- Email: [email protected]
On the nature of cohesion in copper
Solid State Communications,Vol. 15, pp. 957—961, 1974.
Pergamon Press.
Printed in Great Britain
ON THE NATURE OF COHESION IN COPPER G. Solt* International Centre for Theoretical Physics, Trieste, Italy and J. Kollar Central Research Institute for Physics, POB 49, Budapest 114, Hungary (Received 15 May 1974 by S. Lundqvist)
The origin of cohesion in copper is discussed on the basis of calculations performed in a simplified nearly-free (s) electron—tight binding (d) electron model. While the atomic radius dependence of the s-electron energy is obtained similar to earlier ab initio results, the d-electron contribution to the total energy is found not to be the traditional monotonic repulsion, but slightly attractive in the neighbourhood of the observed lattice spacing.
TO EXPLAIN the large binding energy E~= 0.257 Ry, small atomic radius R0 = 2.66 a.u. and low compress-
indicating that the interaction between d shells is probably more complex than a simple hard-core repulsion. Finally,where the same effects isalso in those treatments the energy notappear calculated by first principles but represented by analytical expressions involving some parameters. In these cases,7’8 fitting E~,R~and B to the experimental values leads, as a rule, to unreasonable (imaginary) values for the quantity corresponding to an effective core radius.
ibility (l/~= B3asserts = 1.42 that Mbar) copper, thes electrons traditional theofnearly-free description~ would prefer energetically an even smaller volume, due to the incomplete screening of the nuclear charge by d electrons, and it is the noble-gas-like repulsion of the filled d shells which sets a lower limit for Ra. In contrast with this familiar picture, in most of the ab initio calculations1’4’5 the total energy of the nearly-free s band turned out in fact to reach its minimum at a larger atomic radius than that observed, and the discrepancy was, of course, only increased when taking into account the assumed core—core repulsion.t Similarly, all existing calculations agree in that the nearly-free s-electron energy alone leads to too small cohesive energy (50—70 per cent of Er),
*
The question is then whether any model using the simple picture of nearly-free s electrons and a separate band of tightly-bound d electrons must necessarily fail to reproduce, even qualitatively, the basic features of the cohesion in copper (in which case one must take into account the complex band structure (hybridization) of the metal even in the first approximation)9 or only the energy of the tightly-bound d electrons differs significantly from a monotonic (Born—Mayer-type) repulsive potential.
Permanent address: Central Research Institute for Physics, POB 49, Budapest 114, Hungary.
t The only exception is Kambe’s work,6 the disagree-
The present calculations support the second of these alternatives.~leading to an s-electron energy similar to the Wigner—Seitz results of Fuchs and at
ment of which with Fuchs’ results is, however, not clear, 957
958
ON THE NATURE OF COHESION IN COPPER
the same time to a d-electron energy which is slightly attractive near the equilibrium atomic radius and becomes repulsive only for smaller lattice spacings, The model assumes a half-filled band of orthogonalized plane waves (OPW) for s electrons and tightly-bound Wannier states for d electrons. In this first approximation to the total energy, the structure dependence of the s.band energy is neglected and free (unscattered) OPWs are used. The Wannier states are composed of atomic-like d orbitals at each lattice site, orthogonalized to each other up to first order in the overlap parameter S. The atomic-like orbitals are represented by simple analytical expressions, the parameters of which are adjusted by applying the minimum principle to the total metallic energy. The expectation value of the metal Hamiltonian for the above electronic state is E
=
E
=
5 + Ed T5 +E~~ +E~+E~8,
=
T~+E~d+ E~d+ E~
where T8 is the kinetic energy, ~ (d)
(l)
is the electron—ion (d)
core interaction, and E~,Edd and EdS are the (Coulomb and exchange) interactions between the binding s and dneglecting electrons.non-orthogonality Terms with label between 0 should be calculated orbitals at different sites, while the orthogonalization gives rise10 to terms of order S2 denoted by E~.The term E~is called s energy, as in the limit of “small” d cores (R 0 it becomes the s-band energy for a univalent simple metal. The rest of the total energy, Ed, will be called d-electron energy. Further, the first three terms in E8 can be interpreted as the electronic energy of a hypothetical “alkali” metal, with small —~ 00)
cores (according to the extension of the 3sp shells) in1’~ which, however, the nuclear is that of 2’3 charge for describing thethe R Cu ion. An appropriate form 0 dependence of this energy is therefore* —
~
=
zX
1
R3 + 3r~ —~ + 2.21 —~-X+ 0.284 R + Uc(Ra), 0 Ra) R0 (2) 0
where z = 11 is the core charge, Xis the orthogonality hole factor multiplying and the free-electron kinetic energy,3 and u~,ibothz s the correlation energy for *
The energy unit is Rydberg.
