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Critical specific heat of erbium sesquioxide and the 3-dimensional Ising model
Volume 49A, number 2
PHYSICS LETTERS
26 August 1974
CRITICAL SPECIFIC HEAT OF ERBIUM SESQUIOXIDE AND THE 3-DIMENSIONAL ISING MODEL* H.V. CULBERT, D. HINKS, Z. SUNGAILA and S. SUSMAN Argonne National Laboratory, Argonne, illinois 60439, USA Received 26 July 1974 Specific heat measurements are reported for Er 4K of TN where TN = 3.3707 203 near K. the Good Néel agreement temperature is found TN. with No ‘rounding-off’ theoretical calculations of the data is found for the 3-dimensional to within 4 X Ising i0~ model.
Experimental tests of the 3-dimensional Ising model specific heat in magnetic systems have been of urnited usefulness [11 (excepting perhaps data for DAG [2]). The data were either not taken close enough to the critical temperature T~or the specific heat peak ‘rounded-off” as the critical region was approached. We have made specific heat measurements on erbium sesquioxide near the antiferromagnetic ordering ternperature TN which show no ‘rounding-off’ of the peak and are close enough to TN to make a good comparison with theoretical models. Magnetic ordering in rare-earth compounds holds some promise of applicability to a 3-dimensional Ising model in those cases where there is a large anisotropy in the g-factor and where pure, unstrained crystals can be grown. Erbium sesquioxide (Er 203) is such a system. The compound orders antiferromagnetically at T~ 3.4 K [3, 4]. The g-factor for Er2 03 has been shown to be quite anisotropic [5, 6] with the ordered spins (effective spin = ~)aligned with the local crystal field [6] and good single crystals can be obtained under rather special but experimentally possible conditions. We have grown stoichiometric single crystals of Er203 in a focussed-plasma, float-zone refiner. The stoichiometry of the molten zone was controlled by adjusting the partial pressure of oxygen in the plasma beam. The starting Er203 powder was stated by the manufacturer to be 99.9% pure with Yb203 the principal impurity. 3He cryostat The specific heat was measured in a *
Based on work performed under the auspices of the US Atomic Energy Commission.
using the usual heat-pulse technique. The germanium resistance thermometers were calibrated against the T 4He and the T 3He vapor pressure scales. The 58 were corrected 62 for the lattice specific heat CL data assuming the Debye temperature [7] 8D = 315 K and the low temperature tail of the crystal field specific heat CCF has been determined by fitting our data between 5.5 K and 20K. The total correction (CL+CCF) is only ~ 0.2% of the total specific heat in the critical region. The calorimeter heat capacity was measured in a separate run. Our data between 15 K and 20 K are in good agreement with those of Justice and Westrum [7] where there is overlap. The samples used in our measurements were cut from the as-grown rod (‘-‘ 0.5 cm diameter) into rough cylinders 0.3 cm high. The samples were mounted in a brass calorimeter on 12.5 p gold foils with a thin film of Apiezon ‘N’ grease to reduce the possibility of straining the sample during cool down. A number of theoretical calculations have been made recently for the specific heat near the critical point [8], but here we are interested in the predictions for the three-dimensional Ising model made by Gaunt and Domb [1] for temperatures below the critical point T~and by Sykes et al. [9] above TN. In the general case, the specific heat near the critical point can be written ‘~-
dR ~A~I1 T/TNL°~+B~ (1) where the + refers to temperatures T> TN and the holds for T< TN. For the 3-dimensional Ising model one expects [1, 9] ~+ = = 0.125. The values for A~and B~vary —
—
187
Volume 49A, number 2
PHYSICS LETTERS
3d ISING MODELS
30
~~GAuNT
8 00MB (T~TN) SYKES, HUNTER. McKENZIE
-
S HEAP (T2TN) - 33707K ~ T~T~
-
—
25
Er
“
20
2O3Cp DATA
-
.--,,~
.
-
-
W
•“~l’-~r~,
-
10
-
-
(1
5-
Io~
IO~
IT
TNI/TN
Fig. 1. Specific heat of Er203 versus IT—TNI/TN. 0 — data above TN, s — data below TN, - - - tetragonal 3-d Ising model TN [91.
slightly with coordination number and magnetic lattice structure. The critical region for this model is asymmetric about TN, i.e., the critical region is defined for temperatures in the range t
=
Ii
—
T/T~I~ i0~
for T< Tc
broad anomaly at 1.1 K. The implication is that only one half of the erbium ions are ordering at TN. We have also added a constant term of 0.825 i/mole ErK to both reduced Ising model curves to get a reasonable fit to our data. We conclude the following from fig. 1: 1) Although our data below TN do not extend much into the asymptotic critical region defined by Gaunt and Domb [1], our data agree very well with their interpolation formula up to t 0.1. 2) Above TN, where the critical region begins for r ~ 10—2, our data are in excellent agreement with the 3-dimensional Ising model in its asymptotic form. 3) From 1) and 2) we conclude that our data are consistent with theIsing theoretical prediction [1, 9] for the 3-dimensional model that ~ = = 0.125. 4) We see no ‘rounding-out’ in the data closest to TN, indicating both low impurity content and a relatively strain free crystal. We are now modifying our apparatus in an effort to refine and extend these measurements beyond t = 1 0~.
