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Calculation of the magnetization and total energy of vanadium as a function of lattice parameter
J. Phys. Chem. Solids, 1973,Vol.34, pp. 1627-1638. PergamonPress. Printedin GreatBritain
CALCULATION OF THE MAGNETIZATION AND TOTAL ENERGY OF VANADIUM AS A FUNCTION OF LATTICE PARAMETER*'I T. M. HATTOX~, J. B. CONKLIN, JR., J. C. SLATER and S. B. TRICKEY Quantum Theory Project, Department of Physics arid Astronomy, University of Florida, Gainesville, Florida 32601, U.S.A. (Received 14 August 1972)
Abstract-Self-consistent spin-polarized APW calculations have been performed to determine the energy band structure of metallic vanadium in an assumed ferromagnetic b.c,c, structure as a function of lattice parameter. The statistical exchange ('X~') and muffin-tin approximations were used. At each lattice parameter for which a calculation was performed, the X~ cohesive energy, the pressure, and the magnetization were calculated. The calculated cohesive energy and pressure agree faMy well with experiment. The calculations also correctly predict the absence of a magnetic moment for vanadium at its equilibrium lattice constant. However, a nonmagnetic-to-magnetic transition is found to occur abruptly at a lattice constant which is about a fact9r of 1.25 larger than the equilibrium value, and which is in good qualitative agreement with the appearance of a local magnetic moment in certain vanadium alloys.
1. INTRODUCTION METALLIC vanadium has the bcc crystal structure and is known experimentally to be nonmagnetic. However, the isolated vanadium atom in its ground state has a permanent magnetic moment of 3.0 electrons/atom. Thus, the solid would be expected to transform to a magnetic state at some (hypothetical) lattice constant larger than that found experimentally. The last several years have seen extensive developments in the one-electron theory of solids based on the statistical exchange or 'Xod method[l, 2]. These developments have made it possible [3,4] to obtain at least semiquantitative values of the cohesive energy, pressure, compressibility and magnetization of a solid using methods which are not significantly more complicated than those used simply to obtain energy bands.
It is a property of the Xc~ method that as the lattice spacing of a monatomic crystal is increased to infinity, the problem goes smoothly into that of the ground state of the separated atoms[I,2]. Connolly[5] and others[6,7] have shown that the spin-polarized Xc~-APW (Augmented Plane Wave) method gives an adequate description of the magnetic properties of the transition metals. Thus a set of these spin-polarized APW calctdations at various lattice parameters can be expected to show the transition in vanadium from a non-magnetic state at its equilibrium lattice constant to the magnetic state it must assume at a suitably expanded lattice. With increasing lattice parameter, the energy bands for a hypothetical ordered magnetic arrangement must narrow and separate according to spin type (o~ or/3) as the discrete spin-polarized energy levels of the atomic ground state are approached. The details of this process, and the relation of the energy-band theory to the use of isolated magnetic atoms and the Heisenberg exchange integral used to describe their magnetic interactions, are discussed in detail in a forthcoming book by one of the authors[8]. In
*Supported in part by the National Science Foundation. tBased in part on the dissertation presented by T. M. Hattox to the Dept. of Physics and Astronomy, University of Florida, in partial fulfillment of the requirements of the Ph.D. :~Present Address: Computer Sciences Corp., 8728 Colesville Rd., Silver Spring, Md 20910, U.S.A. 1627
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T . M . H A T T O X et al.
terms of the energy band theory of magnetism, the 3d bands in vanadium must become narrower and narrower as the lattice spacing increases, with the eventual production of a high density of states at the Fermi energy, a condition required for a stable magnetic state [9]. Several of the transition metals near vanadium in the periodic table have magnetic properties at their equilibrium lattice parameters (iron, cobalt and nickel are ferromagnetic; chromium is weakly anti-ferromagnetic). It was hoped that vanadium would undergo a transition to the magnetic state with only a small increase in lattice parameter beyond equilibrium. However, as discussed in Section 3, the magnetic transition was found to occur at an expanded lattice parameter of about 1.25 a0, corresponding to a negative pressure of approximately 240kbar and a vanadium-vanadium nearest neighbor distance of 5.85 a.u. Although this situation is no doubt experimentally unattainable, there exist vanadium compounds in which the vanadium atoms in the solid are separated by distances larger than the calculated transition distance in metallic vanadium, a topic to which we shall return in Section 5. From our calculation of ground state properties we find an equilibrium lattice constant of 5.568 a.u., within 2 per cent of experiment, a cohesive energy o f - 0 ' 3 3 Ryd. (relative to the atomic 3d44s ~ configuration which is the Xa atomic limit) as compared to the experimental value[10] o f - 0 . 3 9 R y d , and an isothermal compressibility of 4.46x10-7cm2/kg vs a 30~ experimental value of 6.05 x 10-7 cm2/ kg[11]. Details of these results are presented in Section 4.
2. SUMMARY OF THE METHOD
The fundamental assumption of the Xa method[l,2] is that the system total energy may be written as
(Ex~) = (Tx~) + (Wx~)
(la)
where
(Tx~) = - ~ n, f
,,*(1)v,~,,(~) dr,
(lb)
(W~.)=~ f p(1)g,~dr~+ l ~, ' g,,~ Izv
+89f p(1)p(2)g,2drldr.z +~1 f [otmUx~r(1)+p+ mUx, ,(1)]
dr~ (lc)
in which the spin-up and spin-down charge densities are
p~(l)
= ~
i1'
n,~u,*~(l)uiz(1)
p+ (1) = Y~ ni+ui](1)u~+(1),
(2a) (2b)
the total charge density is simply p(1) = p t ( l ) + p + ( 1 ) ,
(3)
Greek indices refer to nuclei and Roman to electrons, and the g's are the usual coulomb operators. In (1) and (2) ui(1) is the i th spinorbital at point r~, ni is the occupation number of that orbital, and the local exchange-correlation potentials are Ux~r(l) = -9a[(3/47r)p~ (1)] v3
(4a)
=-9a[(3/47r)p+(1)] v3
(4b)
Ux~(1)
with a a parameter such that 2/3 ~< a ~< 1. The first term in (lc) gives the electron-nucleus interaction energy, the second gives the internuclear repulsion, the third gives the Coulomb interaction among the electrons (including self-energy), and the fourth the exchange-correlation energy. The application of the ordinary variation principle to (Ex~) gives[I,2], in the familiar way, the effective Schr6dinger equation for a spin-up (a) orbital [ - 7 , 2 + ~] g ~ + f 0(2)g,2 dr2+
Vx,~,(1)]u,~ (1) = e~,u;t (1)
(5)
MAGNETIZATION
where Vx~t (1) = ,]Ux~$ (1)
(6)
and similarly for a spin-down (fl) orbital. A valuable property of the X a method is the satisfaction of the virial theorem[12]. Thus, for any lattice spacing in the solid, we may obtain the pressure directly from the terms in the total energy (1), since
P V = [2(Tx~)+(Wx~)]/3.
(7)
At times it is convenient instead to calculate the pressure by means of the ordinary thermodynamic derivative of the energy,
P=
-O(Ex~)IaV.
(8)
Averill[3] has found that there is acceptable agreement numerically between pressures computed via (7) and those found with (9). T h e choice of the parameter a is a subject which has stimulated many lengthy and detailed investigations and much discussion in the literature[I, 2]. Rather than recapitulate that discussion here, we simply present our method of selection of ~ and summarize our motivations. Berrondo and Goscinski [13] first suggested that ~ be chosen to make the virial coefficient "0 = - ( WHF)/2(THE) equal to unity in an atomic calculation. H e r e Tin, and WHF are the H a r t r e e - F o c k total kinetic and potential energy operators and the expectation values in ~ are computed using the X a orbitals. Such a choice is conventionally denoted t~t and has the property that it automatically guarantees the equality of (Ex~) for the atom with the expectation value E~v of the original many-electron Hamiltonian with respect to a single determinant of X a orbitals. Schwarz[14] has determined art for the atoms H through Nb. On the basis of his and similar calculations[15] we have adopted the value a~t = 0.7150 as one which gives a good description of the isolated vanadium atom. 3. MAGNETIC PROPERTIES AND ENERGY BANDS
T w o characteristics have been suggested [9, 16] as necessary for the occurrence of elec-
JPCS Vol. 34, No 10-C
OF VANADIUM
1629
tronic ferromagnetism in a crystalline system. T h e s e are: (1) That an atom of the crystal possess a partially filled inner shell of electrons which overlaps very little with similar shells in the neighboring atoms, so that these shells take little part in the cohesion; and (2) T h a t there be a high density of states at the Fermi energy, with the energy bands of the partially filled states being correspondingly narrow. T h e s e conditions are met for iron, cobalt and nickel at their equilibrium lattice parameters, and they should be obtained, as we remarked in the Introduction, at some suitably expanded lattice parameter for the other transition metals having partially filled atomic 3d levels. F r o m the point of view of the energy bands of a magnetic material, the picture is as follows[17]: T h e exchange potentials for c~and B-spin electrons are different, so that the energy bands for the c~-spin (majority) electrons lie lower than those for the /3-spin electrons by the amount of the exchange effect. Both bands are filled to a common Fermi level; and since the c~-spin bands are lower, they will contain more electrons, producing a net spin density. N e w potentials, which also differ in their c~- and B-spin exchange terms, are then generated from these bands; the calculation can be iterated until a stable magnetization is attained. Self-consistent calculations of this type have been perf o r m e d [ 5 - 7 ] , using the A P W and K K R ( K o r r i n g a - K o h n - R o s t o k e r ) methods, with considerable success. T h e electron energies in a crystal with local magnetic moments depend only very slightly on the relative orientation of these magnetic moments, whether it be a random disorientation (as in the c o m m o n paramagnetic state above the Curie temperature) or an ordered magnetic arrangement. This is because the local magnetic moment is caused primarily by the intra-atomic exchange interaction, rather than the inter-atomic interaction (which is much weaker, as is reflected in the fact that the energies involved in the Curie or Ndel Transitions are very small). It would be un-
1630
T.M. HATTOX et al.
