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The critical exponent for the non-linear paramagnetic effect
Volume 100A, number 5
PHYSICS LETTERS
30 January 1984
THE CRITICAL EXPONENT FOR THE NON-LINEAR PARAMAGNETIC EFFECT ~ B. FUGIEL and J. ZIOLO
Institute of Physics, Silesian University, Uniwersytecka 4, 40-007 Katowice, Poland Received 9 Selgtember 1983
The power exponent 72 determined from measurements of the non-linear paramagnetic effect (NPE) in HgCr2Se4 and CdCr2Se4 was compared with the critical exponent 3'2 of the expression Ax/B 2 against T - Tc found from the scaling laws, i.e. 3"2= 2fl+ 33".For this purpose theoretical and experimental values oft and 3"reported in various articles were used.
In ref. [1] the results are given of measurements of temperature functions for Ax/B 2 in the ferromagnets HgCr2Se4 and CdCr2Se 4, where A× = ×B -- ×0 is the difference of the magnetic susceptibitily (A X < 0) in an external magnetic field where B ~ 0 and a field where B = 0. Experimental conditions were arranged so that values of A× satisfied the condition A× ~ B 2. Hence as a measure of non-linearity of the paramagnetic effect (quadratic form) the magnitude Ax/B 2 was taken. For technical reasons the relation AX ¢cB 2 could only be observed experimentally at temperatures less than T = Tc + 3 K for HgCr2Se 4 and T = Tc + 4 K for CdCr2Se 4. In the temperature ranges Tc + 3 K
< T < Tc + 4O K for HgCr2Se4, and Tc + 4 K < T < T c + 40 K for CdCr2Se 4, experimental temperature curves for Ax/B 2 were found which may be written using the exponential formula: Ax/B 2 ¢c (T - Tc)-¥2,
(1)
where the values o f the exponent are: for HgCr2Se 4 ~2 = 5.07 -+ 0.05 and for CdCr2Se 4 ~2 = 4.94 + 0.05. Since for HgCr2Se4 and CdCr2Se 4 the Curie temperature has the values 105.9 K and 127.2 K, respectivily, for the measurements performed the expression e = ( T - Tc)/T c took values: 2 X 10 - 2 < e < 32 X 10 -2 for HgCr2Se4, and 3 X 10 -2 < e < 38 X 10 -2 for CdCr2Se 4. When making a comparison it should be noted that the range o f e values related to measure¢' Sponsored by the Polish Academy of Science, Problem No. MR.I.9. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ment points used to determine the critical exponent 7 which in ref. [2] is used to express the temperature variations of initial susceptibitfly is l0 -3 ~ e ~ 2 X l0 -2 for HgCr2Se4, and 3 X l0 -3 ~ e ~ 7 X 10 -2 for CdCr2Se 4. The question arises, whether the values o f the power exponent given in ref. [1] for the expression Ax/B2 against T - T c are the same in the close vicinity o f Tc. Due to lack of experimental and theoretical data on temperatures sufficiently close to Tc it is difficult to give a definitive answer supported by the results of measurements or calculations. It is possible, however, to compare the values of the power exponent 72 and the critical exponent 72 when the values of certain other critical exponents are known, having been found either theoretically or experimentally, and when also the relations between these exponents and the exponent 72 are known. In this case it is useful to apply the scaling laws. As is known, the initial magnetic susceptibility and the magnitude Ax/B 2 are proportional to the second and fourth derivative, respectively, of the Gibbs potential G relative to the magnetic field, i.e.: X0 ~(a2G/aH2)T,
AX/B2 ~(34G/aH4)T,
(2)
where the values of derivatives are calculated for H = 0. The following relation also applies [3] :
(a4G/aH4)r ~ (T/Tc - 1)-2a*(aEGDH2)T.
(3)
Applying the scaling laws the exponent A 4 may be expressed in terms o f t and 7 [3] :
259
30 January 1984
PHYSICS LETTERS
Volume 100A, number 5 Table 1
Method of determination
Publication
Value 2# + 37
theoretical, 3-dimensional Heisenberg model, HTSE technique
ref. [5]
5.06
theoretical, 3-dimensional Heisenberg model, HTSE technique
ref. [6]
1.36 -+ 0.04
theoretical, 3-dimensional Heisenberg model, HTSE technique
ref. [7]
0.34 -+0.02
1.30 +-0.02
experimental, magnetic balance, HgCr2Se4
ret: [2]
4.58
0.34 ± 0.02
1.29 -+ 0.02
experimental, magnetic balance, CdCr2Se4
ref. [2]
4.55
Exponent Y 0.385 -+0.025
1.43 ± 0.01
0.365 +-0.035
A 4 =/3 + T.
