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The magnetic phase diagram of pseudo-one-dimensional CsMnBr3·2D2O near the bicritical point
Volume 68A, number 3,4
PHYSICS LETTERS
16 October 1978
THE MAGNETIC PHASE DIAGRAM OF PSEUDO-ONE-DIMENSIONAL CsMnBr3 2D20 NEAR THE BICRITICAL POINT -
J.A.J. BASTEN and E. FRIKKEE Netherlands Energy Research Foundation ECN, Petten (N.H.), The Netherlands
and W.J.M. DE JONGE Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands Received 12 June 1978
The magnetic phase diagram of pseudo-one-dimensional CsMnBr3• 2D20, as determined by means of neutron scattering, is found to deviate markedly from that for three-dimensional systems. Nevertheless, near the bicritical point the phase boundaries are described correctly by extended scaling theory.
In presence of a magnetic field H, the ordering temperature T~in a system of weakly coupled isotropic Heisenberg chains increases considerably, as has been calculated by deanisotropy Jonge andon coworkers The influence of a small the phase[1]. diagrams of several pseudo-one-dimensional (d 1) Heisenberg systems, including the title-compound has been considered by Hijmans et al. [2]. When H is applied along the easy axis, the anomalous behaviour of T~(H)is expected only above the spin-flop field HSF, whree T~(B)may largely exceed TC(H = 0). Although the overall features of such phase diagrams deviate markedly from those for d 3 systems, the critical behaviour near the phase boundaries is expected to be essentially d 3. Consequently, the bicritical behaviour is expected to be not essentially different and bicritical-scaling theories, recently developed [3,4], should also apply to pseudod 1 systems. The crystal structure of CsMnBr 3 2D20 (CMB) is orthorhombic with spacegroup P~.Below TN(AF) 6.30 K the symmetry of the antiferromagnetic ordering is described by the magnetic spacegroup ~c’c’a’ [5]. The b-axis and the c-axis are the easy and intermediate axis, respectively [6].
The neutron scattering experiments on CMB have been performed on a double-axis diffractometer at the Petten HFR-reactor. disk-likeinsingle crystal 3 wasThe mounted a superconof 15 X magnet 15 X 4 (field mm homogeneity 0.15%) with H ducting approximately parallel to the easy axis. After, mounting the crystal to final adjustments could be made. The mismatch angle i,Ii between H and the b-axis has been determined to be 0.5°.Near the bicritical temperature Tb ~ 5.3 K the short-term temperature stability was better than 1 mK with a maximum drift of 5 mK in 24 h. The peak intensities of three magnetic Bragg reflections, viz. (1 0 3), (1 0 1) and (3 0 1), were recorded as a function of H and T. From these data the components of the staggered magnetization M~(H,7) along and M~(H,7) perpendicular to the easy axis have been obtained separately [7]. Some typical results of field scans both for T Tb and T Tb are shown in fig. 1. ‘~
If ui exceeds the can critical angle 7),the nomismatch first-orderangle AF—SF transition be ob~‘c( served [8]. In the present experiment this is the case for all T ~ Tb, since in CMB qi~(o) 0.08°,as can be calculated [81 from the anisotropy field (770 Oe) and the exchange field (273 kOe) [6]. In fact, the 385
Volume 68A, number 3,4
~4O
PHYSICS LETTERS
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observed rapid rotation of M~tfrom an AF-like to an SF-like ordering can be described by molecular field theory, which applies for ç14’ ui~(T)[71. The agreement between the experimental and theoretical curves is striking, as can be seen in fig. lb. The spin-flop line HSF(fl, derived from the points of intersection of the curves for (M~)2 and (M~)2,is shown in fig. 2. The location of the second-order AF—P boundary 7~(H)and SF—P boundary T~(/f)have been obtamed from the disappearance of the order parameters M~and M~,respectively (see fIg. id). Leastsquares fits of I (M~)~ = B c2’k’-i(1) to the intensities of the magnetic reflections have been performed, with,B, /3 and e variable. In eq. (1) e = T/T~(JI)— ii and e = IH/H~(7) ij in temperature and field scans, respectively. In the fits only data with iO—3 ~ 10_i were taken into account. The results for T~(ll)and T~(R)are shown in fig. 2. —
4.5
5.0
5~5
6.0
1(K)
~.