duly optimistic to hope that these orientational energy differences could be calculated accurately enough with presently available techniques to determine the stable configuration of the magnetic moments. F o r all the vanadium calculations presented in this paper, an assumed ferromagnetic a r r a n g e m e n t was chosen for convenience. N o attempt was made to perform calculations using an antiferromagnetic or other arrangement because: (1) the calculated magnetization and other properties should all be quite similar for any choice of orientation and (2) the differences in the numerical errors b e t w e e n calculations for two different magnetic structures would almost surely mask the v e r y small differences in the total energy b e t w e e n the two orientations. T h e magnetic m o m e n t / x was calculated as as the difference between the n u m b e r o f a - and #-spin electrons per unit cell, as determined f r o m the numerically c o m p u t e d weighing factors* described in the Appendix, T h e calculated, self-consistent magnetic m o m e n t t of solid vanadium is shown in Fig. 1 as a function o f the lattice parameter. It is zero for lattice p a r a m e t e r s a ~< 6.3 a.u., in agreement with the experimentally determined absence of a magnetic m o m e n t for vanadium at its equilibrium lattice constant, a0. A b o v e a = 6-3 a.u., the magnetic m o m e n t rises rapidly f r o m zero to a value of 2.99 electrons per atom at a - - - 7 . 3 a.u., then slowly a p p r o a c h e s a computational s e p a r a t e d - a t o m limit of 5.0 electrons/atom. T h i s latter value is that for atomic v a n a d i u m in the (3d 1' )4( 4S '~ )1 configuration, which is the spin-polarized X a ground state for the atom, and hence the A P W s e p a r a t e d - a t o m limit. It
*In order to obtain a convergent sequence of computed magnetic moments, it is imperative that the spin-polarized charge density for the upper valence states be constructed (at each iteration) on the basis of a much finer sampling grid in the Brillouin zone than is commonly used in selfconsistent APW calculations. tFor ease in differentiation, we report theoretical magnetic moments in electrons/atom and quote experimental values in the (essentially) identical Bohr magneton/xB.
is assumed that multiplet effects, which are of course important for the real material, do not b e c o m e significant until the bands are very narrow, at lattice constants larger than those at which the magnetic transition occurs in the crystal. Such multiplet effects are not included in the present model but h a v e been estimated in the atomic limit, for the calculation of cohesive energy. (See Section 4 for further discussion.) A peculiarity discovered at a = 7.0 a.u. is the existence of two distinct self-consistent solutions, leading to the two values of magnetic m o m e n t shown in Fig. 1 for that lattice constant. (See also T a b l e 1.) T h e s e two solutions were obtained by using different starting potentials, one corresponding to an assumed initial magnetic m o m e n t of 3.8 electrons/atom and the other to a smaller initial magnetic m o m e n t of 1.0electrons/atom. In order to obtain evidence that the c o n v e r g e d solution does not depend on the starting potential at other values of the lattice p a r a m e t e r , two complete self-consistent calculations were c a r d e d out at a = 8-5 a.u., using two different starting potentials corresponding, respectively, to the two initial magnetic m o m e n t s used for a -7.0 a.u.; both did indeed c o n v e r g e to the same self-consistent band structure, which corresponds to the magnetic m o m e n t shown for a - 8.5 a.u. Also, at a --- 6.3 a.u., the calculation was initiated with the larger magnetic m o m e n t (3.8 electrons/atom), but it finally converged to a solution having zero magnetic moment. It is, of course, impractical to perform calculations using several different starting potentials for e v e r y value of lattice parameter, especially since the calculations just mentioned indicate that the solutions at a - 6.3 a.u. and a - - 8 . 5 a.u. are unique. In addition, the total energies for the calculations at a = 7.3 a.u. and a = 7.75 a.u. are apparently on the curve of lowest total energy (see Section 4). T h e two solutions at a----7.0a.u, are evidently the result of a double minimum in the total energy vs magnetization curve in the vicinity of this lattice constant. T h e state cor-
MAGNETIZATION OF VANADIUM
1631
5"0-
E
4.0,
~
3"0
2-0
,.0
/,'[~ i I 9
o
5;0
9
o
Oo 6:0
o .r
! !
J i
7.0
s'.o
9:o
o.u.
Fig. 1. The calculated magnetic moment of vanadium metal as a function of lattice parameter. The two points plotted for a = 7.0 a.u. correspond to two distinct self-consistent solutions (see text).
Table 1. The cohesive energy, pressure, enthalpy, and magnetic moment of vanadium at various lattice parameters
Lattice constant (a.u.)
Cohesive en ergy.,[a] (Ry/atom)
Pressure (virial theorem) tha (kbars)
Enthalpyta~ (Ry/atom)
5.25 5.5 5-7225 6.3 7.0 7.0 7-3 7.75 8.5
0.31 1 0-329 0.326 0-259 0' 152 0.167 0.155 0.125 0.083
+518 + 53 -- 149 -366 --299 --246 - 187 --154 - 97
--0.057 --0.300 --0.420 --0.570 --0.500 --0.454 --0.402 --0.369 --0.285
Magnetic moment (electrons/atom) --0 0 1.06"7 2.22-~ 2.99 3.34 3.78
t~The cohesive energy and enthalpy are given relative to the spin-polarized Xc~ total energy of the vanadium atom in the (3d 1' )4(4s 1' )1 configuration. tb~Note that the pressure values given herein for lattice constants a ~ 7.0 a,u. are only semi-quantitative, as discussed in Section 4. r e s p o n d i n g to the larger m a g n e t i c m o m e n t (/x = 2.22 e l e c t r o n s per a t o m ) has a l o w e r total e n e r g y t h a n the o t h e r state (/~ = 1-06 e l e c t r o n s per a t o m ) at a = 7.0 a.u. H e n c e t h e m a g n e t i z a t i o n c u r v e is d r a w n t h r o u g h the p o i n t / z = 2.22 e l e c t r o n s per a t o m at this lattice c o n s t a n t . T h e v a n a d i u m e n e r g y b a n d s in t w o s y m m e t r y d i r e c t i o n s are c o m p a r e d in Fig. 2 for the e q u i l i b r i u m lattice c o n s t a n t (a = 5-7225 a.u.) a n d the first lattice c o n s t a n t (a = 7,0 a.u.) for w h i c h the b a n d s e x h i b i t s p i n - p o l a r i z a t i o n . T h e
similarity b e t w e e n the n o n - s p i n - p o l a r i z e d b a n d s a n d the s p i n - p o l a r i z e d b a n d s of each s p i n - t y p e c a n easily b e s e e n f r o m this figure. A l s o , the figure s h o w s that, for the spinp o l a r i z e d case, the 3d b a n d s h a v e n a r r o w e d in the v i c i n i t y of the F e r m i e n e r g y to p r o d u c e the high d e n s i t y of states r e q u i r e d for a s t a b l e m a g n e t i c state. W i t h i n c r e a s i n g lattice para m e t e r , the b a n d s b e c o m e n a r r o w e r a n d the splitting b e t w e e n the ~- a n d B-spin b a n d s bec o m e s greater. T h u s , a n i n c r e a s i n g p o r t i o n of
1632
T.M.
H A T ' F O X et al.
a-spin /S-spin - - -
02 I
o.i 4 ~
p
/i
" 2 25. ee
-02-
I
J/
tlJ -0"3
-0"4 -0'5-
N
?=57225a.u. I
o= 7.0 a.u.
H
F
Fig. 2. Vanadium energy bands for a non-magnetic lattice constant (a = 5-7225 a.u.) and a magnetiC one (a = 7.0 a.u., ~ = 2.22 electrons/atom).
the a-spin band and a decreasing fraction of the/3-spin band lie below the common Fermi energy as the lattice parameter increases, resulting in the increase of the magnetic moment. In Fig. 3, the s and d band widths for a- and /3-spin electrons are shown as a function of lattice parameter. The d bands are centered around the Fermi energy and are imbedded in
""-.. ~Top
ofs
0.6 04. Ry o.2.
"~'~-~-Top o, a
H',,e
o
",,,--:5::L:' 3a*
...- ..,
""=4s ~
-0.2, -0,4 -0,6Psoloted atom 5~0
ao 6.0
7'.0
8.'0
oo
9~0 O.u.
Fig. 3. Energy levels in vanadium as a function of lattice parameter. Atomic energy levels from a spin-polarized X u calculation for the (3d 1' ) ' ( 4 s ~' )1 configuration.
the broader s bands. The bottoms and tops of the s and d bands are defined here by the selfconsistent eigenvalues for the states T1, H 1 , H12 and H'z~ respectively, as indicated in the figure. The top of the s band, shown as a dashed curve in Fig. 3, was estimated from the appropriate high lying states which were calculated in early iterations for some of the lattice parameters. Figure 3 clearly shows the narrowing of the bands as the lattice parameter is increased. Magnetization begins just below a = 7.0a.u., where each non-spin-polarized state splits into a- and /3-spin states of the same symmetry. The magnetic moment at a given lattice parameter can be estimated crudely using Fig. 3, from the relative portions of the a- and B-spin bands below the Fermi energy. It is seen that the magnetic moment increases quite rapidly from the onset of spinpolarization up to a=7.45a.u., where the H12/3 curve crosses the Fermi level. For larger lattice constants, there are no occupied 3d/3 states (at least as indicated in this figure), and the magnetic moment increases rather slowly with increasing lattice parameter. The in-
MAGNETIZATION
crease in magnetic moment in this region is primarily due to the narrowing of the s bands, which are only weakly split in comparison with their overall width. This description of the magnetic moment in terms of the band splittings and widths corresponds closely to the magnetization curve in Fig. 1. Four selected electronic density of states curves, calculated as described in the Appendix, are presented in Figs. 4-7. The dashed portion of those curves, which always begins above the Fermi energy, indicates that not all the one-electron eigenvalues in that region were calculated. Comparison of these curves illustrates their compression in energy and corresponding increase in height as the lattice is expanded, with relatively small change in shape other than this effective rescaling. (Note that the first two curves are for non-spinpolarized calculations and hence are densities of states of both spins, whereas each curve in Figs. 6 and 7 is the density of states for only one spin.) The Fermi energy is seen to remain at about the same position relative to the structure of the density-of-states curves, for the non-magnetic structures (Figs. 4 and 5), and the density of states at the Fermi energy increases because of the increasing height of the density-of-states curves as the lattice expands, in the non-magnetic regime. For the
40I
=
o
o
95'25 o.u.
30-
g ~
//
20-
w
I0-
\
OF VANADIUM
1633
magnetic density-of-states curves, the /3spin curve is seen to be shifted up in energy with respect to the a-spin curve, so that the common Fermi level does not come at the 70.
60"
50"
a = 6"5
I ~
40-
g ~
30.
