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The low temperature heat capacity of NiCu alloys in a magnetic field
825 THE L O W T E M P E R A T U R E HEAT CAPACITY OF N i - C u ALLOYS IN A MAGNETIC FIELD
P.C. L A N C H E S T E R , N.F. W H I T E H E A D * and P. W E L L S t Physics Department, Southampton University, Southampton S09 5NH, England The heat capacity of nearly ferromagnetic Ni-Cu alloys has been measured from 0.3 to 4.0 K in fields up to 6.5 T. Having subtracted the electronic term in a novel way, the field dependence of the excess heat capacity Cu for T ~> 1.8 K is approximately that expected for superparamagnetic clusters. However as T ~ 0 , CM--'0 linearly rather than exponentially.
1. Introduction
Near the critical concentration for ferromagnetism of alloys such as Ni-Cu and F e - V the low temperature heat capacity is found to be of the form
C = 3"T+AT3+CM,
(1)
where CM is associated with the presence of superparamagnetic clusters. It is usually assumed that CM arises from oscillation of these clusters in a local anisotropy field Ha for which
CM = BE(TE/T),
(2)
where B is a constant proportional to the cluster concentration and E(TE/T) is the Einstein function with kTE = 2/zBHa [1]. In this model B is not expected to vary in an applied field H, but CM should decrease in increasing fields as H begins to dominate Ha causing the effective field H ' and the Einstein temperature T~ to rise. For T , ~ T~, CM should vary exponentially with temperature. Although CM is known to decrease with increasing field, the high field results for F e - V when fitted to eqs. (1) and (2) suggest that B decreases and that 3' too is field dependent [2]. In order to study these unexpected effects we have measured the high field heat capacity of Ni-Cu alloys, which are probably the best characterized alloys of this kind. 2. Experimental Three Nil_xCux samples with x =0.558(A), x = 0.589(B) and x = 0.601(C), have been stu* Post Office Development Department, 100-110 High Holborn, London, England. t Department of Applied Physics, Caulfieid Institute of Technology, Victoria 3145, Australia.
Physica 86-88B (1977) 825-826 © North-Holland
died between 0.3 and 4.0 K in fields up to 6.5 T. The samples were prepared by Metals Research Ltd., of Cambridge from materials of 5 N purity, containing < 1 0 p p m Fe. T h e y were homogenized by annealing in vacuo at I100C for 72h and then water quenched. Measurements were made in a 3He cryostat using a carbon resistance thermometer calibrated in situ against a magnetically shielded standard [3, 4]. 3. Results
As the lattice term was too small to determine reliably this was calculated assuming OD = 390 K [5]. In high fields a small hyperfine term due to the copper was also allowed for. The resultant data C' was then fitted to C ' = 3"T + BE(TE/T) yielding results remarkably similar to those for F e - V [2]. The cluster concentration appeared to vanish above 2 T and 3' increased by as much as 30% up to 2 T, falling slightly at higher fields [3]. However, it is unlikely that the true electronic 3' is so field dependent and a reduction in the cluster concentration contradicts the high field magnetization data [6] which gives cluster concentrations in reasonable accord with the zero field heat capacity. We conclude that eq. (2) is a poor representation of CM- Further evidence of this both for the Ni-Cu and F e - V alloys is apparent in the behaviour of CM at temperatures well below that at which CM peaks, which is always linear in T(CM = 3"T) rather than exponential. In the absence of a satisfactory theory an empirical approach has been adopted to evaluate CM. As the applied field H ~oo, 3" for each sample is observed to tend to a limiting value 3"(oo), which has been taken to be the true electronic 3" for all fields (fig. 1). That is, we assume the electronic 3' is independent of field and that
iI6ca i T
826
~81 ~
--~
~
.~E 4
3'~
0
~
-,
C
4 ~7
%
A
o
~
o
o
o
~c
468
~
"7
,
025
~
Y 05
075
IO
(~c,H)-1(T-11
7
Fig. 1. y' versus (#oH) ' for N i - C u alloys.
there is a cluster component CM linear in T at low temperatures which is suppressed in very high fields. Values of CM = C ' - y ( o o ) T for all samples in zero field are shown in fig. 2(a), whilst fig. 2(b) shows the field dependence of CM for sample A which is typical of all the samples. It is likely that the fall in CM towards 4 K observed in the zero field results for samples A and C [fig. 2(a)] is a general feature resulting from the finite spin of the clusters [5]. The fact
(b)
1T _~o°~°~ ~
~Coo:°
°°~ 4 T o'~"
o o
/:-u ~'
.... ~o/,o oO /.,f . 13) I
2
T(K)
3
4
Fig. 2. (a) CM = C ' - y ( ~ ) T v e r s u s T for H = 0. The solid lines are fits to the Einstein function for T < 1.6 K. (b) CM v e r s u s T for sample A in fields t~oH = 1, 2, 4 and 6 T. The solid lines correspond to the Einstein function with B ' = 6.5 m J m o l ' K ~, t x o H ' = txoH+ 1.1T.
