Bicritical Point Research Articles

Crystal structures of A2FeCl5⋅H2O (A=Rb+, Cs+) and field dependent superconducting susceptometer measurements

Crystal structures and low field principal crystal axis magnetic measurements are reported for the antiferromagnetic iron (III) salts A2FeCl5⋅H2O; A=Rb+, Cs+. The two compounds undergo a zero field paramagnetic to antiferromagnetic phase transition at 6.5 K (A=Cs+) and 10.05 K (A=Rb+). The spin flop magnetic phase is observed in high field measurements. The magnetic field-temperature phase diagrams are reported. All magnetic measurements were recorded on a superconducting susceptometer. The bicritical points are 6.25 K, 14.5 kOe (A=Cs+), 9.75 K, 17.8 KOe (A=Rb+). Crystal data for Cs2FeCl5⋅H2O: space group Cmcm, Z=4, a=7.442(3) Å, b=17.307(7) Å, c=8.077(7) Å, V=1040 Å3, R=5.2%, 561 reflections. Crystal data for Rb2FeCl5⋅H2O, space group Pnma, Z=4, a=13.825(4) Å, b=9.918(3) Å, c=7.100(2) Å, V=4.9 Å3, 5.7%, 712 reflections.

Crossover from first-order to second-order phase transitions in a symmetry-breaking field: Monte Carlo, high-temperature series, and renormalization-group calculations

Certain physical systems are expected classically to exhibit second-order phase transitions but are known to have first-order phase transitions within the renormalization-group approach because no stable fixed point exists. This lack of stable fixed points is caused by fluctuations in the order parameter and depends only upon the symmetry of the system. When a symmetry-breaking field is applied a stable fixed point may emerge and a continuous transition becomes possible. We have studied this crossover from first-order to second-order transitions by three different approaches. Our model has cubic symmetry and the magnetic ground state is similar to that of U${\mathrm{O}}_{2}$ and Nd${\mathrm{Sn}}_{3}$. Firstly, Monte Carlo calculations confirm the existence of the crossover to a second-order transition in the symmetry-breaking field. A small field does not destroy the first-order transition, and the crossover therefore occurs at a finite field indicating true tricritical behavior. Secondly, the model has been analyzed using high-temperature series for the ordering susceptibility. For large enough fields the analysis suggests a second-order phase transition at a temperature which is the same as that determined from the Monte Carlo calculations. In this field region the series analysis predicts a bicritical point. For small fields the series behave irregularly, which is considered as an indication of a crossover to first-order transitions. Thirdly, a semiquantitative renormalization-group calculation is presented. This calculation supports and explains the tricritical behavior discovered by the Monte Carlo calculation. Related experimental investigations are suggested on U${\mathrm{O}}_{2}$.

Virtual Bicritical Point in CsMnF3

The ordering temperature ${T}_{c}$ of the easy-plane antiferromagnet CsMn${\mathrm{F}}_{3}$ was measured as a function of magnetic field $H$. when H is perpendicular to the easy plane, ${T}_{c}$ decreases monotonically with increasing $H$, but the decrease is not proportional to ${H}^{2}$. The latter behavior is explained in terms of a virtual bicritical point which exists mathematically at a negative value of ${H}^{2}$.

Crossover from divergent to non-divergent nature of the ordered to paramagnetic transitions of α-MnCl2⋅4H2O as a function of applied field

The thermodynamic functions CH, (∂M/∂T)H and [(∂M/∂H)T– (∂M/∂H)S] have been determined as a function of temperature over the range 0.5 to 2.0 K for a spherical single crystal of α–MnCl2⋅4H2O with magnetic fields up to 20 kG applied parallel to the c-crystallographic axis. The data clearly show that the lamba anomaly in [(∂M/∂H)T−(∂M/∂H)S] associated with the transition from the ordered to paramagnetic state changes character at the bicritical point and completely disappears for fields greater than 8.4 kG. The region for fileds greater than 8.4 kG also corresponds to apparent non-power law behavior of the heat capacity and (∂M/∂T)H. The bicritical behavior is shown to be consistant with that of a antiferromagnet with orthorhombic anisotropy.

