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Paramagnetic critical exponents
Physica B 149 (1988) 122-124 North-Holland, Amsterdam
PARAMAGNETIC CRITICAL EXPONENTS
D.E.G. WILLIAMS
Physics Department, Loughborough University of Technology, Loughborough LEll 3TU, England Critical point exponents for the paramagnetic susceptibility in Co and Fe have been derived from the measurements published by Develey. The exponent, % for the linear reduced temperature ( 1 - T/Tc) is the same as that for the non-linear reduced temperature ( 1 - Tc/T ). The extension to measurements made on Fe-Si alloys broke down for Si concentrations of more than c = 0.07. In that case, 3' was very different from the value for Fe, it was not possible to judge whether this difference was real or was due to uncertainty in the value of Tc which, for c > 0.07, has three different published values at least.
Magnetic phase transitions between a ferromagnetic phase and a paramagnetic phase are frequently described in terms of critical point exponents [1,2]. These relate, e.g., the behaviour of the magnetization, M, to the reduced t e m p e r a t u r e , e, with e = (1 - (T/Tc), for T < T c where T~ is the ferromagnetic Curie t e m p e r a ture, or the variation of the paramagnetic susceptibility, X, to the reduced t e m p e r a t u r e (--e) = ((T/Tc) -1) for T > T~). The form of the relationship is, for the susceptibility,
and the introduction of a constant ' b a c k g r o u n d ' susceptibility, X0, can increase the range of temperature over which the critical point exponent y ' gives an accurate representation of the behaviour of the paramagnetic susceptibility. Thus, Souletie and Tholence [4] point out that, by using appropriate values of To, X0 and C 1 the paramagnetic susceptibility of Ni can be represented in the t e m p e r a t u r e range from Tc to 3 T¢ by a single exponent 3' defined by
Y=
- l n ( ( x - Xo)T) + In C In*/
(1)
,
where T ' is the critical point exponent. This form is appropriate only in the range of t e m p e r a t u r e close to T¢ since 3" is defined most simply [3] by 3 , ' = lim ln(1/X) T-~rc l n ( - e ) "
This range of t e m p e r a t u r e is that in which the higher order terms in the expansion of 1/X as a power series in ( - e ) can be neglected. T h e r e seems to be general a g r e e m e n t that the u p p e r limit of ( - e ) which will produce an acceptable value of 3" is at most ( - e ) < 0 . 0 5 . Recently it has been p r o p o s e d [4] that a redefinition of the reduced t e m p e r a t u r e variables as non-linear, i.e.,
~7=I-TJT,
for T > T¢,
The constant C was identified with a "Curie constant', and it was suggested that X0 might be a metallic van Vleck susceptibility. The fit given by eq. (1) to the experimental data for Ni was very good. The idea e m b o d i e d in eq. (1) has been applied to some. model systems [5, 6] and the conclusion that a scaling law such as eq. (1) might describe their susceptibilities over a wide range of values of ( T - Tc) was claimed for these systems. This extension of the range of validity of a single critical point exponent might be expected because of the introduction of the disposable p a r a m e t e r X0. In the light of these observations it seemed worthwhile to reexamine some data for the paramagnetic susceptibilities of ferromagnetic materials, if only to find whether the wide range of validity in t e m p e r a t u r e found in the case of Ni
0378-4363/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
D.E.G. Williams / Paramagnetic critical exponents
were reproduced for other materials. The initial ambition was to produce values of 3, for F e - S i alloys, using the susceptibility-temperature table given by Arajs and Miller [7]. As a first step towards attaining this objective an examination was made of the results given by Develey [2] for Co and for Fe in an attempt to find the values of the parameters 3' and X0 which produced the best fit to the experimental data. O t h e r measurements on paramagnetic Co [8, 9] were examined too, but these were not detailed for temperatures close to T~, unlike Develey's experimental resuits. The results of curve fitting using eq. (1) were disappointing; for Co and Fe there was no obviously well-defined value of 3' unless X0 were made very large, and then for T close to T c the 3' value did not agree at all well with those obtained by Develey without the sophistication of introducing X0. In order to check that the data for Co taken from Develey's publication was reasonably precise, the points lowest in temperature were taken and the first four, then the first five, then the first six, and so on up to ten, were fitted in a l o g ( l / x)-log(-e) plot. 3'' was obtained by a mean square straight line fit for each set of points. The first three exponents so found 1.314-+0.014, 1.300 -+ 0.037 and 1.300 + 0.035. A linear regression of the six exponents so found gave a 'zero point' intercept (or a T¢ intercept) for 3,' which had the value 1.335-+0.007, to be compared with Develey's 3' '-value of 1.32 --- 0.02. The justification for the extrapolation procedure is simply that as Tc is more closely approached so the first term in the power series representation of 1/x as a function of ( 1 - e) must become more and more dominant. The same procedure was then used to evaluate the exponent for a ln(1/xT)-ln~7 plot of the data, i.e., X0 was chosen as zero. In this case the 'zero point' intercept 3' '-value was 1.340 -+ 0.008, in good agreement with the 3''-values given above. The neglect of X0 is justified since its expected magnitude (~10 -6) is small compared with the values of X when T is very close to Tc; in this case T < 1.007T~. The use of the extrapolation m e t h o d leads to the conclusion that the use either of 77 or ( - e ) for reduced temperature
123
when T - T c is small makes no practical difference to the value of 3', i.e., T' = T, as might have been expected. Proceeding from this extrapolation method, the data for Co were then fitted by eq. (1) in the temperature range up to 1.023T c. The method used was to fit eq. (1) to the data using the first five points, then points 2-6, and so on, setting X0- This produced a set of 3,'s over the temperature range which decreased with increasing temperature; the highest temperature 3,'s were adjusted to be about equal to the low temperature ones by varying X0. The value of X0 required to 'flatten' the points turned out to be 1.3 × 10 -4, two orders of magnitude bigger than that found by Souletie and Tholence in the case of Ni. It was suggested that X0 was the metallic van Vleck susceptibility proposed by Kubo and Obata [10] and calculated by Shimizu [11] and his collaborators. The van Vleck susceptibility is generally considered as practically temperature independent and of the order of magnitude 10 -6. Is it likely that the band structures of Co and Ni for T > Tc are so different that their )(o'S differ by a fractor of 50? or that X0 for Co falls by a large quantity over a wider range of temperature? With the 5-point fits the value of 3' obtained was 1.30 -+ 0.02; 3, decreased slowly as the number of points in each fit was increased until with 20 points/fit 3, had the value 1.299---0.007, only marginally in agreement with Develey's value. As a further check on the extrapolation method, Develey's results for Fe were also examined. In this case the error bars were not reduced by the extrapolation procedure, perhaps because of the great difficulty of measuring the very large paramagnetic susceptibilities with precision. The values of 3, for the ( - e ) and r/fitting procedures in the temperature range up to 1.02T~ turned out to be 1.337___0.027 and 1 . 3 3 7+- 0.028, respectively, identical with Develey's 1.32-+ 0.02. The Curie constant, C, had the value (2.1 -+ 0.4) × 10 -2 for Fe. Arajs and Miller [7] made measurements on a series of alloys Fe~_cSi c and it seemed of interest to find the critical exponents, 3,, for these alloys. Immediately, a problem presented itself; there are (at least) three separate sets of measure-
124
D.E.G. Williams / Paramagnetic critical exponents
m e n t s of T c for these alloys [12-14] and the spread in T c values for a given alloy c o m p o s i t i o n could be as m u c h as 10 K. A quite arbitrary choice of the Tc values given by Sucksmith, Parsons and T h o m p s o n [12] was m a d e . F o r the first two alloys, with c = 0.0575 and c = 0.0682, the e x t r a p o l a t e d values o f 3' were 1 . 3 1 2 - 0.006 and 1.321---0.015, respectively. T h e 3,-value for the third alloy in the series, with c = 0 . 0 9 4 5 , t u r n e d out to be m u c h closer to 1.1 than 1.3, and this b r o u g h t the analysis to a full stop. Should 3' again be close to 1.3? If that is the case, then the correct value of T c can be f o u n d only by assuming a particular 3,-value. O n the o t h e r h a n d , with c at a b o u t 0.1 in an alloy system which orders crystallographically at c = 0.25, w h a t is the effect of short range o r d e r on T c X or 3'? E v e n the highest d e g r e e o f precision in the m e a s u r e m e n t of t e m p e r a t u r e and magnetic susceptibility m a y not lead to reproducible results w h e n alloys with the same c o m p o s i t i o n m a y have quite different h o m o g e n e i t y (or i n h o m o g e n e i t y ) . M e a s u r e m e n t s m a d e on elements and intermetallic c o m p o u n d s of fixed s t o i c h i o m e t r y should be reproducible e n o u g h to give consistent values of T c and o f 3'. T h e analysis of the results of m e a s u r e m e n t s on n o n s t o i c h i o m e t r i c alloys is beset with pitfalls. W h e n the solute c o n c e n t r a t i o n is small these m a y be relatively u n i m p o r t a n t , but as it b e c o m e s
larger they m a k e it difficult to have confidence in the m e a n i n g of the results of the analysis.
