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Zero-point energy corrections to the interlayer spacing and the C33 elastic constant of graphite
Curhon. 1975. Vol
13. pp. 43X
Pergamon Press.
Printed in Great Britam
ZERO-POINT ENERGY CORRECTIONS TO THE INTERLAYER SPACING AND THE C,, ELASTIC CONSTANT OF GRAPHITE B. T. KELLY and M. ESLICK (Mrss)t UKAEA Reactor Fuel Element Laboratories, SpringfieldsWorks Salwick. Nr Preston, England (Received 14 June 1974)
Abstract-Calculations are made of the effect of the zero point lattice vibrations in the out-of-plane mode on the interlayer spacing and the C,, elastic constant of a graphite crystal. It is found that these vibrations expand the lattice by about 0.5% and contribute 0.24 x 10” dynes/cm’ to C,,.
1. INTRODUCTION
where U, is the potential energy of the lattice due to the interatomic forces, vi is the i’th lattice vibration frequency, h is Plancks constant, k is Boltzmann’s constant and the summation is taken over all modes in unit volume. At 0°K Equation 1 is:
The properties of crystals, such as the elastic constants, are often calculated using interatomic pair potentials[i, 31. The parameters of the potentials are chosen to give zero derivatives with respect to strain at the equilibrium lattice spacing and their validity is assessed by comparing measured and calculated values of the elastic constants. It is occasionally noted that the presence of the zero point lattice vibrations modifies the equilibrium lattice spacing and contributes to the elastic constantsf3-51, but there are very few estimates of the magnitude of these effects. The graphite lattice should be amenable to fairly accurate estimates of the magnitude of these effects as related to interlayer forces since these have been the subject of extensive theoretical and experimental study recently&-81. A simple but realistic model of the lattice vibration spectrum due to Komatsu[9] is available for these calculations. The variation of the elastic constant Ca with temperature, neglecting the zero point effects, has recently been calculated[lO]. In this paper we have made simple analytic calculations of the effect of the zero point vibrations on the interlayer spacing at 0°K and the C,3 elasttc constant,
F=
I&+?$.
(2)
The equilibrium spacing is now determined by:
where the et, are the independent elements of the strain tensor. The usual equilibrium condition considered is (a&/de,,) = 0. If we appfy the theory to interlayer forces in graphite associated with strain e2, parallel to the c-axis, we have: (4) and
2. THEORY
Consider a unit cube of graphite crystal with the edges parallel to the axes of a Cartesian co-ordinate system, the z-axis coinciding with the crystal c-axis. The free energy of the crystal at temperature T “K is [3] F=
U,,+c
4
[~+kTln {1-exp(-hu,/kT)}].
at 0°K where C1, is the usual elastic constant. The terms containing the differentials of the lattice vibration frequencies vi may be evaluated using the lattice dynamics for graphite proposed by Komatsu[9]. Kelly [lo,II] has shown that a strain e,, only affects the ‘out-of-plane’ lattice vibrational modes in this model, In the Komatsu model of the lattice dynamics, the dispersion
(I)
*Sandwich Course Student at the Reactor Fuel Laboratories, from City Univ. London. 43
44
B. T.
KELLY M. ESLICK
relation for the out-of-plane acoustic mode is I/2
C44 C33 22 sin* 7~du, t - ua2 , P P= d
@J)
where Us and F; are wave number components parallel and perpendicular to the basal planes. p is the density, d is the interlayer spacing, C,, and CU are the usual elastic constants and S is the bond bending coefficient defined by Komatsu[9] which measures the flexural rigidity of a single layer plane. The differentials of (6) with respect to strain are found
(~)=-f(4~282~~+~sin2rd~~+~n.i)l’2
and
Ca= (2)
+;l,“/_‘;rd
C3, t~sin’nd~~t--ad_ PP
d
?mr+4a’Bi~. C44
-I,2
P
In this analysis, u,,, is the equivalent radius of either the first or the extended Brillouin zone.
