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Choice of the muffin tin zero in Fermi surface interpolation schemes based on phase shifts
Solid State Communications,
Vol. 9, pp. 1939—1944, 1971. Pergamon Press.
Printed in Great Britain
CHOICE OF THE MUFFIN TIN ZERO IN FERMI SURFACE INTERPOLATION SCHEMES BASED ON PHASE SHIFTS M.A.C. Devillers and A.R. Vroomen Fysisch Laboratorium, Katholieke Universiteit, Nijmegen, The Netherlands
(Received 26 August 1971 by G.W. Rathenau)
It appears, that the diagonal and the important off-diagonal APW matrix elements in the ‘on the Fermi sphere’ approximation are approximately conserved for different values of the Fermi energy parameter E~in an APW interpolation scheme for the Fermi surface of white tin. The same applies for the matrix elements in a KKRZ interpolation scheme for the Fermi surfaces of Cu and Ag. From this empirical conservation law one can understand the functional behaviour of the logarithmic derivative of the l-th radial wave function and of the Friedel sum as a function of E~.ZF is then approximately zero for E~= E~(nearly free electron value) and approximately Z (the valence) for EF = E~./3. One can get some feeling for the role of E~in APW and KKRZ interpolation schemes by focussing on a comparison with local pseudopotential form factors.
IN AN earlier paper’ we already emphasized, that in an AP’X interpolation scheme for geometrical Fermi surface data the parameter E~, i.e. the energy of the electrons on the Fermi surface (FS), relative to the muffin tin zero, is a remarkably weak parameter. To gain more insight into this behaviour it appears to be useful to compare the APW scheme with the KKRZ and the pseudopotential (or OPW) schemes. The APW and 23 KKRZ schemes are equivalent, as is well known. Both use the phase shifts of the muffin tin potential. An exact relationship between an APW matrix element and the corresponding local pseudopotential form factor does not exist, but it will be a reasonable approximation to suppose them to be equal in the ‘on the Fermi sphere’ (OFS) approximation.
by fourteen selected points. For each choice of EF the first four logarithmic derivatives of the radial wave function a~(Er, R~)= RsR~(EF,R~)/ R?(EF. R~)are fitted to these fourteen points by a least square fit procedure (no spin orbit coupling included) [Fig. 1(a), (b)]. In a large range of the muffin tin Fermi energy, 0.5
The weakness of E~as a parameter will be
copper: 0.852, for silver: 0.668.)
demonstrated and discussed for the cases of white tin and of the noble metals. We represent the experimentally determined FS of white tjn4’~
A second illustration of the subject of this paper are the noble metals.6 For these metals 1939
1940
THE MUFFIN TIN ZERO IN FS INTERPOLATION
the phase shifts have been determined in a KKRZ interpolation scheme by fitting them to the very accurate dHvA-data. For the sake of presentation, comparable with that for white tin, twenty points of the interpolated FS with EF = 0.65 c.u. have been taken as representative for the FS and the r.m.s. deviation between the calculated set of points and this defined set of points is plotted in Fig. 3(a) as a function of EF. Note that for the noble metals EF is even less determined than for
Vol. 9, No.22
(c) The individual behaviour of the phase shifts Th of the muffin tin potential corresponding with Fig. 1(b), calculated from a 1 using formula (2). ~3I <0.04 radian for 0
r~W!c.n .6
white tin. .4 ,
2
/
N, S S
O
IO~lr.ms deviatior~(c.u.)
.2
o
10
______________________________
.6
a1 -~
.4
/4 -.2 -4 -.6
C
I’~—~~ £ ‘ / 1.0 1.7 I
E~.2O0cu
1.4
N,
16
2k~ Q
[~~14Oc u / //
1
/
-10
~
-17 -1.4
H
_____________________________ FIG. 2. The experimental APW rnatrix element
i~(rad)
..
.
for the FS of white tin in the OFS approximation for three values of EF. The dots are experimental form factors obtained from a four parameter local pseudopotential fit to the same fourteen points of the Fermi surface of white tin as have been used in the APW fit. (The best local pseudopotential fit has a r.m.s. deviation from experimental calipers of 0.004c.u. for EF= 2.llc.u.) The dotted line stands for the KKRZ matrix element in the OFS approximation for E~=0.77c.u.
