Magnetization steps in a diluted Heisenberg antiferromagnetic chain: Theory and experiments on $({\mathrm{CH}}_3{)}_4{\mathrm{NMn}}_x{\mathrm{Cd}}_{1-x}{\mathrm{Cl}}_3$

A theory for the equilibrium low-temperature magnetization M of a diluted Heisenberg antiferromagnetic chain is presented. Only the nearest-neighbor (NN) exchange interaction is included, and the distribution of the magnetic ions is assumed to be random. Values of the magnetic fields $B_i$ at the magnetization steps (MST’s) from finite chains with two to five spins (pairs, triplets, quartets, and quintets) are given for chains composed of spins $S=5/2.\mathrm{}$ The magnitudes of these MST’s as a function of the fraction, x, of cations that are magnetic are given for any S. An expression for the apparent saturation value of M is derived. The magnetization curve, M versus B, is calculated using the exact contributions of finite chains with one to five spins, and the “rise and ramp approximation” for longer chains. An expression for the low-temperature saturation magnetic field $B_s\mathrm{}\left(n\right)\mathrm{}$ of a finite chain with n spins is given. Some nonequilibrium effects that occur in a rapidly changing B are also considered. Some of these result from the absence of thermal equilibrium within the sample itself, whereas others are caused by the absence of thermal equilibrium between the sample and its environment (e.g., liquid-helium bath). Specific nonequilibrium models based on earlier treatments of the phonon bottleneck, and of spin flips associated with cross relaxation and with level crossings (anticrossings), are discussed. Magnetization data on powders of TMMC diluted with cadmium [i.e., $({\mathrm{CH}}_3{)}_4{\mathrm{NMn}}_x{\mathrm{Cd}}_{1-x}{\mathrm{Cl}}_3,$ with $0.16<~x<~0.50]$ were measured at $0.55\mathrm{K}$ in 18-T superconducting magnets. The field $B_1$ at the first MST from pairs is used to determine the NN exchange constant J. This ${J/k}_{\mathrm{B}}$ changes from $-5.9\mathrm{K}$ to $-6.5\mathrm{K}$ as x increases from $0.16$ to $\mathrm{0.50.}$ The magnetization curves obtained in the superconducting magnets are compared with simulations based on the equilibrium theory. A reasonably good agreement is found. Data for the differential susceptibility, $dM/dB,$ were taken in pulsed magnetic fields (7.4-ms duration) up to $50\mathrm{T}.$ The powder samples were in direct contact with a 1.5-K liquid-helium bath. Nonequilibrium effects, which became more severe as x decreased, were observed. For $x=0.50\mathrm{}$ the nonequilibrium effects are tentatively interpreted using the “inadequate heat flow scenario,” developed earlier in connection with the phonon bottleneck problem. The more severe nonequilibrium effects for $x=0.16\mathrm{}$ and $0.22$ are tentatively attributed to cross relaxation, and to crossings (more accurately, anticrossings) of energy levels, including those of excited states. For $x=0.16\mathrm{}$ (lowest $x),$ no MST’s were observed above $20\mathrm{T},$ which is attributed to a very slow spin relaxation for pairs, compared to a millisecond. A definitive interpretation of this and some other nonequilibrium effects is still lacking.