Local Field Distributions in Systems with Dipolar Interaction *
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ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 29 Number 1 January 2012 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIETY CHIN. PHYS. LETT. Vol. 29, No. 1 (2012) 013203 Local Field Distributions in Systems with Dipolar Interaction * YAN Yue(AT)1, WU Biao(Ç飙)2** 1Institute of Physics, Chinese Academy of Sciences, Beijing 100190 2International Center for Quantum Materials, Peking University, Beijing 100871 (Received and accepted 30 September 2011 by WU Ying) We investigate systematically the local field distribution functions of up-spins for systems of dipolar interaction, with particular emphasis on Ising-type lattice systems. It is found that as the fraction increases, the shape of the distribution function changes from Lorentzian to Gaussian. In addition, sub-peaks can be induced in the distribution function by non-cubic lattice structures. This is in stark contrast with a dilute gas system, where the distribution has only one Lorentzian peak for any up-spin fraction. PACS: 32.10.Dk, 02.30.Nw, 02.50.Ng DOI:10.1088/0256-307X/29/1/013203 The dipolar field interaction is an important however, because the spins are fixed in space, the dis- issue.[1−12] In this Letter, we provide a thorough study tribution function of the system does not depend on of the local field distribution function in Ising-type P(frjg) anymore. In both dilute gas systems and spin systems with particular emphasis on the lattice sys- lattice systems, the probabilities related to the spin tem. degree of freedom can be characterized by the frac- For Ising-type spins, e.g., molecular magnets Fe8 tion of up-spin states and we use F to indicate this [1−3] and Mn12, the dipolar interaction between them fraction. Due to the symmetry of the system, we shall is focus only on the case of F < 1=2 and treat the down- E (1 − 3 cos2 휃)Ω d(r) = D 0 ; (1) spins as the background. r3 We compute the distribution functions of local field where r is the displacement between spins, 휃 the angle for a dilute gas system firstly. This serves for two between r and the easy axis, and ED the strength of purposes. On the one hand, the results for a di- the interaction. For a lattice, Ω0 is the volume of the lute gas may find applications in ultra-cold molecule unit-cell; for a gas, Ω0 is the volume of a particle. systems;[5] on the other hand, these results can be When there is randomness in either spin configu- compared to the previous known results for dipolar ration or spatial configuration or both, each spin feels gases and our results for lattice systems to be pre- a different field strength. In this kind of situation, sented later. rather than to know exactly the field strength felt by We define a concentration parameter as each individual spin, it is usually more important to find the distribution function of the local fields feltby NΩ 휂 = 0 ; (3) the spins. The distribution function can be computed V with the Margenau method,[13−15] where one chooses a test spin and computes how the local field felt by where V is the system volume. For a dilute gas, we the test spin changes with different spin, spatial, or have 휂 ≪ 1. In this dilute limit, the correlation be- both configurations. If there is no correlation between tween the particle positions can be safely ignored and space and spin degrees of freedom, the distribution the particles can be regarded as randomly scattered function can be written as in the space. Therefore, for a dilute dipolar gas, when N the fraction of up-spin is F , Eq. (2) takes the following X X (︁ X )︁ P (휉) = P(frjg) Q(fsjg)훿 휉 − sjdj ; (2) form, frj g fsj g j=1 Z N N 1 Y 3 X n N−n where P(frjg) and Q(fsjg) are the probabilities of a P (휉; F ) = d r F (1 − F ) V N j given spatial configuration and spin configuration, re- j=1 n=0 spectively; the summations are over all possible con- N X X figurations; dj = d(rj) is the dipole field generated by × 훿(휉 − sjdj); (4) a spin at rj; sj = ±1 indicates the spin up or down; fsj gn j=1 and N is the number of particles in the system. In a dilute gas system, the spatial distribution con- where the summation is over all possible spin configu- figuration