Table I Fitted parameters for N i - C u alloys
Sample
y(~) (mJmol~K -2)
B (mJmol 'K-')
/-toll. (T)
B' (mJ tool ' K - ' )
/zoH'(T) = txo(H +/4.')
A B C
4.68±0.07 4.37±0.07 3.~±0.07
5.8 4.0 2.8
0.75 0.66 0.38
6.5 5 4
/~oH + 1.1 txoH + 0.75 t~oH + 0.38
that this does not occur for sample B could be due to a small error in y(oo) for this sample. Below 1.6 K these results fit the Einstein function reasonably well with the values of B and Ha given in table I. An applied field increases the effective cluster concentration so that in moderate fields CM towards 4 K increases before falling again at higher fields [fig. 2(b)]. The behaviour above about 1.5 K can be fitted very approximately to the superparamagnetic model by assuming that for /x0H/> 1 T the cluster concentration is constant, corresponding to a field independent value of B ' > B and that the effective field is H ' = H + Ha' where Ha' = Ha is also field independent. This is illustrated by the continuous curves in fig. 2(b) and the related parameters B' and H', are given in table I. As might be expected the value of B' are in better agreement with the corresponding values deduced from the high field magnetization [5] than are the values of B.
4. Conclusion
The analysis we have used brings the interpretation of the heat capacity into line with the magnetization results, so that both broadly support the superparamagnetic model. However, the detailed behaviour and particularly the linear dependence of CM as T ~ 0 in high fields is not understood although it is qualitatively like that predicted for very dilute impurities by Klein [7]. References [I] K. Schroder, J. Appl. P h y s . 32 (1961) 880. [2] W. Procter and R.G. Scurlock, l i t h Int. Conf. L o w Temp. P h y s . St. Andrews (1968) 1320. [3] N.F. Whitehead, Ph.D. Thesis, University of Southa m p t o n , England (1974). [4] N.F. Whitehead, P.C. L a n c h e s t e r and R.G. Scurlock, J. Sci. Inst. 7 0974) ll7. [5] R.L. Falge, Jr. and N.M. Wolcott, J. Low Temp. P h y s . 5 (1971) 617. [6] F. Acker and R. Huguenin, P h y s . Lett. 38A (1972) 343. [7] M.W. Klein, P h y s . Rev. 188 0969) 933.
P.C. L A N C H E S T E R , N.F. W H I T E H E A D * and P. W E L L S t Physics Department, Southampton University, Southampton S09 5NH, England The heat capacity of nearly ferromagnetic Ni-Cu alloys has been measured from 0.3 to 4.0 K in fields up to 6.5 T. Having subtracted the electronic term in a novel way, the field dependence of the excess heat capacity Cu for T ~> 1.8 K is approximately that expected for superparamagnetic clusters. However as T ~ 0 , CM--'0 linearly rather than exponentially.
1. Introduction
Near the critical concentration for ferromagnetism of alloys such as Ni-Cu and F e - V the low temperature heat capacity is found to be of the form
C = 3"T+AT3+CM,
(1)
where CM is associated with the presence of superparamagnetic clusters. It is usually assumed that CM arises from oscillation of these clusters in a local anisotropy field Ha for which
CM = BE(TE/T),
(2)
where B is a constant proportional to the cluster concentration and E(TE/T) is the Einstein function with kTE = 2/zBHa [1]. In this model B is not expected to vary in an applied field H, but CM should decrease in increasing fields as H begins to dominate Ha causing the effective field H ' and the Einstein temperature T~ to rise. For T , ~ T~, CM should vary exponentially with temperature. Although CM is known to decrease with increasing field, the high field results for F e - V when fitted to eqs. (1) and (2) suggest that B decreases and that 3' too is field dependent [2]. In order to study these unexpected effects we have measured the high field heat capacity of Ni-Cu alloys, which are probably the best characterized alloys of this kind. 2. Experimental Three Nil_xCux samples with x =0.558(A), x = 0.589(B) and x = 0.601(C), have been stu* Post Office Development Department, 100-110 High Holborn, London, England. t Department of Applied Physics, Caulfieid Institute of Technology, Victoria 3145, Australia.