Fig. 1. Field dependence of the scaled peak intensities of three magnetic Bragg reflections near (the continuation of) the spin-flop line, (a) for T ~ Tb and (c) for T ~ Tb. Triangles (1 0 3), squares (1 0 1) and circles (3 0 1). In (b) and (d) the corresponding field dependence of (M0~)2and (M~)2are shown, indicated by open and closed circles, respectively. The solid lines in (b) represent the mean-field prediction of (M~t)2~nd (M~)2for a mismatch angle ‘P 0.5° and ~ as shown by the dash-dotted line.
386
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16 October 1978
features are obvious from the considerable increase of T~ Fig. 2. Magnetic phase diagram of CMB. The pseudo-d = with H. Solid and dash-dotted curves represent best fits of eq. (2) to the data with q variable and ~ = 1.175, respectively. In the fits Tb was fixed at 5.255 K and tmax = 0.15. The theoretical (dashed) and fitted (solid) 7 0 axes are also shown. The insert shows details near Tb.
Along both phase boundaries no systematic variation in 13 was observed. A statistical analysis of the j3values yields ~ = 0.321(6) and ~ = 0.326(7). Both values may be compared with the d = 3 Ising value [9] /3 = 0.325(1), which is theoretically predicted for ~ and i3~in a spin-flop system with orthorhombic anisotropy [3]. Consequently, we expect that the number n of relevant spin components at the bicritical point in this system equals 2. According to extended scaling theory [3] the second-order phase boundaries T~(R)and T~(Ji) meet tangentially at Tb. They are described by =
—w~and +w
1
(2)
,
respectively. Although w,, and w1 are not universal constants, their ratio is. For n = 2, Wj/W// = 1 by ~ymmetry. The optimum scaling axes are2 given by — H~and gt = gT/T~ — pt and t t + gq, where g = H — 1. The j~ 0 scaling axis must be tangent to the spin-flop transition line at Tb, that is p Tb(dH2/dt)b. ~Thevalue of q is non-universal. Starting from mean-field theory Fisher [10] obtained the estimate I ~+2 dT~ q(n) —-~-—~_(——--~) (3) ‘
0
Volume 68A, number 3,4
PHYSICS LETTERS
curves are included in fig. 2. In fact eq. (2) applies only if the magnetic field
‘-I I
1.41-
12F
r
1.61-
0.02
1 ~
o
Tb~5.25K
~
Tb -526K
I
small mismatch angle ‘,li, the phase boundaries ~ and T~will not meet tangentially at Tb. Instead a
•
Tb5.24K
120
has been aligned perfectly along the easy axis. For a
115 ~ jlo
0 T~5.27 ~<
_______________
_________________
16 October 1978
-~
_________________
0.05 0.1 0.2 0.02 tmax
0.05 0.1
0.2 tMax
Fig. 3. Fitted values of ~ and w,~= w
1 as a function of tmax for several choices of Tb. From these results Tb = 5.255(5) K and tmax 0.15 have been estimated,
minute away rounding T~,eq. theis(2) shape expected the T~ phase boundaries can shouldfrom indeed be follow observed in again fig.of2 [3], atat 5.275(5) with aTb. pseudoK.This However, bicritical temperature T, which was checked in the analysis described above. As final estimates for 0 and q we would quote 2. The 00 = 1.22(6) and q = 5.5(2) X l0~~k0e value is in good agreement with the theoretical prediction 0 = 1.175(15). It should be emphasized that the fitted slope q of the 7’ = 0 axis strongly deviates
(see fig. 2) from Fisher’s estimate (eq. (3)). which is expected to be valid for small values of (4 — d), where d is the spatial dimensionality. So, it is
doubtful whether eq. (3) also applies to a pseudod = I system. The theoretical expressions (eq. (2)) for 7~(/1) and T~(ll)have been compared with our 2/d7’)b data in = least-squares In all fits (dH transition 75.4 kOe2/K,fits. obtained fromthethevalue spin-flop line, was used and w 1/w1 was fixed to unity. As the range of t in which eq. (2) applies is not universal, we tried to determine this range in an objective way. The approach is based on the requirement that all fits in which data with t ~ ~ are taken into account must yield the same estimates for any variable, independent of tmax up to an upper