The ion-electron-ion interaction is treated according to the Hartree-. Fock approximation ..... of the net ion charge Ze. ... terms of hydrogen-like wave functions.
LATTICE DYNAMICS AND ELECTRONIC STRUCTURE OF SODIUM, MAGNESIUM AND ALUMINIUM T. SCHNEIDER AND E. STOLL DELEGATION FUR AUSBILDUNG UND HOCHSCHULFORSCHUNG, SWISS FEDERAL INSTITUTE FOR REACTOR RESEARCH, WÜRENLINGEN, SWITZERLAND
Abstract L A T T IC E D Y N A M IC S A N D E LE C T R ON IC S T R U C T U R E O F S O D I U M , M A G N E S IU M A N D A L U M IN I U M . T h e actual interaction between ions in a m etal is divided into the direct interaction of the ion cores and the ion-eiectron-ion interaction.
T h e ion-electron-ion interaction is treated according to the Hartree-
Fock approximation, modified by a screened exchange potential.
Since until now the bare ion potential
has not been known very accurately, the observed phonon frequencies were used to determine a m odel pseudo-potential with the least-squares m etho d.
Using this m odel pseudo-potential, the electronic
band structure, the Fermi surface, the cohesive energy, the crystal stability, the interactomic potential and the electrical resistivity were calculated.
T h e method described enables us to relate the measured
phonon dispersions with the electronic structure and the electrical properties of simple metals.
The neutron scattering technique has now been developed to yield detailed inform ation on phonon dispersion cu rves. The interpretation of the experim ental data is norm ally based on a fo rc e constant m odel. This m odel does not provide the best physically sound interpretation, but seem s to be the best available interpolation schem e fo r fitting the observed frequencies and fo r use in the derivation of a frequency distribution. The first attempt to calculate dispersion curves fo r sim ple metals by a satisfactory method was made by Toya . These calculations, which preceded the experim ental m easurements by some years, proved to be in rather good agreement with them. Toya divided the interactions between the ions in a metal into three parts: the exchange o r overlap potential between the ion co r e s , the Coulomb potential between the ions, and the potential induced by the electron-phonon interaction including e le ctron -electron interactions. In the present paper the potential induced by the electron-phonon interaction is form ulated by the pseudo-potential method. If we know the pseudo-potential, we can calculate the electron ic, e le ctrica l and som e dynamical p rop erties in both the solid and liquid states . It is still a difficult problem to calculate this potential from first principles with any degree of p recision . Consequently, the potential is generally presented in a m odel form , which includes in a sim ple param etric way all the features dictated by the physics of the situation. In the m odel potential method of Heine and Abarenkov  the inform ation on the pseudo-potential is obtained from sp ectroscop ic data pertinent to the free atom. Another approach consists of determining a m odel potential from the m easured phonon dispersion curves. 101
SCHNEIDER and STOLL
Follow ing H arrison  and our ea rlie r work [4 -6 ], we shall show that the m easured phonon dispersion curves can be used to determine a m odel pseudo-potential over the whole range in momentum space. Using this m odel pseudo-potential, we shall discu ss the electron ic band structure, the F erm i surface, the cohesive energy, the crysta l stability, the interatom ic potential and the e le ctrica l conductivity of sodium, magnesium and aluminium,
The problem considered is that of finding the total energy of a sim ple metal in the Born-Oppenheim er adiabatic approximation. The Hamiltonian can be written sym bolically as Н = ^ + Ф (Й ),
3? = E c(R )+ E e(R )
where Ф(Н) is the total potential energy of the crysta l, Ti the kinetic energy of the ions, Ес(Й) the Coulomb energy of the ions embedded in a uniform negative background and Eg(R) is the total energy of the electrons, im m ersed in a uniform background of positive charge. The exchange overlap fo rc e in sodium, magnesium and aluminium is negli gible . The calculation of the Coulomb term E^ fo r the ideal and distorted lattice presents no difficulty using E w a ld 's method . We may now proceed to the evaluation of the total electron ic energy. Here, we p re sent a sim ple calculation, based on perturbation theory, starting from free electron s. Assuming that the electron -ion potential is the sum of individual ionic potentials, the total electron ic energy per ion becom es [8 ] Ее = §Z E p - ^ - k F + Z(0.6781nro - 2 .50 5 )1 0 '^ erg
