8/3/2019 Jack A. Tuszynski- NL2672: Hysteresis
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NL2672 Hysteresis 1
NL2672 Hysteresis
The word hysteresis is derived from the Greek where it meant
shortcoming and in the physical context it
represents a retardation effect when the forces acting upon a
body are changed. In particular, in magnetism this
represents a lagging in the values of the net magnetization in a
magnetic material due to a changing magnetizing
field. In general, hysteresis signifies the historydependence of
physical quantities in systems undergoing a change
in external condition. The term is most commonly applied to
magnetic materials but not exclusively. This lag isrepeatable: the
shape of the loop after the first cycle is roughly the same as it
is after many cycles. There is a class of
metals called shape memory alloys that can be bent or stretched
plastically over large distances back and forth many
times without hardening.
Consider a ferromagnetic material which is originally
unmagnetized. As the external magnetic field (H) is
increased, the induced magnetization (M) also increases. The
induced magnetization eventually saturates. Now, if
the external field is reduced, the induced magnetization also is
reduced, but it does not follow the original curve.
Instead, the material retains a certain permanent magnetization
called the remanent magnetization M r when H=0.
The remanent magnetization is the permanent magnetization that
remains after the external field is removed. If the
external field is reduced more, the remanent magnetization will
eventually be removed. The external field applied in
the opposite direction for which the remanent magnetization goes
to zero is termed the coercivity Hc. The product of
Mr and Hc is termed the strength of the magnet. As the external
field continues to reverse, permanent magnetization
of the opposite sign is created in the magnet. A similar curve
is traced for the negative direction with saturation,
remanent magnetization and coercivity. The hysteresis curve then
retraces the previous points as the field cycles.
Note that the area under the hysteresis loop corresponds to the
work done on the system by an external field that reorients
the magnetization in a single cycle.
In a ferroelectric, when an electric field is applied to a
ferroelectric crystal, the domains that are favorably
oriented with respect to this field grow in size at the expense
of those that are misaligned. In addition, favorably
oriented domains may nucleate and grow until the whole crystal
becomes one domain. The relation between the
resulting polarization P and the electric field E is described
by a hysteresis loop in analogy to the relationship
between M and H for ferromagnets.
Suppose now in general that the system under consideration can
be described by a macroscopic order parameter
. Under the influence of an external field coupled linearly to ,
the state of the system is determined by an equationof state which
expresses a minimum condition of the associated thermodynamic
potential V. Assuming the presence of
at least one control parameter leads directly to the problem of
catastrophes, an area investigated by Ren Thom. The
cusp catastrophe is described by the potential
+= 242
1
4)( aV (1)
where is the control parameter. This potential describes second
order phase transitions both in the absence ( = 0) andin the
presence of external fields ( 0) as proposed by Landau. The
butterfly catastrophe, on the other hand, employs
a
+++= 23462346
1)(
cbaV (2)
and has been used to model first order phase transitions.
To illustrate the related phenomenon of hysteresis we first
investigate the bifurcation effect and minimize
equation (1) which yields the equation of state
(3) =+ a3
A transition between a single stable solution for a > 0 and a
bistable situation takes place when a < 0. However, a new
feature is the phenomenon of external-field induced hysteresis
and metastability. Stability corresponds to a solution for
which 2V/2 > 0 and, if more than one solution of the equation
of state exists that is stable, we call the higher energysolutions
metastable. Fig. 1 shows the difference in the response of the
order parameter to the application of an external
field. In unistable situations, as a function of is a smooth
single-valued function. Multistability results in multi-
