- Email: [email protected]
Temperature and pressure dependence of the isotopic shift in diamonds
22 November 1999
Physics Letters A 263 Ž1999. 123–126 www.elsevier.nlrlocaterphysleta
Temperature and pressure dependence of the isotopic shift in diamonds Robert G. Arkhipov
)
Institute of High Pressure Physics, Russian Academy of Sciences, 142092, Troitsk, Moscow Region, Russia Received 10 September 1999; accepted 27 September 1999 Communicated by V.M. Agranovich
Abstract Thermodynamic calculations have been performed for the isotopic changes of the specific volume and the bulk modulus in diamonds, their temperature and pressure dependence. Effects are determined essentially by zero-point vibrations up to room temperature. The inversion of the sign of the volume isotopic shift with pressure seems to be possible in qualitative agreement with present experimental data. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 35.10.Bg; 64.30.q t; 62.50.q p
Complicated measurements of lattice constants both on 12 C and 13 C specimens were carried out recently at room temperature w1x. The difference occurred to be small, but exhibited drop with pressure. There is also optical data about isotopic changes in some components of the elastic modules tensor w2,3x, which can be recalculated to the bulk modulus changes. This short paper presents thermodynamic calculations for the isotopic shifts of the specific volume and the bulk modulus of diamonds in harmonic approximation for its phonon Žvibration. spectrum. Non-linear effects are only considered via their pressure dependence. More detailed analysis would involve microscopic theory of the elasticity tensor. All effects are treated by the lowest order of the expansion over the isotopic mass and pressure. The linear
dependence of the isotopic shift on concentration has been checked in w4x. One has all the information about thermodynamic properties from the expression for the free energy F Ž V,T, M . as a function of its natural variables – volume V and temperature T. The usual expression w4x is: F Ž V ,T , M . s F0 Ž V . q T f Ž e,T . g Ž e, M ,V . de.
H
The first term is dominant, it corresponds to the covalent bonds and in the adiabatic approximation depends on the specific volume only, the second one includes the contribution to the statistical sum of all vibration states. There are two equivalent expressions for it: T f Ž e,T . g Ž e, M ,V . de s T Ý f Ž e j Ž M ,V . .
H
j )
E-mail: [email protected]
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 0 7 - 0
s T Ý f Ž ej . ,
R.G. ArkhipoÕr Physics Letters A 263 (1999) 123–126
124
where f Ž e j . corresponds to the j-th state of the statistical sum Z j of the phonon spectrum ej 1 Z j s Ý exp n q 12 . s Ž 2T 2sinh Ž e jr2T . n
and
and ej ej f s yln Z j s ln Ž 1 y exp Ž ye jrT . . q . T 2T The energy Žfrequency. e j Ž V, M . depends on the specific volume V and atomic mass M, the index j numerates the states and is not affected by external conditions. The last term corresponds to the zeropoint vibrations, for some effects Že.g. for the heat capacity. it is not important. The integration of f Ž e,T . is performed with the density of states g Ž e,V, M . ŽDOS. which can be calculated from the total vibration spectrum. DOS includes specific root type singularities due to saddle points and edges of each branch of the spectrum. It is essential to perform all differentiations with the same set of variables. T, V, M variables are most handy for calculations of the free energy. The usage of the Gibbs chemical potential GŽT, P, M . s F q PV and the corresponding set of T, P, M variables is less convenient because some terms include cross effects via their pressure dependence. Also one has to be careful with the normalization, having in mind, that total number of states corresponds to N – the number of atoms and is Hg Ž e . de s 3 N, it is possible to avoid most of these problems and replace integration over all states by summation. Later on we shall omit the complete set of arguments and indexes when there is no danger of confusion, also let’s agree, that a small letter below function means differentiation, for example: EF pressure P s y s yFV ; EV EF entropy S s y s yFT . ET The following functions can be compared with experiment and are of interest for us: The isotopic shift of the specific volume V and its pressure dependence are determined by d V s d M Ž VM q PVM P . , EV FM V where VM s sy EM FV V
The bulk modulus Ž B s VFV V . isotopic shift d B s d MVFM V V . Everywhere F s T Ý f ; FM V s T Ý f M V ; FM V V s T Ý f M V V , Žwe have to take d M s13 M y12 M s 1.. All the derivatives must be taken at zero pressure and for pure 12 C. The derivatives of the free energy solely by V involve FoŽ V . only, and no contribution from vibrations:
ž
ž /
/
VM P s
FV V s
E 2V E PE M
B0 V0
;
s
FM V V FV V y FM V FV V V FV3 V
FV V V s
B0 V02
.
