Magnetic properties of an ensemble of rotating ferromagnetic clusters N. Hamamoto Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku Tokyo 153-8902, Japan N. Onishi Department of Ecosocial System Engineering, Yamanashi University, 4-3-11 Takeda, Kofu, Yamanashi 400-8511, Japan G. Bertsch Department of Physics and Institute for Nuclear Theory, Box 351560, University of Washington, Seattle, Washington 98195 ~Received 2 November 1998; revised manuscript received 12 August 1999! We analyze the adiabatic magnetization of ferromagnetic clusters in an intermediate-coupling regime, where the anisotropic potential is comparable to other energy scales. We find a nonmonotonic behavior of the magnetic susceptibility as a function of coupling with a peak. Coriolis coupling effects are calculated; they reduce the susceptibility somewhat. I. INTRODUCTION Magnetic properties of a wide variety of ferromagnetic clusters, e.g., Fe, Ni, and Co in transition metals involving 3 d electrons, 1–4 and Ru, Rh, and Pd associated with 4 d electrons, 3,5,6 are studied using the Stern-Gerlach technique. The observed deflection profile, caused by the interaction of magnetization and the gradient of field strength, provides information about these intrinsic magnetic moments of the cluster as well as its coupling to other degrees of freedom in clusters. The size of the cluster is small enough to be re- garded as a single-domain system, and the electrons involved form a single giant magnetic moment; we will call it the superelectron spin in this paper. It is quite interesting to see that, in such a low-dimensional system, some elements form ferromagnetic clusters, even though the bulk material is nonmagnetic. 6 Also, the magnetization is strongly dependent on the number of atoms in the clusters. 7 The analysis or interpretations of the experimental data have caused signifi- cant discussion and are still debated. Therefore, it is impor- tant in the present stage of study to establish a method of analysis for extracting the intrinsic magnetic moment from the observed deflection profile, and this is the motivation for the present work. Let us consider the experimental setup. First, clusters formed by laser evaporation are cooled by helium gas, and then the clusters are expanded to form a molecular beam. In the present study we assume that the clusters are in the ther- mal equilibrium. The density of clusters in the beam is low, so that the clusters may be assumed to be isolated beyond the equilibration zone. Therefore each cluster stays in a certain quantum state in the beam. Finally, the clusters enter into a Stern-Gerlach magnet and are deflected by the interaction of the gradient of magnetic field and the magnetic polarization of superspin induced by the magnetic field. At the entrance of the magnet, the strength of the field changes gradually in time, and a time-dependent interaction for the superelectron spin causes a transition of the initial quantum state to other quantum states. If the time dependence is sufficiently weak compared with coupling of the spin to other modes, the tran- sition probability to other modes can be neglected. This is called the adiabatic approximation. In the present work, we calculate the profile and the magnetization with this assump- tion. If the electrons were completely decoupled from other degrees of freedom such as rotational motion, the deflection profile would be a flat horizontal distribution independent of the field strength. But this is not actually the case for the observed profiles. A small coupling of the magnetic moment to the internal coordinates of the cluster gives rise to spin relaxation, making the profile different from the flat distribu- tion. Hence it is important to make clear how various cou- plings produce observed deflection profiles. The simplest model is superparamagnetism in which the population of the magnetic states is proportional to a Boltz- mann factor. 8 In other words, the cluster rotation plays a role as a heat bath for the superelectron spin in the magnetic field. In a practical analysis for extracting the giant magnetic mo- ment, the Langevin formula is widely employed. It assumes equilibrium with a thermal reservoir at a temperature which is the same as the source of the cluster beam. However, it predicts a rather sharp deflection profile which is quite dif- ferent from the broad profile that is often observed. Hence, the superparamagnetic model seems to be too simple for the analysis of the experiments. Another simple model is locked-spin model in which the superelectron spin is frozen to the intrinsic orientation of the cluster, which of course is free to rotate. 9–11 This model seems successful in reproducing the small peak observed near the zero deflection angle, which experimentalists call ‘‘superparamagnetism.’’ But it is applicable only to Gd clus- ters and not general. Furthermore, the model ignores the an- gular momentum of the superelectron spin, which is known to give recoil effects in the Einstein–de Haas effect. We propose an intermediate-coupling model as a method to extract the giant magnetic moment from the deflection profile. 11 This model covers the superparamagnetic and locked-moment models as weak- and strong-coupling limits, respectively. This paper is organized as follows: the interme- diate model is explained and formulated in Sec. II, numerical PHYSICAL REVIEW B 1 JANUARY 2000-II VOLUME 61, NUMBER 2 PRB 61 0163-1829/2000/61~2!/1336~15!/$15.00 1336 ©2000 The American Physical Society