References
and
[1] C. Keen, Domb,B.J.Schneider Phys. Cl (1968) 1038. [2] D.S. D.P. Gaunt Landau,and B.E. and W.P. Wolf, t
l0~
for T> T~ -
In fig. 1 we show the specific heat data for Er203, within 0.4 K of the Néel temperature TN plotted versus ln 1 T/TNI (TN = 3.3707 ±0.0003K). We also show the theoretical curves from Gaunt and Domb (T< TN) for a tetragonal lattice and Sykes et al. (T> TN) for the simple cubic lattice. TN was chosen to give a good fit to the curve published by Gaunt and Domb [1]. This curve includes the asymptotitc critical specific heat and an interpolation calculated by the authors to connect to the exact iow ternperature expansion. The theoretical curves shown are reduced by a factor of 2 from the published values [1, 9] to fit our data. This is apparently a result of incomplete ordering in the system at TN. Further order. ing shows in our specific heat measurements as a —
188
26 August 1974
131
Phys. Rev. B3 (1971) 2310; W.P. Wolf, B. Schneider, D.P. Landau and B.E. Keen, Phys. Rev. B5 (1972) 4472. R.E. Brown and W.M. Hubbard, Proc. Fifth Rare Earth
Research Conference, 1965, Book 4, p. 31. H. Bonrath, K.H. Heliwege, K. Nicolay and G. Weber, Phys. Kondens Materie 4 (1966) 382. [5] G. Schafer and S. Scheller, Phys. Kondens. Materie 5 (1966) 48. [6] R.M. Moon, W.C. Koehler, H.R. Child and L.J. Raubenheimer, Phys. Rev. 176 (1968) 722. [7] B.H. Justice and E.F. Westrum, J. Phys. Chem. 67 (1963) 659; 67 (1963) 345. [8] H.E. Stanley, ed., Cooperative phenomena near phase transitions: a bibliography with selected readings (MIT Press, Cambridge, Mass.) 1973. [9] M.F. Sykes, D.L. Hunter, D.S. McKenzie and B.R. Heap, Ploys. AS (1972) 667. [101 To be J. reported elsewhere.
141
PHYSICS LETTERS
26 August 1974
CRITICAL SPECIFIC HEAT OF ERBIUM SESQUIOXIDE AND THE 3-DIMENSIONAL ISING MODEL* H.V. CULBERT, D. HINKS, Z. SUNGAILA and S. SUSMAN Argonne National Laboratory, Argonne, illinois 60439, USA Received 26 July 1974 Specific heat measurements are reported for Er 4K of TN where TN = 3.3707 203 near K. the Good Néel agreement temperature is found TN. with No ‘rounding-off’ theoretical calculations of the data is found for the 3-dimensional to within 4 X Ising i0~ model.
Experimental tests of the 3-dimensional Ising model specific heat in magnetic systems have been of urnited usefulness [11 (excepting perhaps data for DAG [2]). The data were either not taken close enough to the critical temperature T~or the specific heat peak ‘rounded-off” as the critical region was approached. We have made specific heat measurements on erbium sesquioxide near the antiferromagnetic ordering ternperature TN which show no ‘rounding-off’ of the peak and are close enough to TN to make a good comparison with theoretical models. Magnetic ordering in rare-earth compounds holds some promise of applicability to a 3-dimensional Ising model in those cases where there is a large anisotropy in the g-factor and where pure, unstrained crystals can be grown. Erbium sesquioxide (Er 203) is such a system. The compound orders antiferromagnetically at T~ 3.4 K [3, 4]. The g-factor for Er2 03 has been shown to be quite anisotropic [5, 6] with the ordered spins (effective spin = ~)aligned with the local crystal field [6] and good single crystals can be obtained under rather special but experimentally possible conditions. We have grown stoichiometric single crystals of Er203 in a focussed-plasma, float-zone refiner. The stoichiometry of the molten zone was controlled by adjusting the partial pressure of oxygen in the plasma beam. The starting Er203 powder was stated by the manufacturer to be 99.9% pure with Yb203 the principal impurity. 3He cryostat The specific heat was measured in a *
Based on work performed under the auspices of the US Atomic Energy Commission.
