Magnetic cluster excitations

Magnetic clusters, i.e., assemblies of a finite number (between two or three and several hundred) of interacting spin centers which are magnetically decoupled from their environment, can be found in many materials ranging from inorganic compounds and magnetic molecules to artificial metal structures formed on surfaces and metalloproteins. Their magnetic excitation spectra are determined by the nature of the spin centers and of the magnetic interactions, and the particular arrangement of the mutual interaction paths between the spin centers. Small clusters of up to four magnetic ions are ideal model systems in which to examine the fundamental magnetic interactions, which are usually dominated by Heisenberg exchange, but often complemented by anisotropic and/or higher-order interactions. In large magnetic clusters, which may potentially deal with a dozen or more spin centers, there is the possibility of novel many-body quantum states and quantum phenomena. In this review the necessary theoretical concepts and experimental techniques to study the magnetic cluster excitations and the resulting characteristic magnetic properties are introduced, followed by examples of small clusters, demonstrating the enormous amount of detailed physical information that can be retrieved. The current understanding of the excitations and their physical interpretation in the molecular nanomagnets which represent large magnetic clusters is then presented, with a section devoted to the subclass of single-molecule magnets, distinguished by displaying quantum tunneling of the magnetization. Finally, there is a summary of some quantum many-body states which evolve in magnetic insulators characterized by built-in or field-induced magnetic clusters. The review concludes by addressing future perspectives in the field of magnetic cluster excitations.

Figure 1

Sketch of the processes relevant for optical spectroscopies. A, absorption; L, luminescence; and R, radiationless transition.

Figure 2

Temperature dependence of the magnetic susceptibilty of $[({\mathrm{NH}}_3){}_5\mathrm{Cr}\mathrm{OHCr}({\mathrm{NH}}_3{)}_5]{\mathrm{Cl}}_5·{\mathrm{H}}_2\mathrm{O}$. Adapted from 319.

Figure 3

Polarized luminescence spectrum of $[({\mathrm{NH}}_3){}_5\mathrm{Cr}\mathrm{OHCr}({\mathrm{NH}}_3{)}_5]{\mathrm{Cl}}_5·{\mathrm{H}}_2\mathrm{O}$ taken at $T=7\mathrm{K}$. The corresponding transition diagram is shown at the top. Adapted from 90.

Figure 4

(a) Energy spectra of neutrons scattered from deuterated $[({\mathrm{NH}}_3){}_5\mathrm{Cr}\mathrm{OHCr}({\mathrm{NH}}_3{)}_5]{\mathrm{Cl}}_5·{\mathrm{H}}_2\mathrm{O}$. (b) $Q$ dependence of the intensity of the $|0⟩\to |1⟩$ transition observed at $T=4.2\mathrm{K}$ shown in (a). Adapted from (a) 122, and (b) 95.

Figure 5

$Q$ band EPR spectrum of ${\mathrm{Mn}}^{2+}$ ions in $\mathrm{CsMg}{\mathrm{Br}}_3$ at $T=77\mathrm{K}$. Adapted from 208.

Figure 6

Field dependence of the ground-state levels of ${\mathrm{Mn}}^{2+}$ ions in $\mathrm{CsMg}{\mathrm{Br}}_3$. The calculations are based on Eqs. (10, 13), with $D>0$. The double arrows mark the resonances $A$ to $E$ observed in the EPR spectrum of Fig. 5.

Figure 7

(a) Temperature dependence of absorption spectra resulting from a spin-flip process observed for $\mathrm{Cs}{\mathrm{Mn}}_{0.20}{\mathrm{Mg}}_{0.80}{\mathrm{Br}}_3$. The successive population of the ${\mathrm{Mn}}^{2+}$ dimer states is indicated by $S=1$ to $S=4$. (b) Observed intensities of the bands $S=1$ to $S=3$ as a function of temperature. The lines correspond to the Boltzmann populations calculated from Eq. (45) with $J=-0.88\mathrm{meV}$. Adapted from 206.