X[8&(E)&$$(2) t$
(z)rti
(4n2S2~~~+~sin2
rr dgz
3. NUMERICALRESULTS
The integrals in Equations (9) and (10) have been carried out using the following numerical values: d = 3.3535 x lo-* cm S = 6.11 x lo-'cm2/sec2 The summations in Equations (4) and (5) are replaced by an integration over the first Brillouin zone for the acoustic mode and an extended Brillouin zone for the acoustic + optical modes according to the usual prescription. Equations (7) and (8) may be simplified by consideration of the various strain dependent parameters. The elastic constants C,, and C, are determined by the interlayer forces and thus will be sensitive to e,,, but S is a property of the intralayer forces which are practically unaffected by interlayer strain. Thus we assume all differentials of S with respect to ez, to be zero. This approximation has previously been applied to calculations of the temperature variation of Cjj [ lo] and the hexagonal axis expansion coefficient (Y,[lo] with success. If we make this approximation and replace the sum in Equations (4) and (5) by integrals:
C, = 4.0 x 10” dyne/cm’ CXS= 3.65 x 10” dyne/cm*. The following values of (aMae,,) and (aC&err) been obtained by Green et al. [7,8]:
have
(aC,,/ae,,) = -5.5 x lOI dyne/cm’ (aC,/ae,,) = -2.4 x 10” dyne/cm’. The other parameters required are (a*C,,/ae&) and (a2C,/aei,) which have been estimated[lOl to be (a’C,,/aet,) = +9.07x 1013dyne/cm2 (a’C,/ae:,) = +1.44x 10” dyne/cm’. The integrals II and Z2 were evaluated for (r, = 2.46 x IO’cm-’ and 3.47 x 10’cm-‘, which correspond to summation over the acoustic mode and acoustic and
Zero-point energy corrections to the interlayer spacing and the C,, elastic constant of graphite optical modes, respectively. The vaiues obtained were
I, = t~39x?09dynejcm2
for
am =246X 107cm-L
11= 1.95x 10’dyne~cm’ for IZ= I.72 x 10’”dyne/cm2
o;, = 3-47 x IO’cm-
for
u,,, = 2.46 x 10’cm””
Iz = 2.36 x 10” dyne/cm2 for
a,,, = 3.47 x lo7 cm-‘.
4.D~SCUS~ON
The values of It and IZ are principally determined by the out-of-plane acoustic mode. The magnitude of the contribution of the zero point vibrations to CU is 0.24 x 10” dyne/cm’, which is about 6% of the value of C,1. A number of calculations of C33have been made using interatomic pair potentials [6-81, and other methods 112,131, and these should be corrected for the effect described here. The effect of the zero point vibrations in Equation (9) is to expand the interlayer spacing. If the magnitude is small, a simple way of demonstrating the effect of the vibrations is to write (H.J/&h) =
6
=
Cj3enr
=
I ,.
(11)
45
The magnitude of eZ2 obtained in this way by using 1, = 1.95 x 10”dyne/cm’ is 0.53%, and thus the usual neglect of this effect in determining the equilibrium spacing is justified. Ac~~ow~e~ge~e~~s-We are indebted to Mr M. Toes of the UKAEA Central Technical Services for programming the
integralsand to Mr R. V. Moore. UKAEA Member for Reactors for permission to publish. 5.REBECCA 1. Hirschfelder J. O., Curtiss C. F. and Bird R. B., ~o~e~~~ar Theory of Gases and Liquids. Wiley, New York (1954). 2. Pawley G. S., Pkys. Status Sotidi 30, 347 (1%7). 3. Mott N. F. and Jones H., Tkeo~ of the Prope~i~s of Metros and Alloys. Oxford U.P. (1936). 4. Wallace D. C., Solid State Physics. Vol 25 p 301. Academic Press. New York (1970). 5. Kitaigorodsky A. I., Molecular Crystals and Molecules. Academic Press. New York 1973. 6. Kelly B. T. and Duff M. J., CQr6o~ 8, 77 (1970). 7. Green J. F., Bolsaitis P. and Spain I. L., J. Pkys. Ckem. Solids. 34, 1927(1973). 8. Green J. F. and Spain I. L., .I Pkys. Ckem. Solids. 34, 2177 (1973). 9. Komatsu K., J. Pkys. Ckem. Wids. 6 380 (19%). 10.Kelly B. T., Carbon, to be published. 11. Kelly B. T., Carbon 11 379 (1973). 12. Santos E. And Villagra A., Pkys. Reu. B6, p 3134 (1972). 13. Brennan R. O., J. Ckem. Pkys. 20 40 (1951).