(2 ) к < kp where kp = (37r2nZ)l/3
Q? о 1 V iQ'R] F (Q )= -^ e
U e(Q )= Ui (k + Q , k) Vt (k + Q, k) + Ulfk. к + Q) V ¡(к, к + $ ) Е ?+ 3 -Е У к < kp
U j(5 + 3 ,6 ) = n / e
kF Ер Z Ps Ui(Q) V i(3) U J§)
: : : :
U^ (X) e"''^ dX
Ions p er unit volume Number of ions in the crystal Structure factor F(Q) Vp = - Z e 3 /R Vp Self-con sisten t o r screened pseudo-potential Unscreened or bare pseudo-potential F ou rier transform of the electron ic contribution to the effective ion -ion potential
F erm i wave number F erm i energy Valence A tom ic radius
The fir st three term s of Eq. (2) are, resp ectively, the kinetic, exchange, and correlation en ergies, and erg units are used throughout. F o r the latter we used the N o ziè re s -P in es form . The fourth term a rise s from the fir s t-o r d e r contribution to the perturbation expansion. The last term com es from the secon d-order contribution and d escrib es the electron ic band-structure effects. This contribution depends on the detailed configuration of the ions. When the ions disturb the uniform F erm i sea, the electron s under go a change in potential. They redistribute them selves under the in fluence of the bare pseudo-potential and screen this potential. This e f fect has been con sidered by severa l authors [10-13]. If we make the assumption that V] is a loca l pseudo-potential, E q .(4 ), the relation between the bare and screened pseudo-potential reduces to E q.(5 ) where we have used the m odified static d ielectric function of the F e rm i sea introduced by Sham  in the manner suggested by Hubbard: V ¡(S + 3 ,5 ) = v, (Q)
u roi =
S(Q) = ^ ^ [ 1 - C ( Q ) ] X ( Q )
X(Q) = g r f(Q /2 k ,)
1+t 1 -t
- g Q^+pk^
C(Q) represen ts a correction fo r the effect of exchange and correlation as suggested by Sham. Follow ing Geldart and V osko [141, we have chosen p such that we get fo r Q = 0 the com p ressibility of the F erm i sea, determ ined from the fir st three term s in Eq. (2). This leads to P =
(6 ) 1 + 0.153
SCHNEIDER and STOLL
where a¡) is the Bohr radius. becom es
The band structure energy, E q .(3 ), now
lF (Q )p U ^ (Q )
(3) The adiabatic principle enables us to expand the total potential energy $(R ) of the crystal in powers of the ionic displacem ents и(к) as follow s: Ф(Й) = Ф ( Й ° ) + ^ Ф ;( К ) ^ ( К ) + ^ 1.K.Í
3 ij(^ ')U j(K ')
I'.K'.j where Ф(И ) is the static equilibrium energy and
9u¡ (к) 9uj (к* )
"í(K) =Uj(K') = О
In the equilibrium configuration R° the fo rc e on any ion must vanish, therefore Ф ^ ) is zero. The equilibrium position of the ion in the 1^ unit ce ll is denoted by Й°(к). F rom Eqs (1), (2), (3) and (7) we obtain the following expression fo r the static equilibrium energy: -kp + Z(0.6781nro - 2.505) 10-12
+ ^ < 5 I V ; - У р 1 К > + Е ь ( Й ° ) + Е,;(Й0)
h^0 li is a re cip ro ca l lattice v ector. E c(R °) is the electrostatic energy of point ions of valence Z, im m ersed in a uniform compensating back ground of electron s. Е \Í2 - 2тг/а. This assumption is based on B a rd een 's  expression for U j(Q ) on the basis of which the contribution of the Umklapp - p ro ce ss e s to Eq.(21), is underestimated. By using the arguments of Heilman and Kassatotschkin  and of the pseudo-potential method  we expressed the bare m odel potential in the following form [4, 8] : (26) n where
Yn and r„ are the m odel param eters. The first part of E q.(26) r e p r e sents the attractive fo r c e s com ing from the long-range Coulomb field of the net ion charge Z e. The second term is an expansion of all other contributions, including the repulsive part of the pseudo-potential, in term s of hydrogen-like wave functions. Using Eqs (8), (17), (19), (20) and (26), the m odel param eters are readily determined from the experimental data by means of the least squares method. To ensure that the phonon frequencies satisfy the sym m etry of the B rillouin zone, the sum over the re cip ro ca l lattice included the following number of nearest neighbour shells: Na: 16, Al: 20, Mg: 60. The physical constants and the results are listed in Table I. The sou rces of the experim ental results are as follow s: Na, W oods e t a l. , Al; Yarnell et al.  and W alker , Mg: Iyengar et al.  . The fitted and experim ental results are com pared in F igs 1, 2 and 3. F or a quantitative com parison we use the ro o tm ean-square deviation (RMD) of the fit expressed in the form
TABLE I. PHYSICAL CONSTANTS AND MODEL POTENTIAL PARAMETERS Mg
0 .20X 0.01
0.18 ± 0.04
3.77 ± 0.23
FIG.l. Phonon dispersion curves for bcc Na. The circles show some of the experimental points of Woods et al. . The lines represent the four parameter fit. ш in units of sec*'.
