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Pippard's relations in crystalline media
Physica 29
Viswariathan, R,
18-22
1963
PIPPARD's RELATIONS IN CRYSTALLINE MEDIA by R. VISWANATHAN Physics Department, Indian Institute of Science, Bangalore, India.
Synopsis The mare general Pippard's relations for the crystalline media (with six stress and six strain coefficients in place of the usual P and V) have been derived. The scarcity of experimental data for a quantitative test of these relations is discussed and a qualitative verification has been given for the cases of quartz and liquid helium.
The Ehrenfest approach to describe the second order phase transitions is very well known 1)2). This approach fails to give any information in some specific types of second order transitions. For such cases Pip p ar d") has deduced two relations, holding good near the transition point, assuming the entropy and volume surfaces to be cylindrical near the transition point. They are Cp = T"V/l(clP/dT);. + constant. (1 ) A = ~ (dP/dT)" + constant. where Cp is the specific heat at constant pressure, A the coefficient of volume expansion, f3 the cubical compressibility and T;.. the A-point temperature. The above equations are deduced easily for fluids where the variables are only P, V and T. In crystals one has, in place of P and V, six stress (Ti) and six strain (Bi) components. So it is of interest to see whether a more general set of equations can be obtained for the crystalline media. In general, for a second order transition there is a discontinuity in the specific heat and volume expansion coefficient and one has s) 5)
dP/dT
=
l/TV C;-2/111-2
= Al-2/~1-2.
However, crystalline quartz, ammonium chloride, f3-brass, etc. have an infinite discontinuity of Cp and 11 at the transition point. Consequently dP/dT becomes indeterminate. It is here that one thinks of the cylindrical approximation. The specific heat in such cases, tends to infinity as theA-point is approached and the corresponding variation of entropy with temperature, at constant pressure, shows an inflexion point at T},. Also the entropy at T" will be a smoothly varying function of pressure. The entropy surface near the tranI
-
18 -
PIPPARD'S RELATIONS IN CRYSTALLINE MEDIA
19
sition point can here be considered to be approximately cylindrical. Tisza 6) has discussed such a system starting from the Gibbs function. It is relevant to point out here that the approach in the earlier work of the author") is applicable only to the cases where the discontinuities in C'P and A are finite. This is quite evident, since the Taylor series expansion of the Gibbs function breaks down when one of the terms tends to infinity. A clear account of the different types of transitions has been given by Temp e r le y s) U'b b e lo h dev) and Pi pp ar d w). A general discussion of phase transitions has been given by T'i sz a U). The increase in the internal energy is 6
du = T dS - V L; Tt ds t .
(2)
t~l
where
represents the strain energy per unit volume. The Gibb's function is given by G = U - TS
+ V ~ T'lSt.
(3)
i
The equations (2) and (3) give
-5; (BGliJTt)T = VBt; (82GIOTtOTJ)T, r» = VStj,
(8GI8T)Tt
=
(4)
where i, j, k = 1, 2, ... 6 and k =1= i, 7'. The Gibbs function near the A-point be represented by
Ch, T) = Th{rt)
+ I(Tt -
atT).
(5)
where at = (oTt/8T);. and i = 1, 2, ... 6. Let CPi = h - aiT). Eqn. (5) gives (omitting the constant factors like (82hIOTi(lTj)P, etc.)
(oG18Th = - S = - ~ at(8fI8cpi). · (8GI 8Tt)T = VSi = (oflocpt). · (a'!.CI8T2)Ti = -CTIT = ~ ~ aiaj (821Iocpi8epj). · (o2CIOTtOT:J) = VSi:J = (8 2f18cpt 8cpj). · (o2CI8rtoT) = VAt = - :2:: aj(02/18q,t8CP1).
(6)
From the eqns, (6) one gets
V and
~ i
Stj(orjI8T)),
+ constant. ~ Ai ((hi/oT)), + constant.
= -
Cr/TV =
VAt
Eqns. (7) are the Pippard's relations for crystalline media.
(7)
20
R. VISWANATHAN
For the case of hydrostatic pressure (d'T! = el'T2 = dT3 = dp, elT4
=
el'T5 = dT6 = 0, C, = Cp )
the eqns. (7) reduce to
2: Pi dF/dT
= - L; Ai
f3 elF/dT
i.e,
+ constant A dP/dT + constant. A
=
CpjTV =
and
+ constant (8)
These are nothing but the original Pippard's relations (Eqns. 1). The usual Ehrenfest relations can be arrived at quite easily from the above relations. If the superscripts 1 and 2 represent the two phases near the transition point, then 3
~ i= 1
st-
2
dF/dT
=
-Af- 2•
and
Q-2/TV = .,11-2 dPjdT. Eliminating dP/elT fron the above eqns., one has C~-2jTV =
_Al-2A~-2/Pi-2
=
_.I11-2A~-2jf3~-2 =
_.I11-2A~-2/f3~-2.
where 3
Pi
=
b
Sij
i=1
are the linear compressibilities. These expressions are the same as the ones derived earlier by the author 7).