Crossover behavior of the magnetic phase boundaries of isotropic antiferromagnets

Phase boundaries, Tc vs H, of isotropic antiferromagnets are discussed. The Fisher-Nelson-Kosterlitz predictions for the effect of a spin-dimensionally crossover on Tc(H) are summarized. The stringent experimental requirements for testing these predictions are reviewed. Results for Tc(H) in the very-low-anisotropy cubic antiferromagnets RbMnF3 and KNiF3 confirm the theoretical predictions for the case of a Heisenberg-to-XY crossover. The RbMnF3 data give a crossover exponent φ which agrees with theory. Results for Tc(H) in the easy-plane antiferromagnet CsMnF3 are presented. The data when H↘ is in the easy plane are consistent with an XY-to-Ising crossover, and give φ=1.185±0.03. The results when H↘ is perpendicular to the easy plane are interpreted in terms of a virtual bicritical point at some negative value of H2.

The magnetic phase diagram of pseudo-one-dimensional CsMnBr3·2D2O near the bicritical point

Abstract The magnetic phase diagram of pseudo-one-dimensional CsMnBr 3 ·2D 2 O, 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.

Spin-flop multicritical points in systems with random fields and in spin glasses

Mean-field theory and renormalization-group arguments are used to study the phase diagram of an anisotropic $n$-component $d$-dimensional magnetic system with a uniaxially random magnetic field. The resulting phase diagram is shown to be very similar to that of anisotropic antiferromagnets in a uniform field: For small random fields, the system orders along the direction of uniaxial anisotropy, with exponents which are related to those of nonrandom Ising systems in $d\ensuremath{-}2$ dimensions. For larger random fields, parallel to the direction of uniaxial anisotropy, the transverse $n\ensuremath{-}1$ spin components order, with exponents which are unaffected by the random field. The two regions are separated by a spin-flop first-order line, by an intermediate "mixed" phase, and by a tetracritical (or bicritical) point. The exponents at this multicritical point are shown to coincide, near $d=6$, with those of the random-field Ising model. This phase diagram is shown to describe the behavior of random-site spin glasses in a uniform magnetic field. Other types of anisotropic random fields, related experimental realizations and other generalizations are also mentioned. Although some of the quantitative results are found only near $d=6$, qualitative results are believed to apply at $d=3$ as well.

Crossover at a bicritical point: Asymptotic behavior of a field theory with quadratic symmetry breaking

We discuss the case of an O(N)-symmetric theory whose symmetry is broken by a quadratic term which produces two different masses (correlation lengths) one associated with M components and one with N − M. This theory describes physical systems having a bicritical point. To this problem we contribute by showing how within renormalized field theory the calculation of the shift in the critical temperature (leading to a crossover exponent) is separated from the calculation of the crossover function. The computation of the latter is then carried out in terms of the natural temperature variable, relative to the anisotropic critical temperature. This feature allows one to bypass the complicated corrections to scaling generated in earlier approaches by the use of a different variable. Within this frame-work which allows many tests of universality the results are explicit and readily generalizable to the ordered state and to higher orders in ϵ. To field theory we contribute by showing how the mass-dependent renormalization procedure allows the interpolation between asymptotic behaviors with internal O(N)- and O(M)-symmetries. The interpolating parameter is the ratio between the momentum scale and the mass difference breaking the symmetry.

Paramagnetic and antiferromagnetic properties of tris(ethylenediamine)nickel(II) nitrate

Crystals of uniaxial [Ni(NH 2CH 2CH 2NH 2) 3](NO 3) 2 have been examined between 0.3 and 30 K by means of specific-heat and susceptibility measurements. Long-range antiferromagnetic order sets in at 1.25 K, with the preferred axis along the crystal c axis. The magnetic phase diagram has been determined from the field-dependent susceptibilities, the bicritical point occurring at 0.62 ± 0.01 K and 13.9 ± 0.1 kOe. Relaxation effects were observed in the frequency-dependent susceptibilities. The following quantities have also been determined: g = 2.21 ± 0.05, g = 2.24 ± 0.05, D/ k = -1.00 ± 0.05 K, J/ k = -0.19 K, H SF (0) = 12.8 ± 0.2 kOe, H c (0) = 24.0 ± 0.2 kOe, H cl (0) = 37.5 ± 0.2 kOe, H E = 15.4 ± 0.1 kOe, H A = 6.8 ± 0.1 kOe, and b = 9.45 JK/ mol. An exchange path for this weakly anisotropic Heisenberg antiferromagnet is proposed.