Ž BP q 1 . ,
where V0 , B0 and BP s dBO rdP are respectively the specific volume, the bulk modulus and its derivative over pressure. Also the derivatives of f Ž e jrT . over V, T and M can be reduced to those over e: Tf T s yef e ;
fM s y
1 2M
g ef e ;
fV s y
V
ef e
because in the adiabatic approximation all frequencies are proportional to the inverse square root of the atomic mass M; the derivatives of the energy over the specific volume can be expressed via local Gruneisen parameter g j .
g fM V s
g
2 MV
e Ž ef . e s
2 MV
Ž ef e q e 2 f e e . ,
g fM V V s y
2 MV 2
Ž Ž 1 q g . ef e
q Ž 1 q 3g . . e 2 f e e q g e 3 f e e e . . It is convenient to consider dimensionless combinations, so the isotopic shift for the specific volume V and the lattice parameter a, we have:
dV V0
s3
da
dM T sy
a0
2 MV B
Ý g j Ž e j f e q e j2 f e e . , j
we may take
Ý g j Ž e j f e q e j2 f e e . s 3 N²g Ž ef e q e 2 f e e . : j
R.G. ArkhipoÕr Physics Letters A 263 (1999) 123–126
and finally, introducing the mean value V0 Ž V0 N s V .:
dV V0
s3
da
3 d M NT sy
a0
2 MV B0
²g Ž ef e q e 2 f e e . : ,
the same is for the isotopic shift of the bulk modulus:
dB
3d M
T
sy B0
2 M V0 B0
the pressure dependence of the volume shift is determined by B0 d M
3d M T s
V0
2 MV0 B0
²g Ž ef e Ž g y BP .
qe 2 f e e Ž 3g y BP . q g e 3 f e e e . : . One can replace summation by integration with density of states, it is more handy but less precise, because the Gruneisen parameter is not one continuous function of energy. The numerical calculations show that d VrV0 and d BrB0 at the room temperature correspond to the low temperature limit. All effects are practically due to the zero-point vibrations, this conclusion is insensitive to the choice of parameters, the main term Tef e in each sum is simply ye jr2. Other terms including e 2 f e e and e 3 f e e e have no specific effect from zero-point vibrations, the examples for it are expressions for the heat capacity C V s yTFT T s y² e 2 f e e : Žthis short expression for C V is equivalent to the common one, it may be handy sometimes. and for the thermal expansion coefficient
as
1 V
EV
ž / ET
sy
1 FV T V FV V
P
this enables us to calculate mean Gruneisen parameter Grn:
a Grn s B0 V0
s CV
V FT V T FT T
s
²g e 2 f e e : ² e2 fe e :
dV V0
s3
da
,
the average of g means summation over all branches of the phonon spectrum, this time weighted by e 2 f e e ; if g j s const, it produces Grn s const. The measure-
3 d M²g e : sy
a0
dB B0
qe 2 f e e Ž 1 q 3g . q g e 3 f e e e . : ,
VM P
ments give Grn s 1.5. The expressions for the lowtemperature limit have a simple analytical form and are easy to compare with experiment:
sy
²g Ž ef e Ž 1 q g .
125
4 MV0 B0
3 d M²g Ž g q 1 . e : 4 MV0 B0
;
.