References [1] J.E. Noakes, N.E. Tornberg and A. Arrott, J. Appl. Phys. 37 (1966) 1264. [2] G. Develey, Compt. rend. 260 (1965) 4951; ibid. B262 (1966) 103. [3] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971). [4] J. Souletie and J.L. Tholence, Solid State Commun. 48 (1983) 407. [5] M. Fahnle and J. Souletie, J. Phys. C 17 (1984) L469. [6] M. Fahnle, J. Phys. C 18 (1985) 181. [7] S. Arajs and D.R. Miller, J. Appl. Phys. 31 (1960) 986. [8] Y. Nakagawa, J. Phys. Soc. Japan 11 (1956) 855. [9] W. Sucksmith and R.R. Pearce, Proc. Roy. Soc. A 167 (1938) 189. [10] R. Kubo and Y. Obata, J. Phys. Soc. Japan 11 (1956) 547. [11] M. Shimizu, J. Phys. Soc. Japan 16 (1961) 1114; M. Shimizu, T. Takahashi and A. Katsuki, ibid. 17 (1962) 1740; ibid. 18 (1963) 240, 801, 1192; M. Shimizu, D.O. van Ostenburg, D.J. Lam and A. Katsuki, ibid. 18 (1963) 1744; M. Shimizu and A. Katsuki, ibid. 19 (1964) 1135, 1856. [12] W. Sucksmith, D. Parsons and J.E. Thompson, Phil. Mag. 3 (1958) 1174. [13] S. Arajs, Phys. Stat. Sol. 11 (1965) 121. [14] M. Fallot, Ann. de Phys. 6 (1936) 305.
PARAMAGNETIC CRITICAL EXPONENTS
D.E.G. WILLIAMS
Physics Department, Loughborough University of Technology, Loughborough LEll 3TU, England Critical point exponents for the paramagnetic susceptibility in Co and Fe have been derived from the measurements published by Develey. The exponent, % for the linear reduced temperature ( 1 - T/Tc) is the same as that for the non-linear reduced temperature ( 1 - Tc/T ). The extension to measurements made on Fe-Si alloys broke down for Si concentrations of more than c = 0.07. In that case, 3' was very different from the value for Fe, it was not possible to judge whether this difference was real or was due to uncertainty in the value of Tc which, for c > 0.07, has three different published values at least.
Magnetic phase transitions between a ferromagnetic phase and a paramagnetic phase are frequently described in terms of critical point exponents [1,2]. These relate, e.g., the behaviour of the magnetization, M, to the reduced t e m p e r a t u r e , e, with e = (1 - (T/Tc), for T < T c where T~ is the ferromagnetic Curie t e m p e r a ture, or the variation of the paramagnetic susceptibility, X, to the reduced t e m p e r a t u r e (--e) = ((T/Tc) -1) for T > T~). The form of the relationship is, for the susceptibility,
and the introduction of a constant ' b a c k g r o u n d ' susceptibility, X0, can increase the range of temperature over which the critical point exponent y ' gives an accurate representation of the behaviour of the paramagnetic susceptibility. Thus, Souletie and Tholence [4] point out that, by using appropriate values of To, X0 and C 1 the paramagnetic susceptibility of Ni can be represented in the t e m p e r a t u r e range from Tc to 3 T¢ by a single exponent 3' defined by
Y=
- l n ( ( x - Xo)T) + In C In*/
(1)
,
where T ' is the critical point exponent. This form is appropriate only in the range of t e m p e r a t u r e close to T¢ since 3" is defined most simply [3] by 3 , ' = lim ln(1/X) T-~rc l n ( - e ) "
This range of t e m p e r a t u r e is that in which the higher order terms in the expansion of 1/X as a power series in ( - e ) can be neglected. T h e r e seems to be general a g r e e m e n t that the u p p e r limit of ( - e ) which will produce an acceptable value of 3" is at most ( - e ) < 0 . 0 5 . Recently it has been p r o p o s e d [4] that a redefinition of the reduced t e m p e r a t u r e variables as non-linear, i.e.,
~7=I-TJT,
for T > T¢,
The constant C was identified with a "Curie constant', and it was suggested that X0 might be a metallic van Vleck susceptibility. The fit given by eq. (1) to the experimental data for Ni was very good. The idea e m b o d i e d in eq. (1) has been applied to some. model systems [5, 6] and the conclusion that a scaling law such as eq. (1) might describe their susceptibilities over a wide range of values of ( T - Tc) was claimed for these systems. This extension of the range of validity of a single critical point exponent might be expected because of the introduction of the disposable p a r a m e t e r X0. In the light of these observations it seemed worthwhile to reexamine some data for the paramagnetic susceptibilities of ferromagnetic materials, if only to find whether the wide range of validity in t e m p e r a t u r e found in the case of Ni