13. pp. 43X
Pergamon Press.
Printed in Great Britam
ZERO-POINT ENERGY CORRECTIONS TO THE INTERLAYER SPACING AND THE C,, ELASTIC CONSTANT OF GRAPHITE B. T. KELLY and M. ESLICK (Mrss)t UKAEA Reactor Fuel Element Laboratories, SpringfieldsWorks Salwick. Nr Preston, England (Received 14 June 1974)
Abstract-Calculations are made of the effect of the zero point lattice vibrations in the out-of-plane mode on the interlayer spacing and the C,, elastic constant of a graphite crystal. It is found that these vibrations expand the lattice by about 0.5% and contribute 0.24 x 10” dynes/cm’ to C,,.
1. INTRODUCTION
where U, is the potential energy of the lattice due to the interatomic forces, vi is the i’th lattice vibration frequency, h is Plancks constant, k is Boltzmann’s constant and the summation is taken over all modes in unit volume. At 0°K Equation 1 is:
The properties of crystals, such as the elastic constants, are often calculated using interatomic pair potentials[i, 31. The parameters of the potentials are chosen to give zero derivatives with respect to strain at the equilibrium lattice spacing and their validity is assessed by comparing measured and calculated values of the elastic constants. It is occasionally noted that the presence of the zero point lattice vibrations modifies the equilibrium lattice spacing and contributes to the elastic constantsf3-51, but there are very few estimates of the magnitude of these effects. The graphite lattice should be amenable to fairly accurate estimates of the magnitude of these effects as related to interlayer forces since these have been the subject of extensive theoretical and experimental study recently&-81. A simple but realistic model of the lattice vibration spectrum due to Komatsu[9] is available for these calculations. The variation of the elastic constant Ca with temperature, neglecting the zero point effects, has recently been calculated[lO]. In this paper we have made simple analytic calculations of the effect of the zero point vibrations on the interlayer spacing at 0°K and the C,3 elasttc constant,
F=
I&+?$.
(2)
The equilibrium spacing is now determined by:
where the et, are the independent elements of the strain tensor. The usual equilibrium condition considered is (a&/de,,) = 0. If we appfy the theory to interlayer forces in graphite associated with strain e2, parallel to the c-axis, we have: (4) and
2. THEORY
Consider a unit cube of graphite crystal with the edges parallel to the axes of a Cartesian co-ordinate system, the z-axis coinciding with the crystal c-axis. The free energy of the crystal at temperature T “K is [3] F=
U,,+c
4
[~+kTln {1-exp(-hu,/kT)}].
at 0°K where C1, is the usual elastic constant. The terms containing the differentials of the lattice vibration frequencies vi may be evaluated using the lattice dynamics for graphite proposed by Komatsu[9]. Kelly [lo,II] has shown that a strain e,, only affects the ‘out-of-plane’ lattice vibrational modes in this model, In the Komatsu model of the lattice dynamics, the dispersion
(I)
*Sandwich Course Student at the Reactor Fuel Laboratories, from City Univ. London. 43
44
B. T.
KELLY M. ESLICK
relation for the out-of-plane acoustic mode is I/2
C44 C33 22 sin* 7~du, t - ua2 , P P= d
@J)
where Us and F; are wave number components parallel and perpendicular to the basal planes. p is the density, d is the interlayer spacing, C,, and CU are the usual elastic constants and S is the bond bending coefficient defined by Komatsu[9] which measures the flexural rigidity of a single layer plane. The differentials of (6) with respect to strain are found
(~)=-f(4~282~~+~sin2rd~~+~n.i)l’2
and
Ca= (2)
+;l,“/_‘;rd
C3, t~sin’nd~~t--ad_ PP
d
?mr+4a’Bi~. C44
-I,2
P
In this analysis, u,,, is the equivalent radius of either the first or the extended Brillouin zone.