________________________
.2 .4 .6 ~81.0 1,2 1.4 1.6 1.6 2.0 2.2,2.4 E~(cu) FIG. 1. (a) r.m.s. deviation of a four parameter
APW fit to a representative set of fourteen points of the Fermi surface of white tin as a function of the Fermi energy parameter E~ (b) The individual behaviour of the dimensionless parameters a~,i.e. the logarithmic derivatives of the 1-th radial wavefunction of the muffin tin potential times R~.
use To the study OFS approximation. the problem of the In this E~choice approximation we the APW matrix elements are evaluated for = = kF(kF is the radius of the free electron sphere) and become functions of q
=
—
I.
For q> 2k~ k’.~ and k, are taken antiparallel and only kJ is fixed at k~. When we change the Fermi energy E~it appears then (see Fig. 2) that the experimental matrix elements, both diagonal and off-diagonal, are approximately conserved for the most important reciprocal lattice vectors
q,., q~,<2kg, with structure factors unequal to
Vol. 9, No.22
THE MUFFIN TIN ZERO IN FS INTERPOLATION
zero. For the diagonal matrix element this means that for a nearly free electron (NFE) metal
~
k~) E~-_~E~— F~w{k~ q
=
0, a,(E~)].
(1)
rm,s,de,~at,on(Cu,)
1941
of EF, as found in Fig. 1(b), follows immediately. For copper (Fig. 4) the KKRZ matrix element in the OFS approximation is conserved approximately too. Note that in this case the most important matrix elements, determining the FS, are second order terms (q~> 2kv). The latter is also 7 true for the alkali metals. The conservation of diagonal element has an
2
interesting consequence for the Friedel sum of the reduced phase shifts ‘~ of the muffin tin
~
I
I 2,
-i
potential [Fig. 1(c), (d)L The relationship between a 1 and ~ is given by:
I
al(EF, !?~)= x~j/(x)—tg~
20
(2)
1.5
with x
=
277R~(E~)~ and where Il, n1 are the
spherical Bessel- and Neuman-functions. In Fig. 1(d) we show the Friedel sum ZF(EF), defined The Friedel as: sum shows a number of interesting ZF(EP)
~V(2l
=
+
1)~1(E~).
(3)
—5
phenomena. ZF is a rather linear function of EF 0
.1
.2
3
.4
.5
E,lcuI
.6
.7
.6
‘
KKRZ fit to twenty points of the Fermi FIG. 3. (a) r.m.s. deviation of a four parameter surface of copper (solid line) and of silver (broken line), obtained from a KKRZ interpolation of the Fermi surface at EF = 0.65c.u. for copper, resp.EF=0.68c.u. for silver, as a function of EF. (b) The Friedel sum ZF of the phase shifts for copper (heavy solid line) and for silver (broken line) and the curve of formula (7) (thin solid line). E~= 0.61c.u. (From Coenen e a!.,6 to be published.)
in the region E~/3
In the OFS approximation the KKRZ matrix element can be written as:
M(q)=(k_E~)ô(q)+F’~~[k~,q, m(EF)] FKKRZ [kr, q, m(E~)] =
For white tin the constraint (1) on the diagonal APW matrix element is fulfilled to a good approximation: E~—r~w= 2.15 ±0.05c.u. for 0.2
(4)
with
~ ~(2l+
—
~!2~I
E~. Z (E~)~
1)~cotg~ 1(E~)_n1 (x)/j1 (x)~’ 2P
~ji (y)/i1 (x)~
1(cosO~)
(5)
in which x = 27TR8(E~)i,y = 27TR~(E~)i, O~ = 2 arcsin (q/2k~). Then, in analogy with (1), the KKRZ diagonal matrix element should be approximately zero, independent of E~,resulting in
E,~.~ EF
—
r’KKRZ [k~ 0, ~l(EF)].
