Dysprosium-based experimental representatives of an Ising-Heisenberg chain and a decorated Ising ring

It is shown that the bond-decorated Ising model is a realistic model for certain real magnetic compounds containing lanthanide ions. The lanthanide ion plays the role of Ising spin. The required conditions on the crystal-field spectrum of the lanthanide ion for the model to be valid are discussed and found to be in agreement with several recent ab initio calculations on ${\text{Dy}}^{3+}$ centers. Similarities and differences between the spectra of the simple Ising chain and the decorated Ising chain are discussed and illustrated, with attention to level crossings in a magnetic field. The magnetic properties of two actual examples (a ${\left[\text{DyCuMoCu}\right]}_{\infty }$ chain and a ${\text{Dy}}_4{\text{Cr}}_4$ ring) are obtained by a transfer-matrix solution of the decorated Ising model. $g$-factors of the metal ions are directly imported from ab initio results, while exchange coupling constants are fitted to experiment. Agreement with experiment is found to be satisfactory, provided one includes a correction (from ab initio results) for susceptibility and magnetization to account for the presence of excited Kramers doublets on ${\text{Dy}}^{3+}$.

Figure 1

The Ising-Heisenberg chain discussed in Ref. 3 (showing only two unit cells). The bonds of the Ising chain $\left(1-1^{\prime }-1^{″}-\dots \right)$ are decorated with Heisenberg dimers $\left(\left(2-3\right),\left(2^{\prime }-3^{\prime }\right),\dots \right)$. The partition function of this chain is exactly solvable.

Figure 2

Part of the decorated Ising chain. In each Ising bond is inserted an arbitrary statistical mechanical system,[1] called decorating unit, that interacts only with the two Ising spins $s_i$ at the vertices of the bond. The Ising spin variables commute with the Hamiltonian and have definite values in the eigenstates of the chain.

Figure 3

Magnetization along the $z$ direction versus field of the antiferromagnetic Ising chain, defined in Eq. (23). The curve approaches a perfect step as $T\to 0\text{K}$.

Figure 4

Magnetization of a powder sample of antiferromagnetic Ising chains. The low-temperature limiting curve is given in Eq. (25).

Figure 5

Scheme of one unit cell (black) of the ${\left[\text{DyCuMoCu}\right]}_{\infty }$ chain, showing type of exchange interactions and labeling of exchange constants. See Ref. 7 for the complete molecular structure.

Figure 6

Powder magnetization of ${\left[\text{DyCuMoCu}\right]}_{\infty }$.

Figure 7

Powder magnetic susceptibility of ${\left[\text{DyCuMoCu}\right]}_{\infty }$. The theoretical curve contains the correction for the contribution of excited Kramers doublets.

Figure 8

Powder magnetic susceptibility of ${\left[\text{DyCuMoCu}\right]}_{\infty }$. The correction refers to the contribution of excited Kramers doublets, Eq. (26a).

Figure 9

Theoretical magnetization of ${\left[\text{DyCuMoCu}\right]}_{\infty }$. The powder magnetization (see also Fig. 6) is compared with the projections of $\mathbf{M}$ on the field direction [Eq. (21)], for three different directions of the field; $\theta$ is the angle between $\mathbf{B}$ and the $z$ axis.

Figure 10

Theoretical magnetization of ${\left[\text{DyCuMoCu}\right]}_{\infty }$. Same as Fig. 9 but at lower temperature and to higher field.

Figure 11

Eigenvalues of $\widehat{h}\left(s,s^{\prime }\right)$ [Eq. (28)] in a magnetic field parallel with the $z$ axis $\left(\theta =0\right)$. Circles indicate ground state level crossings. The ground state of the chain is AF in zero field (left), switches to F at 0.64 T, and undergoes an internal level crossing at 6.3 T, marked by a change of the internal quantum number $M_S$ from 1/2 to 3/2. Both crossings can be seen in the $\theta =0$ magnetization curve in Fig. 10. Note that the energy curves appear as straight lines, although, with the exception of $M_S=\pm 3/2$, they are not exactly straight, because the Zeeman Hamiltonian does not completely commute with the total Hamiltonian. All ${\epsilon }_k\left(\uparrow \uparrow \right)$ decrease with increasing field strength because the large magnetic moment of ${\text{Dy}}^{3+}$ dominates the smaller magnetic moments of the decorating unit.

Figure 12

Theoretical susceptibility of ${\left[\text{DyCuMoCu}\right]}_{\infty }$, without correction for contribution of excited Kramers doublets. The powder $\chi T$ (see also Fig. 8) is compared with the Cartesian components of $\chi T$. $z$ is the direction of the anisotropy axis of ${\text{Dy}}^{3+}$, $x$ is any direction perpendicular to $z$. $\chi =\left({\chi }_{zz}+2{\chi }_{xx}\right)/3$.

Figure 13

Schematic representation of the ${\text{Dy}}_4{\text{Cr}}_4$ molecule indicating numbering of atoms and exchange coupling constants. The boxes show the orientation of the local anisotropy axes on Dy sites, when viewed from the poles of the $X$ and $Y$ axes. The $Z$ axis points out of the center of the scheme.

Figure 14

Powder magnetic susceptibility of ${\text{Dy}}_4{\text{Cr}}_4$. The correction on the theory refers to Eq. (26).

Figure 15

Powder magnetization of ${\text{Dy}}_4{\text{Cr}}_4$. A linear corrections of $0.5{\mu }_{\text{B}}/\text{T}$ has been added to the theoretical curve, according to Eq. (26b).