Figure 1. Variation of 1/T*( = c/l) against 1/T for dysprosium ethyl sulfate. The points (o) represent experimental results. The curves represent the results of various theoretical models. (a) Molecular field model; (b) Van Vleck expansion to second order (Eq. 2); (c) Ising model with nearest neighbor interactions in a molecular field due to other neighbors; (d) Ising model with nearest and next nearest neighbor interaction in a molecular field. After Ref. 11.
Figure 2. Entropy of dysprosium ethyl sulfate as a function of temperature. The points (o) represent experimental results. The broken line represents a model assuming non-interacting linear Ising chains and the solid lines the results predicted by the Oguchi cluster expansion method. After Ref. 11.
Figure 3. Magnetic specific heat as a function of temperature for DyPO4. The points (o) represent experimental results, the solid line represents the results of a calculation based on high- and low-temperature series expansions with one adjustable constant. After Ref. 17.
Figure 4. Magnetic susceptibility as a function of temperature for DyPO4. The points (o) represent experimental results; the solid line represents the results of a calculation based on high- and low-temperature series expansions with one adjustable constant. After Ref. 17.
Figure 6. Magnetic susceptibility parallel to the axis as a function of temperature for K2CoF4. The points (o) represent experimental results (Ret: 21) corrected for a temperature independent Van Vleck contribution to make c = 0 at T = OK. The solid line represents the series expansion of Sykes and Fisher for the quadratic S = 1/2 Ising antiferromagnet fitted with one adjustable constant. After Ref. 6.
Figure 7. Variation of the magnetic specific heat, as a function of temperature for Rb2CoF4. The solid points (·) are experimental results of optical birefringence measurements shown previously to be proportional to the magnetic specific heat. The solid line is the exact Onsager solution for the two-dimensional Ising model with amplitude and critical temperature adjusted to fit the data, and a small constant background term subtracted. After Ref. 22.
Figure 8.
Magnetic susceptibility as a function of temperature for K
2CoF
4, without any corrections. Note the differences from
Fig. 6 and the large susceptibility perpendicular to the axis reflecting the influence of low-lying states and corresponding deviations from the simple Ising Hamiltonian. After Ref. 21.
Figure 9. Specific heat as a function of temperature for Dy3Al5O12. The points (o) represent experimental results. Curves (a) and (b) were calculated using the mean field approximation, with the constant for (a) estimated on the basis of pure magnetic dipole-dipole interaction, and for (b) by fitting the low-temperature experimental points. After Ref. 24.
Figure 10.
Specific heat of Dy3Al5O12 as a function of temperature near TN, under four decades of temperature resolution. Temperatures are measured relative to an arbitrarily chosen Tmax = 2.544 K. After Ref. 26.
Figure 11.
The reciprocal susceptibility parallel to the crystal c-axis as a function of temperature for LiHoF4. The experimental points correspond to measurements for five different sample shapes, characterized by demagnetizing factors, N. The constant values for T < 1.5 K correspond to a transition to the ferromagnetic state, for which 1/cN = 0 = 0. After Ref.31.
Figure 12.
Possible phase diagrams in the field-temperature plane for antiferromagnets. (a) usual phase diagram with nearest neighbor interactions, in which the antiferromagnetic phase (A) is separated from the paramagnetic phase (P) by a line of second order transitions. (b) phase diagram with competing interactions, in which there are both first order and second order transitions, and a tricritical point where they meet. (c) shows the phase diagram when there is a coupling between the applied field and the antiferromagnetic order parameter. In this case there is only a first order transition ending in a critical point. (d) same as (c) but showing both positive and negative applied fields. The positive field induces one of the two antiferromagnetic states, A+, while the opposite field induces the time-reversed state A-. The phases A+ and A- are separated by a first order line that ends at the Néel point TN. In DyAlG cases (b), (c), and (d) are observed under different conditions.
Figure 13.