0378-4363/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
D.E.G. Williams / Paramagnetic critical exponents
were reproduced for other materials. The initial ambition was to produce values of 3, for F e - S i alloys, using the susceptibility-temperature table given by Arajs and Miller [7]. As a first step towards attaining this objective an examination was made of the results given by Develey [2] for Co and for Fe in an attempt to find the values of the parameters 3' and X0 which produced the best fit to the experimental data. O t h e r measurements on paramagnetic Co [8, 9] were examined too, but these were not detailed for temperatures close to T~, unlike Develey's experimental resuits. The results of curve fitting using eq. (1) were disappointing; for Co and Fe there was no obviously well-defined value of 3' unless X0 were made very large, and then for T close to T c the 3' value did not agree at all well with those obtained by Develey without the sophistication of introducing X0. In order to check that the data for Co taken from Develey's publication was reasonably precise, the points lowest in temperature were taken and the first four, then the first five, then the first six, and so on up to ten, were fitted in a l o g ( l / x)-log(-e) plot. 3'' was obtained by a mean square straight line fit for each set of points. The first three exponents so found 1.314-+0.014, 1.300 -+ 0.037 and 1.300 + 0.035. A linear regression of the six exponents so found gave a 'zero point' intercept (or a T¢ intercept) for 3,' which had the value 1.335-+0.007, to be compared with Develey's 3' '-value of 1.32 --- 0.02. The justification for the extrapolation procedure is simply that as Tc is more closely approached so the first term in the power series representation of 1/x as a function of ( 1 - e) must become more and more dominant. The same procedure was then used to evaluate the exponent for a ln(1/xT)-ln~7 plot of the data, i.e., X0 was chosen as zero. In this case the 'zero point' intercept 3' '-value was 1.340 -+ 0.008, in good agreement with the 3''-values given above. The neglect of X0 is justified since its expected magnitude (~10 -6) is small compared with the values of X when T is very close to Tc; in this case T < 1.007T~. The use of the extrapolation m e t h o d leads to the conclusion that the use either of 77 or ( - e ) for reduced temperature
123
when T - T c is small makes no practical difference to the value of 3', i.e., T' = T, as might have been expected. Proceeding from this extrapolation method, the data for Co were then fitted by eq. (1) in the temperature range up to 1.023T c. The method used was to fit eq. (1) to the data using the first five points, then points 2-6, and so on, setting X0- This produced a set of 3,'s over the temperature range which decreased with increasing temperature; the highest temperature 3,'s were adjusted to be about equal to the low temperature ones by varying X0. The value of X0 required to 'flatten' the points turned out to be 1.3 × 10 -4, two orders of magnitude bigger than that found by Souletie and Tholence in the case of Ni. It was suggested that X0 was the metallic van Vleck susceptibility proposed by Kubo and Obata [10] and calculated by Shimizu [11] and his collaborators. The van Vleck susceptibility is generally considered as practically temperature independent and of the order of magnitude 10 -6. Is it likely that the band structures of Co and Ni for T > Tc are so different that their )(o'S differ by a fractor of 50? or that X0 for Co falls by a large