.10 1 .TAI
FIG.2. Phonon dispersion curves for hep Mg. The circles show some of the experimental points of Iyengar et al. . The lines represent the four parameter fit. и in units of sec"'.
where g¡ = [A t^ ] u¡ are the observed phonon frequen cies, Au¡ the e x perim ental e r r o r s and the calculated frequen cies. The num erical RMD values are listed in Table I. The values of and (21) fo r hexagonal Na, Mg and A l are given in Table II. It is known that the
SCHNEIDER and STOLL
FIG.3. Phonon dispersion curves for fee Al. The circles show some of the experimental points of Yarnell et al.  and Walker . The lines represent the four parameter fit. и in units of sec**.
IN THE  DIRECTION
ч/Зтах Na LA:
strengths of the electron-phonon interaction in Na, Mg and A l are d if ferent, i . e . v ery weak, weak, interm ediate. This differen ce m ani fests itself in many of their p rop erties, e .g . their electron ic con tri bution to the sp ecific heat which is influenced by the electron-phonon interaction . (Na: C /C o = 1 .2 7 , Mg: 1.31 and A l: 1.45, where Co is the free electron value. ) The distinction is even m ore striking when one com pares (Table II). Since represents an important con tri bution to it is clea r that phonon dispersion curves can be used to obtain inform ation about the m odel potential fo r a large continuous range of Q. Figure 4 shows the screened m odel potentials U¡ (Q) fo r Na, Mg and A l, derived from the observed phonon frequen cies. 4.
EFFECTIVE ION-ION INTERACTION
Using Eqs (8), (23), (25), (26) and Table I we calculated the e ffe c tive ion interaction U(R) and the amplitudes of the F ried el oscillation s B(2kp). Figure 5 shows the results we have obtained fo r Na, Mg and A l. A s expected (Table III), the F ried el oscillation s in Na are too sm all to produce a visib le effect. With regard to Al, Y arn ell et al.  have pointed out that their data indicates interactions up to 15 neighbours. This a grees with our findings fo r the m ore pronounced F ried el o s c illa tions in this metal (Table III, F ig. 5). UtRMo"
FIG.5. Effective ion-ion interaction in Na, Mg and Al. U(R) in units of erg.
(R) is the asymptotic form given by Eq.(24).
SCHNEIDER and STOLL
1 1 2
TABLE III. AMPLITUDES OF THE FRIEDEL OSCILLATIONS IN UNITS OF 10*^2 erg
BAND STRUCTURE AND FERMI SURFACE
We computed the energy shifts of electron states by the perturba tion theory, E q .(2 ). In sim ple m etals the m atrix elements are sm all, but the computed shifts are not always sm all since the energy deno m inators E y ^ - E ^ , which enter the perturbation expansion, can b e com e zero near the Bragg planes ^ =Й/2. To im prove upon this ap proach, it is n ecessary to use perturbation theory fo r degenerate s y s tem s. One uses a few -plane-w ave basis set fo r the expansion of the electron wave function. The plane-wave expansion (27) may be substi tuted into the on e-electron Schrödinger equation (28). This leads in the usual way to a set of coupled equations (29): Ц) = ^
Ti [T + U j(? ) ]¡g ) = E j g )
^ L -(!? -R )3 -E (g ,H )
F (H -H ')U t (!i? -H '!)a ^ = 0
F o r a consistent solution the determinant (29) must vanish: det
(k -h )
-E (k ,h )
0^ . + F (R -K ')U i(IH -& l)
F o r each к there are many solutions of the secu lar equation; these solutions correspon d to the many-band en ergies. F rom the F erm i energy defined by
N(E) dE = Z
where N(E) =
dag lgradE (k )lg
and the secu lar equation (30), it is now possible to calculate the shape of the F erm i surface E(kp)=Ep
TABLE IV. FERMI ENERGY. Ep IS THE FREE ELECTRON AND Ep THE EFFECTIVE FERMI ENERGY IN UNITS OF 10'12 erg
Using Eqs (30), (31), (32) and Table I, we calculated the F erm i energy and the radial distortion of the F erm i su rface. The F erm i energies are listed in Table IV. The calculated angular distortions of the F erm i surface in sodium are shown in F ig. 6. F o r com parison we have also plotted the experim ental results of Lee . The agreement between the calculated and m easured distortions is satisfactory. The band structure and the F erm i surface fo r Mg w ill be published e ls e where .