Figure 8

(a) Energy spectra of neutrons scattered from ${\mathrm{Mn}}^{2+}$ pairs in $\mathrm{Cs}{\mathrm{Mn}}_{0.28}{\mathrm{Mg}}_{0.72}{\mathrm{Br}}_3$ at $T=30\mathrm{K}$. (b) Energy-level sequence of an antiferromagnetically coupled spin pair. Adapted from 89.

Figure 9

Energy-level splittings of Mn trimers in $\mathrm{Cs}{\mathrm{Mn}}_{0.28}{\mathrm{Mg}}_{0.72}{\mathrm{Br}}_3$. The arrows denote the observed transitions. Adapted from 87.

Figure 10

Energy spectra of neutrons scattered from $\mathrm{Cs}{\mathrm{Mn}}_x{\mathrm{Mg}}_{1-x}{\mathrm{Br}}_3$. The energy resolution amounts to $55\mu \mathrm{eV}$. For clarity, the data for $x=0.14$ and 0.28 are shifted by 10 and 20 intensity units, respectively. The lines refer to Gaussian peak fits with equal linewidths for both transitions $A$ and $B$. Adapted from 98.

Figure 11

High-field differential magnetization of $\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\}$. The circles represent the experimental data taken at 0.44 K, and the line a calculation based on the Hamiltonian equations (10, 11, 66), with the model parameters listed in the text. Adapted from 268.

Figure 12

Energy levels of $S=1$ tetramers calculated from Eq. (67) with $J<0$. The lowest states of a given $S$ value are marked by bold lines. The $|S_{34}{S}_{234}S⟩$ states are identified on the right-hand side. The inset shows the coupling parameters in a slightly distorted tetrahedron as realized for the compound $\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\}$.

Figure 13

Energy spectra of neutrons scattered from polycrystalline $\{{\mathrm{Ni}}_4{\mathrm{Mo}}_{12}\}$ with an incident neutron energy of 3.27 meV. The inset attributes the observed transitions to the low-energy part of the splitting pattern. Adapted from 226.

Figure 14

Pressure dependence of the $|2⟩\to |1⟩$ transition associated with ${\mathrm{Mn}}^{2+}$ dimers in ${\mathrm{Mn}}_{0.02}{\mathrm{Zn}}_{0.98}\mathrm{Te}$ measured by INS. The lines denote Gaussian fits to the data. Adapted from 173.

Figure 15

Doping dependence of the $|2⟩\to |1⟩$ transition associated with ${\mathrm{Mn}}^{2+}$ dimers in ${\mathrm{Mn}}_x{\mathrm{Zn}}_{1-x}\mathrm{Te}$ measured by INS. The lines denote Gaussian fits to the data. Adapted from 165.

Figure 16

Energy spectrum of neutrons scattered from ${\mathrm{K}}_{10}[{\mathrm{Co}}_4({\mathrm{D}}_2\mathrm{O}{)}_2({\mathrm{PW}}_9{\mathrm{O}}_{34}{)}_2]·20{\mathrm{D}}_2\mathrm{O}$ at $T=1.7\mathrm{K}$. The insets show a sketch of the tetrameric ${\mathrm{Co}}^{2+}$ unit and the resulting energy-level scheme, labeled with the dominant component $|S_{12}{S}_{34}S,\pm M⟩$ of the wave function. The highest states $|1,1,1,0⟩$ and $|1,1,0,0⟩$ located at 5.04 and 5.79 meV, respectively, are not shown. Adapted from 67.

Figure 17

Energy spectrum of neutrons scattered from ${\mathrm{Fe}}_4$ at $T=6.5\mathrm{K}$. The transitions $|\pm M⟩\to |\pm M^{\prime }⟩$ within the $S=5$ ground-state multiplet are marked for each observed peak. Adapted from 8.

Figure 18

Structural arrangement of the ${\mathrm{Cu}}^{2+}$ dimers (balls connected with solid black lines) in $\mathrm{Sr}{\mathrm{Cu}}_2({\mathrm{BO}}_3{)}_2$ including the Dzyaloshinski-Moriya interactions whose vectors are perpendicular to the $(a,b)$ plane. The arrows show the order of the spins in the expression $\mathbf{d}({\widehat{\mathbf{s}}}_i\times {\widehat{\mathbf{s}}}_j)$. Adapted from 56.