Vol. 15, No. 5
s electrons. The only free parameter in this expression is the “Ashcroft core radius”, r~,which is simply related to the average non-coulombic interaction between plane waves and the ion core.11 The OPW formalism gives for X:3 = ~I ~ I(kIw~)II iav’ c, core’
where (kI~~) is the matrix element between the core state ç1~and the plane wave k, and the expression is averaged over the Fermi sea. Neglecting the k dependence of the matrix element and putting k = 0 (which, due to the opposite trends in the k dependence for S and p states,12 is a reasonable approximation) one has ~ If~drI2 r~= 4~score (3) Here the “OPW core radius”, r~,is evidently an atomic parameter determined by the shape of the ~‘°
=
I
—
1 (rc/Ra)3’
~
—~—
outermost (3s) orbital. Obviously, for any reasonable model one should expect that the fitted value ofr~he close to r~,and this.wiIl indeed be seen to be the case. As to E~5and Ed, the approximation of neutral spheres was used to calculate the Coulomb parts of E~8,E~dand Efld. Besides the OPW density of s electrons, the d density in each atomic sphere was assumed to be of the form: 1max 13 () = p~(r)+ ~ c 21, r~
(5
where EE is now understood to include, besides the orthogonality terms, also interatomic d—d exchange
Vol. 15, No. 5
ON THE NATURE OF COHESION IN COPPER
rc1~a
~
r
5
/
(au) 2,5
—
0,6
E (ry)
0.5
0.1
-
0,02 0
-
959
~ -
0,4
‘C
I 2.5
2.0
3.
-
—0,02
.
Ed_Ed
—0.1 1,5
-
E -0,2 1,0
I 1<
I I I I I I I ScTi V CrMr,FeCoNi Cu
FIG. 1. The OPW core radii for the iron group (equation 3). The r, values determined from lattice equilibrium conditions (for K taken from reference 19, for Cu taken from the present result) are mdicated by crosses.2°for The K. dotThe shows there/Ra “electronic” ratio is plotted by the dotted line. Ashcroft radius
(which is also of order ~2 or higher). In the definition of the overlap parameterS, R2 is the radial part of the d orbital, Rn,, is the distance between neighbouring sites, and C is14 an between amplitude parameter. to the different termsDue involved large cancellation in E~the calculation of C is rather difficult, it is therefore included here as the second parameter of the model which, together with r~,will be determined from the conditions of lattice equilibrium. One notices that, in contrast with the usual exchange charge treatments14 where E~is taken to represent all overlap effects in the energy, here the Coulomb part of the d energy was calculated separately in the closed framework of the neutral sphere model. Also, in determining E~ 8,the readjustment of the d density (as described by Pd) could be properly taken into account. Despite the ratheruse crude simplifications (neglect of hybridization, of free OPW orthogonalized only to inner s states, etc.) the model is thought to contain most of the essential features of the R0 dependence of the total energy and at the same time it yields a very simply analytical framework for calculation. In fact, taking ford orbitals a combination of two Slater-type functions,’5 2e~a1r +A R2(r)
=
A~r
2e~2r, 2r
FIG. 2. Total energy vsR0 for copper. BesidesE, the contributions E~and Ed are also plotted. ~ is the experimental cohesive energy 0.257 Ri’, I = 0.5677 Ry is the first ionization potential and E,~is the value of Ed for separate atoms. with three parameters a 1, a2 and A1/A2, all terms in the total energy (I) reduce to combinations of exponential and polynomial functions. The procedure is then to use some starting values (e.g. atomic) for (a1 A,) to calculate the pressure and the bulk modulus and, by fitting these to the experimental data, the parameters r, and C are determined. The “metallic” values for (a, A) are then found by varying them to minimize the total energy. The procedure should, in principle, be repeated but the first variation already determines the wave-function parameters with adequate accuracy. The results can be summarized as follows: (a) The atomic-like orbitals tend to spread out slightly in the metal, as compared with the atomic case, as may be expected. One may notice, incidentally, the equally slight shrinking of 6the original atomic for an alkali halide.’ (b) orbitals The fitting at lattice equilibrium leads to r 1 = 1 .2852 a.u. which correlates remarkably well with the ab nitio OPW core radius r~= 1 .2695 a.u. This is seen in Fig. 1, where the OPW core radii, as defined by (3) and calculated by using simple 5 are plotted from K to Cu analytical s functions,’ and compared with the r 1’s found from the lattice equilibrium requirements. The fact that the
960
ON THE NATURE OF COHESION IN COPPER
“dynamical” value of the Ashcroft radius r1 is reasonably close to the “atomic” r~not only for an alkali metal but for copper as well, despite the large change in the scale, suggests that the definition of the s energy E~in its present form (2) is meaningful. As to the exchange amplitude fitting leads to Ccharge = 261.7 a.u. Thisparameter, value seemsthe to be realistic when compared with the traditional results,’4 taking into account the fact that here E~does not contain the Coulomb-type overlap energy. (c) The main result of the present calculations can be seen in Fig. 2, where the total energy and its components E~and Ed are plotted. The scale is such that E~is measured relative to the first ionization potential —Ill, and from Ed the corresponding atomic value (obtained by a Hartree—Fock calculation using the same set of analytical d orbitals) is subtracted. One sees that takingE 5 alone would result in a slightly too largeR0 and a too small curvature (bulk modulus) and cohesive energy, precisely as in Fuchs’ result. The fact that the s electrons actually would adjust an R0 which is rather close to the observed atomic radius, and not a significantly smaller one, is already indicated by the value of r~/R0(Fig. I), this ratio being almost the same for both K and Cu. (Incidentally, we notice the characteristic deviation of r~/R0from this value for real transition elements, where the band-width contribution d electrons 7 the atomic radius.)ofAs to the d-electendsenergy, to shrink’ tron we see that Ed is not the commonly assumed monotonic repulsion; on the contrary, it has
Vol. 15, No. 5
a minimum near the equilibrium R0 and it contributes to E~,making Rasomewhat smaller and E~larger than E8 alone would suggest. Physically, this behaviour of Ed is a result of the competition between,the attractive Coulomb energy of the d density in the field 1’ ions on the one hand, and thePd“exchange of Cu” charge” (orthogonality) repulsion, E~,on the other. Although to get any attraction at all due to the d shells of copper would seem surprising, in view of the well-known repulsion between filled shell atoms, one must realize that Ed is, by definition, not the interaction energy between two neutral noble gas atoms, due to the enhanced charge on the Cu11~ion, and that the balance of attractive and repulsive components in these cases is rather delicate. In this connection, one may refer to the example of the He— Li~system, where the interaction potential18 was also found to be slightly attractive. Details of the calculation and further results such as, for example, the equation of state for copper and the dependence of one-electron levels on lattice spacing will be published later.
Acknowledgements One of us (SG) is indebted to Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the Prof. 1. Wailer continuous interest in Interthis work, national Centrefor forhisTheoretical Physics, Trieste, to and to Profs. S. Lundqvist and N.H. March for their valuable comments. —
REFERENCES 1.
FUCHS K.,&oc. R. Soc. (London) A151, 585 (1935).
2.
BROOKS H., Nuovo Cimento Suppl. 7, 165 (1958).
3. 4.
