u03c0 electrons."
    },
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      "unstructured": "We have already emphasized in Section 17 that weakening the potential by pseudiz-ing away the core states is always a mathematically valid procedure even in NaCl for example190 but what remains may or may not be small enough for perturbation theory to be a good approximation. The potentials encountered in covalent structures are \u201clarge\u201d in this latter sense."
    },
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      "series-title": "\u201cNeutron Inelastic Scattering.\u201d",
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      "unstructured": "Ewald energies of other structures of less general interest may be found in Harrison, 32 and Table III and Figs. 96 and 97 of this review."
    },
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      "unstructured": "W. Kohn and D. Schechter (unpublished) quoted by W. J. Carr, Jr. 201"
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      "unstructured": "For further details of this procedure see Shaw,23 Harrison,32 Tosi,207 and Johnson.208 The method is based on the work of Epstein.209,210"
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      "unstructured": "For the details of this expansion see Harrison,32 Chapter 7."
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      "unstructured": "For a review of attempts to fit phonon dispersion relations by the alternative procedure of adjustment of a parameterized form for \u03c0be (q) see Cohen and Heine.34"
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    {
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      "doi-asserted-by": "crossref",
      "first-page": "473",
      "DOI": "10.1016/0022-3697(65)90121-6",
      "volume": "26",
      "author": "Freeman",
      "year": "1965",
      "journal-title": "J. Phys. Chem. Solids"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib228",
      "unstructured": "Strictly speaking, such structures are close-packed only if the interplanar spacingh, which we assume constant although this is not dictated by symmetry in all cases, is related to the interatomic spacinga of the close-packed layers byh = (f)1/2a as in fee. The theory as presented here is not restricted to this case, although some of the qualitative remarks made about it are not valid if the structure is not reasonably near to close-packing."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib229",
      "volume": "5",
      "author": "Harrison",
      "year": "1970"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib230",
      "unstructured": "One pseudopotential calculation reported by Krasko et al 223a in stands rather puzzling contradiction of this conclusion since a large positive stacking fault energy is obtained for fee copper."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib231",
      "first-page": "2414",
      "volume": "9",
      "author": "Krasko",
      "year": "1968",
      "journal-title": "Soviet Phys.-Solid State"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib232",
      "author": "Gallagher",
      "year": "1970",
      "journal-title": "Trans. AIME"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib233",
      "unstructured": "The unusual form of the energy-wavenumber characteristic in this calculation derives from the fact that the various structure-dependent terms were grouped in a way rather different from what is now usual, leading to a different definition of effective charges and a form for the energy-wavenumber characteristic which was not negative-definite as it usually is."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib234",
      "unstructured": "Sham also included an estimate of Born-Mayer repulsion between ion cores in his calculation of phonon dispersion in Na. Effects due to Born-Mayer repulsion appear to be quite small and of uncertain magnitude for the simple metals due to the small sizes of the core, and are ignored in this review."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib235",
      "doi-asserted-by": "crossref",
      "first-page": "539",
      "DOI": "10.1051/jphys:01967002807053900",
      "volume": "28",
      "author": "Pick",
      "year": "1967",
      "journal-title": "J. Phys. (France)"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib236",
      "doi-asserted-by": "crossref",
      "first-page": "769",
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      "volume": "156",
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib237",
      "unstructured": "The equations used by Roy and Venkataraman228 for the calculation of phonon frequencies are incorrect. An unpublished erratum gives corrected results in qualitative agreement with the observed dispersion curves, with quantitative discrepancies of up to 20%."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib238",
      "doi-asserted-by": "crossref",
      "first-page": "289",
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      "journal-title": "Phys. Letters"
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    {
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      "doi-asserted-by": "crossref",
      "first-page": "295",
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      "journal-title": "Solid State Commun."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib240",
      "unstructured": "See Section 47."
    },
    {
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      "doi-asserted-by": "crossref",
      "first-page": "699",
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    {
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    },
    {
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      "series-title": "\u201cNeutron Inelastic Scattering.\u201d",
      "author": "Brovman",
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      "doi-asserted-by": "crossref",
      "first-page": "1105",
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      "journal-title": "Phil. Mag."