_~ 20. U.I
I0"
0
, -0.30
E
-0.15
i
F
0.15
Ry
Fig 5. V a n a d i u m density of states, a = 6.3 a.u. IOO
90 80 70 60 50 40 30 ~
2o
L
io
--
I0
~
20
~
30
( ~ - s p i ~
~.40
/
/
9
~ 5o ~
W
6o 70 8O
a-7.0o.u.
9O
-0.45
-0"30
-0.15
i
E
0.15
Ry Fig. 4. Vanadium density of states, a = 5.25 a.u.
I00
-0~'36 -0~24 -0'q2
E,,-
C;'12 0~'24
Ry F i g . 6. V a n a d i u m
d e n s i t y o f s t a t e s , a = 7 . 0 a . u . (pt = 2 . 2 2 electrons/atom).
1634
T.M.
HATTOX
I10I0090807060. 50" n." 4030~ i -~
cx-spin
20I00
9~
J
20/3-spin
~2 40,7, 5060708090a = 8"5o.u.
IO0 I10
i
-0.24
-0"12
EF
0.12
0.24
0'~6
Ry Fig. 7. V a n a d i u m d e n s i t y o f s t a t e s , a = 8.5 a.u.
same relative position on the o~- and /i-spin curves. T h e approximately rigid band shifting of these two curves is a measure o f the difference in exchange energy for electrons of majority and minority spins, T h e deviation from rigid-band shifting is an indication of the inaccuracy of simpler approaches which merely superimpose an exchange energy difference on a single spin-independent band structure and density of states. T h e iteration-to-iteration convergence of our calculations was relatively slow both for a = 6.3 a.u. and for a = 7.3 a.u.; it was extremely slow for a = 7.0 a.u. It will be recalled that it was for a = 7.0 a.u. that two, magnetically and energetically distinct, selfconsistent solutions were found. Because of the convergence slowness, additional selfconsistent calculations were not attempted in the region 6.3 < a < 7.3 a.u. F o r this reason the detailed behavior of the magnetic m o m e n t in that region is not known; however, it must
e t al.
certainly be bracketed by the two curves we have displayed in Fig. 1. T h e r e are two features of the electronic structure which very likely contribute to the slow convergence in the transition region. T h e first is that the Fermi level lies very nearly at a peak on the electronic density of states curves for both a- and /i-spin electrons (see Fig. 6), which is not the case for other lattice parameters. This means that small changes in the eigenvalues near the Fermi energy significantly affect the weighting factors for those states, so that the potentials change considerably from iteration to iteration, thus hindering convergence. (The high density of states at the Fermi level for a- and /t-spin electrons also causes the magnetic moment to be very sensitive to small changes in the eigenvalues, so that a high degree of convergence in energy, is required in order to achieve acceptable convergence in the magnetic moment.) A second factor which likely influences the slow convergence is the expected small dependence of the total energy on the magnetic moment in the transition region. T h e total energy curve as a function of magnetic moment should be monotonically increasing with its minimum at zero magnetic moment for a non-magnetic state. F o r a magnetic state, the minimum will occur for some non-zero magnetic moment. Therefore, one might expect the energy dependence upon magnetic moment to be rather weak in the transition region (/x ~ 0), causing the convergence to the magnetic m o m e n t which minimizes the energy to be very slow. Papaconstantopoulos et al.[18] and Yasui et al.[19] have performed self-consistent A P W and O P W energy band calculations, respectively, on V. Neither group of workers investigated the total energy of crystalline vanadium. T h o u g h there are differences in detail, our calculated band structure (Fig. 2) and Fermi surface are in satisfactory agreement with those found by the aforementioned workers. In particular, the band structure and Fermi surface at the normal lattice constant agree quite well with the self-consistent re-
MAGNETIZATION
suits obtained by Yasui et al. using a = 0-725, and fairly well with the self-consistent results obtained by Papaconstantopoulos et al. using a = 2/3. We therefore refer the interested reader to those papers for details of the band structure and Fermi surface, other than those we have already discussed.
OF VANADIUM
1635
energy) branch of the total energy curve. T h e equilibrium lattice constant, determined as the minimum of the total energy curve (or the zero of the pressure curve), was found to occur at a = 5.568 a.u., 2.6 per cent smaller 0-
4. COHESIVE P R O P E R T I E S
We have calculated the total energy and pressure* at each lattice parameter for which a self-consistent band calculation was performed, with the results given in Table 1. T h e total energy as a function o f lattice parameter is plotted in Fig. 8, relative to the spinpolarized X a ( S P X a ) total energy of the vanadium atom in the (3d~')4(4s~' )1 configuration, which is the spin-polarized A P W separated-atom limit (see discussion below). This curve is drawn through the calculated points such that its slope at each of these points corresponds to the calculated pressure for the corresponding point. T h e resulting kink in the curve comes just at the onset of magnetization (recall Section 3). T h e pressure is plotted as a function of lattice parameter in Fig. 9, where the dashed portion of this curve is drawn to match the slope along the kind in Fig. 8. T h e additional point plotted for a = 7.0 a.u. on Figs. 8 and 9 corresponds to the second self-consistent solution of higher total energy at that lattice constant, which was discussed in Section 3. T h e pressure from Fig. 9 agrees well with the slope indicated in Fig. 8 for that additional point, providing further evidence that the point represents a self-consistent solution for a second (higher *The pressure is obtained from the virial theorem, equation (7), applied to the kinetic and potential energy contributions calculated separately in the final self-consistency iteration for each lattice constant. Its error is a first-order quantity in the error of the w a v e function (as contrasted to that of the total energy), so the convergence of the computed pressures is considerably poorer t h a n that of the other quantities reported, particularly for the spin-polarized calculations (a /> 7.0 a.u.) for which the pressure values given in Table 1 and Fig. 9 should be considered only semiquantitative.
-0.I z =1.06- . ~ ' ~
oE P
-0"2
fie - 0"3
Experimental
-0.4
5:0
bo o'o
7!o
6'o
9'.00.U
Fig. 8. T h e total energy of v a n a d i u m as a function of lattice parameter, relative to the SPXc~ total energy of the the (3d 1' )4(4s T )1 configuration.
600" 500400300200100-
-I00-
-200-
k~._
-300-
//,"~"- /.t. = 1-06
-400-
510
' Go 610
7.'0
810
910 O.U.
Fig. 9. T h e pressure of v a n a d i u m as a function of lattice parameter, as obtained from the virial theorem.
1636
T.M. HATTOX et al.
than the experimental value[20] (at 25~ of a = 5.715 a.u. T h e cohesive energy was calculated as the difference between the Xo~ total energy per unit cell of the crystal and the X a total energy of the isolated atom in its ground state, using the same value for ot in the crystal and atomic calculations. H o w e v e r , the multiplet structure o f atomic vanadium complicates matters considerably. Experimentally [21] there are lowlying multiplets from both the (3d)3(4s) 2 configuration and the (3d)4(4s) 1 configuration. T h e ground state is the 4F multiplet of the (3d)4(4s) 2 configuration, but the energy of the 6D multiplet of the ( 3 d) 4 (4s) 1 configuration is only 0 . 0 1 8 R y a b o v e the ground state energy. In contrast, S P X a atomic calculations (with a = 0.715) for which the multiplet splittings are taken into account, place this 6/9 multiplet lower in energy than the 4F multiplet b y 0.124 Ry. U s e of the S P X a total energy of the ~D multiplet as the ground state energy of the isolated atom leads to a calculated cohesive energy of 0.33 Ry, while use o f the S P X ~ total energy of the 4F multiplet leads to a cohesive energy of 0.45 Ry. T h u s the experimental cohesive energy[10] of 0.39 Ry falls half w a y between the cohesive energy obtained using the S P X a atomic ground state total energy and the SPXot total energy of the experimental atomic ground state.* T h e isothermal compressibility is defined as K0 = - V -1 (av/a/') r
(9)
which with (8) gives, for the b.c.c, structure,
*Since the 6/9 multiplet is the only sextet for the configuration (3d)4(4s) 1, it must be the only multiplet corresponding to the configuration (3d t )4(4s 1' )~, and the SPXa total energy for this configuration is -1885.867 Ry. However, there are two quartet states (4F and 4p) which correspond to the configuration (3d1')3(4s 1' )1(4s $ )1, so that the S PXa total energy for this configuration represents a weighted average energy of the two multiplets. The energies of these two multiplets relative to their average energy may be computed straightforwardly[21] with a result of-1885.743 Ry for the energy of the 4F multiplet.