value of tmax. This requirement can be fulfilled only if the correct value of Tb is used. The procedure is illustrated in fig. 3, where for several choices of Tb the fitted values of 0 and w,1 = w1 are shown as a function of tm~.The estimates for q and ~b are not shown as these hardly vary with Tb and tmax. From the results in fig. 3 we estimated Tb = 5.255(5) K. Systematic deviations of the data from eq. (2) become apparent for i’ > tmax 0.15. With Tb fixed at 5.255 K, a least-squares fit to the data with t ~ = 0.15 gives Hb = 26.541(17) 2 and kOe, 0 = 1.226(9), q = 5.38(10) X iO~ k0e w 2. If in addition 0 is fixed = 7033(52) kOe at1 thew1 theoretical prediction 0 1.175, the values = 26.555(21) kOe, q 5.57(12)X 10~ k0e2 and w 2 result. Both fitted
8
=
w1
=
In conclusion it may be stated that also in the
pseudo-d = 1 system CMB the shape of the phase boundaries near the bicritical point is predicted correctly by extended scaling theory with n = 2. Fisher’s estimate of the non-universal value q is not appropriate, as might used be expected from the mean field type of approach in the determination, which clearly is not applicable in pseudo-d = 1 systems. Besides the crossover exponent 0 also the critical exponents ~3,and fl~ are found in good agreement with the theoretical predictions for an antiferromagnet with weak orthorhombic anisotropy. References [1] W.J.M. de Jonge et al., Phys. Rev. B17 (1978), and references therein. [2] J.P.A.M. Hijmans et al., Phys. Rev. Lett. 40 (1978) 1108. [31 J.M. Kosterlitz, D.R. Nelson and M.E. Fisher, Phys. Rev. B13 (1976) 412, and references therein. [4] E. Domany and M.E. Fisher, Phys. Rev. B15 (1977)
3510, and references therein. [5] C.H.W. Swüste, W.J.M. de Jonge and J.A.G.W. van Meyel, Physica 76 (1974) 21. [6] AC. Botterman, Ph.D. Thesis (Eindhoven, 1976), unpublished. [7] J.A.J. Basten, E. Frikkee and W.J.M. de Jonge, to be published. [8] H. Rohrer and H. Thomas, J. AppL Phys. 40 (1969) 1025. [9] J.C. le Guillou and 3. Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95.
[10] M.E. Fisher, Phys. Rev. Lett. 34 (1975) 1634.
6226(38) kOe 387
PHYSICS LETTERS
16 October 1978
THE MAGNETIC PHASE DIAGRAM OF PSEUDO-ONE-DIMENSIONAL CsMnBr3 2D20 NEAR THE BICRITICAL POINT -
J.A.J. BASTEN and E. FRIKKEE Netherlands Energy Research Foundation ECN, Petten (N.H.), The Netherlands
and W.J.M. DE JONGE Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands Received 12 June 1978
The magnetic phase diagram of pseudo-one-dimensional CsMnBr3• 2D20, as determined by means of neutron scattering, is found to deviate markedly from that for three-dimensional systems. Nevertheless, near the bicritical point the phase boundaries are described correctly by extended scaling theory.
In presence of a magnetic field H, the ordering temperature T~in a system of weakly coupled isotropic Heisenberg chains increases considerably, as has been calculated by deanisotropy Jonge andon coworkers The influence of a small the phase[1]. diagrams of several pseudo-one-dimensional (d 1) Heisenberg systems, including the title-compound has been considered by Hijmans et al. [2]. When H is applied along the easy axis, the anomalous behaviour of T~(H)is expected only above the spin-flop field HSF, whree T~(B)may largely exceed TC(H = 0). Although the overall features of such phase diagrams deviate markedly from those for d 3 systems, the critical behaviour near the phase boundaries is expected to be essentially d 3. Consequently, the bicritical behaviour is expected to be not essentially different and bicritical-scaling theories, recently developed [3,4], should also apply to pseudod 1 systems. The crystal structure of CsMnBr 3 2D20 (CMB) is orthorhombic with spacegroup P~.Below TN(AF) 6.30 K the symmetry of the antiferromagnetic ordering is described by the magnetic spacegroup ~c’c’a’ [5]. The b-axis and the c-axis are the easy and intermediate axis, respectively [6].