(6)
1942
THE MUFFIN TIN ZERO IN FS INTERPOLATION
(c
‘1
I
I
I
(tli] (2x]
(220)
I
I
(311] (722]
I I I (4O~) (331)1470]
Vol. 9, No.22
(422]
7-
I
.—,.t,.,0s I
/
/
4
\
/
3,
/
12
-u
7.4
1.0
11
2.0
72
2~2i
21
3.7
32
——
—s
FIG. 4. The KKRZ matrix element in the OFS approximation for three values of the Fermi energy for 14 from first principles copper (solid lines), the pseudopotential form factor as calculated by Moriarty (broken line) and the APW matrix element in the OFS approximation for Ef’ = 0.20c.u. (dotted line). For ~ E,~. we know experimentally that the phase shifts are small. Then cot g~ ~ 1/Th>>
depends on EF and therefore a will be a function of E~,the experimental phase shifts in
n 1 (x)/)3(x) and equations (5) and (6) reduce to:
Zp(EF)/Z~—(EF-E~)/~E~.
(7)
The experimental Z,/(Ep) for white tin [Fig. 1(d)] shows very nearly the behaviour expressed by (7) over a large range of E~. However it is remarkable that (7) is obeyed over a much larger range of EF than is allowed by the various approximations leading to (7). Probably this is caused by the fact that the Friedel sum is of a more fundamental nature than its introduction from the formula above would suggest. Though somewhat less rigorously, relation copper and silver too [Fig. 3(b)].
C7)
holds for
=
azE~~’2
=
with —a0/2T
=
a0 (E~)~
(9)
0.300 ±0.003c.u. (For corn-
parison: the radius of the Slater sphere R~= 0.259c.u. and the radius of the Wigner—Seitz sphere R~5 = 0.319c.u.) The implication of (9) is not clear to the present authors. We can gain some further insight into the role of E~ in APW and KKRZ interpolation schemes for Fermi surfaces b~-a comparison with pseudopotential form factors. A mathematically
The individual behaviour of the phase shifts in Fig. 1(c) demands some remarks. - From scattering theory we know that the limit of 77~for a fixed potential lim0 7)1(E) E-’
Fig. 1(c) do behave like (8). Further, it is remarkable that in the whole region 0
* ?72~7T
(8)
with m1 the number of bound 1-states and —a0/277 the ‘scattering-length’ of the scattering potential. Though (8) is not strictly applicable in our case, because the muffin tin potential
exact relationship between APW matrix elements and local pseudopotential (LP) form factors doesn’t exist but a good approximation will be v~(Es) = [‘APW [kF q, a1(E~)]. (10) In Fig. 2 we show the experimental LP form factors for E,~.= 2.11 c.u. as determined 10 by a direct four parameter fit to the same fourteen points of the FS of white tin, that have been used in the APW fit. The Fermi energy is used as a fifth parameter in the LP form factor fit. The
Vol. 9, No. 22
THE MUFFIN TIN ZERO IN FS INTERPOLATION
value EF = 2.11 ±0.01 c.u. agrees very well with EF = 2.lOc.u. in the APW fit for which FAPW [kF,0,al(Ep)] = 0. Note that the direct LP form factors agree rather well with FAPW [kr, q, a1 (E~)1for EF = E~-/3 (see reference 9). We cannot expect this agreement to be exact for at least two reasons. First, the OFS approximation itself is an approximation of the exact APW matrix element. Secondly, all LP 2kF have been set equal form factors with q~> to zero in the four parameter LP fit, whereas
fAPW
1943
An additional remark concerns the behaviour of F’~ [kr, q, a 1(Ep)} for white tin [Fig. 2] and of Fl~RZ[kp,q, 771(EF)I for the noble metals [Fig. 4] at q = q0, i.e. the point at q/2kp~.0.9 where the matrix element has a node. (For the physical meaning of q0, see for example reference 11.) It is suggested by this behaviour, that q0 and the slope of FAPW, resp. FI~RZ,in q0 are conservedparameters quantitiesintoo, these explicit the although formalisms. In are the not ‘empty core’ model’1 conservation of q 0 means conservation of the core radius and in this sense it seems to be a satisfying result physically.