Physica 86-88B (1977) 825-826 © North-Holland
died between 0.3 and 4.0 K in fields up to 6.5 T. The samples were prepared by Metals Research Ltd., of Cambridge from materials of 5 N purity, containing < 1 0 p p m Fe. T h e y were homogenized by annealing in vacuo at I100C for 72h and then water quenched. Measurements were made in a 3He cryostat using a carbon resistance thermometer calibrated in situ against a magnetically shielded standard [3, 4]. 3. Results
As the lattice term was too small to determine reliably this was calculated assuming OD = 390 K [5]. In high fields a small hyperfine term due to the copper was also allowed for. The resultant data C' was then fitted to C ' = 3"T + BE(TE/T) yielding results remarkably similar to those for F e - V [2]. The cluster concentration appeared to vanish above 2 T and 3' increased by as much as 30% up to 2 T, falling slightly at higher fields [3]. However, it is unlikely that the true electronic 3' is so field dependent and a reduction in the cluster concentration contradicts the high field magnetization data [6] which gives cluster concentrations in reasonable accord with the zero field heat capacity. We conclude that eq. (2) is a poor representation of CM- Further evidence of this both for the Ni-Cu and F e - V alloys is apparent in the behaviour of CM at temperatures well below that at which CM peaks, which is always linear in T(CM = 3"T) rather than exponential. In the absence of a satisfactory theory an empirical approach has been adopted to evaluate CM. As the applied field H ~oo, 3" for each sample is observed to tend to a limiting value 3"(oo), which has been taken to be the true electronic 3" for all fields (fig. 1). That is, we assume the electronic 3' is independent of field and that
iI6ca i T
826
~81 ~
--~
~
.~E 4
3'~
0
~
-,
C
4 ~7
%
A
o
~
o
o
o
~c
468
~
"7
,
025
~
Y 05
075
IO
(~c,H)-1(T-11
7
Fig. 1. y' versus (#oH) ' for N i - C u alloys.
there is a cluster component CM linear in T at low temperatures which is suppressed in very high fields. Values of CM = C ' - y ( o o ) T for all samples in zero field are shown in fig. 2(a), whilst fig. 2(b) shows the field dependence of CM for sample A which is typical of all the samples. It is likely that the fall in CM towards 4 K observed in the zero field results for samples A and C [fig. 2(a)] is a general feature resulting from the finite spin of the clusters [5]. The fact
(b)
1T _~o°~°~ ~
~Coo:°
°°~ 4 T o'~"
o o
/:-u ~'
.... ~o/,o oO /.,f . 13) I
2
T(K)
3
4
Fig. 2. (a) CM = C ' - y ( ~ ) T v e r s u s T for H = 0. The solid lines are fits to the Einstein function for T < 1.6 K. (b) CM v e r s u s T for sample A in fields t~oH = 1, 2, 4 and 6 T. The solid lines correspond to the Einstein function with B ' = 6.5 m J m o l ' K ~, t x o H ' = txoH+ 1.1T.
Table I Fitted parameters for N i - C u alloys
Sample
y(~) (mJmol~K -2)
B (mJmol 'K-')
/-toll. (T)
B' (mJ tool ' K - ' )
/zoH'(T) = txo(H +/4.')
A B C
4.68±0.07 4.37±0.07 3.~±0.07
5.8 4.0 2.8
0.75 0.66 0.38
6.5 5 4
/~oH + 1.1 txoH + 0.75 t~oH + 0.38
that this does not occur for sample B could be due to a small error in y(oo) for this sample. Below 1.6 K these results fit the Einstein function reasonably well with the values of B and Ha given in table I. An applied field increases the effective cluster concentration so that in moderate fields CM towards 4 K increases before falling again at higher fields [fig. 2(b)]. The behaviour above about 1.5 K can be fitted very approximately to the superparamagnetic model by assuming that for /x0H/> 1 T the cluster concentration is constant, corresponding to a field independent value of B ' > B and that the effective field is H ' = H + Ha' where Ha' = Ha is also field independent. This is illustrated by the continuous curves in fig. 2(b) and the related parameters B' and H', are given in table I. As might be expected the value of B' are in better agreement with the corresponding values deduced from the high field magnetization [5] than are the values of B.
4. Conclusion
The analysis we have used brings the interpretation of the heat capacity into line with the magnetization results, so that both broadly support the superparamagnetic model. However, the detailed behaviour and particularly the linear dependence of CM as T ~ 0 in high fields is not understood although it is qualitatively like that predicted for very dilute impurities by Klein [7]. References [I] K. Schroder, J. Appl. P h y s . 32 (1961) 880. [2] W. Procter and R.G. Scurlock, l i t h Int. Conf. L o w Temp. P h y s . St. Andrews (1968) 1320. [3] N.F. Whitehead, Ph.D. Thesis, University of Southa m p t o n , England (1974). [4] N.F. Whitehead, P.C. L a n c h e s t e r and R.G. Scurlock, J. Sci. Inst. 7 0974) ll7. [5] R.L. Falge, Jr. and N.M. Wolcott, J. Low Temp. P h y s . 5 (1971) 617. [6] F. Acker and R. Huguenin, P h y s . Lett. 38A (1972) 343. [7] M.W. Klein, P h y s . Rev. 188 0969) 933.