Research Letters in Physics

VolumeΒ 2008, Article IDΒ 157070, 4 pages

http://dx.doi.org/10.1155/2008/157070

## Molecular Field Calculation of Magnetization on Single Crystal

^{1}Graduate School of Science and Engineering, Yamaguchi University, Yamaguchi 753-8511, Japan^{2}General Culture Department, Ube National College of Technology, Ube 755-8555, Japan

Received 17 April 2008; Accepted 20 May 2008

Academic Editor: SeanΒ Cadogan

Copyright Β© 2008 A. Himori et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Calculation of magnetization of the ternary single crystal compound has been carried out by using the wave-like molecular field model to explain the complex magnetic behavior. The field-induced magnetic structures having the propagation vectors, , , , and (= the field-induced ferromagnetic phase) were proposed. Calculation on the basis of these structures and the antiferromagnetic phase with well reproduces the experimental magnetization processes and - magnetic phase diagram.

#### 1. Introduction

Ternary rare
earth compounds RM_{2}X_{2} (R = rare earth, M = Rh, Ru, X = Si or
Ge) crystallize in the tetragonal ThCr_{2}Si_{2}-type structure
(I4/mmm) [1, 2]. In the numerous series, the RRh_{2}Ge_{2} is
worth studying because of the great diversity of its magnetic properties [3]. Recently,
we reported very interesting magnetic behavior of NdRh_{2}Ge_{2} single crystal [4] as follows.

(i)The compound shows an antiferromagnetic behavior with neel temperature
of 50βK.(ii)In the temperature dependence of magnetic susceptibility, there
is another anomaly at 37βK which indicates a magnetic transition. A magnetic transition is also suggested at 20βK from
magnetization measurements.(iii)There is
a strong uniaxial magnetic anisotropy with the easy *c*-axis which leads to an Ising-like
behavior in the compound.(iv)At a low
temperature, a four-step metamagnetic process appears along the easy *c*-axis
(see Figure 1).(v)The *H-T* magnetic phase diagram, where there
are five magnetic phases, was constructed (see Figure 2).

In the present study, in order to explain this complex magnetic behavior, calculations of the magnetization and an analysis of moment arrangements at various temperatures and under various magnetic fields have been carried out with a wave-like molecular field model [5].

#### 2. Wave-Like Molecular Field Model

The s-f
interaction may be an important interaction in metallic rare earth compounds.
Considering this interaction of *i*th
atom as the effective Hamiltonian in terms of the molecular field
where is a good quantum number.

The molecular
field acting on an atom on the *i*th
c-plane is given by introducing molecular field coefficient depended on Fourier *q* component
as
where is the Fourier *q* component
of These equations are to be
solved self-consistently. The details have been reported by Iwata [5].

#### 3. Results and Discussion

##### 3.1. Assumption of the Propagation Vectors at Each Magnetic Phase

For the sake
of calculating the magnetic processes by the wave-like molecular field model, knowledge
of magnetic structures is required. Only antiferromagnetic structure has been
reported on the NdRh_{2}Ge_{2} compound;
it is a simple structure having the propagation vector and magnetic moments along the *c*-axis [6]. On
the basis of the facts mentioned below, we can, now, propose magnetic
structures for the field-induced magnetic phases.

On most of compounds which show an Ising-like
multistep metamagnetic process along the *c*-axis and have the simple
antiferromagnetic structure with , the field-induced
phases have propagation vectors for
example, on PrCo_{2}Si_{2} [7], NdCo_{2}Si_{2} [8], and so on. Thus, we assume that the field-induced magnetic phases of NdRh_{2}Ge_{2 } compound have
Wave numbers βs are
supposed from the magnetization process as shown in Figure 1. There are five
magnetic phases in the metamagnetic process though it is hardly seen in the
figure. The magnetization in each phase is 0 0.06 0.32 0.58 and 2.59 (= saturation
value ), corresponding to and respectively. This large common denominator suggests a
long period structure. Metamagnetic transitions should be responsible for a
spin-flip due to the strong uniaxial magnetic anisotropy. The induced -phase requires 40 unit
cells. Then, we can propose that the field-induced phases have the propagation
vectors, (= the
field-induced ferromagnetic phase). Of course, the antiferromagnetic phase has

##### 3.2. Determination of the Wave-Dependent Molecular Field Coefficients

Molecular-field
coefficients are estimated by
finding the best fit of the calculated values with the experimental data of the
magnetic susceptibility, specific heat, and the magnetization process. The
values of obtained in this
study are plotted in Figure 2. The similar figure of this characteristic curve
has been seen in PrCo_{2}Si_{2} and NdCo_{2}Si_{2}.

##### 3.3. Moment Arrangements in Nonexternal Magnetic Field

The calculated
Nd magnetic moment arrangements at various temperatures in nonexternal magnetic
field together with the wave-like molecular fields are illustrated in Figures 3(a)β3(c). The -structure
(= phase I) is expressed by only one function with since the magnetic structure is AFI-type [6], whereas
the -, -structure
(= phase II, phase III) are expressed by a sum of forty one harmonic functions
with since the magnetic phases are intermediate phases below the induced
ferromagnetic phase. Obviously, these moment arrangements have been
corresponded with the wave-like molecular fields. It is very interesting
results that some paramagnetic Nd ions appear in the - and -structure.
It is caused by the (=
0) around the moments and it is seen in TbRu_{2}Ge_{2} [9].

##### 3.4. Moment Arrangements Under Various Applied Field

Figures 4(a)β4(e) illustrate the calculated Nd magnetic moment arrangements and the behaviors of the total field at 4.2βK. It is seen that the moments flips together with the change of a total fieldβs sign. The magnitudes of the moments are smaller than for all phases because of the CEF effect. It is noticed that no paramagnetic Nd ions appear in this case.

##### 3.5. Magnetization Process and Magnetic Phase Diagram

The calculated magnetization process at 4.2βK under applied fields is shown in Figure 1. The magnetizations are 0/f.u., 0.068/f.u., 0.32/f.u., 0.58/f.u., and 2.59/f.u. at 0T, 1.4T, 8.9T, 10.1T, and 13.7T, respectively. These values are in good agreement with the experimental values of 0/f.u., 0.06/f.u., 0.32/f.u., 0.58/f.u., and 2.59/f.u. It is clear that the magnetization curve is reproduced by the calculated values fairly well at 4.2βK.

The calculated *H-T* diagram
together with the experimental points obtained from the magnetization
measurements are illustrated in Figure 5. It is shown that the calculations reproduce the main feature of the experimental *H*-*T* phase diagram.

#### 4. Summary

The
interesting magnetic behavior on the NdRh_{2}Ge_{2} single crystal had been
reported; successive magnetic phase transitions occur at 20βK, 37βK, and 50βK At low temperatures, a four-step metamagnetic process
appears. We try to explain this complex magnetic behavior by the wave-like
molecular field model. For the sake of calculation, the field-induced magnetic
structures having the propagation vectors, and (= the field-induced ferromagnetic
phase) were proposed. On the basis of these structures and the antiferromagnetic structure reported, calculation of magnetization
for various temperatures and fields has been performed. The calculation well
reproduces main features of the experimental magnetization processes and *H-T* magnetic phase diagram. So, we
believe that the magnetic structures proposed for the field-induced phases are
right. To confirm the magnetic structures proposed, neutron diffraction study
is needed.

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