[ks, a.~,a1 (EF)] is obviously unequal to
zero for these q~. A way of thinking that throws some light upon the problem of the choice of EF is the following. Suppose we have a muffin tin potential V?~which generates the FS correctly for EF= ~ We then add a relatively small potential ~V, which equals c in the interstitial region and whichthat hasthe such a form withintransthe Slater spheres, crystal Fourier forms (~V)qr, = 0 for those few important reciprocal lattice vectors q~for which the structure factor is unequal to zero. This new potential will leave the FS approximately unaltered but with a new value of the Fermi energy, = ~ — c, and with new values of the phase shifts. The more Fourier transforms are important for the FS of a metal (low symmetry and many electrons per primitive cell), the more difficult it will be to fulfill the constraint that (V)~,~ remains the same and in our opinion a ‘better’ determination of E~by FS data may be expected.
Finally it should be noted that FAPW and are not exactly equal in the OFS approximation for the same choice of EF (Fig. 2 and 4). The constraint (1) is somewhat better obeyed than the constraint (6) for both white tin and the noble metals, and moreover, oscil2kF has the a smaller lating tail of FAPW for q > amplitude than that of fKKRZ• This seems to be FKKRZ
consistent who argue converges the KKRZ
with the opinion of several authors ~ that the plane wave representation faster for the APW formalism than for formalism.
E~
Acknowledgements — We wish to thank Mr. N. Coenen for providing us with his results on the noble metals, Mr. MC. Knoope, who performed part of the APW calculations, and Mr. M.J.G. Lee for sending us a preprint.
REFERENCES 1.
DEVILLERS MAC. and DE VROOMEN A.R., Phys. Leit. 30A, 159 (1969).
2.
ZIMAN J.M., Proc. Phys. Soc.,
3.
LLOYD P., Proc. Phys. Soc., 86, 825 (1965).
4.
MATTHEY M.M.M.P. and DE VROOMEN AR., Solid Slate Commun. 9, 1329 (1971).
5.
DEVILLERS MAC., MATTHEY M.M.M.P. and DE VROOMEN A.R., to be published.
6.
COENEN N.J. and DE VROOMEN AR., to be published.
7.
LEE M.J.G., Computational Methods in Band Theory, p. 63, Plenum Press, New York (1971).
8.
BALL M.A., J. Phys. C (Solid State Phys.), 2, 1248 (1g70).
9.
COHEN ML. and HEINE V., Solid State Physics, Vol. 24, p. 243, Academic Press, New York and London (1970).
91, 701 (1967).
1944
THE MUFFIN TIN ZERO IN FS INTERPOLATION
Vol. 9, No.22
10.
DEVILLERS M.A.C. and DE VROOMEN AR., to be published.
11.
ASHCROFT N.W., Phys. Leti.
12.
JOHNSON K.H., Phys. Rev., 150, 429 (1966).
13.
SEGALL B. and HAM S., Methods in Computational Physics, Vol.8, p. 267, Academic Press, New York and London (1968).
14.
MORIARTY J.A., Phys. Rev. B, 1, 1363 (1970).
23, 48 (1966).
Die diagonalen und die wichtigen nicht diagonalen APW-Matrixelemente bleiben in der ‘on the Fermi sphere’ — Näherung für verschiedene Werte des Parameters der Fermienergie E~in einem
APW-Interpolationsschema für die Ferrnifläche von weissem Zinn nãherungsweise erhalten. Dasselbe gilt für die Matrixelemente in einem KKRZ-Interpolationsschema für die Fermiflächen von Cu und Ag. Auf Grund theses empirischen Erhaltungsgesetzes kann man das Verhalten der logarithmischen Ableitung der i-ten radialen Wellenfunktion und der Friedelsumme ZF als Funktion von E~verstehen. Z,- ist danro näherungsweise null für E~-= E~-(Wert für quasifreie Elektronen) und etwa Z (Anzahl der Valenzelektronen) für EF = E~./3. Man erhält einen Eindruck, weiche Rolle E~ in den APW- und KKRZ-Interpolationsschemata spielt, wenn man deren Matrixelemente mit den lokalen Formfaktoren des Pseudopotentials vergleicht.