LATTICE ENERGY AND CRYSTAL STABILITY
The lattice energy, E q.(15), provides a wealth of inform ation in cluding the relative stability of different phases, their lattice constants and the cohesive energy. Th erefore this is one of the central problem s in understanding the nature of simple m etals. H arrison  and the authors  have applied the pseudo-potential approach to problem s
FIG. 6. Deviation of the Fermi surface from spherity in Na. The circles show some experimental data from Ref..
CONTRIBUTIONS TO THE LATTICE ENERGY E^ (11, 33) IN UNITS OF 10*^ erg
Eçprr = (0.678-lnr„ -2.505)
and STO LL
such as the relative stability of different lattices. These effects are expected to be insensitive to e r r o r s arising from the little-known abso lute energy. Kleinman , how ever, has claim ed that a p rop er a c counting of the energy contributions can lead to reasonable values of the cohesive energy in the O PW -pseudo-potential method. Here we examine a sim ple m odel for the lattice energy based on perturbation theory. Using Eqs (11), (13), (15) and Table I we c a l culated the fre e energy per electron at 0°K. This energy is the sum of the sublimation heat Es and the magnitude of the first Z ionization en e r gies per electron I E c=3?(R °)+E N = Es + lll
In these calculations we neglected the zero-point energy. The results are listed in Table V . The agreem ent with the experim ental values is reasonable. In spite of these uncertainties, there are other p rop erties which we may reasonably treat. These are properties which involve r e arrangem ents at constant volum e, such as the energy differen ce between different structures. In the follow ing we specify the volum e of the m etal and examine the relative energies of different structures at that volum e. We have done this fo r the three sim ple m etallic structures b cc, fee and hep, the latter as a function of the с /a ratio. Table VI gives these values at constant volume and 0°K relative to the m ost stable structure. From experim ent  it is known that Na undergoes partial transition to the hep structure below 35°K. The co r re c t structure is th erefore obtained fo r Na, Mg and A l. A lso, the computed axial ratios are close to those observed. To investigate the m icro sco p ic stability , we calculated the phonon dispersion cu rves of each m etal in all three stru ctu res. Some resu lts are listed in Table II. We have found that in all ca s e s, except b c c A l, u2(a, 3) > 0 so that the m icr o sc o p ic stability condition is ful fille d . Judging from our lim ited resu lts, it is unlikely that the m artensitic transform ation of Na could be caused by instability of phonons. Our c a l culations indicate that the lowest transverse branch is higher in the fee and hep than in the b cc phase. This is consistent with a tem peratureinduced phase transition. TABLE VI. RELATIVE ENERGIES OF THE CRYSTAL STRUCTURES IN UNITS OF 1 0 is erg PER ION Na
1 3 .9
1 .6 3 3
1 .6 3 3
1 .6 3 3
'/ " 'e x p
1 .6 3 4
1 .6 2 2
SCHNEIDER and STOLL
FIG. 7. Calculated valence-charge density on a  plane in Na. electron model, p (R) in units of
p is the charge density of the free
FIG.8. Calculated valence-charge density on a  plane in Al. p is the charge density of the free electron model. p%)inunitsof7S"^.
We w ill now consider one further property which sheds some light on the m etallic binding. Since we have computed the valence wave func tions to first ord er, we may also obtain the v a len ce-electron charge densi ty fo r any crysta l structure of interest. Figure 7 shows the calculated charge density on a  plane in Na and F ig. 8 the charge density on a  plane in A l. F or Na we find a nearly uniform density between the ions, whereas A l shows m oderate non-uniform ity. This partially explains the deviations between the experim ental and calculated lattice energies.
The variational expression fo r the ele ctrica l resistivity obtained, using the sim plest trial function, is given in the first Born approximation by R ef.  : P = 2 f i k B T e ^ k '^ J 7 Y
Q i I < -P? I U , I !' i ^ e -й"