HEINE V. and WEAIRE D.,Solid State Phys. 24, 249 (1970) (edited by EHRENREICH H., SEITZ F. and TURNBULL D.). HAYES T.M. and YOUNG W.H., Phil. Mag. 18, 965 (1968).
5.
LADANY! K.,ActaPhys. Acad. Sci. Hung. 5, 361 (1955).
6.
KAMBE K., Phys. Rev. 99, 419 (1955).
7.
JASWAL S.S. and GIRIFALCO L.A.,J. Phys. Chem. Solids 28,457 (1967). HSIEH K. and BOLSAITIS P.J., J. Phys. Chem. Solids 33, 1838 (1972).
8. 9. 10.
WATSON R.E. and EHRENREICH H., Comments on Solid State Phys. 3, 109 (1970). LOWDIN P.O.,Adv. Phys. 5, 1(1956).
Vol. 15, No. 5
ON THE NATURE OF COHESION IN COPPER
961
11.
ASHCROFT N.W. and LANGRETH D.C.,Phys. Rev. 155, 682 (1967).
12.
KLEINMAN L.,Phys. Rev. 146,472 (1966).
13.
STERN F.,Phys. Rev. 116, 1399 (1959).
14. .15.
DICK B.G. and OVERHAUSER A.W.,Phys. Rev. 112,90(1958); HAFEMEISTER D.W.,J. Phys. Chem. Solids 30, 117 (1969). SOLT G. and KOLLAR J.,J. Phys. B 5, Ll24 (1972); KOLLAR I. and SOLT G.,J Phys. B 6,329 (1973).
16.
LUNDQVIST S.,Ark. Fys. 8, 177 (1954).
17.
KOLLAR J. and SOLT G.,J. Phys. Chem. Solids 33, 651 (1972).
18. 19.
SCHNEIDERMAN S.B. and MICHELS H.H.,J. Chem. Phys. 42, 3706 (1965). BROVMAN E.G., KAGAN YU. and HOLAS A.,SovietPhys. Solid State 12, 1001 (1970).
20.
ASHCROFT N.W.,J. Phys. Cl, 232 (1968).
Pergamon Press.
Printed in Great Britain
ON THE NATURE OF COHESION IN COPPER G. Solt* International Centre for Theoretical Physics, Trieste, Italy and J. Kollar Central Research Institute for Physics, POB 49, Budapest 114, Hungary (Received 15 May 1974 by S. Lundqvist)
The origin of cohesion in copper is discussed on the basis of calculations performed in a simplified nearly-free (s) electron—tight binding (d) electron model. While the atomic radius dependence of the s-electron energy is obtained similar to earlier ab initio results, the d-electron contribution to the total energy is found not to be the traditional monotonic repulsion, but slightly attractive in the neighbourhood of the observed lattice spacing.
TO EXPLAIN the large binding energy E~= 0.257 Ry, small atomic radius R0 = 2.66 a.u. and low compress-
indicating that the interaction between d shells is probably more complex than a simple hard-core repulsion. Finally,where the same effects isalso in those treatments the energy notappear calculated by first principles but represented by analytical expressions involving some parameters. In these cases,7’8 fitting E~,R~and B to the experimental values leads, as a rule, to unreasonable (imaginary) values for the quantity corresponding to an effective core radius.
ibility (l/~= B3asserts = 1.42 that Mbar) copper, thes electrons traditional theofnearly-free description~ would prefer energetically an even smaller volume, due to the incomplete screening of the nuclear charge by d electrons, and it is the noble-gas-like repulsion of the filled d shells which sets a lower limit for Ra. In contrast with this familiar picture, in most of the ab initio calculations1’4’5 the total energy of the nearly-free s band turned out in fact to reach its minimum at a larger atomic radius than that observed, and the discrepancy was, of course, only increased when taking into account the assumed core—core repulsion.t Similarly, all existing calculations agree in that the nearly-free s-electron energy alone leads to too small cohesive energy (50—70 per cent of Er),
*
The question is then whether any model using the simple picture of nearly-free s electrons and a separate band of tightly-bound d electrons must necessarily fail to reproduce, even qualitatively, the basic features of the cohesion in copper (in which case one must take into account the complex band structure (hybridization) of the metal even in the first approximation)9 or only the energy of the tightly-bound d electrons differs significantly from a monotonic (Born—Mayer-type) repulsive potential.