T"
"'
Q
~o
.,'
Q
><
-c
2
-I-I
cpX 10 CAL.. G • OEG
37 3~~----=----"='7""------:;"" 35 36
Fig. 1. Cp-A relation near the transition point for quartz. Data taken from Moser 12) and J ay13).
Pi pp ar d himself has clearly pointed out how, because of the insufficient data available at present, one fails to test these relations quantitatively.
PIPPARD'S RELATIONS IN CRYSTALLINE MEDIA
21
Little experimental study of the variation of TJ. with pressure has been done. Also in cases like liquid helium the slight error in the measurement of temperature (which is not unlikely) introduces a large error in the quantitative calculations. Hence one is left with only a possible qualitative test for the above relations. The eqns. (1) and (8) represent two straight line graphs, one between Cp and A and the other between A and p. 3'45
VI
...
Ie> 0
"'0
x: en <
2'45
-I -I C 5 JOULE.DEG. G
1095~1-5;------~'----------'=-' 13-5
Fig. 2. C8
-
As relation near the transition point for liquid helium. Data taken from Atkins and Edwa.rd s-s) and Hill and Lounasama 15). ,....5
'.... 4
T...
?: 1.43,,0.
:IE u
",'
1.42
0
'"
Ill.
3
1040
18
-I
II X 10 • DEG 24
30
36
Fig. 3. A - ~ relation near the transition point for liquid helium. Data taken from Atkins and Edwards 14).
The data for quartz have been taken from the papers of Moser 12) and J ay13). Figure 1 shows the straight line relationship between Cp and A. It is to be remembered that as one moves quite far away from the transition point, the points will be found to fall away from the straight line, thereby showing that this cylindrical approximation is valid only near the transition point. The data for liquid helium are those of Atkins and Edwards-s) and Hill and Lo unasama-s). The Cs - As (for the specific heat and volume expansion at constant saturated vapour pressure one can refer Keesom 16) )
22
PIPPARD'S RELATIONS IN CRYSTALLINE MEDIA
and the A - (3 straight lines are seen in fig. 2 and fig. 3 respectively*). Thus one can arrive at more general Pippard's relations applicable to the crystalline media starting from the Gibbs function. However, these relations can be tested only qualitatively due to lack of experimental data. Acknowledgement. The author is grateful to Professor Dr. R. S. Krishnan for encouragement and Dr. E. S. Raj agopal **) for the suggestion of the problem. The author thanks the Council of Scientific and Industrial Research for the financial assistance. Received 6-6-62 REFERENCES t) Ehrenfest, P., Communications from the Kamerlingh Ormes Lab., Leiden. Suppl. 75b (1933). 2} Gu gge n h e i m, E. A., Thermodynamics - An advanced treatment for Physicists and Chemists, North Holland Publ, Camp. (1950) 287. 3} Pip pard, A. B., Phil. Mag. 8 (1956) 473. 4} Zemansky, M. W., Heat and Thermodynamics, McGraw HilI, New York, (195!) 323. 5} Atkins, K. R, Liquid Helium, Cambridge University Press, (1959) 43. 6) Tisza, L., Annals of Physics 13 (1961) I. 7} Viswanathan, R. and Rajagopal, E. S., Physica 27 (1961) 765. 8} Temperley, H. N. V., Changes of State, Cleaver Hume, London, (1956) 8. 9) Ub b el o h d e, A. R., Quart. Rev. (London) 11 (1957) 246. 10) Pip pard, A. B., The Elements of Classical Thermodynamics, Cambridge University Press, (1960) Chapter IX. It) Tisza, L., Phase Transformations in Solids, Eds. Souluchowski Mayer, J. E., and We yl, W. A., Wiley, New York, (1954). 12} Moser, H., Physik. Z. 37 (1936) 737. 13) Jay, A. H., Proc, Royal. Soc. H2A (1933) 237. 14) Atkins, K. R. and Edwards, M. A., Phys, Rev. 97 (1955) 1429. 15) Hill, R. W. and Lo u n a s a m a, O. V., Phil. Mag. 2 (1954) 144. 16) Keesom, W. H., Comrn, Kamerlingh Onnes Lab., Lelden, Suppl. 75a (1933).