X[8&(E)&$$(2) t$
(z)rti
(4n2S2~~~+~sin2
rr dgz
3. NUMERICALRESULTS
The integrals in Equations (9) and (10) have been carried out using the following numerical values: d = 3.3535 x lo-* cm S = 6.11 x lo-'cm2/sec2 The summations in Equations (4) and (5) are replaced by an integration over the first Brillouin zone for the acoustic mode and an extended Brillouin zone for the acoustic + optical modes according to the usual prescription. Equations (7) and (8) may be simplified by consideration of the various strain dependent parameters. The elastic constants C,, and C, are determined by the interlayer forces and thus will be sensitive to e,,, but S is a property of the intralayer forces which are practically unaffected by interlayer strain. Thus we assume all differentials of S with respect to ez, to be zero. This approximation has previously been applied to calculations of the temperature variation of Cjj [ lo] and the hexagonal axis expansion coefficient (Y,[lo] with success. If we make this approximation and replace the sum in Equations (4) and (5) by integrals:
C, = 4.0 x 10” dyne/cm’ CXS= 3.65 x 10” dyne/cm*. The following values of (aMae,,) and (aC&err) been obtained by Green et al. [7,8]:
have
(aC,,/ae,,) = -5.5 x lOI dyne/cm’ (aC,/ae,,) = -2.4 x 10” dyne/cm’. The other parameters required are (a*C,,/ae&) and (a2C,/aei,) which have been estimated[lOl to be (a’C,,/aet,) = +9.07x 1013dyne/cm2 (a’C,/ae:,) = +1.44x 10” dyne/cm’. The integrals II and Z2 were evaluated for (r, = 2.46 x IO’cm-’ and 3.47 x 10’cm-‘, which correspond to summation over the acoustic mode and acoustic and
Zero-point energy corrections to the interlayer spacing and the C,, elastic constant of graphite optical modes, respectively. The vaiues obtained were
I, = t~39x?09dynejcm2
for
am =246X 107cm-L
11= 1.95x 10’dyne~cm’ for IZ= I.72 x 10’”dyne/cm2
o;, = 3-47 x IO’cm-
for
u,,, = 2.46 x 10’cm””
Iz = 2.36 x 10” dyne/cm2 for
a,,, = 3.47 x lo7 cm-‘.
4.D~SCUS~ON
The values of It and IZ are principally determined by the out-of-plane acoustic mode. The magnitude of the contribution of the zero point vibrations to CU is 0.24 x 10” dyne/cm’, which is about 6% of the value of C,1. A number of calculations of C33have been made using interatomic pair potentials [6-81, and other methods 112,131, and these should be corrected for the effect described here. The effect of the zero point vibrations in Equation (9) is to expand the interlayer spacing. If the magnitude is small, a simple way of demonstrating the effect of the vibrations is to write (H.J/&h) =
6
=
Cj3enr
=
I ,.
(11)
45
The magnitude of eZ2 obtained in this way by using 1, = 1.95 x 10”dyne/cm’ is 0.53%, and thus the usual neglect of this effect in determining the equilibrium spacing is justified. Ac~~ow~e~ge~e~~s-We are indebted to Mr M. Toes of the UKAEA Central Technical Services for programming the
integralsand to Mr R. V. Moore. UKAEA Member for Reactors for permission to publish. 5.REBECCA 1. Hirschfelder J. O., Curtiss C. F. and Bird R. B., ~o~e~~~ar Theory of Gases and Liquids. Wiley, New York (1954). 2. Pawley G. S., Pkys. Status Sotidi 30, 347 (1%7). 3. Mott N. F. and Jones H., Tkeo~ of the Prope~i~s of Metros and Alloys. Oxford U.P. (1936). 4. Wallace D. C., Solid State Physics. Vol 25 p 301. Academic Press. New York (1970). 5. Kitaigorodsky A. I., Molecular Crystals and Molecules. Academic Press. New York 1973. 6. Kelly B. T. and Duff M. J., CQr6o~ 8, 77 (1970). 7. Green J. F., Bolsaitis P. and Spain I. L., J. Pkys. Ckem. Solids. 34, 1927(1973). 8. Green J. F. and Spain I. L., .I Pkys. Ckem. Solids. 34, 2177 (1973). 9. Komatsu K., J. Pkys. Ckem. Wids. 6 380 (19%). 10.Kelly B. T., Carbon, to be published. 11. Kelly B. T., Carbon 11 379 (1973). 12. Santos E. And Villagra A., Pkys. Reu. B6, p 3134 (1972). 13. Brennan R. O., J. Ckem. Pkys. 20 40 (1951).