The total volume shift is proportional to Ž VM q PVM P ., it goes down with pressure and must pass zero at Pc s
B0 ² g e : ² g e : BP y ² g 2 e :
.
For these expressions the average is via the integration with DOS multiplied by e. The dimensionless coefficient in each expression is a product of two small quantities: an isotope mass difference and a parameter of the adiabatic approximation – the ratio of the phonon spectrum energy scale to that of the valence electrons. According to the review w5x, local Gruneisen parameters for all six frequency branches in diamond spectrum is between 0.5 and 2.0. This is a real source of errors. We lack the information about d 2 e jrdV 2 , d 3 e jrdV 3 , . . . and have to take the approximation g j s const for all higher order derivatives, but this increases uncertainty. The higher is the order of the derivative the bigger is the error, so VM is good, but VM P is the less reliable. There is no direct contradiction with experiment, but unfortunately the precision of the experimental data is not high enough, so ² e : s 1400 K, ²g 2 e :r² e : f 2, and ²g e :r² e : f 1.5 which produce d VrV s 0.0007 and d BrB s 0.0017 seem acceptable. The main difficulty is with Pc , which should be s 50 GPa for BP s 4.5.
Acknowledgements The author is grateful to S.M. Stishov, for proposing this problem and helpful discussions and to V.A. Sidorov and V.V. Brazhkin for valuable help.
126
R.G. ArkhipoÕr Physics Letters A 263 (1999) 123–126
References w1x H. Fujihisa, V.A. Sidorov, K. Takemura, H. Kanda, S.M. Stishov, Sov. Phys. JETP Lett. 63 Ž1996. 83. w2x A.K. Ramdas, S. Rodriguez, M. Grimsditch, T.R. Antony, Physical Review Letters 71 Ž1993. 189.
w3x M. Muinov, H. Kanda, S.M. Stishov, Phys. Rev. B 50 Ž1994. 13860. w4x P. Pavone, S. Baroni, Solid State Communications 90 Ž1994. 295. w5x A.F. Goncharov, Uspekhi Phys. Nauk 152 Ž1987. 317.
Physics Letters A 263 Ž1999. 123–126 www.elsevier.nlrlocaterphysleta
Temperature and pressure dependence of the isotopic shift in diamonds Robert G. Arkhipov
)
Institute of High Pressure Physics, Russian Academy of Sciences, 142092, Troitsk, Moscow Region, Russia Received 10 September 1999; accepted 27 September 1999 Communicated by V.M. Agranovich
Abstract Thermodynamic calculations have been performed for the isotopic changes of the specific volume and the bulk modulus in diamonds, their temperature and pressure dependence. Effects are determined essentially by zero-point vibrations up to room temperature. The inversion of the sign of the volume isotopic shift with pressure seems to be possible in qualitative agreement with present experimental data. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 35.10.Bg; 64.30.q t; 62.50.q p
Complicated measurements of lattice constants both on 12 C and 13 C specimens were carried out recently at room temperature w1x. The difference occurred to be small, but exhibited drop with pressure. There is also optical data about isotopic changes in some components of the elastic modules tensor w2,3x, which can be recalculated to the bulk modulus changes. This short paper presents thermodynamic calculations for the isotopic shifts of the specific volume and the bulk modulus of diamonds in harmonic approximation for its phonon Žvibration. spectrum. Non-linear effects are only considered via their pressure dependence. More detailed analysis would involve microscopic theory of the elasticity tensor. All effects are treated by the lowest order of the expansion over the isotopic mass and pressure. The linear
dependence of the isotopic shift on concentration has been checked in w4x. One has all the information about thermodynamic properties from the expression for the free energy F Ž V,T, M . as a function of its natural variables – volume V and temperature T. The usual expression w4x is: F Ž V ,T , M . s F0 Ž V . q T f Ž e,T . g Ž e, M ,V . de.
H
The first term is dominant, it corresponds to the covalent bonds and in the adiabatic approximation depends on the specific volume only, the second one includes the contribution to the statistical sum of all vibration states. There are two equivalent expressions for it: T f Ž e,T . g Ž e, M ,V . de s T Ý f Ž e j Ž M ,V . .
