Neutron scattering and magnetic observables for $S=1∕2$ spin clusters and molecular magnets

In this paper we report results for magnetic observables of finite spin clusters composed of $S=1∕2$ ions. We consider clusters of two, three, and four spins in distinct spatial arrangements, with isotropic Heisenberg interactions of various strengths between ion pairs. In addition to the complete set of energy eigenvalues and eigenvectors, specific heat and magnetic susceptibility, we also quote results for the single crystal and powder average inelastic neutron scattering structure factors. Examples of the application of these results to experimental systems are also discussed.

Figure 1

The geometry and energy levels of a Heisenberg spin dimer.

Figure 2

The magnetic contribution to the specific heat of a spin dimer, Eq. (23) (dimensionless units).

Figure 3

The magnetic susceptibility of a spin dimer, Eq. (24) (dimensionless units).

Figure 4

A fit of the dimer susceptibility formula of Eq. (24) to the measured susceptibility of $\mathrm{V}\mathrm{O}\left(\mathrm{H}\mathrm{P}{\mathrm{O}}_4\right)\bullet 0.5{\mathrm{H}}_2\mathrm{O}$. A defect term was also included.

Figure 5

The powder average, unpolarized neutron structure factor $\overline{S}\left(q\right)$ for a spin dimer with pointlike magnetic ions.

Figure 6

Geometries and energy levels of (a) symmetric, (b) isosceles, and (c) general trimer systems. These systems have one $S_{\mathit{tot}}=3∕2$ multiplet and two $S_{\mathit{tot}}=1∕2$ multiplets.

Figure 7

The magnetic susceptibility of a symmetric trimer. The inset shows $\chi T$ vs $T$ for ferromagnetic (dashed) and antiferromagnetic (solid) couplings.

Figure 8

The unpolarized structure factors $\left\{S\left(\vec{q}\right)\right\}$ (proportional to the angular scattering intensities) predicted for inelastic neutron scattering from the $\mid \lambda ⟩$ ground state of a single crystal of an isosceles trimer material (see text).

Figure 9

The powder average inelastic neutron structure factor $\overline{S}\left(q\right)$ for the three allowed transitions of an isosceles trimer, with $b∕a=2$.

Figure 10

Energy level diagrams for (a) the tetrahedron, (b) the rectangular tetramer, and (c) the linear (dimer-pair) tetramer.

Figure 11

Magnetic contribution to the specific heat of a regular tetrahedron.

Figure 12

Susceptibility of a regular tetrahedron.

Figure 13

Powder average, unpolarized structure factors $\overline{S}\left(q\right)$ for the excitation of the three $S_{\mathit{tot}}=1$ excited states of the rectangular spin tetramer from the $S_{\mathit{tot}}=0$ ground state $\mid {\mathrm{\Psi }}_{0,1}⟩$. The interdimer coupling strength is $\alpha =0.3$, and the side length ratio is $b∕a=1.5$. The structure factor for the transition to the third state is scaled up by a factor of 10 for visibility.

Figure 14

Variation of the $\overline{S}\left(q\right)$ structure factors for excitation of the lowest $S_{\mathit{tot}}=1$ state of the rectangular spin tetramer with side length ratio $b∕a$. This illustrates the use of powder structure factors in establishing the internal geometry of spin clusters. The dimer coupling is $\alpha =0.3$, however, this only affects the overall normalization.

Figure 15

Powder average, unpolarized structure factors $\overline{S}\left(q\right)$ for the excitation of the three $S_{\mathit{tot}}=1$ states of the linear tetramer, including the ionic form factor, with magnetic coupling ratio $\alpha =0.4$ and ion positions taken from the $\mathrm{Na}\mathrm{Cu}\mathrm{As}{\mathrm{O}}_4$ structure (Ref. 28). The structure factor for the transition to the third state is scaled up by a factor of 10 for visibility.

Figure 16

Contours of equal intensity of the powder average, unpolarized structure factors of the linear tetramer, with parameters appropriate to $\mathrm{Na}\mathrm{Cu}\mathrm{As}{\mathrm{O}}_4$, as in Fig. 15. The contours are normalized to the peak of the first transition, and intervals of 0.2 in intensity are shown for the two lower states. The much weaker third transition is displayed with intervals of 0.01.