Figure 19

(a) Magnetic field dependence ($B\Vert c$) of the singlet-triplet excitations observed in $\mathrm{Sr}{\mathrm{Cu}}_2({\mathrm{BO}}_3{)}_2$ by EPR (open dots) and INS (solid dots) experiments. (b) INS spectrum of $\mathrm{Sr}{\mathrm{Cu}}_2({\mathrm{BO}}_3{)}_2$ at $\mathbf{Q}=(1,0,0)$ and $B=6\mathrm{T}$ $(B\Vert c)$. Adapted from 229, and 56.

Figure 20

(a) Energy distribution of the $|S=0,M=0⟩\to |S=1,M=1⟩$ ${\mathrm{Mn}}^{2+}$ dimer transition observed for $\mathrm{Cs}{\mathrm{Mn}}_{0.1}{\mathrm{Mg}}_{0.9}{\mathrm{Br}}_3$ at $T=1.6\mathrm{K}$. (b) Sketch of statistical distributions of ${\mathrm{Mn}}^{2+}$ and ${\mathrm{Mg}}^{2+}$ ions along the $c$ axis. $m$ is the number of ${\mathrm{Mn}}^{2+}$ ions replacing ${\mathrm{Mg}}^{2+}$ ions outside the central ${\mathrm{Mn}}^{2+}$ dimer embedded in the shaded area. Adapted from 101.

Figure 21

Molecular structures of the five even-membered antiferromagnetic wheels ${\mathrm{NaFe}}_6$, ${\mathrm{Cr}}_8$, ${\mathrm{CsFe}}_8$, ${\mathrm{Fe}}_{10}$, and ${\mathrm{Fe}}_{18}$ (for the chemical compositions see the text).

Figure 22

(a) Magnetization vs field and (b) differential magnetization $dM/dB$ at a temperature of 0.6 K of the antiferromagnetic wheel ${\mathrm{Fe}}_{10}$. The data show the sequence of magnetization steps, traced here for fields up to 40 T. Adapted from 292.

Figure 23

Sketch of the low-lying energies in even antiferromagnetic molecular wheels. The Heisenberg interactions give rise to spin multiplets $S=0,1,2,\dots$ which split in a magnetic field and lead to level crossings in the ground state, which changes from $|S=0,M=0⟩$ to $|1,-1⟩$, $|1,-1⟩\to |2,-2⟩$, etc., and steps in the magnetization curve. The magnetic anisotropy splits the spin multiplets in zero field, as indicated to the left. Some allowed INS transitions are indicated by arrows.

Figure 24

Neutron energy-loss spectrum of ${\mathrm{NaFe}}_6$ at the indicated temperatures ($Q=1.0{\AA }^{-1}$). The transition $i$ could be assigned to an impurity species in the sample. Adapted from 336.

Figure 25

(a) Simulated energy spectrum for a $N=8$, $s=\frac{3}{2}$ AFMHR with $2J=-1.46\mathrm{meV}$ as for ${\mathrm{Cr}}_8$. The low-lying levels form rotational bands. The $L$ band can be rationalized as (quantized) rotation of the Néel vector. The next higher-lying bands are denoted collectively as the $E$ band and may be regarded as (discrete) antiferromagnetic spin waves. Some INS transitions are indicated by arrows with associated labels. The numbers in brackets indicate the shift quantum number $q$ for some spin multiplets. (b) Simulated spin-spin autocorrelation function $C_0^z(\omega )$ in units of $2|J|$ for various temperatures. The $L$- and $E$-band transitions indicated in (a) are identified. Adapted from 323, and 334.

Figure 26

Neutron energy-loss spectrum of ${\mathrm{Cr}}_8$ at the indicated temperatures. Adapted from 334.