Permanent address: Central Research Institute for Physics, POB 49, Budapest 114, Hungary.
t The only exception is Kambe’s work,6 the disagree-
The present calculations support the second of these alternatives.~leading to an s-electron energy similar to the Wigner—Seitz results of Fuchs and at
ment of which with Fuchs’ results is, however, not clear, 957
958
ON THE NATURE OF COHESION IN COPPER
the same time to a d-electron energy which is slightly attractive near the equilibrium atomic radius and becomes repulsive only for smaller lattice spacings, The model assumes a half-filled band of orthogonalized plane waves (OPW) for s electrons and tightly-bound Wannier states for d electrons. In this first approximation to the total energy, the structure dependence of the s.band energy is neglected and free (unscattered) OPWs are used. The Wannier states are composed of atomic-like d orbitals at each lattice site, orthogonalized to each other up to first order in the overlap parameter S. The atomic-like orbitals are represented by simple analytical expressions, the parameters of which are adjusted by applying the minimum principle to the total metallic energy. The expectation value of the metal Hamiltonian for the above electronic state is E
=
E
=
5 + Ed T5 +E~~ +E~+E~8,
=
T~+E~d+ E~d+ E~
where T8 is the kinetic energy, ~ (d)
(l)
is the electron—ion (d)
core interaction, and E~,Edd and EdS are the (Coulomb and exchange) interactions between the binding s and dneglecting electrons.non-orthogonality Terms with label between 0 should be calculated orbitals at different sites, while the orthogonalization gives rise10 to terms of order S2 denoted by E~.The term E~is called s energy, as in the limit of “small” d cores (R 0 it becomes the s-band energy for a univalent simple metal. The rest of the total energy, Ed, will be called d-electron energy. Further, the first three terms in E8 can be interpreted as the electronic energy of a hypothetical “alkali” metal, with small —~ 00)
cores (according to the extension of the 3sp shells) in1’~ which, however, the nuclear is that of 2’3 charge for describing thethe R Cu ion. An appropriate form 0 dependence of this energy is therefore* —
~
=
zX
1
R3 + 3r~ —~ + 2.21 —~-X+ 0.284 R + Uc(Ra), 0 Ra) R0 (2) 0
where z = 11 is the core charge, Xis the orthogonality hole factor multiplying and the free-electron kinetic energy,3 and u~,ibothz s the correlation energy for *
The energy unit is Rydberg.
Vol. 15, No. 5
s electrons. The only free parameter in this expression is the “Ashcroft core radius”, r~,which is simply related to the average non-coulombic interaction between plane waves and the ion core.11 The OPW formalism gives for X:3 = ~I ~ I(kIw~)II iav’ c, core’
where (kI~~) is the matrix element between the core state ç1~and the plane wave k, and the expression is averaged over the Fermi sea. Neglecting the k dependence of the matrix element and putting k = 0 (which, due to the opposite trends in the k dependence for S and p states,12 is a reasonable approximation) one has ~ If~drI2 r~= 4~score (3) Here the “OPW core radius”, r~,is evidently an atomic parameter determined by the shape of the ~‘°
=
I
—
1 (rc/Ra)3’
~
—~—
outermost (3s) orbital. Obviously, for any reasonable model one should expect that the fitted value ofr~he close to r~,and this.wiIl indeed be seen to be the case. As to E~5and Ed, the approximation of neutral spheres was used to calculate the Coulomb parts of E~8,E~dand Efld. Besides the OPW density of s electrons, the d density in each atomic sphere was assumed to be of the form: 1max 13 () = p~(r)+ ~ c 21, r~
(5
where EE is now understood to include, besides the orthogonality terms, also interatomic d—d exchange
Vol. 15, No. 5
ON THE NATURE OF COHESION IN COPPER
rc1~a
~
r
5
/
(au) 2,5
—
0,6
E (ry)
0.5
0.1
-
0,02 0
-
959
~ -
0,4
‘C
I 2.5
2.0
3.