K o = ( q / 2 ) a [ O 2 E / O a Z ] -1
(10)
where a is, as usual, the lattice parameter. T h e second derivative in equation (10) was evaluated for a parabola fitted to the three points (a = 5-25, 5.5, 5.7225 a.u.) on the total energy curve (Fig. 8) nearest its minimum. T h e resulting calculated value for the zero-pressure isothermal compressibility is K = 4.46 • l0 -7 cm2/Kg, as c o m p a r e d to the experimental value [ 1 1] (at 30~ o f K = 6-05 • 10-7 cm2/Kg. 5. DISCUSSION T h e magnetic transition we have found in the present calculation occurs, as expected, at a lattice p a r a m e t e r which is outside the range of experimentally attainable values. H o w e v e r , there are alloys of vanadium with other metals which display an interesting variety of magnetic properties[23,24]. While it would be unwarranted to suggest that our calculations of the properties of crystalline vanadium are intrinsically applicable to these alloys, some cautious simple c o m p a r i s o n s can be made. First, the usual interpretation of the experimental evidence at present is that the vanadium m o m e n t is localized in ordered ferromagnetic Au4V with a magnitude near 1 Bohr magneton, as opposed to an essentially vanishing m o m e n t in disordered Au4V. Further, a m o m e n t of about 0.32 Bohr m a g n e t o n has been m e a s u r e d in AI3V (which, however, is not ferromagnetic). N o w , if we order these c o m p o u n d s by the v a n a d i u m - v a n a d i u m distance dvv we find: (1) vanadium metal, dvv = 4.98 a.u., zero m o m e n t ; (2) disordered Au4V, dv~ --- 5.38 a.u., zero moment; (3) A13V, d~v = 7.19 a.u., 0.32 izB; (4) ordered Au4V, dv~ = 7.57 a.u., 1 tz~. It would a p p e a r from these data that the onset of a local vanadium m o m e n t in a solid s y s t e m requires 5.38 a.u.-,_< d~,, ~ 7.19 a.u. As noted in Section 3, we were unable to define precisely the threshold value of d ~ for the beginning of the transition to a magnetized state in the expanded metal. Reference to Fig. 1 shows that a plausible (and convenient) value to assign to this calculated critical separation
MAGNETIZATION OF VANADIUM is d~,~.~- 5.85 a.u. ( = V3 a/2, with a ~ 6-75 a.u.). O u r calculations also clearly yield a zero m o m e n t for d~,,, <~ 5-45 a.u. T h e s e results are in good qualitative a g r e e m e n t with the experimental evidence we have just summarized. We regard this agreement as suggestive but not compelling evidence for the validity of the notion that a prime requisite for magnetization in v a n a d i u m alloys is a V - V separation greater than some minimum distance[25]. It may be that the magnetic properties [26] of the ferrimagnetic garnets Ca~Bi~-~,xFes-xVxOlz also owe their existence, at least in part, to the kind of b e h a v i o r we h a v e found for vanadium. T h e features of the calculated E ( a ) and P ( a ) curves (see Figs. 8 and 9) n e a r a = 7.1 a.u. (which we associate with the magnetic transition) are worthy of s o m e tentative comment in connection with the magnetic vanadium alloys. T h e y o c c u r for a d~v corresponding to physically unrealizable negative pressures for vanadium metal. H o w e v e r , the pressures needed to attain the same d ~ in the magnetic v a n a d i u m intermetallic c o m p o u n d s are positive (or zero). I f the E ( a ) and P ( a ) curves in these alloys were to exhibit features associated with the magnetic transitions similar to those we have c o m p u t e d for metallic vanadium, then the rapid change of slope in E ( a ) and attendant peak in P ( a ) would occur at some physically realizable pressure. T h e s e features of E ( a ) and P ( a ) would correspond, in such a case, to an isomorphic phase transition in which the onset o f magnetization is a c c o m p a n i e d by a discontinuous change in lattice p a r a m e t e r (unless some polymorphic transition, which this calculation forbade by construction, were to dominate). Obviously there are m a n y subtle complexities in these systems which we have not mentioned, much the less treated, and our c o m m e n t s are not in any m a n n e r intended to c o n v e y a sense of conclusiveness. On the basis of the present w o r k it would seem that spin-polarized APW-Xo~ calculations can yield meaningful results for the total energy, pressure, and compressibility of
1637
transition metals. W e believe that the spinpolarized method should be used for both the solid and the isolated atoms, when applicable, in determining the cohesive energy, because the differences b e t w e e n the spin-polarized and non-spin polarized total energies (in the atom or the solid or both) may be large compared to the cohesive energy itself. It is also necessary to consider the multiplet structure of the atom. In the particular case of vanadium, for example, the spin-polarized X a total energy of the (3d ~' )4(4S '~ )1 atomic configuration is lower in energy than the non-spinpolarized X a total energy of the (3d)3(4s) ' configuration (averaged o v e r multiplets) by 0.33 Ry, a very significant difference compared to the experimental cohesive energy [10] of 0.39Ry. F u r t h e r m o r e , there is no reason to believe that these energy differences in the a t o m and in the solid will cancel; in fact, there is no spin-polarized lowering of energy in the (non-magnetic) vanadium solid at equilibrium to cancel the quite large lowering of energy in the atom. Acknowledgements-We are grateful to the members of the Qauntum Theory Project of the University of Florida, especially to Prof. J. W. D. Connolly and to Drs. F. W. Averill and K. Schwarz, for many useful discussions. We also gratefully acknowledge support of a portion of the computing costs for the work reported here from the Computing Center of the University of Florida. REFERENCES
1. SLATER J. C. and JOHNSON K. H., Phys. Rev. BS, 844 (1972), and references therein. 2. SLATER J. C., in Advances in Quantum Chemistry (Edited by P.-O. L6wdin), Vol, 6, p. 1. Academic Press, New York (1972), and references therein. 3. AVER1LL F. W., Phys. Rev. B4, 3315 (1971), and references therein. 4. CONKLIN J. B., Jr., AVERILL F. W. and HATTOX T. M.,J. de Phys. 33, C3-213 (1972). 5. CONNOLLY J. W., Phys. Rev. 159, 415 (1967); Int. J. Quant. Chem. IS, 615 (1967); Int. J. Quant. Chem, IIS, 257 (1968). 6. WAKO H S., J. phys. Soc. Japan 20, 1894 (1965), and references therein. 7. DECICCO P. and KITZ A., M.I.T. Quant. Prog. Report No. 63, 15 Jan. 1967, p. 2 (unpublished). 8. SLATER, J. C., The Self-Consistent Field for Molecules and Solids, Vok 4 of the series Quantum
T. M. HATTOX et al.
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9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
22. 23. 24. 25.
26. 27. 28.
29. 30.
31.
Theory of Molecules and Solids. McGraw-Hill, New York (to be published 1973). SLATER J. C., Phys. Rev. 49,537 (1936). G S C H N E I D E R K. A., Jr., in Solid State Physics (Edited by F. Seitz and D. Turnbull), Vol. 16, p. 275. Academic Press, New York (1964). B R I D G M A N P. W., Phys. Rev. 57, 235 (1940); SLATER J. C., Phys. Rev. 57,744 (1940). SLATER J. C.,J. chem. Phys. 57, 2389 (1972). BERRONDO M. and G O S C I N S K I O., Phys. Rev. 184, 10(1969). SCHWARZ K., Phys. Rev. BS, 2466 (1972). HATTOX THOMAS M., Ph.D. Thesis, Univ. of Fla., 1972 (unpublished). SLATER J. C., Phys. Rev. 36, 57 (1930). SLATER J. C.,J. appl. Phys. 39,761 (1968). P A P A C O N S T A N T O P O U L O S D. A., A N D E R SON J. R. and M C C A F F R E Y J. W., Phys. Rev. BS, 1214(1972). YASUI M., H A Y A S H I E. and S H I M I Z U M. J., J. phys. Soc.Japan 29, 1446 (1970). JAMES W. J. and S T A U M A N I S M. E., Z. Phys. Chem. 29, 134(1961). MOORE C. E., Atomic Energy Levels as Derived from the Analyses of Optical Spectra, Vol. 1, p. 291. Circular of the N.B.S. 467 (1949). SLATER J. C., Quantum Theory of Atomic Structure, Vol. 2, Appendix 21. McGraw-Hill, New York (1960). C R E V E L I N G L., Jr. and LUO H. L., Phys. Rev. 176, 614 (1968), and references therein. C R E V E L I N G L., Jr. and LUO H. L., Phys. Lett. 28A, 772 (1969). See G O O D E N O U G H J. B., Magnetism and the Chemical Bond. Wiley, New York (1963), Chapter I., for highly detailed discussion of this problem. G E L L E R S., ESPINOSA G. P., W I L L I A M S H. J., SHERWOOD R. C. and NESBITT E. A., J. appl. Phys. 35,570 (1964). SLATER J. C., Phys. Rev. 51,846 (1937). MATTHEISS L. F., WOOD J. H. and SWlTENDICK A. C., in Methods in Computational Physics (Edited by B. J. Alder, S. Fernback and M. Rotenberg), Vol. 8, p. 63. Academic Press, New York (1968). H E R M A N F. and S K I L L M A N S.,Atomic Structure Calculations. Prentice-Hall, New Jersey (1963). SCHWARZ K. and C O N K L I N J. B., Jr., University of Florida, Quantum Theory. Project Technical Report No. 213, 27 Jan. 1971, and to be published. MATTHEISS L. F., Phys. Rev. 139, A 1893 (1965). APPENDIX
The self-consistent energy bands of metallic vanadium were calculated at eight lattice parameters by solving the one-electron Xc~ equation using the non-relativistic APW method within the framework of the muffin-tin approximation [27]. A general discussion of the A PW method and the numerical details relating to the APW program (of which the program used in this work is a descendant) are given in an excellent article by Mattheiss, Wood and Switendick[28]. The spin-polarized method, using an assumed ferromagnetic arrangement, was used at all
lattice parameters except for the two smallest (a = 5.25 a.u. and a = 5.5 a.u.), t:or these two lattice parameters, it was assumed that a spin-polarized calculation would reduce to the non-spin-polarized result, as is the case for the next two larger lattice parameters (a = 5.7225 a.u. and a = 6.3 a.u.). The spir polarized calculations were initiated with a net magnr moment obtained (except at a = 5.7225a.u.) by superposing spin-polarized Xct atomic potentials which had been calculated for a set of fractional occupation numbers corresponding to various assumed values of the atomic magnetic moment. For a = 5.7225 a.u., the spin-polarized calculation was initiated by artificially inducing a spin-splitting in the nonspin-polarized energy bands at this lattice constant to give a starting magnetic moment of 1.25 electrons/atom. The b.c.c, crystal structure was used, and the muffin-tin sphere radius was chosen to be equal to half the nearest neighbor distance. The value of the parameter c~ was always chosen to be 0.7150, the value determined for atomic vanadium (recall Section 2). The energy eigenvalues and eigenfunctions of the true core levels (1 s, 2s, 2p) were calculated self-consistently in atomic fashion using essentially the Schrrdinger subroutine of the Herman-Skillman atomic program[29]. Since the 3s and 3p bands were far below the valence bands and had practically all of their charge within the APW spheres, they were calculated on a grid of only 16 equally spaced points in the b.c.c. Brillouin zone, as suggested by Schwarz and Conklin[30]. Th6 valence bands were calculated on a grid of 128 equally spaced points in the b.c.c. Brillouin zone. The wave functions calculated for the occupied bands at these 128 points were used to generate the charge densities for self-consistency iterations. However, for states near the Fermi energy, these representative states were not simply weighted by their degeneracies. Instead, additional interpolation of the bands to a grid of 8192 points in the (full) zone was performed using a method somewhat akin to that of Mattheiss [31]. In addition to the common practice of using these interpolated points for the determination of density-ofstates curves, they were also used in this calculation to determine modified weighting factors for the representative states near the Fermi surface, for the calculation of charge density and total energy. In this way, portions of a band which are in fact closer in k-space to a state on the 128-point grid which lies above the Fermi energy (but in that band), are actually represented in the charge-density calculation by the wave functions of that state (which is given the appropriate fractional occupancy as determined from the 8192-point interpolation grid), and the weighting factor of some state(s) below the Fermi energy is correspondingly reduced. This procedure improves convergence for non-spin-polarized calculations and allows use of coarser computational grids than would otherwise be possible; it is absolutely essential for calculation of magnetic moments when the Fermi energy is near a sharp peak in the density of states for one or both spins, because the moments depend on the differences of numbers of electrons with majority and minority spins. All calculations were performed on the HermanSkillman radial mesh [3, 28]. The calculations were carried out on an IBM 360/65 computer, using double precision for the critical sections of all programs.