*) The curves for liquid helium are drawn assuming the normal second order transition. The log. nature of variation of C" with T is not considered. *") Now fit Clarendon Laboratory, Oxford,
Viswariathan, R,
18-22
1963
PIPPARD's RELATIONS IN CRYSTALLINE MEDIA by R. VISWANATHAN Physics Department, Indian Institute of Science, Bangalore, India.
Synopsis The mare general Pippard's relations for the crystalline media (with six stress and six strain coefficients in place of the usual P and V) have been derived. The scarcity of experimental data for a quantitative test of these relations is discussed and a qualitative verification has been given for the cases of quartz and liquid helium.
The Ehrenfest approach to describe the second order phase transitions is very well known 1)2). This approach fails to give any information in some specific types of second order transitions. For such cases Pip p ar d") has deduced two relations, holding good near the transition point, assuming the entropy and volume surfaces to be cylindrical near the transition point. They are Cp = T"V/l(clP/dT);. + constant. (1 ) A = ~ (dP/dT)" + constant. where Cp is the specific heat at constant pressure, A the coefficient of volume expansion, f3 the cubical compressibility and T;.. the A-point temperature. The above equations are deduced easily for fluids where the variables are only P, V and T. In crystals one has, in place of P and V, six stress (Ti) and six strain (Bi) components. So it is of interest to see whether a more general set of equations can be obtained for the crystalline media. In general, for a second order transition there is a discontinuity in the specific heat and volume expansion coefficient and one has s) 5)
dP/dT
=
l/TV C;-2/111-2
= Al-2/~1-2.
However, crystalline quartz, ammonium chloride, f3-brass, etc. have an infinite discontinuity of Cp and 11 at the transition point. Consequently dP/dT becomes indeterminate. It is here that one thinks of the cylindrical approximation. The specific heat in such cases, tends to infinity as theA-point is approached and the corresponding variation of entropy with temperature, at constant pressure, shows an inflexion point at T},. Also the entropy at T" will be a smoothly varying function of pressure. The entropy surface near the tranI
-
18 -
PIPPARD'S RELATIONS IN CRYSTALLINE MEDIA
19
sition point can here be considered to be approximately cylindrical. Tisza 6) has discussed such a system starting from the Gibbs function. It is relevant to point out here that the approach in the earlier work of the author") is applicable only to the cases where the discontinuities in C'P and A are finite. This is quite evident, since the Taylor series expansion of the Gibbs function breaks down when one of the terms tends to infinity. A clear account of the different types of transitions has been given by Temp e r le y s) U'b b e lo h dev) and Pi pp ar d w). A general discussion of phase transitions has been given by T'i sz a U). The increase in the internal energy is 6
du = T dS - V L; Tt ds t .
(2)
t~l
where
represents the strain energy per unit volume. The Gibb's function is given by G = U - TS
+ V ~ T'lSt.
(3)
i
The equations (2) and (3) give
-5; (BGliJTt)T = VBt; (82GIOTtOTJ)T, r» = VStj,
(8GI8T)Tt
=
(4)
where i, j, k = 1, 2, ... 6 and k =1= i, 7'. The Gibbs function near the A-point be represented by
Ch, T) = Th{rt)
+ I(Tt -
atT).
(5)
where at = (oTt/8T);. and i = 1, 2, ... 6. Let CPi = h - aiT). Eqn. (5) gives (omitting the constant factors like (82hIOTi(lTj)P, etc.)
(oG18Th = - S = - ~ at(8fI8cpi). · (8GI 8Tt)T = VSi = (oflocpt). · (a'!.CI8T2)Ti = -CTIT = ~ ~ aiaj (821Iocpi8epj). · (o2CIOTtOT:J) = VSi:J = (8 2f18cpt 8cpj). · (o2CI8rtoT) = VAt = - :2:: aj(02/18q,t8CP1).
(6)
From the eqns, (6) one gets
V and
~ i
Stj(orjI8T)),
+ constant. ~ Ai ((hi/oT)), + constant.
= -
Cr/TV =
VAt
Eqns. (7) are the Pippard's relations for crystalline media.
(7)
20
R. VISWANATHAN
For the case of hydrostatic pressure (d'T! = el'T2 = dT3 = dp, elT4
=
el'T5 = dT6 = 0, C, = Cp )
the eqns. (7) reduce to
2: Pi dF/dT
= - L; Ai
f3 elF/dT
i.e,
+ constant A dP/dT + constant. A
=
CpjTV =
and
+ constant (8)
These are nothing but the original Pippard's relations (Eqns. 1). The usual Ehrenfest relations can be arrived at quite easily from the above relations. If the superscripts 1 and 2 represent the two phases near the transition point, then 3
~ i= 1
st-
2
dF/dT
=
-Af- 2•
and
Q-2/TV = .,11-2 dPjdT. Eliminating dP/elT fron the above eqns., one has C~-2jTV =
_Al-2A~-2/Pi-2
=
_.I11-2A~-2jf3~-2 =
_.I11-2A~-2/f3~-2.
where 3
Pi
=
b
Sij
i=1
are the linear compressibilities. These expressions are the same as the ones derived earlier by the author 7).