-
—0,02
.
Ed_Ed
—0.1 1,5
-
E -0,2 1,0
I 1<
I I I I I I I ScTi V CrMr,FeCoNi Cu
FIG. 1. The OPW core radii for the iron group (equation 3). The r, values determined from lattice equilibrium conditions (for K taken from reference 19, for Cu taken from the present result) are mdicated by crosses.2°for The K. dotThe shows there/Ra “electronic” ratio is plotted by the dotted line. Ashcroft radius
(which is also of order ~2 or higher). In the definition of the overlap parameterS, R2 is the radial part of the d orbital, Rn,, is the distance between neighbouring sites, and C is14 an between amplitude parameter. to the different termsDue involved large cancellation in E~the calculation of C is rather difficult, it is therefore included here as the second parameter of the model which, together with r~,will be determined from the conditions of lattice equilibrium. One notices that, in contrast with the usual exchange charge treatments14 where E~is taken to represent all overlap effects in the energy, here the Coulomb part of the d energy was calculated separately in the closed framework of the neutral sphere model. Also, in determining E~ 8,the readjustment of the d density (as described by Pd) could be properly taken into account. Despite the ratheruse crude simplifications (neglect of hybridization, of free OPW orthogonalized only to inner s states, etc.) the model is thought to contain most of the essential features of the R0 dependence of the total energy and at the same time it yields a very simply analytical framework for calculation. In fact, taking ford orbitals a combination of two Slater-type functions,’5 2e~a1r +A R2(r)
=
A~r
2e~2r, 2r
FIG. 2. Total energy vsR0 for copper. BesidesE, the contributions E~and Ed are also plotted. ~ is the experimental cohesive energy 0.257 Ri’, I = 0.5677 Ry is the first ionization potential and E,~is the value of Ed for separate atoms. with three parameters a 1, a2 and A1/A2, all terms in the total energy (I) reduce to combinations of exponential and polynomial functions. The procedure is then to use some starting values (e.g. atomic) for (a1 A,) to calculate the pressure and the bulk modulus and, by fitting these to the experimental data, the parameters r, and C are determined. The “metallic” values for (a, A) are then found by varying them to minimize the total energy. The procedure should, in principle, be repeated but the first variation already determines the wave-function parameters with adequate accuracy. The results can be summarized as follows: (a) The atomic-like orbitals tend to spread out slightly in the metal, as compared with the atomic case, as may be expected. One may notice, incidentally, the equally slight shrinking of 6the original atomic for an alkali halide.’ (b) orbitals The fitting at lattice equilibrium leads to r 1 = 1 .2852 a.u. which correlates remarkably well with the ab nitio OPW core radius r~= 1 .2695 a.u. This is seen in Fig. 1, where the OPW core radii, as defined by (3) and calculated by using simple 5 are plotted from K to Cu analytical s functions,’ and compared with the r 1’s found from the lattice equilibrium requirements. The fact that the
960
ON THE NATURE OF COHESION IN COPPER
“dynamical” value of the Ashcroft radius r1 is reasonably close to the “atomic” r~not only for an alkali metal but for copper as well, despite the large change in the scale, suggests that the definition of the s energy E~in its present form (2) is meaningful. As to the exchange amplitude fitting leads to Ccharge = 261.7 a.u. Thisparameter, value seemsthe to be realistic when compared with the traditional results,’4 taking into account the fact that here E~does not contain the Coulomb-type overlap energy. (c) The main result of the present calculations can be seen in