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    {
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      "first-page": "795",
      "DOI": "10.1088/0022-3719/2/4/126",
      "volume": "2",
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    {
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      "doi-asserted-by": "crossref",
      "first-page": "1285",
      "DOI": "10.1088/0022-3719/2/7/320",
      "volume": "2",
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      "journal-title": "J. Phys. C"
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    {
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      "first-page": "1293",
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      "journal-title": "J. Phys. C"
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    {
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      "doi-asserted-by": "crossref",
      "first-page": "1395",
      "DOI": "10.1016/0001-6160(69)90001-7",
      "volume": "17",
      "author": "Inglesfield",
      "year": "1969",
      "journal-title": "Acta Met."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib249",
      "doi-asserted-by": "crossref",
      "first-page": "820",
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      "author": "Cousins",
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    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib252",
      "unstructured": "Apart from this the formalism is identical with that described in Part I"
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      "key": "10.1016/S0081-1947(08)60071-5_bib253",
      "doi-asserted-by": "crossref",
      "first-page": "500",
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      "first-page": "487",
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      "first-page": "489",
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    {
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    {
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      "unstructured": "D. Taylor (1970) (to be published)."
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      "first-page": "330",
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      "volume": "178",
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      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib270",
      "unstructured": "P. S. Ho (to be published)."
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      "volume": "II",
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      "unstructured": "We will generally refer to Pearson,263 for well-known structures, and will give the appropriate detailed references only in unfamiliar cases."
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      "unstructured": "R. Jacobs, thesis, University of Cambridge (1969) and to be published 1970."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib282",
      "series-title": "\u201cMagnesium and Its Alloys.\u201d",
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib285",
      "unstructured": "See the preface to Pearson, 263 also Roberts.271"
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      "key": "10.1016/S0081-1947(08)60071-5_bib286",
      "doi-asserted-by": "crossref",
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    },
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      "first-page": "1083",
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    {
      "key": "10.1016/S0081-1947(08)60071-5_bib290",
      "unstructured": "Because the inclusion of effective mass corrections in the crude form employed here destroys the cancellation of the Ewald and band structure terms at lowq, which is irrelevant to the comparison of simple crystal structures but would lead to pair interactions of an unusual form, an extra factor of (mKmE)-1 was arbitrarily included in E in the calculations of Fig. 24 to rectify this while making little difference at highq."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib291",
      "unstructured": "The idea that the distorted structures of Zn and Cd could be explained in terms of pairwise central forces of reasonable form, provided that there were additional large volume-dependent energy terms, appears to have been first suggested by Nabarro and Varley.281 However, the idea did not gain general acceptance (see e.g. Wallace282)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib292",
      "doi-asserted-by": "crossref",
      "first-page": "316",
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    {
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      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib294",
      "unstructured": "This discrepancy, which casts some doubt on the discussion of axial ratios given in this section, may be at least partially attributable to the fact that Shaw did not use effective mass corrections as did Weaire.64 Such corrections are rather small for Na, Mg, and Al, for which Shaw's work gives such excellent results, but are large enough to be significant for most other simple metals (see Shaw154 and Weaire156)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib295",
      "doi-asserted-by": "crossref",
      "first-page": "240",
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      "year": "1966",
      "journal-title": "Proc. Roy. Soc."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib296",
      "doi-asserted-by": "crossref",
      "first-page": "201",
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      "year": "1966",
      "journal-title": "Phil. Mag."
    },
    {
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      "doi-asserted-by": "crossref",
      "first-page": "392",
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      "unstructured": "R. Pynns, private communication (1969). See footnote 283."
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    {
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      "unstructured": "J. S. Abell, thesis, Univ. of Surrey (1969); private communication."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib305",
      "doi-asserted-by": "crossref",
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    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib306",
      "unstructured": "See e.g. Atoji.291"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib307",
      "unstructured": "According to Animalu, and Heine,47Ra = 2.1 a.u.,Ra = 3.35 a.u. The nearest neighbor distance in \u03b1-Hg is 5.7 a.u. ~ 1.4 (2Re)."
    },
    {
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      "doi-asserted-by": "crossref",
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      "unstructured": "M. Appapillai (to be published)."