CALCULATION OF THE MAGNETIZATION AND TOTAL ENERGY OF VANADIUM AS A FUNCTION OF LATTICE PARAMETER*'I T. M. HATTOX~, J. B. CONKLIN, JR., J. C. SLATER and S. B. TRICKEY Quantum Theory Project, Department of Physics arid Astronomy, University of Florida, Gainesville, Florida 32601, U.S.A. (Received 14 August 1972)
Abstract-Self-consistent spin-polarized APW calculations have been performed to determine the energy band structure of metallic vanadium in an assumed ferromagnetic b.c,c, structure as a function of lattice parameter. The statistical exchange ('X~') and muffin-tin approximations were used. At each lattice parameter for which a calculation was performed, the X~ cohesive energy, the pressure, and the magnetization were calculated. The calculated cohesive energy and pressure agree faMy well with experiment. The calculations also correctly predict the absence of a magnetic moment for vanadium at its equilibrium lattice constant. However, a nonmagnetic-to-magnetic transition is found to occur abruptly at a lattice constant which is about a fact9r of 1.25 larger than the equilibrium value, and which is in good qualitative agreement with the appearance of a local magnetic moment in certain vanadium alloys.
1. INTRODUCTION METALLIC vanadium has the bcc crystal structure and is known experimentally to be nonmagnetic. However, the isolated vanadium atom in its ground state has a permanent magnetic moment of 3.0 electrons/atom. Thus, the solid would be expected to transform to a magnetic state at some (hypothetical) lattice constant larger than that found experimentally. The last several years have seen extensive developments in the one-electron theory of solids based on the statistical exchange or 'Xod method[l, 2]. These developments have made it possible [3,4] to obtain at least semiquantitative values of the cohesive energy, pressure, compressibility and magnetization of a solid using methods which are not significantly more complicated than those used simply to obtain energy bands.
It is a property of the Xc~ method that as the lattice spacing of a monatomic crystal is increased to infinity, the problem goes smoothly into that of the ground state of the separated atoms[I,2]. Connolly[5] and others[6,7] have shown that the spin-polarized Xc~-APW (Augmented Plane Wave) method gives an adequate description of the magnetic properties of the transition metals. Thus a set of these spin-polarized APW calctdations at various lattice parameters can be expected to show the transition in vanadium from a non-magnetic state at its equilibrium lattice constant to the magnetic state it must assume at a suitably expanded lattice. With increasing lattice parameter, the energy bands for a hypothetical ordered magnetic arrangement must narrow and separate according to spin type (o~ or/3) as the discrete spin-polarized energy levels of the atomic ground state are approached. The details of this process, and the relation of the energy-band theory to the use of isolated magnetic atoms and the Heisenberg exchange integral used to describe their magnetic interactions, are discussed in detail in a forthcoming book by one of the authors[8]. In
*Supported in part by the National Science Foundation. tBased in part on the dissertation presented by T. M. Hattox to the Dept. of Physics and Astronomy, University of Florida, in partial fulfillment of the requirements of the Ph.D. :~Present Address: Computer Sciences Corp., 8728 Colesville Rd., Silver Spring, Md 20910, U.S.A. 1627
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T . M . H A T T O X et al.
terms of the energy band theory of magnetism, the 3d bands in vanadium must become narrower and narrower as the lattice spacing increases, with the eventual production of a high density of states at the Fermi energy, a condition required for a stable magnetic state [9]. Several of the transition metals near vanadium in the periodic table have magnetic properties at their equilibrium lattice parameters (iron, cobalt and nickel are ferromagnetic; chromium is weakly anti-ferromagnetic). It was hoped that vanadium would undergo a transition to the magnetic state with only a small increase in lattice parameter beyond equilibrium. However, as discussed in Section 3, the magnetic transition was found to occur at an expanded lattice parameter of about 1.25 a0, corresponding to a negative pressure of approximately 240kbar and a vanadium-vanadium nearest neighbor distance of 5.85 a.u. Although this situation is no doubt experimentally unattainable, there exist vanadium compounds in which the vanadium atoms in the solid are separated by distances larger than the calculated transition distance in metallic vanadium, a topic to which we shall return in Section 5. From our calculation of ground state properties we find an equilibrium lattice constant of 5.568 a.u., within 2 per cent of experiment, a cohesive energy o f - 0 ' 3 3 Ryd. (relative to the atomic 3d44s ~ configuration which is the Xa atomic limit) as compared to the experimental value[10] o f - 0 . 3 9 R y d , and an isothermal compressibility of 4.46x10-7cm2/kg vs a 30~ experimental value of 6.05 x 10-7 cm2/ kg[11]. Details of these results are presented in Section 4.
2. SUMMARY OF THE METHOD
The fundamental assumption of the Xa method[l,2] is that the system total energy may be written as
(Ex~) = (Tx~) + (Wx~)
(la)
where
(Tx~) = - ~ n, f
,,*(1)v,~,,(~) dr,
(lb)
(W~.)=~ f p(1)g,~dr~+ l ~, ' g,,~ Izv
+89f p(1)p(2)g,2drldr.z +~1 f [otmUx~r(1)+p+ mUx, ,(1)]
dr~ (lc)
in which the spin-up and spin-down charge densities are
p~(l)
= ~
i1'
n,~u,*~(l)uiz(1)
p+ (1) = Y~ ni+ui](1)u~+(1),
(2a) (2b)
the total charge density is simply p(1) = p t ( l ) + p + ( 1 ) ,
(3)
Greek indices refer to nuclei and Roman to electrons, and the g's are the usual coulomb operators. In (1) and (2) ui(1) is the i th spinorbital at point r~, ni is the occupation number of that orbital, and the local exchange-correlation potentials are Ux~r(l) = -9a[(3/47r)p~ (1)] v3
(4a)
=-9a[(3/47r)p+(1)] v3
(4b)
Ux~(1)
with a a parameter such that 2/3 ~< a ~< 1. The first term in (lc) gives the electron-nucleus interaction energy, the second gives the internuclear repulsion, the third gives the Coulomb interaction among the electrons (including self-energy), and the fourth the exchange-correlation energy. The application of the ordinary variation principle to (Ex~) gives[I,2], in the familiar way, the effective Schr6dinger equation for a spin-up (a) orbital [ - 7 , 2 + ~] g ~ + f 0(2)g,2 dr2+
Vx,~,(1)]u,~ (1) = e~,u;t (1)
(5)
MAGNETIZATION
where Vx~t (1) = ,]Ux~$ (1)
(6)
and similarly for a spin-down (fl) orbital. A valuable property of the X a method is the satisfaction of the virial theorem[12]. Thus, for any lattice spacing in the solid, we may obtain the pressure directly from the terms in the total energy (1), since
P V = [2(Tx~)+(Wx~)]/3.
(7)
At times it is convenient instead to calculate the pressure by means of the ordinary thermodynamic derivative of the energy,
P=
-O(Ex~)IaV.
(8)
Averill[3] has found that there is acceptable agreement numerically between pressures computed via (7) and those found with (9). T h e choice of the parameter a is a subject which has stimulated many lengthy and detailed investigations and much discussion in the literature[I, 2]. Rather than recapitulate that discussion here, we simply present our method of selection of ~ and summarize our motivations. Berrondo and Goscinski [13] first suggested that ~ be chosen to make the virial coefficient "0 = - ( WHF)/2(THE) equal to unity in an atomic calculation. H e r e Tin, and WHF are the H a r t r e e - F o c k total kinetic and potential energy operators and the expectation values in ~ are computed using the X a orbitals. Such a choice is conventionally denoted t~t and has the property that it automatically guarantees the equality of (Ex~) for the atom with the expectation value E~v of the original many-electron Hamiltonian with respect to a single determinant of X a orbitals. Schwarz[14] has determined art for the atoms H through Nb. On the basis of his and similar calculations[15] we have adopted the value a~t = 0.7150 as one which gives a good description of the isolated vanadium atom. 3. MAGNETIC PROPERTIES AND ENERGY BANDS
T w o characteristics have been suggested [9, 16] as necessary for the occurrence of elec-
JPCS Vol. 34, No 10-C
OF VANADIUM
1629
tronic ferromagnetism in a crystalline system. T h e s e are: (1) That an atom of the crystal possess a partially filled inner shell of electrons which overlaps very little with similar shells in the neighboring atoms, so that these shells take little part in the cohesion; and (2) T h a t there be a high density of states at the Fermi energy, with the energy bands of the partially filled states being correspondingly narrow. T h e s e conditions are met for iron, cobalt and nickel at their equilibrium lattice parameters, and they should be obtained, as we remarked in the Introduction, at some suitably expanded lattice parameter for the other transition metals having partially filled atomic 3d levels. F r o m the point of view of the energy bands of a magnetic material, the picture is as follows[17]: T h e exchange potentials for c~and B-spin electrons are different, so that the energy bands for the c~-spin (majority) electrons lie lower than those for the /3-spin electrons by the amount of the exchange effect. Both bands are filled to a common Fermi level; and since the c~-spin bands are lower, they will contain more electrons, producing a net spin density. N e w potentials, which also differ in their c~- and B-spin exchange terms, are then generated from these bands; the calculation can be iterated until a stable magnetization is attained. Self-consistent calculations of this type have been perf o r m e d [ 5 - 7 ] , using the A P W and K K R ( K o r r i n g a - K o h n - R o s t o k e r ) methods, with considerable success. T h e electron energies in a crystal with local magnetic moments depend only very slightly on the relative orientation of these magnetic moments, whether it be a random disorientation (as in the c o m m o n paramagnetic state above the Curie temperature) or an ordered magnetic arrangement. This is because the local magnetic moment is caused primarily by the intra-atomic exchange interaction, rather than the inter-atomic interaction (which is much weaker, as is reflected in the fact that the energies involved in the Curie or Ndel Transitions are very small). It would be un-
1630
T.M. HATTOX et al.