H
j )
E-mail: [email protected]
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 0 7 - 0
s T Ý f Ž ej . ,
R.G. ArkhipoÕr Physics Letters A 263 (1999) 123–126
124
where f Ž e j . corresponds to the j-th state of the statistical sum Z j of the phonon spectrum ej 1 Z j s Ý exp n q 12 . s Ž 2T 2sinh Ž e jr2T . n
and
and ej ej f s yln Z j s ln Ž 1 y exp Ž ye jrT . . q . T 2T The energy Žfrequency. e j Ž V, M . depends on the specific volume V and atomic mass M, the index j numerates the states and is not affected by external conditions. The last term corresponds to the zeropoint vibrations, for some effects Že.g. for the heat capacity. it is not important. The integration of f Ž e,T . is performed with the density of states g Ž e,V, M . ŽDOS. which can be calculated from the total vibration spectrum. DOS includes specific root type singularities due to saddle points and edges of each branch of the spectrum. It is essential to perform all differentiations with the same set of variables. T, V, M variables are most handy for calculations of the free energy. The usage of the Gibbs chemical potential GŽT, P, M . s F q PV and the corresponding set of T, P, M variables is less convenient because some terms include cross effects via their pressure dependence. Also one has to be careful with the normalization, having in mind, that total number of states corresponds to N – the number of atoms and is Hg Ž e . de s 3 N, it is possible to avoid most of these problems and replace integration over all states by summation. Later on we shall omit the complete set of arguments and indexes when there is no danger of confusion, also let’s agree, that a small letter below function means differentiation, for example: EF pressure P s y s yFV ; EV EF entropy S s y s yFT . ET The following functions can be compared with experiment and are of interest for us: The isotopic shift of the specific volume V and its pressure dependence are determined by d V s d M Ž VM q PVM P . , EV FM V where VM s sy EM FV V
The bulk modulus Ž B s VFV V . isotopic shift d B s d MVFM V V . Everywhere F s T Ý f ; FM V s T Ý f M V ; FM V V s T Ý f M V V , Žwe have to take d M s13 M y12 M s 1.. All the derivatives must be taken at zero pressure and for pure 12 C. The derivatives of the free energy solely by V involve FoŽ V . only, and no contribution from vibrations:
ž
ž /
/
VM P s
FV V s
E 2V E PE M
B0 V0
;
s
FM V V FV V y FM V FV V V FV3 V
FV V V s
B0 V02
.
Ž BP q 1 . ,
where V0 , B0 and BP s dBO rdP are respectively the specific volume, the bulk modulus and its derivative over pressure. Also the derivatives of f Ž e jrT . over V, T and M can be reduced to those over e: Tf T s yef e ;
fM s y
1 2M
g ef e ;
fV s y
V
ef e
because in the adiabatic approximation all frequencies are proportional to the inverse square root of the atomic mass M; the derivatives of the energy over the specific volume can be expressed via local Gruneisen parameter g j .
g fM V s
g
2 MV
e Ž ef . e s
2 MV
Ž ef e q e 2 f e e . ,
g fM V V s y
2 MV 2
Ž Ž 1 q g . ef e
q Ž 1 q 3g . . e 2 f e e q g e 3 f e e e . . It is convenient to consider dimensionless combinations, so the isotopic shift for the specific volume V and the lattice parameter a, we have:
dV V0
s3
da
dM T sy
a0
2 MV B
Ý g j Ž e j f e q e j2 f e e . , j
we may take
Ý g j Ž e j f e q e j2 f e e . s 3 N²g Ž ef e q e 2 f e e . : j
R.G. ArkhipoÕr Physics Letters A 263 (1999) 123–126
and finally, introducing the mean value V0 Ž V0 N s V .:
dV V0
s3
da
3 d M NT sy
a0
2 MV B0
²g Ž ef e q e 2 f e e . : ,
the same is for the isotopic shift of the bulk modulus:
dB
3d M
T
sy B0
2 M V0 B0
the pressure dependence of the volume shift is determined by B0 d M
3d M T s
V0
2 MV0 B0
²g Ž ef e Ž g y BP .