Vol. 9, pp. 1939—1944, 1971. Pergamon Press.
Printed in Great Britain
CHOICE OF THE MUFFIN TIN ZERO IN FERMI SURFACE INTERPOLATION SCHEMES BASED ON PHASE SHIFTS M.A.C. Devillers and A.R. Vroomen Fysisch Laboratorium, Katholieke Universiteit, Nijmegen, The Netherlands
(Received 26 August 1971 by G.W. Rathenau)
It appears, that the diagonal and the important off-diagonal APW matrix elements in the ‘on the Fermi sphere’ approximation are approximately conserved for different values of the Fermi energy parameter E~in an APW interpolation scheme for the Fermi surface of white tin. The same applies for the matrix elements in a KKRZ interpolation scheme for the Fermi surfaces of Cu and Ag. From this empirical conservation law one can understand the functional behaviour of the logarithmic derivative of the l-th radial wave function and of the Friedel sum as a function of E~.ZF is then approximately zero for E~= E~(nearly free electron value) and approximately Z (the valence) for EF = E~./3. One can get some feeling for the role of E~in APW and KKRZ interpolation schemes by focussing on a comparison with local pseudopotential form factors.
IN AN earlier paper’ we already emphasized, that in an AP’X interpolation scheme for geometrical Fermi surface data the parameter E~, i.e. the energy of the electrons on the Fermi surface (FS), relative to the muffin tin zero, is a remarkably weak parameter. To gain more insight into this behaviour it appears to be useful to compare the APW scheme with the KKRZ and the pseudopotential (or OPW) schemes. The APW and 23 KKRZ schemes are equivalent, as is well known. Both use the phase shifts of the muffin tin potential. An exact relationship between an APW matrix element and the corresponding local pseudopotential form factor does not exist, but it will be a reasonable approximation to suppose them to be equal in the ‘on the Fermi sphere’ (OFS) approximation.
by fourteen selected points. For each choice of EF the first four logarithmic derivatives of the radial wave function a~(Er, R~)= RsR~(EF,R~)/ R?(EF. R~)are fitted to these fourteen points by a least square fit procedure (no spin orbit coupling included) [Fig. 1(a), (b)]. In a large range of the muffin tin Fermi energy, 0.5
The weakness of E~as a parameter will be
copper: 0.852, for silver: 0.668.)
demonstrated and discussed for the cases of white tin and of the noble metals. We represent the experimentally determined FS of white tjn4’~
A second illustration of the subject of this paper are the noble metals.6 For these metals 1939
1940
THE MUFFIN TIN ZERO IN FS INTERPOLATION
the phase shifts have been determined in a KKRZ interpolation scheme by fitting them to the very accurate dHvA-data. For the sake of presentation, comparable with that for white tin, twenty points of the interpolated FS with EF = 0.65 c.u. have been taken as representative for the FS and the r.m.s. deviation between the calculated set of points and this defined set of points is plotted in Fig. 3(a) as a function of EF. Note that for the noble metals EF is even less determined than for
Vol. 9, No.22
(c) The individual behaviour of the phase shifts Th of the muffin tin potential corresponding with Fig. 1(b), calculated from a 1 using formula (2). ~3I <0.04 radian for 0
r~W!c.n .6
white tin. .4 ,
2
/
N, S S
O
IO~lr.ms deviatior~(c.u.)
.2
o
10
______________________________
.6
a1 -~
.4
/4 -.2 -4 -.6
C
I’~—~~ £ ‘ / 1.0 1.7 I
E~.2O0cu
1.4
N,
16
2k~ Q
[~~14Oc u / //
1
/
-10
~
-17 -1.4
H
_____________________________ FIG. 2. The experimental APW rnatrix element
i~(rad)
..
.
for the FS of white tin in the OFS approximation for three values of EF. The dots are experimental form factors obtained from a four parameter local pseudopotential fit to the same fourteen points of the Fermi surface of white tin as have been used in the APW fit. (The best local pseudopotential fit has a r.m.s. deviation from experimental calipers of 0.004c.u. for EF= 2.llc.u.) The dotted line stands for the KKRZ matrix element in the OFS approximation for E~=0.77c.u.