duly optimistic to hope that these orientational energy differences could be calculated accurately enough with presently available techniques to determine the stable configuration of the magnetic moments. F o r all the vanadium calculations presented in this paper, an assumed ferromagnetic a r r a n g e m e n t was chosen for convenience. N o attempt was made to perform calculations using an antiferromagnetic or other arrangement because: (1) the calculated magnetization and other properties should all be quite similar for any choice of orientation and (2) the differences in the numerical errors b e t w e e n calculations for two different magnetic structures would almost surely mask the v e r y small differences in the total energy b e t w e e n the two orientations. T h e magnetic m o m e n t / x was calculated as as the difference between the n u m b e r o f a - and #-spin electrons per unit cell, as determined f r o m the numerically c o m p u t e d weighing factors* described in the Appendix, T h e calculated, self-consistent magnetic m o m e n t t of solid vanadium is shown in Fig. 1 as a function o f the lattice parameter. It is zero for lattice p a r a m e t e r s a ~< 6.3 a.u., in agreement with the experimentally determined absence of a magnetic m o m e n t for vanadium at its equilibrium lattice constant, a0. A b o v e a = 6-3 a.u., the magnetic m o m e n t rises rapidly f r o m zero to a value of 2.99 electrons per atom at a - - - 7 . 3 a.u., then slowly a p p r o a c h e s a computational s e p a r a t e d - a t o m limit of 5.0 electrons/atom. T h i s latter value is that for atomic v a n a d i u m in the (3d 1' )4( 4S '~ )1 configuration, which is the spin-polarized X a ground state for the atom, and hence the A P W s e p a r a t e d - a t o m limit. It
*In order to obtain a convergent sequence of computed magnetic moments, it is imperative that the spin-polarized charge density for the upper valence states be constructed (at each iteration) on the basis of a much finer sampling grid in the Brillouin zone than is commonly used in selfconsistent APW calculations. tFor ease in differentiation, we report theoretical magnetic moments in electrons/atom and quote experimental values in the (essentially) identical Bohr magneton/xB.
is assumed that multiplet effects, which are of course important for the real material, do not b e c o m e significant until the bands are very narrow, at lattice constants larger than those at which the magnetic transition occurs in the crystal. Such multiplet effects are not included in the present model but h a v e been estimated in the atomic limit, for the calculation of cohesive energy. (See Section 4 for further discussion.) A peculiarity discovered at a = 7.0 a.u. is the existence of two distinct self-consistent solutions, leading to the two values of magnetic m o m e n t shown in Fig. 1 for that lattice constant. (See also T a b l e 1.) T h e s e two solutions were obtained by using different starting potentials, one corresponding to an assumed initial magnetic m o m e n t of 3.8 electrons/atom and the other to a smaller initial magnetic m o m e n t of 1.0electrons/atom. In order to obtain evidence that the c o n v e r g e d solution does not depend on the starting potential at other values of the lattice p a r a m e t e r , two complete self-consistent calculations were c a r d e d out at a = 8-5 a.u., using two different starting potentials corresponding, respectively, to the two initial magnetic m o m e n t s used for a -7.0 a.u.; both did indeed c o n v e r g e to the same self-consistent band structure, which corresponds to the magnetic m o m e n t shown for a - 8.5 a.u. Also, at a --- 6.3 a.u., the calculation was initiated with the larger magnetic m o m e n t (3.8 electrons/atom), but it finally converged to a solution having zero magnetic moment. It is, of course, impractical to perform calculations using several different starting potentials for e v e r y value of lattice parameter, especially since the calculations just mentioned indicate that the solutions at a - 6.3 a.u. and a - - 8 . 5 a.u. are unique. In addition, the total energies for the calculations at a = 7.3 a.u. and a = 7.75 a.u. are apparently on the curve of lowest total energy (see Section 4). T h e two solutions at a----7.0a.u, are evidently the result of a double minimum in the total energy vs magnetization curve in the vicinity of this lattice constant. T h e state cor-
MAGNETIZATION OF VANADIUM
1631
5"0-
E
4.0,
~
3"0
2-0
,.0
/,'[~ i I 9
o
5;0
9
o
Oo 6:0
o .r
! !
J i
7.0
s'.o
9:o
o.u.
Fig. 1. The calculated magnetic moment of vanadium metal as a function of lattice parameter. The two points plotted for a = 7.0 a.u. correspond to two distinct self-consistent solutions (see text).
Table 1. The cohesive energy, pressure, enthalpy, and magnetic moment of vanadium at various lattice parameters
Lattice constant (a.u.)
Cohesive en ergy.,[a] (Ry/atom)
Pressure (virial theorem) tha (kbars)
Enthalpyta~ (Ry/atom)
5.25 5.5 5-7225 6.3 7.0 7.0 7-3 7.75 8.5
0.31 1 0-329 0.326 0-259 0' 152 0.167 0.155 0.125 0.083
+518 + 53 -- 149 -366 --299 --246 - 187 --154 - 97
--0.057 --0.300 --0.420 --0.570 --0.500 --0.454 --0.402 --0.369 --0.285
Magnetic moment (electrons/atom) --0 0 1.06"7 2.22-~ 2.99 3.34 3.78
t~The cohesive energy and enthalpy are given relative to the spin-polarized Xc~ total energy of the vanadium atom in the (3d 1' )4(4s 1' )1 configuration. tb~Note that the pressure values given herein for lattice constants a ~ 7.0 a,u. are only semi-quantitative, as discussed in Section 4. r e s p o n d i n g to the larger m a g n e t i c m o m e n t (/x = 2.22 e l e c t r o n s per a t o m ) has a l o w e r total e n e r g y t h a n the o t h e r state (/~ = 1-06 e l e c t r o n s per a t o m ) at a = 7.0 a.u. H e n c e t h e m a g n e t i z a t i o n c u r v e is d r a w n t h r o u g h the p o i n t / z = 2.22 e l e c t r o n s per a t o m at this lattice c o n s t a n t . T h e v a n a d i u m e n e r g y b a n d s in t w o s y m m e t r y d i r e c t i o n s are c o m p a r e d in Fig. 2 for the e q u i l i b r i u m lattice c o n s t a n t (a = 5-7225 a.u.) a n d the first lattice c o n s t a n t (a = 7,0 a.u.) for w h i c h the b a n d s e x h i b i t s p i n - p o l a r i z a t i o n . T h e
similarity b e t w e e n the n o n - s p i n - p o l a r i z e d b a n d s a n d the s p i n - p o l a r i z e d b a n d s of each s p i n - t y p e c a n easily b e s e e n f r o m this figure. A l s o , the figure s h o w s that, for the spinp o l a r i z e d case, the 3d b a n d s h a v e n a r r o w e d in the v i c i n i t y of the F e r m i e n e r g y to p r o d u c e the high d e n s i t y of states r e q u i r e d for a s t a b l e m a g n e t i c state. W i t h i n c r e a s i n g lattice para m e t e r , the b a n d s b e c o m e n a r r o w e r a n d the splitting b e t w e e n the ~- a n d B-spin b a n d s bec o m e s greater. T h u s , a n i n c r e a s i n g p o r t i o n of
1632
T.M.
H A T ' F O X et al.
a-spin /S-spin - - -
02 I
o.i 4 ~
p
/i
" 2 25. ee
-02-
I
J/
tlJ -0"3
-0"4 -0'5-
N
?=57225a.u. I
o= 7.0 a.u.
H
F
Fig. 2. Vanadium energy bands for a non-magnetic lattice constant (a = 5-7225 a.u.) and a magnetiC one (a = 7.0 a.u., ~ = 2.22 electrons/atom).
the a-spin band and a decreasing fraction of the/3-spin band lie below the common Fermi energy as the lattice parameter increases, resulting in the increase of the magnetic moment. In Fig. 3, the s and d band widths for a- and /3-spin electrons are shown as a function of lattice parameter. The d bands are centered around the Fermi energy and are imbedded in
""-.. ~Top
ofs
0.6 04. Ry o.2.
"~'~-~-Top o, a
H',,e
o
",,,--:5::L:' 3a*
...- ..,
""=4s ~
-0.2, -0,4 -0,6Psoloted atom 5~0
ao 6.0
7'.0
8.'0
oo
9~0 O.u.
Fig. 3. Energy levels in vanadium as a function of lattice parameter. Atomic energy levels from a spin-polarized X u calculation for the (3d 1' ) ' ( 4 s ~' )1 configuration.
the broader s bands. The bottoms and tops of the s and d bands are defined here by the selfconsistent eigenvalues for the states T1, H 1 , H12 and H'z~ respectively, as indicated in the figure. The top of the s band, shown as a dashed curve in Fig. 3, was estimated from the appropriate high lying states which were calculated in early iterations for some of the lattice parameters. Figure 3 clearly shows the narrowing of the bands as the lattice parameter is increased. Magnetization begins just below a = 7.0a.u., where each non-spin-polarized state splits into a- and /3-spin states of the same symmetry. The magnetic moment at a given lattice parameter can be estimated crudely using Fig. 3, from the relative portions of the a- and B-spin bands below the Fermi energy. It is seen that the magnetic moment increases quite rapidly from the onset of spinpolarization up to a=7.45a.u., where the H12/3 curve crosses the Fermi level. For larger lattice constants, there are no occupied 3d/3 states (at least as indicated in this figure), and the magnetic moment increases rather slowly with increasing lattice parameter. The in-
MAGNETIZATION
crease in magnetic moment in this region is primarily due to the narrowing of the s bands, which are only weakly split in comparison with their overall width. This description of the magnetic moment in terms of the band splittings and widths corresponds closely to the magnetization curve in Fig. 1. Four selected electronic density of states curves, calculated as described in the Appendix, are presented in Figs. 4-7. The dashed portion of those curves, which always begins above the Fermi energy, indicates that not all the one-electron eigenvalues in that region were calculated. Comparison of these curves illustrates their compression in energy and corresponding increase in height as the lattice is expanded, with relatively small change in shape other than this effective rescaling. (Note that the first two curves are for non-spinpolarized calculations and hence are densities of states of both spins, whereas each curve in Figs. 6 and 7 is the density of states for only one spin.) The Fermi energy is seen to remain at about the same position relative to the structure of the density-of-states curves, for the non-magnetic structures (Figs. 4 and 5), and the density of states at the Fermi energy increases because of the increasing height of the density-of-states curves as the lattice expands, in the non-magnetic regime. For the
40I
=
o
o
95'25 o.u.
30-
g ~
//
20-
w
I0-
\
OF VANADIUM
1633
magnetic density-of-states curves, the /3spin curve is seen to be shifted up in energy with respect to the a-spin curve, so that the common Fermi level does not come at the 70.
60"
50"
a = 6"5
I ~
40-
g ~
30.
_~ 20. U.I
I0"
0
, -0.30
E
-0.15
i
F
0.15
Ry
Fig 5. V a n a d i u m density of states, a = 6.3 a.u. IOO
90 80 70 60 50 40 30 ~
2o
L
io
--
I0
~
20
~
30
( ~ - s p i ~
~.40
/
/
9
~ 5o ~
W
6o 70 8O
a-7.0o.u.
