Figure 1

Schematic Zeeman diagram of a cluster ($S=2$) for various couplings. (a) Spin 2 cluster coupled to a heat bath causing transitions at a rate $\nu$ between the $2S+1$ Zeeman levels (examples are indicated by arrows). If the transition time $\tau =1/\nu$ is such that $\tau \gg t_{\mathrm{mag}}$ (the time of passage through the magnet) then the SG deflection pattern consist of $2S+1$ symmetrically positioned deflection maxima as in the uncoupled case. However, if $\tau \ll t_{\mathrm{mag}}$, the $2S+1$ maxima collapse in a single deflection maximum in the direction of increasing field [the width of the peak $\Delta M/\mu \approx \sqrt{(\tau /t_{\mathrm{mag}})}$], and peak position follows the Brillouin function (that converges to the Langevin function for large $S$). (b) Spin coupled to the rotations (all levels in this schematic diagram have the same $J_z=R_z+S_z$). If the spin is uncoupled from the rotations then the deflections are as in (a). (c) Now the spin is coupled to the rotations causing avoided crossings. Note that all of the adiabatic Zeeman levels tend downward with increasing field, indicating increasing magnetization with increasing field causing single-sided deflections. The Zeeman levels are canonically populated in the source (temperature $T$; $B=0$). The levels follow their adiabatic paths into the magnet ($B=B_{\mathrm{mag}}$) and the clusters deflect according to their magnetizations $m$, i.e., the slope of the levels at $B_{\mathrm{mag}}$ (for real clusters, the separation between avoided crossings is so small that the measurement averages over several of them.) The average magnetizations for large $S$ are Langevin-like: for low fields, $M={\mu }^{2}B/3kT$ ($\mu =2{\mu }_{B}S$) and for large fields $M=\mu$ independent of the density of states. The magnetization distribution width depends primarily on the density of states (see text).Reuse & Permissions

Figure 2

Deflections and magnetization distributions of ${\mathrm{Co}}_N$ at $T=40\mathrm{K}$ and $B=2\mathrm{T}$. (a) Position sensitive TOF mass peak of ${\mathrm{Co}}_{20}$ showing the field off (dashed line) and the field on ($B=2\mathrm{T}$, solid line) deflections (the entire spectrum is composed of about 200 distinct mass peaks). Note the single-sided deflections. (b) (inset) The normalized magnetization probability distribution determined from the deflections in (a) $N=20$; $M/N=0.83\mu B$; $\mu /N=2.3{\mu }_B$; $x=1.5$; $\Delta M/\mu \approx 0.4$. (c) Normalized magnetization distributions of ${\mathrm{Co}}_N$ ($12\le N\le 200$, $T=40\mathrm{K}$, $B=2\mathrm{T}$). Amplitudes are represented in color (blue: low; red: high). The magnetization is linear with $N$ for small $N$ and saturates at about $\mu (N)\approx 2N{\mu }_B$ for large $N$.Reuse & Permissions

Figure 3

Magnetic moments ${\mu }_N$ and normalized magnetizations $M_N/{\mu }_N$ of ${\mathrm{Co}}_N$. (a) $M_N/{\mu }_N$ of ${\mathrm{Co}}_{100}$ for $25\mathrm{K}\le T\le 100\mathrm{K}$ and $0<B\le 2\mathrm{T}$, corresponding to $x$ ranging from 0.4 to 12. The data scale with $x$. Note the linear increase for small $x$: $M_{100}/{\mu }_{100}\approx 0.3x$ and $M_{100}/{\mu }_{100}=1$ for large $x$. The trend is consistent with the Langevin function (bold line); however, the Langevin function approaches saturation more slowly. (b) $M_N/{\mu }_N$ for $12\le N\le 200$, $20\mathrm{K}\le T\le 100\mathrm{K}$ and $B\le 2\mathrm{T}$ measured in 63 data sets (9 temperatures from 25 K to 100 K and 7 fields from 0 to 2 T, representing about 10 000 data points), plotted as a function of $x$. (c) Magnetic moments per atom for ${\mathrm{Co}}_N$. Note that ${\mu }_{12}/12\approx 2{\mu }_B$, ${\mu }_N/N$ increases to a maximum at $N=37$ followed by a gradual decrease with weak oscillations converting to $2{\mu }_B$ for $N=150$.Reuse & Permissions