Dynamic scaling and corrections to scaling near bicritical and tetracritical points

We study the critical dynamics of stochastic models appropriate for describing bicritical and tetracritical points in anisotropic antiferromagnetic systems. The dynamic exponents and the transient exponents are calculated by renormalized field theory up to two-loop order. In the presence of reversible mode-coupling terms, two-loop contributions establish bicritical dynamic scaling in the restricted sense and invalidate recent predictions based on mode-coupling arguments. In the case of an n-component relaxational model total dynamic scaling is found to O (ε2) both at the bicritical (n≲3) and the tetracritical (4≲n<11) points.

Study of the bicritical region of the biaxial antiferromagnet MnCl2⋅4H2O

The bicritical region of the phase diagram of manganous chloride tetrahydrate has been investigated by the determination of the heat capacity as a function of temperature at constant applied magnetic field. The experiments were repeated at field increments of 10 to 50 G in the region of the bicritical point. Plots of the heat capacity maxima in the H-T plane clearly exhibit an umbilicus at the bicritical point. These data have been fit to the expression given by Fisher and Nelson following the procedure for locating the appropriate scaling axes given by Fisher. The ratio of the amplitudes of the slope of the phase boundaries in the region of the bicritical point has been determined to be 1.000±0.006, which is not consistent with the theoretical prediction of 2.51 for a uniaxial antiferromagnet. These results and also the variation of the behavior of the phase boundaries in the bicritical region as a function of the angle between the applied magnetic field and the spin axis is discussed.

Magnetic phase boundaries near the bicritical and Néel points ofCr2O3

The magnetic phase diagram of the easy-axis uniaxial antiferromagnet ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$ was measured in magnetic fields $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ up to 180 kOe. Phase transitions were determined from anomalies in the attenuation of 30- and 50-MHz longitudinal ultrasonic waves. Temperatures $T$ were measured with a precision of \ensuremath{\sim}10 mK, corresponding to $\frac{\ensuremath{\Delta}T}{T}\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$. Attention was focused on the phase boundaries near both the N\'eel point (${T}_{N}=307.3$ K) and the bicritical point [${T}_{b}={T}_{N}\ensuremath{-}(0.505\ifmmode\pm\else\textpm\fi{}0.01)$ K, ${H}_{b}=120.04\ifmmode\pm\else\textpm\fi{}0.6$ kOe]. Data were taken for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ parallel to the easy axis, perpendicular to the easy axis, and for some intermediate orientations. Near the bicritical point, the phase boundaries in the $T\ensuremath{-}{H}_{\ensuremath{\parallel}}\ensuremath{-}{H}_{\ensuremath{\perp}}$ space (where $\ensuremath{\parallel}$ and $\ensuremath{\perp}$ indicate parallel and perpendicular to the easy axis, respectively) show marked deviations from mean-field theory. In contrast, the recent theory of Fisher, Nelson, and Kosterlitz (FNK) explains all the salient features of the phase boundries near the bicritical point. The crossover exponent $\ensuremath{\varphi}$ was determined from numerical fits of the data near the bicritical point taken (i) with $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ parallel to the easy axis, and (ii) with $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ gradually tipped away from the easy axis, keeping ${H}_{\ensuremath{\parallel}}={H}_{b}=\mathrm{const}$. The values for $\ensuremath{\varphi}$ were consistent with the FNK theory. However, it appears that Fisher's estimate for one of the best scaling axes ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}=0$, for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ parallel to the easy axis) is not applicable to ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$. Near ${T}_{N}$ and with $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ parallel to the easy axis, the ordering temperature ${T}_{c}$ decreases linearly with ${H}^{2}$. However, for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ perpendicular to the easy axis, ${T}_{c}$ increases with increasing $H$ in fields up to 180 kOe. This increase is attributed to the extremely low anisotropy of ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$ which causes the behavior in perpendicular fields to qualitatively resemble the behavior predicted by FNK for the completely isotropic antiferromagnet. Ultrasonic methods for determining phase boundaries of antiferromagnets are reviewed briefly.

Antiferromagnetic ordering in the three-state Potts model

The three-state Potts model on a triangular lattice with nearest neighbour interaction K is studied. Using a simple position-space renormalisation group transformation it is found that, in addition to the usual continuous transition to the ferromagnetic state which occurs for positive K, a continuous transition to an antiferromagnetic state occurs for negative K. The transformation introduces a three-spin interaction. In the space of these two interactions, the lines of ferromagnetic and antiferromagnetic transitions meet at a bicritical point. These results differ from those predicted by Landau theory.