________________________
.2 .4 .6 ~81.0 1,2 1.4 1.6 1.6 2.0 2.2,2.4 E~(cu) FIG. 1. (a) r.m.s. deviation of a four parameter
APW fit to a representative set of fourteen points of the Fermi surface of white tin as a function of the Fermi energy parameter E~ (b) The individual behaviour of the dimensionless parameters a~,i.e. the logarithmic derivatives of the 1-th radial wavefunction of the muffin tin potential times R~.
use To the study OFS approximation. the problem of the In this E~choice approximation we the APW matrix elements are evaluated for = = kF(kF is the radius of the free electron sphere) and become functions of q
=
—
I.
For q> 2k~ k’.~ and k, are taken antiparallel and only kJ is fixed at k~. When we change the Fermi energy E~it appears then (see Fig. 2) that the experimental matrix elements, both diagonal and off-diagonal, are approximately conserved for the most important reciprocal lattice vectors
q,., q~,<2kg, with structure factors unequal to
Vol. 9, No.22
THE MUFFIN TIN ZERO IN FS INTERPOLATION
zero. For the diagonal matrix element this means that for a nearly free electron (NFE) metal
~
k~) E~-_~E~— F~w{k~ q
=
0, a,(E~)].
(1)
rm,s,de,~at,on(Cu,)
1941
of EF, as found in Fig. 1(b), follows immediately. For copper (Fig. 4) the KKRZ matrix element in the OFS approximation is conserved approximately too. Note that in this case the most important matrix elements, determining the FS, are second order terms (q~> 2kv). The latter is also 7 true for the alkali metals. The conservation of diagonal element has an
2
interesting consequence for the Friedel sum of the reduced phase shifts ‘~ of the muffin tin
~
I
I 2,
-i
potential [Fig. 1(c), (d)L The relationship between a 1 and ~ is given by:
I
al(EF, !?~)= x~j/(x)—tg~
20
(2)
1.5
with x
=
277R~(E~)~ and where Il, n1 are the
spherical Bessel- and Neuman-functions. In Fig. 1(d) we show the Friedel sum ZF(EF), defined The Friedel as: sum shows a number of interesting ZF(EP)
~V(2l
=
+
1)~1(E~).
(3)
—5
phenomena. ZF is a rather linear function of EF 0
.1
.2
3
.4
.5
E,lcuI
.6
.7
.6
‘
KKRZ fit to twenty points of the Fermi FIG. 3. (a) r.m.s. deviation of a four parameter surface of copper (solid line) and of silver (broken line), obtained from a KKRZ interpolation of the Fermi surface at EF = 0.65c.u. for copper, resp.EF=0.68c.u. for silver, as a function of EF. (b) The Friedel sum ZF of the phase shifts for copper (heavy solid line) and for silver (broken line) and the curve of formula (7) (thin solid line). E~= 0.61c.u. (From Coenen e a!.,6 to be published.)
in the region E~/3
In the OFS approximation the KKRZ matrix element can be written as:
M(q)=(k_E~)ô(q)+F’~~[k~,q, m(EF)] FKKRZ [kr, q, m(E~)] =
For white tin the constraint (1) on the diagonal APW matrix element is fulfilled to a good approximation: E~—r~w= 2.15 ±0.05c.u. for 0.2
(4)
with
~ ~(2l+
—
~!2~I
E~. Z (E~)~
1)~cotg~ 1(E~)_n1 (x)/j1 (x)~’ 2P
~ji (y)/i1 (x)~
1(cosO~)
(5)
in which x = 27TR8(E~)i,y = 27TR~(E~)i, O~ = 2 arcsin (q/2k~). Then, in analogy with (1), the KKRZ diagonal matrix element should be approximately zero, independent of E~,resulting in
E,~.~ EF
—
r’KKRZ [k~ 0, ~l(EF)].