The magnetization as a function of magnetic field at fixed temperature for a two-dimensional super-exchange antiferromagnet. The dashed curve is the locus of transition points. The curves are labeled by appropriate values of the reduced temperature. Note the continuity of all curves except for zero temperature, and the infinite derivative at the transition points. After Ref. 48.
Figure 14.
The magnetization of a spherical sample of DyAlG as a function of field along [111] at temperatures above and below the Néel temperature. In (a) the magnetization is plotted as a function of the externally applied field. In (b) the magnetization is plotted as a function of the internal field, calculated by Hint = Hext- NM, with N = 4p/3 for a sphere. After Ref. 49.
Figure 15.
Results for the staggered magnetization, Ms, for DyAlG as a function of the internal field with Hi|| [111]. (a) Experimental results of Blume et al. [Ref. 53]; (b) results of the cluster calculation for the same value of T/TN. After Ref. 55.
Figure 16.
Lattice and bond structure of the staggered interaction model shown in a [001] projection. The structure is body centered cubic, with 6 sites in the primitive unit cell. The figure shows the conventional unit cell. The numbers give the heights above the z = 0 plane in terms of the unit-cell edge length. Ferromagnetic bonds are shown as solid lines and antiferromagnetic bonds are shown as broken lines. Note that only half the sites, marked as (·), form triangles with three ferromagnetic bonds, whereas the sites marked (o) have only two ferromagnetic bonds. The structure shown corresponds to one half of the sites in DyAlG. The omitted sites form a similar lattice related by simple translations and do not share any nearest neighbor bonds with the sites shown. After Ref.55.
Figure 17.
(a) A schematic representation of the pyrochlore lattice, showing the positions of the magnetic ions. (b) The ground state of a single tetrahedron of spins coupled ferromagnetically with local Ising anisotropy. (c) Local proton arrangement in ice, showing the oxygen atoms (·) and hydrogen atoms (-), and with the displacement of the hydrogen atoms trom the mid-points of the oxygen-oxygen bonds marked by arrows. The similarity to (b) has led to the concept of 'spin-ice.' After Refs. 67 and 73.
Figure 18.
Specific heat and entropy of Dy2Ti2O7 and Pauling's prediction for ice. (a) Specific heat divided by temperature for H=0 (o) and H=0.5T (·). The dashed line is a Monte Carlo simulation of the zero-field C(T)/T. (b) Entropy of Dy2Ti2O7 found by integrating C/T trom 0.2 to 14K. The value of R(ln2-1/2 ln3/2) is that found for ice (Ih), and ln2 is the usual full spin entropy. After Ref. 71.
Figure 19.
Specific heat as a function of temperature for Dy2Ti2O7 in various applied fields. The broad H = 0 feature is suppressed on increasing H and replaced by three sharp features at 0.34, 0.47 and 1.12K. Inset (a) shows the constancy of these transition temperatures with field. Inset (b) shows the result of finite-field Monte Carlo simulations of C/T. After Ref. 71.
Figure 20.
A plot ot the out-of-phase susceptibility c¢¢ as a function of the in-phase c¢ for DyES for various frequencies at four different temperatures. The frequency of measurement, in Hz, is written beside each symbol. The curves are circles which pass through the origin and best fit the data. The extrapolated values of c¢ for c¢¢ = 0 are clearly different from the measured d.c. value, cd.c.. After Ref. 75.
Figure 21.
Field and temperature dependence of one of the relaxation times, IHF, of DyAlG. The arrow indicates the location of the phase boundary Hc at 1.8 K; at the other temperatures shown Hc is not very different. All the results therefore correspond to measurements in the antiferromagnetic phase. The broken line at 10-5 indicates the lower limit of the measurements. After Ref. 83.
TABLE II.
Critical entropy and energy parametersa
TABLE III.
Critical exponents and amplitudes for Dy3Al5O12 for T > TN, showing the effect of choosing different values for TN, together with theoretical estimates for three cubic Ising models.
TABLE IV.
Spin-spin interactions in Dy3Al5O12a, showing the relative importance of several shells of near neighbors. Similar competing terms may be expected in many other materials, but are often not considered.