9O
-0.45
-0"30
-0.15
i
E
0.15
Ry Fig. 4. Vanadium density of states, a = 5.25 a.u.
I00
-0~'36 -0~24 -0'q2
E,,-
C;'12 0~'24
Ry F i g . 6. V a n a d i u m
d e n s i t y o f s t a t e s , a = 7 . 0 a . u . (pt = 2 . 2 2 electrons/atom).
1634
T.M.
HATTOX
I10I0090807060. 50" n." 4030~ i -~
cx-spin
20I00
9~
J
20/3-spin
~2 40,7, 5060708090a = 8"5o.u.
IO0 I10
i
-0.24
-0"12
EF
0.12
0.24
0'~6
Ry Fig. 7. V a n a d i u m d e n s i t y o f s t a t e s , a = 8.5 a.u.
same relative position on the o~- and /i-spin curves. T h e approximately rigid band shifting of these two curves is a measure o f the difference in exchange energy for electrons of majority and minority spins, T h e deviation from rigid-band shifting is an indication of the inaccuracy of simpler approaches which merely superimpose an exchange energy difference on a single spin-independent band structure and density of states. T h e iteration-to-iteration convergence of our calculations was relatively slow both for a = 6.3 a.u. and for a = 7.3 a.u.; it was extremely slow for a = 7.0 a.u. It will be recalled that it was for a = 7.0 a.u. that two, magnetically and energetically distinct, selfconsistent solutions were found. Because of the convergence slowness, additional selfconsistent calculations were not attempted in the region 6.3 < a < 7.3 a.u. F o r this reason the detailed behavior of the magnetic m o m e n t in that region is not known; however, it must
e t al.
certainly be bracketed by the two curves we have displayed in Fig. 1. T h e r e are two features of the electronic structure which very likely contribute to the slow convergence in the transition region. T h e first is that the Fermi level lies very nearly at a peak on the electronic density of states curves for both a- and /i-spin electrons (see Fig. 6), which is not the case for other lattice parameters. This means that small changes in the eigenvalues near the Fermi energy significantly affect the weighting factors for those states, so that the potentials change considerably from iteration to iteration, thus hindering convergence. (The high density of states at the Fermi level for a- and /t-spin electrons also causes the magnetic moment to be very sensitive to small changes in the eigenvalues, so that a high degree of convergence in energy, is required in order to achieve acceptable convergence in the magnetic moment.) A second factor which likely influences the slow convergence is the expected small dependence of the total energy on the magnetic moment in the transition region. T h e total energy curve as a function of magnetic moment should be monotonically increasing with its minimum at zero magnetic moment for a non-magnetic state. F o r a magnetic state, the minimum will occur for some non-zero magnetic moment. Therefore, one might expect the energy dependence upon magnetic moment to be rather weak in the transition region (/x ~ 0), causing the convergence to the magnetic m o m e n t which minimizes the energy to be very slow. Papaconstantopoulos et al.[18] and Yasui et al.[19] have performed self-consistent A P W and O P W energy band calculations, respectively, on V. Neither group of workers investigated the total energy of crystalline vanadium. T h o u g h there are differences in detail, our calculated band structure (Fig. 2) and Fermi surface are in satisfactory agreement with those found by the aforementioned workers. In particular, the band structure and Fermi surface at the normal lattice constant agree quite well with the self-consistent re-
MAGNETIZATION
suits obtained by Yasui et al. using a = 0-725, and fairly well with the self-consistent results obtained by Papaconstantopoulos et al. using a = 2/3. We therefore refer the interested reader to those papers for details of the band structure and Fermi surface, other than those we have already discussed.
OF VANADIUM
1635
energy) branch of the total energy curve. T h e equilibrium lattice constant, determined as the minimum of the total energy curve (or the zero of the pressure curve), was found to occur at a = 5.568 a.u., 2.6 per cent smaller 0-
4. COHESIVE P R O P E R T I E S
We have calculated the total energy and pressure* at each lattice parameter for which a self-consistent band calculation was performed, with the results given in Table 1. T h e total energy as a function o f lattice parameter is plotted in Fig. 8, relative to the spinpolarized X a ( S P X a ) total energy of the vanadium atom in the (3d~')4(4s~' )1 configuration, which is the spin-polarized A P W separated-atom limit (see discussion below). This curve is drawn through the calculated points such that its slope at each of these points corresponds to the calculated pressure for the corresponding point. T h e resulting kink in the curve comes just at the onset of magnetization (recall Section 3). T h e pressure is plotted as a function of lattice parameter in Fig. 9, where the dashed portion of this curve is drawn to match the slope along the kind in Fig. 8. T h e additional point plotted for a = 7.0 a.u. on Figs. 8 and 9 corresponds to the second self-consistent solution of higher total energy at that lattice constant, which was discussed in Section 3. T h e pressure from Fig. 9 agrees well with the slope indicated in Fig. 8 for that additional point, providing further evidence that the point represents a self-consistent solution for a second (higher *The pressure is obtained from the virial theorem, equation (7), applied to the kinetic and potential energy contributions calculated separately in the final self-consistency iteration for each lattice constant. Its error is a first-order quantity in the error of the w a v e function (as contrasted to that of the total energy), so the convergence of the computed pressures is considerably poorer t h a n that of the other quantities reported, particularly for the spin-polarized calculations (a /> 7.0 a.u.) for which the pressure values given in Table 1 and Fig. 9 should be considered only semiquantitative.
-0.I z =1.06- . ~ ' ~
oE P
-0"2
fie - 0"3
Experimental
-0.4
5:0
bo o'o
7!o
6'o
9'.00.U
Fig. 8. T h e total energy of v a n a d i u m as a function of lattice parameter, relative to the SPXc~ total energy of the the (3d 1' )4(4s T )1 configuration.
600" 500400300200100-
-I00-
-200-
k~._
-300-
//,"~"- /.t. = 1-06
-400-
510
' Go 610
7.'0
810
910 O.U.
Fig. 9. T h e pressure of v a n a d i u m as a function of lattice parameter, as obtained from the virial theorem.
1636
T.M. HATTOX et al.
than the experimental value[20] (at 25~ of a = 5.715 a.u. T h e cohesive energy was calculated as the difference between the Xo~ total energy per unit cell of the crystal and the X a total energy of the isolated atom in its ground state, using the same value for ot in the crystal and atomic calculations. H o w e v e r , the multiplet structure o f atomic vanadium complicates matters considerably. Experimentally [21] there are lowlying multiplets from both the (3d)3(4s) 2 configuration and the (3d)4(4s) 1 configuration. T h e ground state is the 4F multiplet of the (3d)4(4s) 2 configuration, but the energy of the 6D multiplet of the ( 3 d) 4 (4s) 1 configuration is only 0 . 0 1 8 R y a b o v e the ground state energy. In contrast, S P X a atomic calculations (with a = 0.715) for which the multiplet splittings are taken into account, place this 6/9 multiplet lower in energy than the 4F multiplet b y 0.124 Ry. U s e of the S P X a total energy of the ~D multiplet as the ground state energy of the isolated atom leads to a calculated cohesive energy of 0.33 Ry, while use o f the S P X ~ total energy of the 4F multiplet leads to a cohesive energy of 0.45 Ry. T h u s the experimental cohesive energy[10] of 0.39 Ry falls half w a y between the cohesive energy obtained using the S P X a atomic ground state total energy and the SPXot total energy of the experimental atomic ground state.* T h e isothermal compressibility is defined as K0 = - V -1 (av/a/') r
(9)
which with (8) gives, for the b.c.c, structure,
*Since the 6/9 multiplet is the only sextet for the configuration (3d)4(4s) 1, it must be the only multiplet corresponding to the configuration (3d t )4(4s 1' )~, and the SPXa total energy for this configuration is -1885.867 Ry. However, there are two quartet states (4F and 4p) which correspond to the configuration (3d1')3(4s 1' )1(4s $ )1, so that the S PXa total energy for this configuration represents a weighted average energy of the two multiplets. The energies of these two multiplets relative to their average energy may be computed straightforwardly[21] with a result of-1885.743 Ry for the energy of the 4F multiplet.