Figure 27

$Q$ dependence of the integrated neutron scattering intensity of ${\mathrm{Cr}}_8$ for the peaks (a) $L^0$ and (b) $E_1$. Both peaks arise from transitions from the $S=0$ ground state into the $S=1$ sector, but the final spin multiplets involved differ by their shift quantum number $q$, as indicated in the panels. The solid lines represent the theoretical curves. Adapted from 334.

Figure 28

Physical properties of the antiferromagnetic Heisenberg ring as a function of its size $N$ and the spin length $s$. In the shaded area quantum fluctuations are weak and the spin structure is essentially classical. Here the $L$- and $E$-band concept becomes valid and ${\widehat{\mathbf{H}}}_{AB}$ is a good effective spin Hamiltonian. This classical regime is reached when size $N$ is significantly smaller than a characteristic correlation length $\xi$, which roughly increases exponentially with $s$. Adapted from 323.

Figure 29

Neutron energy-loss spectrum in ${\mathrm{CsFe}}_8$, after background correction, at $T=17\mathrm{K}$, and best-fit curve calculated from Eq. (77). Adapted from 330.

Figure 30

Experimental (crosses) and simulated (dotted lines) EPR transitions as observed in angle-resolved EPR experiments on single crystals of ${\mathrm{CsFe}}_8$ at frequencies of (a) the $Q$ band (35 GHz) ($T=25\mathrm{K}$) and (b) 190 GHz ($T=5$ and 15 K). Adapted from 81.

Figure 31

Neutron energy-loss spectrum in ${\mathrm{CsFe}}_8$ ($T=5\mathrm{K}$). (a) $Q$-sliced INS data (open circles), with curves offset for clarity (the $Q$ ranges are $1.0–2.5{\AA }^{-1}$, $2.5–4.0{\AA }^{-1}$, and $4.0–5.5{\AA }^{-1}$ for the lower, middle, and upper curves, respectively). The solid circles represent the INS data recorded at $T=58\mathrm{K}$ after the Bose correction. The solid line represents the best-fit simulation of the INS spectrum. (b) $S(Q,\omega )$ plot. Adapted from 82.

Figure 32

190 GHz EPR spectrum of a single crystal of ${\mathrm{CsFe}}_8$. The intramultiplet transition $\alpha$ is clearly observed. The inset to the left depicts the three lowest energy levels and the allowed INS transitions I, II, and $\alpha$. The inset to the right shows INS data recorded at $T=9.7\mathrm{K}$ in which the transition $\alpha$ should have been observed if it were of appreciable intensity. Adapted from 81.

Figure 33

Zero-temperature excitation spectrum of the $N=8$, $s=\frac{5}{2}$ AFMHR, as a function of the shift quantum number $q$ calculated using exact numerical diagonalization (stars) and the indicated SWTs (lines) (in the latter $q$ was assumed as continuous for clarity). LSWT indicates linear SWT, ISWT interacting SWT, IMSWT full-diagonalization interacting modified SWT, and $\mathrm{LSWT}+{\Delta }_c$ linear SWT with a shift. Adapted from 82.

Figure 34

Classical ground-state spin configuration for an (a) $N=8$, (b) and an $N=18$ antiferromagnetic wheel with the Néel vector (long arrow) pointing along (a) $\mathbf{z}$ and (b) $\mathbf{y}$. (c) Shape of the potential $V(\mathbf{n})$ in the case (B). The two tunneling paths from $\mathbf{n}=+y$ to $-y$ are indicated. Simulated low-lying energies of an $N=8$, $s=\frac{5}{2}$ antiferromagnetic wheel vs $D/(2J)$ in zero field ($M$ is then an exact quantum number). Transitions I and II are observed in INS experiments (see Fig. 35). Energy scheme as given by semiclassical theory for (e) ${\mathrm{Fe}}_{18}$, (f) $\mathrm{Cs}{\mathrm{Fe}}_8$, and (g) ${\mathrm{Fe}}_{10}$. Adapted from 331, 337, and 176.