function is uniform. In a lattice system, rations when each spin is fixed in space. To compute *Supported by the National Natural Science Foundation of China under Grant No 10825417. **Corresponding author. Email: [email protected] ○c 2012 Chinese Physical Society and IOP Publishing Ltd 013203-1 CHIN. PHYS. LETT. Vol. 29, No. 1 (2012) 013203 the distribution function, it is convenient to compute We have also investigated the distribution function first its Fourier transform, with Monte Carlo simulation and they match almost Z perfectly with our analytical results in Eq. (9). P¯(k; F ) = P (휉; F ) exp(−ik휉)푑휉 For the general case, the shape of the system only shifts the position of the peak and we find perfect Z [︀ 1 agreement between our theoretical results[17] and the = [F exp(−ikdj) V Monte Carlo simulation. 3 ]︀N + (1 − F ) exp(ikdj)]d rj : (5) 0.03 0.028 0.025 In the limit of V ! 1, for a fixed gas concentration 0.026 N=V = 휂, using Fresnel sine and cosine integral,[16] 0.02 F↽ 0.024 ֒ ξ 0.015 we obtain ↼ 0.022 P 휂 0.01 h 1 iV Ω 0.02 ¯ 0 -8 -6 -4 -2 0 2 4 P (k; F ) = lim 1 − (A1jkjΩ0 − iA2kΩ0) 0.005 V !1 V (︁ )︁ 0 = exp − A1jkj휂 + iA2k휂 ; (6) -200 -100 0 100 200 300 400 500 600 ξ with Fig. 1. The local dipolar field distribution of a dilute gas with F = 0:015 and 휂 = 0:01. The red solid line is the 8휋2E A = D ; (7) theoretical result of Eq. (9) and the blue area is the Monte 1 35=2 Carlo simulation result. The inset shows an enlarged por- tion of the distribution peak, clearly indicating that the (1 − 2 F )2휋ED (︁4 4p p )︁ A = − 3 ln (︀2 + 3)︀ : peak center deviates from zero. 2 3 3 9 (8) We now turn to lattice systems, where each lattice After a simple reverse Fourier transform, we find that site has one Ising-spin. Molecular magnets such as Fe8 [1−3] the distribution function P (휉; F ) is Lorentzian and Mn12 are this type of spin lattice system: at low temperature, only two ground states jsz = ±10i A 휂/휋 are occupied in these systems. In these molecular sys- P (휉; F ) = 1 : (9) A2휂2 + (휉 + A 휂)2 tems, Ω0 is the volume of the unit cell of the lattice 1 2 2 2휇0 (gS휇B ) and ED = 4휋 Ω , where 휇B is Bohr magneton. [11;12] 0 This recovers the result by Berkov and Meshkov. For Fe8 and Mn12, S = 10 and g ≈ 2. Similar spin However, we want to emphasize that our result lattice systems have been investigated for the dipolar here is in a sense more general than Berkov and line width.[9] Meshkov’s although we consider only Ising-type spin. Because all spins are fixed in space in a lattice In Refs. [11,12], the orientation of magnetic moment of system, there is only one spatial configuration. As a the particle is random, equivalent to F = 1=2 in our result, the distribution function Eq. (2) becomes[3] system. Consequently, it is unclear how the distribu- tion function would change if the magnetic moment N X m N−m X (︁ X )︁ (or spin) has a favored direction. In contrast, it is D(휉; F ) = F (1−F ) 훿 휉 − sjdj(r) : clear from our result in Eq. (9) that the distribution m=0 fsj g j=1 does not change its height and width as F changes, (11) only the peak position shifts with F . We use D(휉; F ) to distinguish the distribution func- In deriving Eq. (9), we have implicitly assumed tion in a spin lattice system from the one in a dilute that the shape of the system is spherical. For other gas system. Similarly, we compute its Fourier trans- shapes, the peak position is shifted and the distribu- form first tion function becomes Z D¯(k; F ) ≡ 푑휉D(휉; F ) exp(−ik휉) A1휂/휋 P (휉; F ) = 2 2 2 ; N A1휂 + (휉 + A2휂 + 2CED휂(2F − 1)) Y (10) = [F exp(−ikdj)+(1−F ) exp(ikdj)]: where the parameter C is the shape coefficient of the j=1 (12) sample (Generally, C = 2휋( 1 −N ) in which N is the 3 g g We have used this equation to compute numerically demagnetization parameter of the sample. For ellip- how the distribution function D(휉; F ) changes with soid sample with three axes a b and c located in the F . The results are shown in Fig. 2. When F is very x, y and z axes respectively, the demagnetization pa- 1 R 1 dx small, the distribution function is a Lorentzian. As rameter Ng = abc p for 2 0 (x+c2) (x+a2)(x+b2)(x+c2) the F increases, the distribution function is broadened the magnetic field along the z axis).