K o = ( q / 2 ) a [ O 2 E / O a Z ] -1
(10)
where a is, as usual, the lattice parameter. T h e second derivative in equation (10) was evaluated for a parabola fitted to the three points (a = 5-25, 5.5, 5.7225 a.u.) on the total energy curve (Fig. 8) nearest its minimum. T h e resulting calculated value for the zero-pressure isothermal compressibility is K = 4.46 • l0 -7 cm2/Kg, as c o m p a r e d to the experimental value [ 1 1] (at 30~ o f K = 6-05 • 10-7 cm2/Kg. 5. DISCUSSION T h e magnetic transition we have found in the present calculation occurs, as expected, at a lattice p a r a m e t e r which is outside the range of experimentally attainable values. H o w e v e r , there are alloys of vanadium with other metals which display an interesting variety of magnetic properties[23,24]. While it would be unwarranted to suggest that our calculations of the properties of crystalline vanadium are intrinsically applicable to these alloys, some cautious simple c o m p a r i s o n s can be made. First, the usual interpretation of the experimental evidence at present is that the vanadium m o m e n t is localized in ordered ferromagnetic Au4V with a magnitude near 1 Bohr magneton, as opposed to an essentially vanishing m o m e n t in disordered Au4V. Further, a m o m e n t of about 0.32 Bohr m a g n e t o n has been m e a s u r e d in AI3V (which, however, is not ferromagnetic). N o w , if we order these c o m p o u n d s by the v a n a d i u m - v a n a d i u m distance dvv we find: (1) vanadium metal, dvv = 4.98 a.u., zero m o m e n t ; (2) disordered Au4V, dv~ --- 5.38 a.u., zero moment; (3) A13V, d~v = 7.19 a.u., 0.32 izB; (4) ordered Au4V, dv~ = 7.57 a.u., 1 tz~. It would a p p e a r from these data that the onset of a local vanadium m o m e n t in a solid s y s t e m requires 5.38 a.u.-,_< d~,, ~ 7.19 a.u. As noted in Section 3, we were unable to define precisely the threshold value of d ~ for the beginning of the transition to a magnetized state in the expanded metal. Reference to Fig. 1 shows that a plausible (and convenient) value to assign to this calculated critical separation
MAGNETIZATION OF VANADIUM is d~,~.~- 5.85 a.u. ( = V3 a/2, with a ~ 6-75 a.u.). O u r calculations also clearly yield a zero m o m e n t for d~,,, <~ 5-45 a.u. T h e s e results are in good qualitative a g r e e m e n t with the experimental evidence we have just summarized. We regard this agreement as suggestive but not compelling evidence for the validity of the notion that a prime requisite for magnetization in v a n a d i u m alloys is a V - V separation greater than some minimum distance[25]. It may be that the magnetic properties [26] of the ferrimagnetic garnets Ca~Bi~-~,xFes-xVxOlz also owe their existence, at least in part, to the kind of b e h a v i o r we h a v e found for vanadium. T h e features of the calculated E ( a ) and P ( a ) curves (see Figs. 8 and 9) n e a r a = 7.1 a.u. (which we associate with the magnetic transition) are worthy of s o m e tentative comment in connection with the magnetic vanadium alloys. T h e y o c c u r for a d~v corresponding to physically unrealizable negative pressures for vanadium metal. H o w e v e r , the pressures needed to attain the same d ~ in the magnetic v a n a d i u m intermetallic c o m p o u n d s are positive (or zero). I f the E ( a ) and P ( a ) curves in these alloys were to exhibit features associated with the magnetic transitions similar to those we have c o m p u t e d for metallic vanadium, then the rapid change of slope in E ( a ) and attendant peak in P ( a ) would occur at some physically realizable pressure. T h e s e features of E ( a ) and P ( a ) would correspond, in such a case, to an isomorphic phase transition in which the onset o f magnetization is a c c o m p a n i e d by a discontinuous change in lattice p a r a m e t e r (unless some polymorphic transition, which this calculation forbade by construction, were to dominate). Obviously there are m a n y subtle complexities in these systems which we have not mentioned, much the less treated, and our c o m m e n t s are not in any m a n n e r intended to c o n v e y a sense of conclusiveness. On the basis of the present w o r k it would seem that spin-polarized APW-Xo~ calculations can yield meaningful results for the total energy, pressure, and compressibility of
1637
transition metals. W e believe that the spinpolarized method should be used for both the solid and the isolated atoms, when applicable, in determining the cohesive energy, because the differences b e t w e e n the spin-polarized and non-spin polarized total energies (in the atom or the solid or both) may be large compared to the cohesive energy itself. It is also necessary to consider the multiplet structure of the atom. In the particular case of vanadium, for example, the spin-polarized X a total energy of the (3d ~' )4(4S '~ )1 atomic configuration is lower in energy than the non-spinpolarized X a total energy of the (3d)3(4s) ' configuration (averaged o v e r multiplets) by 0.33 Ry, a very significant difference compared to the experimental cohesive energy [10] of 0.39Ry. F u r t h e r m o r e , there is no reason to believe that these energy differences in the a t o m and in the solid will cancel; in fact, there is no spin-polarized lowering of energy in the (non-magnetic) vanadium solid at equilibrium to cancel the quite large lowering of energy in the atom. Acknowledgements-We are grateful to the members of the Qauntum Theory Project of the University of Florida, especially to Prof. J. W. D. Connolly and to Drs. F. W. Averill and K. Schwarz, for many useful discussions. We also gratefully acknowledge support of a portion of the computing costs for the work reported here from the Computing Center of the University of Florida. REFERENCES
1. SLATER J. C. and JOHNSON K. H., Phys. Rev. BS, 844 (1972), and references therein. 2. SLATER J. C., in Advances in Quantum Chemistry (Edited by P.-O. L6wdin), Vol, 6, p. 1. Academic Press, New York (1972), and references therein. 3. AVER1LL F. W., Phys. Rev. B4, 3315 (1971), and references therein. 4. CONKLIN J. B., Jr., AVERILL F. W. and HATTOX T. M.,J. de Phys. 33, C3-213 (1972). 5. CONNOLLY J. W., Phys. Rev. 159, 415 (1967); Int. J. Quant. Chem. IS, 615 (1967); Int. J. Quant. Chem, IIS, 257 (1968). 6. WAKO H S., J. phys. Soc. Japan 20, 1894 (1965), and references therein. 7. DECICCO P. and KITZ A., M.I.T. Quant. Prog. Report No. 63, 15 Jan. 1967, p. 2 (unpublished). 8. SLATER, J. C., The Self-Consistent Field for Molecules and Solids, Vok 4 of the series Quantum
T. M. HATTOX et al.
1638
9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
22. 23. 24. 25.
26. 27. 28.
29. 30.
31.
Theory of Molecules and Solids. McGraw-Hill, New York (to be published 1973). SLATER J. C., Phys. Rev. 49,537 (1936). G S C H N E I D E R K. A., Jr., in Solid State Physics (Edited by F. Seitz and D. Turnbull), Vol. 16, p. 275. Academic Press, New York (1964). B R I D G M A N P. W., Phys. Rev. 57, 235 (1940); SLATER J. C., Phys. Rev. 57,744 (1940). SLATER J. C.,J. chem. Phys. 57, 2389 (1972). BERRONDO M. and G O S C I N S K I O., Phys. Rev. 184, 10(1969). SCHWARZ K., Phys. Rev. BS, 2466 (1972). HATTOX THOMAS M., Ph.D. Thesis, Univ. of Fla., 1972 (unpublished). SLATER J. C., Phys. Rev. 36, 57 (1930). SLATER J. C.,J. appl. Phys. 39,761 (1968). P A P A C O N S T A N T O P O U L O S D. A., A N D E R SON J. R. and M C C A F F R E Y J. W., Phys. Rev. BS, 1214(1972). YASUI M., H A Y A S H I E. and S H I M I Z U M. J., J. phys. Soc.Japan 29, 1446 (1970). JAMES W. J. and S T A U M A N I S M. E., Z. Phys. Chem. 29, 134(1961). MOORE C. E., Atomic Energy Levels as Derived from the Analyses of Optical Spectra, Vol. 1, p. 291. Circular of the N.B.S. 467 (1949). SLATER J. C., Quantum Theory of Atomic Structure, Vol. 2, Appendix 21. McGraw-Hill, New York (1960). C R E V E L I N G L., Jr. and LUO H. L., Phys. Rev. 176, 614 (1968), and references therein. C R E V E L I N G L., Jr. and LUO H. L., Phys. Lett. 28A, 772 (1969). See G O O D E N O U G H J. B., Magnetism and the Chemical Bond. Wiley, New York (1963), Chapter I., for highly detailed discussion of this problem. G E L L E R S., ESPINOSA G. P., W I L L I A M S H. J., SHERWOOD R. C. and NESBITT E. A., J. appl. Phys. 35,570 (1964). SLATER J. C., Phys. Rev. 51,846 (1937). MATTHEISS L. F., WOOD J. H. and SWlTENDICK A. C., in Methods in Computational Physics (Edited by B. J. Alder, S. Fernback and M. Rotenberg), Vol. 8, p. 63. Academic Press, New York (1968). H E R M A N F. and S K I L L M A N S.,Atomic Structure Calculations. Prentice-Hall, New Jersey (1963). SCHWARZ K. and C O N K L I N J. B., Jr., University of Florida, Quantum Theory. Project Technical Report No. 213, 27 Jan. 1971, and to be published. MATTHEISS L. F., Phys. Rev. 139, A 1893 (1965). APPENDIX
The self-consistent energy bands of metallic vanadium were calculated at eight lattice parameters by solving the one-electron Xc~ equation using the non-relativistic APW method within the framework of the muffin-tin approximation [27]. A general discussion of the A PW method and the numerical details relating to the APW program (of which the program used in this work is a descendant) are given in an excellent article by Mattheiss, Wood and Switendick[28]. The spin-polarized method, using an assumed ferromagnetic arrangement, was used at all
lattice parameters except for the two smallest (a = 5.25 a.u. and a = 5.5 a.u.), t:or these two lattice parameters, it was assumed that a spin-polarized calculation would reduce to the non-spin-polarized result, as is the case for the next two larger lattice parameters (a = 5.7225 a.u. and a = 6.3 a.u.). The spir polarized calculations were initiated with a net magnr moment obtained (except at a = 5.7225a.u.) by superposing spin-polarized Xct atomic potentials which had been calculated for a set of fractional occupation numbers corresponding to various assumed values of the atomic magnetic moment. For a = 5.7225 a.u., the spin-polarized calculation was initiated by artificially inducing a spin-splitting in the nonspin-polarized energy bands at this lattice constant to give a starting magnetic moment of 1.25 electrons/atom. The b.c.c, crystal structure was used, and the muffin-tin sphere radius was chosen to be equal to half the nearest neighbor distance. The value of the parameter c~ was always chosen to be 0.7150, the value determined for atomic vanadium (recall Section 2). The energy eigenvalues and eigenfunctions of the true core levels (1 s, 2s, 2p) were calculated self-consistently in atomic fashion using essentially the Schrrdinger subroutine of the Herman-Skillman atomic program[29]. Since the 3s and 3p bands were far below the valence bands and had practically all of their charge within the APW spheres, they were calculated on a grid of only 16 equally spaced points in the b.c.c. Brillouin zone, as suggested by Schwarz and Conklin[30]. Th6 valence bands were calculated on a grid of 128 equally spaced points in the b.c.c. Brillouin zone. The wave functions calculated for the occupied bands at these 128 points were used to generate the charge densities for self-consistency iterations. However, for states near the Fermi energy, these representative states were not simply weighted by their degeneracies. Instead, additional interpolation of the bands to a grid of 8192 points in the (full) zone was performed using a method somewhat akin to that of Mattheiss [31]. In addition to the common practice of using these interpolated points for the determination of density-ofstates curves, they were also used in this calculation to determine modified weighting factors for the representative states near the Fermi surface, for the calculation of charge density and total energy. In this way, portions of a band which are in fact closer in k-space to a state on the 128-point grid which lies above the Fermi energy (but in that band), are actually represented in the charge-density calculation by the wave functions of that state (which is given the appropriate fractional occupancy as determined from the 8192-point interpolation grid), and the weighting factor of some state(s) below the Fermi energy is correspondingly reduced. This procedure improves convergence for non-spin-polarized calculations and allows use of coarser computational grids than would otherwise be possible; it is absolutely essential for calculation of magnetic moments when the Fermi energy is near a sharp peak in the density of states for one or both spins, because the moments depend on the differences of numbers of electrons with majority and minority spins. All calculations were performed on the HermanSkillman radial mesh [3, 28]. The calculations were carried out on an IBM 360/65 computer, using double precision for the critical sections of all programs.