Figure 35

Neutron energy-loss spectra for (a) ${\mathrm{Fe}}_{10}$ ($T=2$, 5, and 10 K); (b) ${\mathrm{CsFe}}_8$ ($T=2.4\mathrm{K}$); and (c) ${\mathrm{Fe}}_{18}$ ($T=1.9$ and 4.2 K) showing transition I, which in the QTNV regime corresponds to the Néel-vector tunneling splitting ${\Delta }_{01}$. Adapted from 263, and 331, 337.

Figure 36

Magnetic torque vs field for $B$ nearly perpendicular to $z$ ($\mathrm{\text{angle}}=80.5°$, $T=0.1\mathrm{K}$), showing wiggles due to the oscillations of the Néel-vector tunneling gap. The inset shows the simulated field-dependent torque and the tunneling gap as calculated quantum mechanically and semiclassically. Adapted from 337.

Figure 37

Molecular structures of the doped wheel ${\mathrm{Cr}}_7\mathrm{Ni}$, short chain ${\mathrm{Cr}}_6$, $\mathrm{Mn}\mathrm{\text{−}}[3\times 3]$ grid representing a wheel with magnetic center, and disk ${\mathrm{Fe}}_7$ (for the chemical compositions see the text).

Figure 38

(a) Neutron energy-loss spectra of $\mathrm{Mn}\mathrm{\text{−}}[3\times 3]$ (2) at 2.5 K in two different energy ranges. (b) Simulated energy spectrum with anisotropy neglected. The observed INS transitions are indicated by arrows. The inset sketches the observed transitions within the $S=\frac{5}{2}$ ground and first excited $S=\frac{7}{2}$ multiplet. Adapted from 124.

Figure 39

Magnetic torque vs field (upper curve) and the field derivative (lower curve) for $\mathrm{Mn}\mathrm{\text{−}}[3\times 3]$ ($\mathbf{1}$) with $B$ nearly perpendicular to the magnetic $z$ axis ($\mathrm{\text{angle}}=80.5°$, $T=0.1\mathrm{K}$), showing the oscillatory torque behavior. The inset shows the energies of the spin multiplets as extracted from the level-crossing fields. From 338.

Figure 40

(a) Neutron energy-loss spectra of ${\mathrm{Cr}}_7\mathrm{Ni}$ at $T=2$ and 12 K. The circles represent the experimental data; the lines represent the simulation result. Adapted from 41. (b) Simulated energy spectra with anisotropy neglected. Observed INS transitions are indicated.

Figure 41

(a) High-resolution neutron energy-loss spectra on a crystal sample of ${\mathrm{Cr}}_7\mathrm{Ni}$ as a function of the magnetic field ($T=66\mathrm{mK}$, $\theta =50°$). (b) Simulated low-lying energy levels as a function of the magnetic field (lines) with all observed INS transition energies included (squares). Adapted from 50.

Figure 42

(a) Neutron energy-loss spectra on ${\mathrm{Cr}}_6$. The solid lines represent simulations. The inset shows the formation of standing spin waves in a chain and the link to the gap in the spin-wave spectrum. (b) Simulated energy spectra for ${\mathrm{Cr}}_6$ and a hypothetical $N=6$, $s=\frac{3}{2}$ antiferromagnetic wheel. The observed INS transitions and the gap in the $E$ band are indicated. Adapted from 231.

Figure 43

(a) Spin-wave dispersion as obtained in linear SWT for ${\mathrm{Cr}}_6$ and a hypothetical $N=6$, $s=\frac{3}{2}$ antiferromagnetic wheel. (b) Exact excitation energies in the $S=1$ sector of ${\mathrm{Cr}}_6$ and comparison to the results of various variants of SWT. LSWT, linear SWT; ISWT, interacting SWT; MSWT, modified linear SWT; MISWT, modified interacting SWT; SLSWT, spin-level SWT. Adapted from 231.

Figure 44

(a) Sketch of the energy-level scheme of an antiferromagnetic spin-$\frac{1}{2}$ triangle. (b) Structure of the anion of ${\mathrm{V}}_{15}$. The thick lines indicate the two hexagons, and the triangular area the spin triangle. (c) Sketch of the exchange interactions in the ${\mathrm{V}}_{15}$ molecule. Adapted from 59, and 298.