Fig. 2, where the total energy and its components E~and Ed are plotted. The scale is such that E~is measured relative to the first ionization potential —Ill, and from Ed the corresponding atomic value (obtained by a Hartree—Fock calculation using the same set of analytical d orbitals) is subtracted. One sees that takingE 5 alone would result in a slightly too largeR0 and a too small curvature (bulk modulus) and cohesive energy, precisely as in Fuchs’ result. The fact that the s electrons actually would adjust an R0 which is rather close to the observed atomic radius, and not a significantly smaller one, is already indicated by the value of r~/R0(Fig. I), this ratio being almost the same for both K and Cu. (Incidentally, we notice the characteristic deviation of r~/R0from this value for real transition elements, where the band-width contribution d electrons 7 the atomic radius.)ofAs to the d-electendsenergy, to shrink’ tron we see that Ed is not the commonly assumed monotonic repulsion; on the contrary, it has
Vol. 15, No. 5
a minimum near the equilibrium R0 and it contributes to E~,making Rasomewhat smaller and E~larger than E8 alone would suggest. Physically, this behaviour of Ed is a result of the competition between,the attractive Coulomb energy of the d density in the field 1’ ions on the one hand, and thePd“exchange of Cu” charge” (orthogonality) repulsion, E~,on the other. Although to get any attraction at all due to the d shells of copper would seem surprising, in view of the well-known repulsion between filled shell atoms, one must realize that Ed is, by definition, not the interaction energy between two neutral noble gas atoms, due to the enhanced charge on the Cu11~ion, and that the balance of attractive and repulsive components in these cases is rather delicate. In this connection, one may refer to the example of the He— Li~system, where the interaction potential18 was also found to be slightly attractive. Details of the calculation and further results such as, for example, the equation of state for copper and the dependence of one-electron levels on lattice spacing will be published later.
Acknowledgements One of us (SG) is indebted to Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the Prof. 1. Wailer continuous interest in Interthis work, national Centrefor forhisTheoretical Physics, Trieste, to and to Profs. S. Lundqvist and N.H. March for their valuable comments. —
REFERENCES 1.
FUCHS K.,&oc. R. Soc. (London) A151, 585 (1935).
2.
BROOKS H., Nuovo Cimento Suppl. 7, 165 (1958).
3. 4.
HEINE V. and WEAIRE D.,Solid State Phys. 24, 249 (1970) (edited by EHRENREICH H., SEITZ F. and TURNBULL D.). HAYES T.M. and YOUNG W.H., Phil. Mag. 18, 965 (1968).
5.
LADANY! K.,ActaPhys. Acad. Sci. Hung. 5, 361 (1955).
6.
KAMBE K., Phys. Rev. 99, 419 (1955).
7.
JASWAL S.S. and GIRIFALCO L.A.,J. Phys. Chem. Solids 28,457 (1967). HSIEH K. and BOLSAITIS P.J., J. Phys. Chem. Solids 33, 1838 (1972).
8. 9. 10.
WATSON R.E. and EHRENREICH H., Comments on Solid State Phys. 3, 109 (1970). LOWDIN P.O.,Adv. Phys. 5, 1(1956).
Vol. 15, No. 5
ON THE NATURE OF COHESION IN COPPER
961
11.
ASHCROFT N.W. and LANGRETH D.C.,Phys. Rev. 155, 682 (1967).
12.
KLEINMAN L.,Phys. Rev. 146,472 (1966).
13.
STERN F.,Phys. Rev. 116, 1399 (1959).
14. .15.
DICK B.G. and OVERHAUSER A.W.,Phys. Rev. 112,90(1958); HAFEMEISTER D.W.,J. Phys. Chem. Solids 30, 117 (1969). SOLT G. and KOLLAR J.,J. Phys. B 5, Ll24 (1972); KOLLAR I. and SOLT G.,J Phys. B 6,329 (1973).
16.
LUNDQVIST S.,Ark. Fys. 8, 177 (1954).
17.
KOLLAR J. and SOLT G.,J. Phys. Chem. Solids 33, 651 (1972).
18. 19.
SCHNEIDERMAN S.B. and MICHELS H.H.,J. Chem. Phys. 42, 3706 (1965). BROVMAN E.G., KAGAN YU. and HOLAS A.,SovietPhys. Solid State 12, 1001 (1970).
20.
ASHCROFT N.W.,J. Phys. Cl, 232 (1968).