(6)
1942
THE MUFFIN TIN ZERO IN FS INTERPOLATION
(c
‘1
I
I
I
(tli] (2x]
(220)
I
I
(311] (722]
I I I (4O~) (331)1470]
Vol. 9, No.22
(422]
7-
I
.—,.t,.,0s I
/
/
4
\
/
3,
/
12
-u
7.4
1.0
11
2.0
72
2~2i
21
3.7
32
——
—s
FIG. 4. The KKRZ matrix element in the OFS approximation for three values of the Fermi energy for 14 from first principles copper (solid lines), the pseudopotential form factor as calculated by Moriarty (broken line) and the APW matrix element in the OFS approximation for Ef’ = 0.20c.u. (dotted line). For ~ E,~. we know experimentally that the phase shifts are small. Then cot g~ ~ 1/Th>>
depends on EF and therefore a will be a function of E~,the experimental phase shifts in
n 1 (x)/)3(x) and equations (5) and (6) reduce to:
Zp(EF)/Z~—(EF-E~)/~E~.
(7)
The experimental Z,/(Ep) for white tin [Fig. 1(d)] shows very nearly the behaviour expressed by (7) over a large range of E~. However it is remarkable that (7) is obeyed over a much larger range of EF than is allowed by the various approximations leading to (7). Probably this is caused by the fact that the Friedel sum is of a more fundamental nature than its introduction from the formula above would suggest. Though somewhat less rigorously, relation copper and silver too [Fig. 3(b)].
C7)
holds for
=
azE~~’2
=
with —a0/2T
=
a0 (E~)~
(9)
0.300 ±0.003c.u. (For corn-
parison: the radius of the Slater sphere R~= 0.259c.u. and the radius of the Wigner—Seitz sphere R~5 = 0.319c.u.) The implication of (9) is not clear to the present authors. We can gain some further insight into the role of E~ in APW and KKRZ interpolation schemes for Fermi surfaces b~-a comparison with pseudopotential form factors. A mathematically
The individual behaviour of the phase shifts in Fig. 1(c) demands some remarks. - From scattering theory we know that the limit of 77~for a fixed potential lim0 7)1(E) E-’
Fig. 1(c) do behave like (8). Further, it is remarkable that in the whole region 0
* ?72~7T
(8)
with m1 the number of bound 1-states and —a0/277 the ‘scattering-length’ of the scattering potential. Though (8) is not strictly applicable in our case, because the muffin tin potential
exact relationship between APW matrix elements and local pseudopotential (LP) form factors doesn’t exist but a good approximation will be v~(Es) = [‘APW [kF q, a1(E~)]. (10) In Fig. 2 we show the experimental LP form factors for E,~.= 2.11 c.u. as determined 10 by a direct four parameter fit to the same fourteen points of the FS of white tin, that have been used in the APW fit. The Fermi energy is used as a fifth parameter in the LP form factor fit. The
Vol. 9, No. 22
THE MUFFIN TIN ZERO IN FS INTERPOLATION
value EF = 2.11 ±0.01 c.u. agrees very well with EF = 2.lOc.u. in the APW fit for which FAPW [kF,0,al(Ep)] = 0. Note that the direct LP form factors agree rather well with FAPW [kr, q, a1 (E~)1for EF = E~-/3 (see reference 9). We cannot expect this agreement to be exact for at least two reasons. First, the OFS approximation itself is an approximation of the exact APW matrix element. Secondly, all LP 2kF have been set equal form factors with q~> to zero in the four parameter LP fit, whereas
fAPW
1943
An additional remark concerns the behaviour of F’~ [kr, q, a 1(Ep)} for white tin [Fig. 2] and of Fl~RZ[kp,q, 771(EF)I for the noble metals [Fig. 4] at q = q0, i.e. the point at q/2kp~.0.9 where the matrix element has a node. (For the physical meaning of q0, see for example reference 11.) It is suggested by this behaviour, that q0 and the slope of FAPW, resp. FI~RZ,in q0 are conservedparameters quantitiesintoo, these explicit the although formalisms. In are the not ‘empty core’ model’1 conservation of q 0 means conservation of the core radius and in this sense it seems to be a satisfying result physically.