The neutron scattering experiments on CMB have been performed on a double-axis diffractometer at the Petten HFR-reactor. disk-likeinsingle crystal 3 wasThe mounted a superconof 15 X magnet 15 X 4 (field mm homogeneity 0.15%) with H ducting approximately parallel to the easy axis. After, mounting the crystal to final adjustments could be made. The mismatch angle i,Ii between H and the b-axis has been determined to be 0.5°.Near the bicritical temperature Tb ~ 5.3 K the short-term temperature stability was better than 1 mK with a maximum drift of 5 mK in 24 h. The peak intensities of three magnetic Bragg reflections, viz. (1 0 3), (1 0 1) and (3 0 1), were recorded as a function of H and T. From these data the components of the staggered magnetization M~(H,7) along and M~(H,7) perpendicular to the easy axis have been obtained separately [7]. Some typical results of field scans both for T Tb and T Tb are shown in fig. 1. ‘~
If ui exceeds the can critical angle 7),the nomismatch first-orderangle AF—SF transition be ob~‘c( served [8]. In the present experiment this is the case for all T ~ Tb, since in CMB qi~(o) 0.08°,as can be calculated [81 from the anisotropy field (770 Oe) and the exchange field (273 kOe) [6]. In fact, the 385
Volume 68A, number 3,4
~4O
PHYSICS LETTERS
I
,
~
~
Io6~,~
,,30
.
~
~20
.~ C dl
-~
4O~~
I
2000
(a)
H(kOe)
(b)
20~
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i
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~
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i
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..
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27 28 H (kOe)
U AF I \I ~ N ______________________________
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25
26
27 28 H(kOe)
(d)
observed rapid rotation of M~tfrom an AF-like to an SF-like ordering can be described by molecular field theory, which applies for ç14’ ui~(T)[71. The agreement between the experimental and theoretical curves is striking, as can be seen in fig. lb. The spin-flop line HSF(fl, derived from the points of intersection of the curves for (M~)2 and (M~)2,is shown in fig. 2. The location of the second-order AF—P boundary 7~(H)and SF—P boundary T~(/f)have been obtamed from the disappearance of the order parameters M~and M~,respectively (see fIg. id). Leastsquares fits of I (M~)~ = B c2’k’-i(1) to the intensities of the magnetic reflections have been performed, with,B, /3 and e variable. In eq. (1) e = T/T~(JI)— ii and e = IH/H~(7) ij in temperature and field scans, respectively. In the fits only data with iO—3 ~ 10_i were taken into account. The results for T~(ll)and T~(R)are shown in fig. 2. —
4.5
5.0
5~5
6.0
1(K)
~.
Fig. 1. Field dependence of the scaled peak intensities of three magnetic Bragg reflections near (the continuation of) the spin-flop line, (a) for T ~ Tb and (c) for T ~ Tb. Triangles (1 0 3), squares (1 0 1) and circles (3 0 1). In (b) and (d) the corresponding field dependence of (M0~)2and (M~)2are shown, indicated by open and closed circles, respectively. The solid lines in (b) represent the mean-field prediction of (M~t)2~nd (M~)2for a mismatch angle ‘P 0.5° and ~ as shown by the dash-dotted line.
386
o
~ dl 1500 “ii
~
500
-
~
/ /
r
~
C
C
I
SF
AF ~ SF 10 3.736 K . ___________________ •—‘‘~~4_ ‘ ~- ~‘°°°° 23.5 24.0 24.5 250 23.5 24.0 24.5 25.0
—I IC
I
800
t
V
.
I
~
— —
16 October 1978
features are obvious from the considerable increase of T~ Fig. 2. Magnetic phase diagram of CMB. The pseudo-d = with H. Solid and dash-dotted curves represent best fits of eq. (2) to the data with q variable and ~ = 1.175, respectively. In the fits Tb was fixed at 5.255 K and tmax = 0.15. The theoretical (dashed) and fitted (solid) 7 0 axes are also shown. The insert shows details near Tb.