quantity over a wider range of temperature? With the 5-point fits the value of 3' obtained was 1.30 -+ 0.02; 3, decreased slowly as the number of points in each fit was increased until with 20 points/fit 3, had the value 1.299---0.007, only marginally in agreement with Develey's value. As a further check on the extrapolation method, Develey's results for Fe were also examined. In this case the error bars were not reduced by the extrapolation procedure, perhaps because of the great difficulty of measuring the very large paramagnetic susceptibilities with precision. The values of 3, for the ( - e ) and r/fitting procedures in the temperature range up to 1.02T~ turned out to be 1.337___0.027 and 1 . 3 3 7+- 0.028, respectively, identical with Develey's 1.32-+ 0.02. The Curie constant, C, had the value (2.1 -+ 0.4) × 10 -2 for Fe. Arajs and Miller [7] made measurements on a series of alloys Fe~_cSi c and it seemed of interest to find the critical exponents, 3,, for these alloys. Immediately, a problem presented itself; there are (at least) three separate sets of measure-
124
D.E.G. Williams / Paramagnetic critical exponents
m e n t s of T c for these alloys [12-14] and the spread in T c values for a given alloy c o m p o s i t i o n could be as m u c h as 10 K. A quite arbitrary choice of the Tc values given by Sucksmith, Parsons and T h o m p s o n [12] was m a d e . F o r the first two alloys, with c = 0.0575 and c = 0.0682, the e x t r a p o l a t e d values o f 3' were 1 . 3 1 2 - 0.006 and 1.321---0.015, respectively. T h e 3,-value for the third alloy in the series, with c = 0 . 0 9 4 5 , t u r n e d out to be m u c h closer to 1.1 than 1.3, and this b r o u g h t the analysis to a full stop. Should 3' again be close to 1.3? If that is the case, then the correct value of T c can be f o u n d only by assuming a particular 3,-value. O n the o t h e r h a n d , with c at a b o u t 0.1 in an alloy system which orders crystallographically at c = 0.25, w h a t is the effect of short range o r d e r on T c X or 3'? E v e n the highest d e g r e e o f precision in the m e a s u r e m e n t of t e m p e r a t u r e and magnetic susceptibility m a y not lead to reproducible results w h e n alloys with the same c o m p o s i t i o n m a y have quite different h o m o g e n e i t y (or i n h o m o g e n e i t y ) . M e a s u r e m e n t s m a d e on elements and intermetallic c o m p o u n d s of fixed s t o i c h i o m e t r y should be reproducible e n o u g h to give consistent values of T c and o f 3'. T h e analysis of the results of m e a s u r e m e n t s on n o n s t o i c h i o m e t r i c alloys is beset with pitfalls. W h e n the solute c o n c e n t r a t i o n is small these m a y be relatively u n i m p o r t a n t , but as it b e c o m e s
larger they m a k e it difficult to have confidence in the m e a n i n g of the results of the analysis.
References [1] J.E. Noakes, N.E. Tornberg and A. Arrott, J. Appl. Phys. 37 (1966) 1264. [2] G. Develey, Compt. rend. 260 (1965) 4951; ibid. B262 (1966) 103. [3] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971). [4] J. Souletie and J.L. Tholence, Solid State Commun. 48 (1983) 407. [5] M. Fahnle and J. Souletie, J. Phys. C 17 (1984) L469. [6] M. Fahnle, J. Phys. C 18 (1985) 181. [7] S. Arajs and D.R. Miller, J. Appl. Phys. 31 (1960) 986. [8] Y. Nakagawa, J. Phys. Soc. Japan 11 (1956) 855. [9] W. Sucksmith and R.R. Pearce, Proc. Roy. Soc. A 167 (1938) 189. [10] R. Kubo and Y. Obata, J. Phys. Soc. Japan 11 (1956) 547. [11] M. Shimizu, J. Phys. Soc. Japan 16 (1961) 1114; M. Shimizu, T. Takahashi and A. Katsuki, ibid. 17 (1962) 1740; ibid. 18 (1963) 240, 801, 1192; M. Shimizu, D.O. van Ostenburg, D.J. Lam and A. Katsuki, ibid. 18 (1963) 1744; M. Shimizu and A. Katsuki, ibid. 19 (1964) 1135, 1856. [12] W. Sucksmith, D. Parsons and J.E. Thompson, Phil. Mag. 3 (1958) 1174. [13] S. Arajs, Phys. Stat. Sol. 11 (1965) 121. [14] M. Fallot, Ann. de Phys. 6 (1936) 305.