[ks, a.~,a1 (EF)] is obviously unequal to
zero for these q~. A way of thinking that throws some light upon the problem of the choice of EF is the following. Suppose we have a muffin tin potential V?~which generates the FS correctly for EF= ~ We then add a relatively small potential ~V, which equals c in the interstitial region and whichthat hasthe such a form withintransthe Slater spheres, crystal Fourier forms (~V)qr, = 0 for those few important reciprocal lattice vectors q~for which the structure factor is unequal to zero. This new potential will leave the FS approximately unaltered but with a new value of the Fermi energy, = ~ — c, and with new values of the phase shifts. The more Fourier transforms are important for the FS of a metal (low symmetry and many electrons per primitive cell), the more difficult it will be to fulfill the constraint that (V)~,~ remains the same and in our opinion a ‘better’ determination of E~by FS data may be expected.
Finally it should be noted that FAPW and are not exactly equal in the OFS approximation for the same choice of EF (Fig. 2 and 4). The constraint (1) is somewhat better obeyed than the constraint (6) for both white tin and the noble metals, and moreover, oscil2kF has the a smaller lating tail of FAPW for q > amplitude than that of fKKRZ• This seems to be FKKRZ
consistent who argue converges the KKRZ
with the opinion of several authors ~ that the plane wave representation faster for the APW formalism than for formalism.
E~
Acknowledgements — We wish to thank Mr. N. Coenen for providing us with his results on the noble metals, Mr. MC. Knoope, who performed part of the APW calculations, and Mr. M.J.G. Lee for sending us a preprint.
REFERENCES 1.
DEVILLERS MAC. and DE VROOMEN A.R., Phys. Leit. 30A, 159 (1969).
2.
ZIMAN J.M., Proc. Phys. Soc.,
3.
LLOYD P., Proc. Phys. Soc., 86, 825 (1965).
4.
MATTHEY M.M.M.P. and DE VROOMEN AR., Solid Slate Commun. 9, 1329 (1971).
5.
DEVILLERS MAC., MATTHEY M.M.M.P. and DE VROOMEN A.R., to be published.
6.
COENEN N.J. and DE VROOMEN AR., to be published.
7.
LEE M.J.G., Computational Methods in Band Theory, p. 63, Plenum Press, New York (1971).
8.
BALL M.A., J. Phys. C (Solid State Phys.), 2, 1248 (1g70).
9.
COHEN ML. and HEINE V., Solid State Physics, Vol. 24, p. 243, Academic Press, New York and London (1970).
91, 701 (1967).
1944
THE MUFFIN TIN ZERO IN FS INTERPOLATION
Vol. 9, No.22
10.
DEVILLERS M.A.C. and DE VROOMEN AR., to be published.
11.
ASHCROFT N.W., Phys. Leti.
12.
JOHNSON K.H., Phys. Rev., 150, 429 (1966).
13.
SEGALL B. and HAM S., Methods in Computational Physics, Vol.8, p. 267, Academic Press, New York and London (1968).
14.
MORIARTY J.A., Phys. Rev. B, 1, 1363 (1970).
23, 48 (1966).
Die diagonalen und die wichtigen nicht diagonalen APW-Matrixelemente bleiben in der ‘on the Fermi sphere’ — Näherung für verschiedene Werte des Parameters der Fermienergie E~in einem
APW-Interpolationsschema für die Ferrnifläche von weissem Zinn nãherungsweise erhalten. Dasselbe gilt für die Matrixelemente in einem KKRZ-Interpolationsschema für die Fermiflächen von Cu und Ag. Auf Grund theses empirischen Erhaltungsgesetzes kann man das Verhalten der logarithmischen Ableitung der i-ten radialen Wellenfunktion und der Friedelsumme ZF als Funktion von E~verstehen. Z,- ist danro näherungsweise null für E~-= E~-(Wert für quasifreie Elektronen) und etwa Z (Anzahl der Valenzelektronen) für EF = E~./3. Man erhält einen Eindruck, weiche Rolle E~ in den APW- und KKRZ-Interpolationsschemata spielt, wenn man deren Matrixelemente mit den lokalen Formfaktoren des Pseudopotentials vergleicht.