Figure 45

(a) Field dependence of the INS peak energies observed in ${\mathrm{V}}_{15}$ at $T=45\mathrm{mK}$. Lines are fits to the data. (b) Derived energy spectrum and assignment of INS transitions. (c) $Q$ dependence of the intensities of peaks $\mathrm{III}+\mathrm{IV}+\mathrm{V}$ in zero field. (d) $Q$ dependence of the intensity of transitions $\mathrm{III}+\mathrm{IV}+\mathrm{V}$ at 0 T (circles) and transition I at 1 T (squares). Lines are fits to the data. Adapted from 59.

Figure 46

(a) Molecular structure of ${\mathrm{Fe}}_{30}$. (b) ${\mathrm{Fe}}^{3+}$ core forming an icosidodecahedron. The spin orientations in the classical ground state are represented by the arrows, the colors refer to the sublattices $A$, $B$, and $C$, respectively. (c) Low-lying energy spectrum as a function of total spin $S$ as predicted in the rotational-band model equation (88), showing the $L$ and $E$ bands. (d) Lowest energy in each sector $S_z$ as calculated by DMRG (squares) and comparison to the $L$ band in the rotational-band model (line). The two curves are essentially superimposed, confirming the $S(S+1)$ energy dependence of the lowest eigenstates in ${\mathrm{Fe}}_{30}$. The inset shows the experimental magnetization curve. (a), (b), and (d) courtesy of J. Schnack; (c) adapted from 103.

Figure 47

Neutron energy-loss spectrum of ${\mathrm{Fe}}_{30}$ at $T=65\mathrm{mK}$. The solid line represents the estimated background, and the determined magnetic scattering is displayed in the inset. Adapted from 103.

Figure 48

(a) Low-temperature excitation spectrum of ${\mathrm{Fe}}_{30}$ as predicted by linear SWT (LSWT), modified linear SWT (MSWT), and spin-level SWT (SLSWT). (b) Experimental INS spectrum of ${\mathrm{Fe}}_{30}$ (points) and simulated spectrum using SLSWT (line). Inset: Experimental minus simulated spectrum, providing evidence for magnetic scattering at ca. 0.3 meV, which the SLSWT model does not take into account. Adapted from 327.

Figure 49

Molecular structures of three prominent SMMs: (a) ${\mathrm{Mn}}_{12}$, (c) ${\mathrm{Fe}}_8$, and (d) ${\mathrm{Mn}}_6$ (for the chemical compositions see the text). (b)The metal core in the ${\mathrm{Mn}}_{12}$ molecule, the exchange coupling paths, and the classical ground-state spin configuration.

Figure 50

Inelastic neutron scattering spectrum of ${\mathrm{Fe}}_8$ at 9.6 K. Circles represent the experimental data, the dashed lines represent Gaussian fits, and the solid line represents a simulation. The transitions are labeled according to $|\pm M⟩\to |\pm M^{\prime }⟩$. The inset shows the parabolic energy spectrum with the allowed INS transitions indicated by arrows. Adapted from 40.

Figure 51

Neutron energy-loss spectra of ${\mathrm{Mn}}_{12}$ recorded in two energy regimes. Data were recorded at the direct TOF spectrometer MARI at Rutherford Appleton Laboratory ISIS. Adapted from 60.

Figure 52

Energy-level scheme as derived from the experimental INS data. (a) A zoom into the low-energy spectrum. Adapted from 60.

Figure 53

(a) Neutron energy-loss spectra in ${\mathrm{Mn}}_6$ recorded for two energy regimes up to 6 meV at the indicated temperatures (symbols). The lines represent simulations using Eq. (91) and the parameters given in the text. (b) Calculated energy spectrum as a function of $M$. The shading represents the value of the expectation value $⟨{\widehat{\mathbf{S}}}^2⟩$. Adapted from 46.

Figure 54

The core of ${\mathrm{Mn}}_6$ with the labeling of the ${\mathrm{Mn}}^{3+}$ sites and the assumed coupling paths indicated. Adapted from 46.