(4)
Hence substituting from (2), (3) and (4) we obtain: T2 = 2/3 + 37.
(5)
Table 1 sets out theoretical and experimental values of/3 and 7 as reported by various authors. From table 1 it may be seen that, especially for/3 and 3' determined theoretically by the HTSE (high temperature series expansion) method, expression (5) gives good agreement between ~2 and ")'2 values. Certain deviations may be found when using in formula (5) the experimental data given in the table. For the molecular field approximation (MFA) relation (5) gives a value for the exponent o f 72 = 4 03 = 0.5, 7 = 1), which differs markedly from the value o f the power exponent ~2 in expression (1). This may be explained by the fact that when measuring the non-linear paramagnetic effect in HgCr2Se 4 and CdCr2Se 4 the range o f temperature applied in determining ~2 extends considerably further than the region in which the tested magnetic system may be described by the MFA. For HgCr2Se 4 and CdCr2Se 4 the values o f p a -
260
ramagnetic Curie temperature are 200 K and 204 K, respectively [7]. Basing on relation (5) and data given in the table it may be concluded that for HgCr2Se 4 and CdCr2Se 4 there is no appreciable divergence between the value o f the critical exponent for the relation A×/B 2 against T - T c, and the value o f the power exponent 52 which characterises the temperature dependence o f Ax/B 2 in a relatively wide range of temperatures from a few up to some tens o f degrees above T c.
References [ 1 ] B. Fugiel, J. ZioIo and M. Drzazga, Phys. Rev. B, to be published. [2] W. Zarek, Acta Phys. Poton. A52 (1977) 657. [ 3 ] H.E Stanley, Introduction to phase transitions and critical phenomena (Clarendon, Oxford, 1971). [4] Baker et al., Phys. Rev. 164 (1967) 800. [5] Ferer et al., Phys. Rev. B4 (1971) 3954. [6] Lee and H.E. Stanley, Proc. Intern. Conf. on Magnetism (Grenoble, 1970). [7] P.K. Baltzer, P.J. Wojtowicz, M. Robbins and E. Lopatin, Phys. Rev. 151 (1966) 367.
PHYSICS LETTERS
30 January 1984
THE CRITICAL EXPONENT FOR THE NON-LINEAR PARAMAGNETIC EFFECT ~ B. FUGIEL and J. ZIOLO
Institute of Physics, Silesian University, Uniwersytecka 4, 40-007 Katowice, Poland Received 9 Selgtember 1983
The power exponent 72 determined from measurements of the non-linear paramagnetic effect (NPE) in HgCr2Se4 and CdCr2Se4 was compared with the critical exponent 3'2 of the expression Ax/B 2 against T - Tc found from the scaling laws, i.e. 3"2= 2fl+ 33".For this purpose theoretical and experimental values oft and 3"reported in various articles were used.
In ref. [1] the results are given of measurements of temperature functions for Ax/B 2 in the ferromagnets HgCr2Se4 and CdCr2Se 4, where A× = ×B -- ×0 is the difference of the magnetic susceptibitily (A X < 0) in an external magnetic field where B ~ 0 and a field where B = 0. Experimental conditions were arranged so that values of A× satisfied the condition A× ~ B 2. Hence as a measure of non-linearity of the paramagnetic effect (quadratic form) the magnitude Ax/B 2 was taken. For technical reasons the relation AX ¢cB 2 could only be observed experimentally at temperatures less than T = Tc + 3 K for HgCr2Se 4 and T = Tc + 4 K for CdCr2Se 4. In the temperature ranges Tc + 3 K
< T < Tc + 4O K for HgCr2Se4, and Tc + 4 K < T < T c + 40 K for CdCr2Se 4, experimental temperature curves for Ax/B 2 were found which may be written using the exponential formula: Ax/B 2 ¢c (T - Tc)-¥2,
(1)