Along both phase boundaries no systematic variation in 13 was observed. A statistical analysis of the j3values yields ~ = 0.321(6) and ~ = 0.326(7). Both values may be compared with the d = 3 Ising value [9] /3 = 0.325(1), which is theoretically predicted for ~ and i3~in a spin-flop system with orthorhombic anisotropy [3]. Consequently, we expect that the number n of relevant spin components at the bicritical point in this system equals 2. According to extended scaling theory [3] the second-order phase boundaries T~(R)and T~(Ji) meet tangentially at Tb. They are described by =
—w~and +w
1
(2)
,
respectively. Although w,, and w1 are not universal constants, their ratio is. For n = 2, Wj/W// = 1 by ~ymmetry. The optimum scaling axes are2 given by — H~and gt = gT/T~ — pt and t t + gq, where g = H — 1. The j~ 0 scaling axis must be tangent to the spin-flop transition line at Tb, that is p Tb(dH2/dt)b. ~Thevalue of q is non-universal. Starting from mean-field theory Fisher [10] obtained the estimate I ~+2 dT~ q(n) —-~-—~_(——--~) (3) ‘
0
Volume 68A, number 3,4
PHYSICS LETTERS
curves are included in fig. 2. In fact eq. (2) applies only if the magnetic field
‘-I I
1.41-
12F
r
1.61-
0.02
1 ~
o
Tb~5.25K
~
Tb -526K
I
small mismatch angle ‘,li, the phase boundaries ~ and T~will not meet tangentially at Tb. Instead a
•
Tb5.24K
120
has been aligned perfectly along the easy axis. For a
115 ~ jlo
0 T~5.27 ~<
_______________
_________________
16 October 1978
-~
_________________
0.05 0.1 0.2 0.02 tmax
0.05 0.1
0.2 tMax
Fig. 3. Fitted values of ~ and w,~= w
1 as a function of tmax for several choices of Tb. From these results Tb = 5.255(5) K and tmax 0.15 have been estimated,
minute away rounding T~,eq. theis(2) shape expected the T~ phase boundaries can shouldfrom indeed be follow observed in again fig.of2 [3], atat 5.275(5) with aTb. pseudoK.This However, bicritical temperature T, which was checked in the analysis described above. As final estimates for 0 and q we would quote 2. The 00 = 1.22(6) and q = 5.5(2) X l0~~k0e value is in good agreement with the theoretical prediction 0 = 1.175(15). It should be emphasized that the fitted slope q of the 7’ = 0 axis strongly deviates
(see fig. 2) from Fisher’s estimate (eq. (3)). which is expected to be valid for small values of (4 — d), where d is the spatial dimensionality. So, it is
doubtful whether eq. (3) also applies to a pseudod = I system. The theoretical expressions (eq. (2)) for 7~(/1) and T~(ll)have been compared with our 2/d7’)b data in = least-squares In all fits (dH transition 75.4 kOe2/K,fits. obtained fromthethevalue spin-flop line, was used and w 1/w1 was fixed to unity. As the range of t in which eq. (2) applies is not universal, we tried to determine this range in an objective way. The approach is based on the requirement that all fits in which data with t ~ ~ are taken into account must yield the same estimates for any variable, independent of tmax up to an upper value of tmax. This requirement can be fulfilled only if the correct value of Tb is used. The procedure is illustrated in fig. 3, where for several choices of Tb the fitted values of 0 and w,1 = w1 are shown as a function of tm~.The estimates for q and ~b are not shown as these hardly vary with Tb and tmax. From the results in fig. 3 we estimated Tb = 5.255(5) K. Systematic deviations of the data from eq. (2) become apparent for i’ > tmax 0.15. With Tb fixed at 5.255 K, a least-squares fit to the data with t ~ = 0.15 gives Hb = 26.541(17) 2 and kOe, 0 = 1.226(9), q = 5.38(10) X iO~ k0e w 2. If in addition 0 is fixed = 7033(52) kOe at1 thew1 theoretical prediction 0 1.175, the values = 26.555(21) kOe, q 5.57(12)X 10~ k0e2 and w 2 result. Both fitted
8
=
w1
=
In conclusion it may be stated that also in the
pseudo-d = 1 system CMB the shape of the phase boundaries near the bicritical point is predicted correctly by extended scaling theory with n = 2. Fisher’s estimate of the non-universal value q is not appropriate, as might used be expected from the mean field type of approach in the determination, which clearly is not applicable in pseudo-d = 1 systems. Besides the crossover exponent 0 also the critical exponents ~3,and fl~ are found in good agreement with the theoretical predictions for an antiferromagnet with weak orthorhombic anisotropy. References [1] W.J.M. de Jonge et al., Phys. Rev. B17 (1978), and references therein. [2] J.P.A.M. Hijmans et al., Phys. Rev. Lett. 40 (1978) 1108. [31 J.M. Kosterlitz, D.R. Nelson and M.E. Fisher, Phys. Rev. B13 (1976) 412, and references therein. [4] E. Domany and M.E. Fisher, Phys. Rev. B15 (1977)
3510, and references therein. [5] C.H.W. Swüste, W.J.M. de Jonge and J.A.G.W. van Meyel, Physica 76 (1974) 21. [6] AC. Botterman, Ph.D. Thesis (Eindhoven, 1976), unpublished. [7] J.A.J. Basten, E. Frikkee and W.J.M. de Jonge, to be published. [8] H. Rohrer and H. Thomas, J. AppL Phys. 40 (1969) 1025. [9] J.C. le Guillou and 3. Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95.
[10] M.E. Fisher, Phys. Rev. Lett. 34 (1975) 1634.
6226(38) kOe 387