Figure 55

(a) Magnetic susceptibility measured for ${\mathrm{KCuCl}}_3$ with $B=0.5\mathrm{T}$ oriented perpendicular to the cleavage plane $(1,0,-2)$. Adapted from 294. (b) Magnetization measured for ${\mathrm{KCuCl}}_3$ at $T=1.7\mathrm{K}$ with $B$ perpendicular to the cleavage plane. Adapted from 232.

Figure 56

Selected results obtained from INS experiments on ${\mathrm{TlCuCl}}_3$. (a) Field dependence of the magnetic excitation energies measured at $\mathbf{Q}=(0,4,0)$. The solid lines reflect a linear Zeeman model. The critical field is $B_c=5.7\mathrm{T}$. Adapted from 253. (b) Splitting of the singlet-triplet excitation measured at $\mathbf{Q}=(-0.5,0,2)$ and $T=1.5\mathrm{K}$. The asymmetric line shapes are typical resolution effects. Adapted from 55.

Figure 57

(a) Energy dispersion of the triplons observed in ${\mathrm{TlCuCl}}_3$ at $T=1.5\mathrm{K}$ and $B=5.5\mathrm{T}$. The lines correspond to model expectations based on a three-dimensional extension of Eq. (92). Adapted from 55. (b) Energy dispersion of the low-lying magnetic excitations observed in ${\mathrm{TlCuCl}}_3$ at different temperatures, fields, and pressures. The lines denote the linear behavior of the Goldstone mode according to Eq. (97). Adapted from 253, 257.

Figure 58

Excitation spectra of copper nitrate at $T=0.12\mathrm{K}$. (a), (c) Background-subtracted two- and one-magnon INS data, respectively, and (b), (d) the simulated $T=0$ spectra. Adapted from 301.

Figure 59

Schematic representation of the structure of ${\mathrm{CuGeO}}_3$. The arrows denote the AFM spin alignment of the ${\mathrm{Cu}}^{2+}$ ions. The dashed lines mark the Cu-O(2)-Cu superexchange bonds. (a) $T>T_c=14\mathrm{K}$. (b) Copper dimerization for $T<T_c$ indicated by the dashed boxes.

Figure 60

(a) Energy scan profiles of the magnetic excitation observed for ${\mathrm{CuGeO}}_3$ at $\mathbf{Q}=(0,1,0.52)$ for various temperatures. The solid and dashed lines denote Gaussian fits to the data obtained at $T=4$ and 12 K, respectively. (b) Dispersion of the magnetic excitation observed in ${\mathrm{CuGeO}}_3$ along the $c$ direction at $T=4\mathrm{K}$. Adapted from 227.

Figure 61

Schematic representation of the Ce-Al zigzag chains in ${\mathrm{CeRu}}_2{\mathrm{Al}}_{10}$. The ${\mathrm{Ce}}^{3+}$ dimerization for $T<T_0$ is indicated by the dashed boxes.

Figure 62

(a) Temperature dependence of the energy spectra of neutrons scattered from ${\mathrm{CeRu}}_2{\mathrm{Al}}_{10}$ for $Q=1.5{\AA }^{-1}$. The lines denote least-squares fits. (b) $Q$ dependence of the intensity of the peak at 8 meV. The line corresponds to the dimer cross section Eq. (3.9) with $R=5.2\AA$. Adapted from 248.

Figure 63

Energy spectra of neutrons scattered from ${\mathrm{La}}_{0.998}{\mathrm{Sr}}_{0.002}{\mathrm{CoO}}_3$ at $T=1.5$ and 10 K. The open circles correspond to the nonmagnetic reference compound ${\mathrm{LaCoO}}_3$. For clarity, the intensities of the $T=10\mathrm{K}$ data are shifted by 400 neutron counts. Adapted from 241.

Figure 64

$Q$ dependence of the intensity of the transition observed at 0.75 meV in ${\mathrm{La}}_{0.998}{\mathrm{Sr}}_{0.002}{\mathrm{CoO}}_3$. The inset sketches different types of Co multimers. The lines are the result of structure factor calculations based on Eq. (101). Adapted from 241.