where the values o f the exponent are: for HgCr2Se 4 ~2 = 5.07 -+ 0.05 and for CdCr2Se 4 ~2 = 4.94 + 0.05. Since for HgCr2Se4 and CdCr2Se 4 the Curie temperature has the values 105.9 K and 127.2 K, respectivily, for the measurements performed the expression e = ( T - Tc)/T c took values: 2 X 10 - 2 < e < 32 X 10 -2 for HgCr2Se4, and 3 X 10 -2 < e < 38 X 10 -2 for CdCr2Se 4. When making a comparison it should be noted that the range o f e values related to measure¢' Sponsored by the Polish Academy of Science, Problem No. MR.I.9. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ment points used to determine the critical exponent 7 which in ref. [2] is used to express the temperature variations of initial susceptibitfly is l0 -3 ~ e ~ 2 X l0 -2 for HgCr2Se4, and 3 X l0 -3 ~ e ~ 7 X 10 -2 for CdCr2Se 4. The question arises, whether the values o f the power exponent given in ref. [1] for the expression Ax/B2 against T - T c are the same in the close vicinity o f Tc. Due to lack of experimental and theoretical data on temperatures sufficiently close to Tc it is difficult to give a definitive answer supported by the results of measurements or calculations. It is possible, however, to compare the values of the power exponent 72 and the critical exponent 72 when the values of certain other critical exponents are known, having been found either theoretically or experimentally, and when also the relations between these exponents and the exponent 72 are known. In this case it is useful to apply the scaling laws. As is known, the initial magnetic susceptibility and the magnitude Ax/B 2 are proportional to the second and fourth derivative, respectively, of the Gibbs potential G relative to the magnetic field, i.e.: X0 ~(a2G/aH2)T,
AX/B2 ~(34G/aH4)T,
(2)
where the values of derivatives are calculated for H = 0. The following relation also applies [3] :
(a4G/aH4)r ~ (T/Tc - 1)-2a*(aEGDH2)T.
(3)
Applying the scaling laws the exponent A 4 may be expressed in terms o f t and 7 [3] :
259
30 January 1984
PHYSICS LETTERS
Volume 100A, number 5 Table 1
Method of determination
Publication
Value 2# + 37
theoretical, 3-dimensional Heisenberg model, HTSE technique
ref. [5]
5.06
theoretical, 3-dimensional Heisenberg model, HTSE technique
ref. [6]
1.36 -+ 0.04
theoretical, 3-dimensional Heisenberg model, HTSE technique
ref. [7]
0.34 -+0.02
1.30 +-0.02
experimental, magnetic balance, HgCr2Se4
ret: [2]
4.58
0.34 ± 0.02
1.29 -+ 0.02
experimental, magnetic balance, CdCr2Se4
ref. [2]
4.55
Exponent Y 0.385 -+0.025
1.43 ± 0.01
0.365 +-0.035
A 4 =/3 + T.
(4)
Hence substituting from (2), (3) and (4) we obtain: T2 = 2/3 + 37.
(5)
Table 1 sets out theoretical and experimental values of/3 and 7 as reported by various authors. From table 1 it may be seen that, especially for/3 and 3' determined theoretically by the HTSE (high temperature series expansion) method, expression (5) gives good agreement between ~2 and ")'2 values. Certain deviations may be found when using in formula (5) the experimental data given in the table. For the molecular field approximation (MFA) relation (5) gives a value for the exponent o f 72 = 4 03 = 0.5, 7 = 1), which differs markedly from the value o f the power exponent ~2 in expression (1). This may be explained by the fact that when measuring the non-linear paramagnetic effect in HgCr2Se 4 and CdCr2Se 4 the range o f temperature applied in determining ~2 extends considerably further than the region in which the tested magnetic system may be described by the MFA. For HgCr2Se 4 and CdCr2Se 4 the values o f p a -
260
ramagnetic Curie temperature are 200 K and 204 K, respectively [7]. Basing on relation (5) and data given in the table it may be concluded that for HgCr2Se 4 and CdCr2Se 4 there is no appreciable divergence between the value o f the critical exponent for the relation A×/B 2 against T - T c, and the value o f the power exponent 52 which characterises the temperature dependence o f Ax/B 2 in a relatively wide range of temperatures from a few up to some tens o f degrees above T c.
References [ 1 ] B. Fugiel, J. ZioIo and M. Drzazga, Phys. Rev. B, to be published. [2] W. Zarek, Acta Phys. Poton. A52 (1977) 657. [ 3 ] H.E Stanley, Introduction to phase transitions and critical phenomena (Clarendon, Oxford, 1971). [4] Baker et al., Phys. Rev. 164 (1967) 800. [5] Ferer et al., Phys. Rev. B4 (1971) 3954. [6] Lee and H.E. Stanley, Proc. Intern. Conf. on Magnetism (Grenoble, 1970). [7] P.K. Baltzer, P.J. Wojtowicz, M. Robbins and E. Lopatin, Phys. Rev. 151 (1966) 367.