T"
"'
Q
~o
.,'
Q
><
-c
2
-I-I
cpX 10 CAL.. G • OEG
37 3~~----=----"='7""------:;"" 35 36
Fig. 1. Cp-A relation near the transition point for quartz. Data taken from Moser 12) and J ay13).
Pi pp ar d himself has clearly pointed out how, because of the insufficient data available at present, one fails to test these relations quantitatively.
PIPPARD'S RELATIONS IN CRYSTALLINE MEDIA
21
Little experimental study of the variation of TJ. with pressure has been done. Also in cases like liquid helium the slight error in the measurement of temperature (which is not unlikely) introduces a large error in the quantitative calculations. Hence one is left with only a possible qualitative test for the above relations. The eqns. (1) and (8) represent two straight line graphs, one between Cp and A and the other between A and p. 3'45
VI
...
Ie> 0
"'0
x: en <
2'45
-I -I C 5 JOULE.DEG. G
1095~1-5;------~'----------'=-' 13-5
Fig. 2. C8
-
As relation near the transition point for liquid helium. Data taken from Atkins and Edwa.rd s-s) and Hill and Lounasama 15). ,....5
'.... 4
T...
?: 1.43,,0.
:IE u
",'
1.42
0
'"
Ill.
3
1040
18
-I
II X 10 • DEG 24
30
36
Fig. 3. A - ~ relation near the transition point for liquid helium. Data taken from Atkins and Edwards 14).
The data for quartz have been taken from the papers of Moser 12) and J ay13). Figure 1 shows the straight line relationship between Cp and A. It is to be remembered that as one moves quite far away from the transition point, the points will be found to fall away from the straight line, thereby showing that this cylindrical approximation is valid only near the transition point. The data for liquid helium are those of Atkins and Edwards-s) and Hill and Lo unasama-s). The Cs - As (for the specific heat and volume expansion at constant saturated vapour pressure one can refer Keesom 16) )
22
PIPPARD'S RELATIONS IN CRYSTALLINE MEDIA
and the A - (3 straight lines are seen in fig. 2 and fig. 3 respectively*). Thus one can arrive at more general Pippard's relations applicable to the crystalline media starting from the Gibbs function. However, these relations can be tested only qualitatively due to lack of experimental data. Acknowledgement. The author is grateful to Professor Dr. R. S. Krishnan for encouragement and Dr. E. S. Raj agopal **) for the suggestion of the problem. The author thanks the Council of Scientific and Industrial Research for the financial assistance. Received 6-6-62 REFERENCES t) Ehrenfest, P., Communications from the Kamerlingh Ormes Lab., Leiden. Suppl. 75b (1933). 2} Gu gge n h e i m, E. A., Thermodynamics - An advanced treatment for Physicists and Chemists, North Holland Publ, Camp. (1950) 287. 3} Pip pard, A. B., Phil. Mag. 8 (1956) 473. 4} Zemansky, M. W., Heat and Thermodynamics, McGraw HilI, New York, (195!) 323. 5} Atkins, K. R, Liquid Helium, Cambridge University Press, (1959) 43. 6) Tisza, L., Annals of Physics 13 (1961) I. 7} Viswanathan, R. and Rajagopal, E. S., Physica 27 (1961) 765. 8} Temperley, H. N. V., Changes of State, Cleaver Hume, London, (1956) 8. 9) Ub b el o h d e, A. R., Quart. Rev. (London) 11 (1957) 246. 10) Pip pard, A. B., The Elements of Classical Thermodynamics, Cambridge University Press, (1960) Chapter IX. It) Tisza, L., Phase Transformations in Solids, Eds. Souluchowski Mayer, J. E., and We yl, W. A., Wiley, New York, (1954). 12} Moser, H., Physik. Z. 37 (1936) 737. 13) Jay, A. H., Proc, Royal. Soc. H2A (1933) 237. 14) Atkins, K. R. and Edwards, M. A., Phys, Rev. 97 (1955) 1429. 15) Hill, R. W. and Lo u n a s a m a, O. V., Phil. Mag. 2 (1954) 144. 16) Keesom, W. H., Comrn, Kamerlingh Onnes Lab., Lelden, Suppl. 75a (1933).
*) The curves for liquid helium are drawn assuming the normal second order transition. The log. nature of variation of C" with T is not considered. *") Now fit Clarendon Laboratory, Oxford,