Phase-space formulation of the nonlinear longitudinal relaxation of the magnetization
in quantum spin systems
Yuri P. Kalmykov,
1
William T. Coffey,
2
and Serguey V. Titov
2,3
1
Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 52, Avenue de Paul Alduy,
66860 Perpignan Cedex, France
2
Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
3
Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino,
141190, Russian Federation
Received 30 April 2007; revised manuscript received 18 June 2007; published 8 November 2007
Nonlinear longitudinal relaxation of a spin in a uniform external dc magnetic field is treated using a master
equation for the quasiprobability distribution function of spin orientations in the configuration space of polar
and azimuthal angles analogous to the Wigner phase space distribution for translational motion. The solution
of the corresponding classical problem of the rotational Brownian motion of a magnetic moment in an external
magnetic field essentially carries over to the quantum regime yielding in closed form the dependence of the
longitudinal spin relaxation on the spin size S as well as an expression for the integral relaxation time, which
in linear response reduces to that previously given by D. A. Garanin Phys. Rev. E 55, 2569 1997 using the
density matrix approach. The nonlinear relaxation is dominated by a single exponential having as time constant
the integral relaxation time. Thus a simple description in terms of a Bloch equation holds even for the nonlinear
response of a giant spin.
DOI: 10.1103/PhysRevE.76.051104 PACS numbers: 05.40.-a, 03.65.Yz, 76.20.+q
I. INTRODUCTION
Spin relaxation is fundamental in the physics and chem-
istry of condensed phases, e.g., on an atomic level, nuclear
magnetic and related spin resonance experiments probe the
time evolution of the elementary spins of nuclei, electrons,
muons, etc. 1,2. On a larger scale the time evolution of
magnetic molecular clusters exhibiting relatively large quan-
tum effects 3 with spins of order 15–25
B
is currently of
interest in the context of molecular magnets. Finally, on a
nanoscale level, we have magnetic fluids composed of single
domain ferromagnetic particles constituting a single giant
spin of magnitude 10
4
–10
5
B
in a colloidal suspension.
Here relaxation experiments detect 4,5 both the Arrhenius
or solid-state-like Néel mechanism 6 of relaxation of the
magnetization, which may overcome via thermal agitation
anisotropy, potential barriers inside the particle and the De-
bye or Brownian relaxation 7 due to physical rotation of
the suspended particles in the presence of an applied field
and the heat bath. Here quantum effects are expected to be
much smaller.
Spin relaxation experiments in nuclear magnetic or elec-
tron spin resonance are usually interpreted via the phenom-
enological Bloch 8 equations and their later modifications
1,2. They describe relaxation of an assembly of elementary
spins in a sample subjected to an external magnetic field and
coupled to a heat bath. These simple linear equations of mo-
tion for the nuclear magnetization were originally proposed
on phenomenological grounds. The main assumption is that
the effects of the heat bath can be described by two time
constants, the so-called relaxation times. They provide a sub-
stantially correct 1 quantitative description for liquid
samples. Microscopic theories of the relaxation in quantum
spin systems have been developed by Bloembergen, Purcell,
and Pound 9, and other authors see, e.g., 10–12.
Proceeding to larger scales, in magnetic molecular clus-
ters comprising a few spins the relaxation behavior as a func-
tion of spin is of paramount importance as strong quantum
effects are expected to manifest themselves as the spin de-
creases, while in single domain giant spin nanoparticles
suspended in a fluid carrier the relaxation is usually assumed
to be classical. Thus the Néel mechanism of the magnetiza-
tion reversal 6 occurring inside the ferromagnetic particles
is described by a classical Langevin equation for the time
evolution of the magnetization as adapted to magnetic mo-
ments by Brown 13,14 while the Debye theory 7 of di-
electric relaxation of polar molecules is used to describe the
relaxation by physical rotation of the suspended particles
4,15. In the description of the Néel mechanism 13,14, the
Langevin equation is the phenomenological Landau-Lifshitz
16 or Gilbert equation 17 for the magnetization Mt
used originally to study the motion of a domain wall aug-
mented by random magnetic fields due to the heat bath 18.
This equation leads 4,13,14 to the Fokker-Planck equation
in the space of polar angles for the surface distribution of the
magnetic moment orientations. For simplicity it is commonly
assumed that the solid state and Brownian relaxation mecha-
nisms may be treated independently. A discussion of the limi-
tations of that assumption has been given in Refs. 4,15.
Moreover, memory effects are ignored, however, they may
also be included as in 19–21. It is immediately apparent
that treating magnetization relaxation via the Landau-
Lifshitz equation augmented by stochastic terms is essen-
tially just another problem concerning the rotational Brown-
ian motion under the combined effect of an external field and
the internal magnetocrystalline anisotropy. Many particular
cases have been treated 5,13,14 by using the Kramers es-
cape rate 22 as adapted to magnetic relaxation 14,23,24 in
order to calculate the reversal time of the magnetization over
the internal potential barrier. Moreover, the results have been
PHYSICAL REVIEW E 76, 051104 2007
1539-3755/2007/765/05110411 ©2007 The American Physical Society 051104-1