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      "series-title": "\u201cTheory of Alloy Phases.\u201d",
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    {
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      "first-page": "894",
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    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib317",
      "unstructured": "D. B. McWhan, private communication (1968)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib318",
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      "volume": "84",
      "author": "Curien",
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    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib319",
      "first-page": "378",
      "volume": "14",
      "author": "Reichardt",
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    },
    {
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      "volume": "141",
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      "year": "1963",
      "journal-title": "Science"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib321",
      "unstructured": "Not to be confused with the hexagonal close-packed structure."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib322",
      "first-page": "833",
      "volume": "239",
      "author": "Giessen",
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    },
    {
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    },
    {
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      "first-page": "111",
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      "journal-title": "Acta Met."
    },
    {
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      "series-title": "\u201cThe Mechanism of Phase Transformations in Crystalline Solids,\u201d",
      "author": "Weaire",
      "year": "1969"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib327",
      "first-page": "365",
      "volume": "25",
      "author": "Brovman",
      "year": "1967",
      "journal-title": "Soviet Phys. JETP"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib328",
      "doi-asserted-by": "crossref",
      "first-page": "194",
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      "volume": "57B",
      "author": "Bonsignori",
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      "journal-title": "Nuovo Cimento"
    },
    {
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    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib330",
      "unstructured": "Added in proof. Coulthard241 has recently published such a calculation"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib331",
      "doi-asserted-by": "crossref",
      "first-page": "185",
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      "volume": "10",
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      "year": "1962",
      "journal-title": "Acta Met."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib332",
      "doi-asserted-by": "crossref",
      "first-page": "215",
      "DOI": "10.1147/rd.83.0215",
      "volume": "8",
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      "journal-title": "IBM J. Res. Develop."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib333",
      "unstructured": "See Mott and Jones,3 p. 167."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib334",
      "author": "Williams",
      "year": "1970",
      "journal-title": "J. Phys. Chem. Sol."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib335",
      "doi-asserted-by": "crossref",
      "first-page": "441",
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      "volume": "142",
      "author": "Lin",
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      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib336",
      "doi-asserted-by": "crossref",
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      "DOI": "10.1103/PhysRevLett.22.1251",
      "volume": "22",
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      "year": "1969",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib337",
      "unstructured": "See below Eq. (4.2), or Eq. (4.4) of Cohen and Heine.34 In the present context we identifyre of the empty core model with the radiusRe of the ion core."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib338",
      "doi-asserted-by": "crossref",
      "first-page": "1821",
      "DOI": "10.1103/PhysRev.122.1821",
      "volume": "122",
      "author": "Cohen",
      "year": "1961",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib339",
      "unstructured": "The (R) for 29 elements has been calculated by R. Bowley from the model potential (Section 9) using the \u2018local\u2019 form of the theory as in Section 5. The results are not being published in full because of the criticisms of such an approximation (Section 11), and because full nonlocal calculations have been started by M. Appapillai using Shaw's optimized form of the model potential (Section 9). We are indebted to Mr. Bowley for making his results available to us."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib340",
      "unstructured": "Readers unfamiliar with this item should note that it is a hard boiled egg, shelled and surrounded by a layer of sausage meat approximately one inch thick."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib341",
      "series-title": "\u201cPhase Stability in Metals and Alloys\u201d",
      "first-page": "3",
      "author": "Hume-Rothery",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib342_1",
      "doi-asserted-by": "crossref",
      "first-page": "607",
      "DOI": "10.1051/jphysrad:019620023010060700",
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      "author": "Massalski",
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      "journal-title": "Phys. Radium"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib342_2",
      "series-title": "\u201cMetallic Solid Solutions\u201d",
      "author": "Friedel",
      "year": "1963"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib343",
      "first-page": "1",
      "volume": "10",
      "author": "Massalski",
      "year": "1961",
      "journal-title": "Progr. Mater. Sci."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib344",
      "series-title": "\u201cPhase Stability in Metals and Alloys\u201d",
      "first-page": "281",