qe 2 f e e Ž 3g y BP . q g e 3 f e e e . : . One can replace summation by integration with density of states, it is more handy but less precise, because the Gruneisen parameter is not one continuous function of energy. The numerical calculations show that d VrV0 and d BrB0 at the room temperature correspond to the low temperature limit. All effects are practically due to the zero-point vibrations, this conclusion is insensitive to the choice of parameters, the main term Tef e in each sum is simply ye jr2. Other terms including e 2 f e e and e 3 f e e e have no specific effect from zero-point vibrations, the examples for it are expressions for the heat capacity C V s yTFT T s y² e 2 f e e : Žthis short expression for C V is equivalent to the common one, it may be handy sometimes. and for the thermal expansion coefficient
as
1 V
EV
ž / ET
sy
1 FV T V FV V
P
this enables us to calculate mean Gruneisen parameter Grn:
a Grn s B0 V0
s CV
V FT V T FT T
s
²g e 2 f e e : ² e2 fe e :
dV V0
s3
da
,
the average of g means summation over all branches of the phonon spectrum, this time weighted by e 2 f e e ; if g j s const, it produces Grn s const. The measure-
3 d M²g e : sy
a0
dB B0
qe 2 f e e Ž 1 q 3g . q g e 3 f e e e . : ,
VM P
ments give Grn s 1.5. The expressions for the lowtemperature limit have a simple analytical form and are easy to compare with experiment:
sy
²g Ž ef e Ž 1 q g .
125
4 MV0 B0
3 d M²g Ž g q 1 . e : 4 MV0 B0
;
.
The total volume shift is proportional to Ž VM q PVM P ., it goes down with pressure and must pass zero at Pc s
B0 ² g e : ² g e : BP y ² g 2 e :
.
For these expressions the average is via the integration with DOS multiplied by e. The dimensionless coefficient in each expression is a product of two small quantities: an isotope mass difference and a parameter of the adiabatic approximation – the ratio of the phonon spectrum energy scale to that of the valence electrons. According to the review w5x, local Gruneisen parameters for all six frequency branches in diamond spectrum is between 0.5 and 2.0. This is a real source of errors. We lack the information about d 2 e jrdV 2 , d 3 e jrdV 3 , . . . and have to take the approximation g j s const for all higher order derivatives, but this increases uncertainty. The higher is the order of the derivative the bigger is the error, so VM is good, but VM P is the less reliable. There is no direct contradiction with experiment, but unfortunately the precision of the experimental data is not high enough, so ² e : s 1400 K, ²g 2 e :r² e : f 2, and ²g e :r² e : f 1.5 which produce d VrV s 0.0007 and d BrB s 0.0017 seem acceptable. The main difficulty is with Pc , which should be s 50 GPa for BP s 4.5.
Acknowledgements The author is grateful to S.M. Stishov, for proposing this problem and helpful discussions and to V.A. Sidorov and V.V. Brazhkin for valuable help.
126
R.G. ArkhipoÕr Physics Letters A 263 (1999) 123–126
References w1x H. Fujihisa, V.A. Sidorov, K. Takemura, H. Kanda, S.M. Stishov, Sov. Phys. JETP Lett. 63 Ž1996. 83. w2x A.K. Ramdas, S. Rodriguez, M. Grimsditch, T.R. Antony, Physical Review Letters 71 Ž1993. 189.
w3x M. Muinov, H. Kanda, S.M. Stishov, Phys. Rev. B 50 Ž1994. 13860. w4x P. Pavone, S. Baroni, Solid State Communications 90 Ž1994. 295. w5x A.F. Goncharov, Uspekhi Phys. Nauk 152 Ž1987. 317.