Thermodynamic properties of ferromagnets with uniaxial and biaxial anisotropy

The thermodynamic properties of spin-one Heisenberg ferromagnets with uniaxial and biaxial anisotropy are investigated in the molecular field approximation (MFA) and random phase approximation (RPA). In MFA a full phase diagram comprising three lines of bicritical points, is obtained. Using the standard-basis operator method, the collective excitation spectrum is studied in detail; moreover the softening of the excitations at the phase boundaries is discussed. A self-consistent version of RPA with emphasis on the role of kinematic restrictions with regard to the standard-basis operators is analyzed. Moreover, the phase transitions are considered in the Ising model, where a line of tricritical points occurs, and the planar model, where a line of bicritical points, two lines of tricritical points and a line of triple points, occur.

Partial Differential Approximants for Multicritical Singularities

A recently proposed approach---partial differential approximants---for analyzing power series in two (or more) variables for functions which exhibit singularities of multicritical, scaling character is tested by analyzing the susceptibility series for the bicritical point describing the Ising-Heisenberg-$\mathrm{XY}$ crossover in a three-dimensional classical ferromagnet. Encouraging results are obtained for the unbiased estimation of the bicritical point, of the multicritical exponents $\ensuremath{\gamma}$ and $\ensuremath{\varphi}$, and of the slopes of the scaling axes.

Theory of rare-earth alloys

A mean-field random alloy theory combined with a simple calculation of the exchange interaction J (c,Q) is shown to quantitatively account for the phase diagrams for alloys of rare-earth metals with Y, Lu, Sc, and other rare-earth metals. A concentration-dependent J (c,Q) explains the empirical 2/3 law and reconstitutes the idea of de Gennes scaling. The phase diagrams exhibit examples of both bicritical and tetracritical points. No significant deviations from the mean-field calculation can be detected with the present experimental accuracy. A linear interpolation of J (Q) for Gd and Er is found to account for all alloys except the Sc based. The exceptional behavior of the Sc alloys is due to a low density of states for Sc. A brief discussion is given of the effect on the mean-field results of changes in volume or c/a ratio and of critical fluctuations. Since the physical mechanisms of these ideal alloys are well documented they may serve as good candidates for studies of statistical effects such as multicritical phenomena or spin-glass phenomena.

Bicritical lines and crossover in the non-uniaxial spin-flop magnet Mn(Nb 0.5Ta 0.5) 2O 6

By means of neutron diffraction and magnetization, the H x - H z - T phase diagram of the anisotropic Heisenberg antiferromagnet Mn(Nb 0.5Ta 0.5) 2O 6 was measured. Three bicritical lines were detected which meet at a multicritical point of a new kind( H x = 2.3 T; H z = 1.2 T; T = 5.0 K).The critical exponent β was measured along the critical lines for H ∥ x, H ∥ y and H ∥ z, especially the crossover phenomena at the three bicritical points B 1, B 2 and B 3. Additionally, at B 1 and B 2, we measured the critical exponent ∼β and at B 1, the critical exponents γ and ν, in order to test the scaling relations at B 1 and to determine the crossover exponents ∥ø for B 1 and B 2. The absolute values of the critical exponent β are higher than the critical exponent β of MnNb 2O 6. This can be explained by the statistical distribution of the Nb- and Ta-ions.

Critical behavior of the heat capacity in SrTiO3

Using an ac calorimetric method, heat-capacity measurements were made for a "monodomain" and a "polydomain" sample. The former exhibited a clear critical behavior which was analyzed with an expression containing a critical-exponent term. The exponent has an Ising value within a standard deviation. On the other hand, the polydomain sample showed a smaller and different anomaly. Using a special sample-setting method, the above behaviors were verified directly by measuring the heat capacity under externally applied stress conditions. The differences in heat capacities observed leads us to imagine that crossover from Heisenberg to Ising behavior with external stress takes place due to the existence of the bicritical point.

New Type of Multicritical Behavior in a Triangular Lattice Gas Model

A Monte Carlo study of a triangular lattice gas model appropriate to ${\mathrm{He}}^{4}$ monolayers adsorbed on the basal plane of graphite reveals a rich variety of critical and multicritical points including a new, special type of bicritical point.

Equations of state for bicritical points. II. Ising-like ordered phases

Calculations of bicritical crossover scaling functions are extended into the Ising-like bicritical ordered phases. Closed-form expressions are derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for the specific heat, nonordering susceptibility, and longitudinal, susceptibilities in this regime. Scaling functions in ordered and disordered regions are displayed graphically and compared with series-expansion results.