Magnetism and spin dynamics in the cluster compound † Cr 4 S„O 2 CCH 3 … 8 „H 2 O… 4 ‡„NO 3 … 2 "H 2 O Y. Furukawa,* M. Luban, F. Borsa, † D. C. Johnston, A. V. Mahajan, ‡ and L. L. Miller Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011 D. Mentrup and J. Schnack Fachbereich Physik, Universita ¨t Osnabru ¨ck, Barabarastrasse 7, D-49069 Osnabru ¨ck, Germany A. Bino Department of Inorganic and Analytical Chemistry, Hebrew University, 91904 Jerusalem, Israel Received 1 November 1999 The magnetism and spin dynamics of  Cr 4 SO 2 CCH 3  8 H 2 O 4 NO 3  2 •H 2 O have been investigated by mag- netic susceptibility and proton NMR measurements and by theoretical calculations. The proton-spin lattice relaxation rate 1/T 1 , as a function of the temperature T and external magnetic field B, provides a very useful probe of the dynamical behavior of the four Cr +3 spins s = 3 2  paramagnetic ions which are ferromagnetically coupled via isotropic Heisenberg exchange interaction. From our derived formulas for the two-ion time cor- relation functions, we find that 1/T 1 may be expressed as a function of the single scaling variable  B B /( k B T ), where  B is the Bohr magneton and k B is Boltzmann’s constant, for the regime of our measurements. Our experimental data are in good agreement with this prediction and our derived form of the scaling function. The great progress 1–4 made in recent years in synthesizing bulk samples of identical molecular-size magnetic com- plexes containing relatively small numbers of mutually inter- acting paramagnetic ions ‘‘spins’’ has provided the oppor- tunity to systematically investigate magnetism at the mesoscopic scale. It is noteworthy that the intercomplex in- teractions between spins are in many cases negligible over the temperature range of typical measurements compared to those within a given complex. This property greatly simpli- fies the analysis since the magnetic properties of a bulk sample are determined in such cases by intracomplex spin- spin interactions alone. However, the diversity in the practi- cal choices of spins and their placement within a complex is matched by the great diversity of their properties. In many cases the Heisenberg model of isotropic exchange interaction between the spins of a given complex provides a satisfactory theoretical framework. In this paper we summarize the major results of our com- bined experimental and theoretical study of a magnetic clus- ter compound,  Cr 4 SO 2 CCH 3  8 H 2 O 4 NO 3  2 •H 2 O, to be abbreviated as Cr 4 -NO 3 , which may be pictured as four Cr +3 ions individual spins s = 3 2  situated at the vertices of a nearly regular tetrahedron that is embedded within the host complex. The primary motivation of this work is to elucidate the spin dynamics of a cluster compound that is amenable to a comprehensive theoretical as well as experimental study. The understanding derived from a study of a benchmark ma- terial such as this may further the understanding of other diverse magnetic complexes. It should be noted that the present compound differs from one investigated previously, 5 to be abbreviated as Cr 4 -BF 4 , which featured a BF 4 radical rather than the NO 3 . The crystal structure and dimensions of the two compounds are essentially identical. We have measured the magnetic susceptibility and the proton spin-lattice relaxation rate 1/T 1 of Cr 4 -NO 3 . Using our susceptibility data we determine the strength of the ex- change interaction between Cr ions as well as the value of the total spin S of the ground state. We find that our 1/T 1 data are in good agreement with the predictions of our theory briefly described below which is based on a first-principles quantum-mechanical calculation of the equilibrium two-spin time correlation functions for the Heisenberg Hamiltonian of Eq. 1. In addition to providing information on the low- frequency portion of the dynamics of the interacting spins, that can be accessed by measuring 1/T 1 using NMR tech- niques, the time correlation functions are invaluable for giv- ing detailed predictions for eventual inelastic neutron- scattering measurements. For the regime of temperatures (1.5T 30 K) and external magnetic fields (0.5B 5.5 T) of our measurements, we find that 1/T 1 may be expressed as a function of a single scaling variable  B B /( k B T ), a prediction in good agreement with our experi- mental data. Here  B is the Bohr magneton and k B is Bolt- zmann’s constant. The scaling property is an immediate consequence of the fact that in this regime of T and B, only the S =6 ground-state manifold of the Heisenberg Hamil- tonian is relevant. In fact, we find that 1/T 1 is proportional to T  ( T , B ), where  ( T , B ) = M ( T , B )/  B is the field- dependent differential paramagnetic susceptibility per mole of formula units mol FU described as independent S =6 spins. We idealize the structure as four Cr ions which are situ- ated at the vertices of a regular 6 tetrahedron and which in- teract with each other via isotropic Heisenberg exchange and with a uniform external magnetic field B whose direction defines the z axis. 7 The model Hamiltonian reads H =-J  m=1 3  n =m+1 4 s m • s n +  n =1 4 g  B Bs nz , 1 where the individual spin operators are measured in units of , the exchange energy J is positive for ferromagnetic inter- PHYSICAL REVIEW B 1 APRIL 2000-I VOLUME 61, NUMBER 13 PRB 61 0163-1829/2000/6113/86354/$15.00 8635 ©2000 The American Physical Society