using the usual heat-pulse technique. The germanium resistance thermometers were calibrated against the T 4He and the T 3He vapor pressure scales. The 58 were corrected 62 for the lattice specific heat CL data assuming the Debye temperature [7] 8D = 315 K and the low temperature tail of the crystal field specific heat CCF has been determined by fitting our data between 5.5 K and 20K. The total correction (CL+CCF) is only ~ 0.2% of the total specific heat in the critical region. The calorimeter heat capacity was measured in a separate run. Our data between 15 K and 20 K are in good agreement with those of Justice and Westrum [7] where there is overlap. The samples used in our measurements were cut from the as-grown rod (‘-‘ 0.5 cm diameter) into rough cylinders 0.3 cm high. The samples were mounted in a brass calorimeter on 12.5 p gold foils with a thin film of Apiezon ‘N’ grease to reduce the possibility of straining the sample during cool down. A number of theoretical calculations have been made recently for the specific heat near the critical point [8], but here we are interested in the predictions for the three-dimensional Ising model made by Gaunt and Domb [1] for temperatures below the critical point T~and by Sykes et al. [9] above TN. In the general case, the specific heat near the critical point can be written ‘~-
dR ~A~I1 T/TNL°~+B~ (1) where the + refers to temperatures T> TN and the holds for T< TN. For the 3-dimensional Ising model one expects [1, 9] ~+ = = 0.125. The values for A~and B~vary —
—
187
Volume 49A, number 2
PHYSICS LETTERS
3d ISING MODELS
30
~~GAuNT
8 00MB (T~TN) SYKES, HUNTER. McKENZIE
-
S HEAP (T2TN) - 33707K ~ T~T~
-
—
25
Er
“
20
2O3Cp DATA
-
.--,,~
.
-
-
W
•“~l’-~r~,
-
10
-
-
(1
5-
Io~
IO~
IT
TNI/TN
Fig. 1. Specific heat of Er203 versus IT—TNI/TN. 0 — data above TN, s — data below TN, - - - tetragonal 3-d Ising model
slightly with coordination number and magnetic lattice structure. The critical region for this model is asymmetric about TN, i.e., the critical region is defined for temperatures in the range t
=
Ii
—
T/T~I~ i0~
for T< Tc
broad anomaly at 1.1 K. The implication is that only one half of the erbium ions are ordering at TN. We have also added a constant term of 0.825 i/mole ErK to both reduced Ising model curves to get a reasonable fit to our data. We conclude the following from fig. 1: 1) Although our data below TN do not extend much into the asymptotic critical region defined by Gaunt and Domb [1], our data agree very well with their interpolation formula up to t 0.1. 2) Above TN, where the critical region begins for r ~ 10—2, our data are in excellent agreement with the 3-dimensional Ising model in its asymptotic form. 3) From 1) and 2) we conclude that our data are consistent with theIsing theoretical prediction [1, 9] for the 3-dimensional model that ~ = = 0.125. 4) We see no ‘rounding-out’ in the data closest to TN, indicating both low impurity content and a relatively strain free crystal. We are now modifying our apparatus in an effort to refine and extend these measurements beyond t = 1 0~.
References
and
[1] C. Keen, Domb,B.J.Schneider Phys. Cl (1968) 1038. [2] D.S. D.P. Gaunt Landau,and B.E. and W.P. Wolf, t
l0~
for T> T~ -
In fig. 1 we show the specific heat data for Er203, within 0.4 K of the Néel temperature TN plotted versus ln 1 T/TNI (TN = 3.3707 ±0.0003K). We also show the theoretical curves from Gaunt and Domb (T< TN) for a tetragonal lattice and Sykes et al. (T> TN) for the simple cubic lattice. TN was chosen to give a good fit to the curve published by Gaunt and Domb [1]. This curve includes the asymptotitc critical specific heat and an interpolation calculated by the authors to connect to the exact iow ternperature expansion. The theoretical curves shown are reduced by a factor of 2 from the published values [1, 9] to fit our data. This is apparently a result of incomplete ordering in the system at TN. Further order. ing shows in our specific heat measurements as a —
188
26 August 1974
131
Phys. Rev. B3 (1971) 2310; W.P. Wolf, B. Schneider, D.P. Landau and B.E. Keen, Phys. Rev. B5 (1972) 4472. R.E. Brown and W.M. Hubbard, Proc. Fifth Rare Earth
Research Conference, 1965, Book 4, p. 31. H. Bonrath, K.H. Heliwege, K. Nicolay and G. Weber, Phys. Kondens Materie 4 (1966) 382. [5] G. Schafer and S. Scheller, Phys. Kondens. Materie 5 (1966) 48. [6] R.M. Moon, W.C. Koehler, H.R. Child and L.J. Raubenheimer, Phys. Rev. 176 (1968) 722. [7] B.H. Justice and E.F. Westrum, J. Phys. Chem. 67 (1963) 659; 67 (1963) 345. [8] H.E. Stanley, ed., Cooperative phenomena near phase transitions: a bibliography with selected readings (MIT Press, Cambridge, Mass.) 1973. [9] M.F. Sykes, D.L. Hunter, D.S. McKenzie and B.R. Heap, Ploys. AS (1972) 667. [101 To be J. reported elsewhere.
141