      "author": "Nevitt",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib345",
      "unstructured": "See Section 41."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib346",
      "unstructured": "See e.g. Mott and Jones,3 Chapter IV."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib347",
      "volume": "2",
      "author": "Slater",
      "year": "1965"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib348",
      "doi-asserted-by": "crossref",
      "first-page": "117",
      "DOI": "10.1016/0022-3697(69)90344-8",
      "volume": "30",
      "author": "Hafemeister",
      "year": "1969",
      "journal-title": "J. Phys. Chem. Solids"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib349",
      "doi-asserted-by": "crossref",
      "first-page": "553",
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      "volume": "141",
      "author": "Higgins",
      "year": "1966",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib350",
      "doi-asserted-by": "crossref",
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      "volume": "24",
      "author": "Fujiwara",
      "year": "1968",
      "journal-title": "J. Phys. Soc. Japan"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib351",
      "series-title": "\u201cQuantum Theory of Solids,\u201d",
      "author": "Kittel",
      "year": "1963"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib352",
      "doi-asserted-by": "crossref",
      "first-page": "446",
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      "journal-title": "Advan. Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib353_1",
      "series-title": "\u201cSoft X-Ray Band Spectra\u201d",
      "author": "Rooke",
      "year": "1968"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib353_2",
      "doi-asserted-by": "crossref",
      "first-page": "245",
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      "volume": "12",
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      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib353_3",
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      "first-page": "523",
      "DOI": "10.1016/0375-9601(69)90284-9",
      "volume": "30A",
      "author": "Jacobs",
      "year": "1969",
      "journal-title": "Phys. Letters"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib354",
      "doi-asserted-by": "crossref",
      "unstructured": "W. Kohn and C. Majumdar, Phys. Rev. 138, A1617",
      "DOI": "10.1103/PhysRev.138.A1617"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib355",
      "unstructured": "See Cohen,25 p. 11."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib356",
      "series-title": "\u201cStructure of Metals,\u201d",
      "author": "Barrett",
      "year": "1966"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib357",
      "unstructured": "Rather similar results have been obtained in recent calculations by Shaw,23,222 who performs lattice sums using the asymptotic form of the interatomic potential for the hep, fee, and bee structures."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib358_1",
      "doi-asserted-by": "crossref",
      "first-page": "637",
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      "journal-title": "J. Phys. Radium"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib358_2",
      "series-title": "\u201cMetallic Solid Solutions\u201d",
      "author": "Friedel",
      "year": "1963"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib359",
      "doi-asserted-by": "crossref",
      "first-page": "395",
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      "volume": "7",
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      "year": "1958",
      "journal-title": "Advan. Phys."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib360",
      "doi-asserted-by": "crossref",
      "first-page": "55",
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      "year": "1961",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib361",
      "doi-asserted-by": "crossref",
      "first-page": "863",
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      "volume": "11",
      "author": "Lettington",
      "year": "1965",
      "journal-title": "Phil. Mag."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib362",
      "unstructured": "We are grateful to T. B. Massalski for pointing this out to us."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib363",
      "unstructured": "See e.g. Jacobs.181,182"
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib364",
      "unstructured": "For instance, in the case of Ga alloys unusual structures are found351,352 which may be interpreted in terms of the tendency of Ga atoms to favor an interatomic distance rather shorter than that which would be normally dictated by considerations of atomic volumes, exactly as in the case of the pure metal (Section 5 and Part VIII)."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib365",
      "doi-asserted-by": "crossref",
      "first-page": "549",
      "DOI": "10.1107/S0567740869002536",
      "volume": "B25",
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    },
    {
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      "doi-asserted-by": "crossref",
      "first-page": "554",
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      "volume": "B25",
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      "year": "1969",
      "journal-title": "Acta Crystall."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib367",
      "unstructured": "The definition ofequivalence for these purposes is that every site has the sameradial distribution of other sites."
    },
    {
      "key": "10.1016/S0081-1947(08)60071-5_bib368",
      "doi-asserted-by": "crossref",
      "first-page": "5278",
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    },
    {
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      "series-title": "\u201cConstitution of Binary Alloys,\u201d",
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      "year": "1958"
    },
    {
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      "series-title": "\u201cThermochemical Data of Alloys,\u201d",
      "author": "Kubaschewski",
      "year": "1956"
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      "journal-title": "Proc. Phys. Soc. (London) (Solid State Phys.)"
    },
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