Molecular Field Calculation of Magnetization on NdRh 2 Ge 2 Single Crystal

Calculation of magnetization of the ternary single crystal compound NdRh 2 Ge 2 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, Q 2 = (0,0,39 / 40), Q 3 = (0,0,35 / 40), Q 4 = (0,0,31 / 40), and Q 5 = (0,0,0 / 40) ( = the ﬁeld-induced ferromagnetic phase) were proposed. Calculation on the basis of these structures and the antiferromagnetic phase with Q 1 = (0,0,1) well reproduces the experimental magnetization processes and H - T magnetic phase diagram. any medium, provided the original work is


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 2type 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 50 K.
(ii) In the temperature dependence of magnetic susceptibility, there is another anomaly at 37 K which indicates a magnetic transition. A magnetic transition is also suggested at 20 K 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].

WAVE-LIKE MOLECULAR FIELD MODEL
The s-f interaction may be an important interaction in metallic rare earth compounds. Considering this interaction of ith atom as the effective Hamiltonian in terms of the molecular field H m (i), where J i is a good quantum number. The molecular field acting on an atom on the ith cplane is given by introducing molecular field coefficient λ(q) depended on Fourier q component as where J q is the Fourier q component of J i . These equations are to be solved self-consistently. The details have been reported by Iwata [5].

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 Q 1 = (0, 0, 1) 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 Q 1 , the fieldinduced phases have propagation vectors Q z = (0, 0, q z ) (z = 1, 2, . . .), 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 Q z . Wave numbers q z 's are supposed from the magnetization process as shown in

Determination of the wave-dependent molecular field coefficients
Molecular-field coefficients λ(Q z ) 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 λ(Q z ) 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 .

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 (H m ) are illustrated in Figures 3(a)-3(c). The Q 1 -structure (= phase I) is expressed by only one function with q z = 1 since the magnetic structure is AFI-type [6], whereas the Q 2 -, Q 3 -structure (= phase II, phase III) are expressed by a sum of forty A. Himori et al. 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 Q 2 -and Q 3 -structure. It is caused by the H m (= 0) around the moments and it is seen in TbRu 2 Ge 2 [9].

Moment arrangements under various applied field
Figures 4(a)-4(e) illustrate the calculated Nd magnetic moment arrangements and the behaviors of the total field (H + H m ) at 4.2 K. It is seen that the moments flips together with the change of a total field's sign. The magnitudes of the moments are smaller than gJμ B for all phases because of the CEF effect. It is noticed that no paramagnetic Nd ions appear in this case.

Magnetization process and magnetic phase diagram
The calculated magnetization process at 4.2 K under applied fields is shown in Figure 1. 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.

SUMMARY
The interesting magnetic behavior on the NdRh 2 Ge 2 single crystal had been reported; successive magnetic phase transitions occur at 20 K, 37 K, and 50 K (= T N ). At low temperatures, a four-step metamagnetic process appears. We try to explain this complex magnetic behavior by the wavelike molecular field model. For the sake of calculation, the field-induced magnetic structures having the propagation vectors, Q 2 = (0, 0, 39/40), Q 3 = (0, 0, 35/40), Q 4 = (0, 0, 31/40), and Q 5 = (0, 0, 0/40) (= 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.