MAGNETISM ELECTRON TRANSPORT MAGNETORESISTIVE LANTHANUM CALCIUM MANGANITE

60 Chapter 3
Arrott plot), will then have only two curves: one branch for the T < T C data
and another for T > T C .
The magnetization near T C is predicted in the SCR theory, described i n
4/3 4/3 1/2
section 3.2.2.2.3, to behave as M = (TC -T ) [84, 91] which reducecs to β = 1
for T near T C .
Table 3-1 Theoretical 3-dimentional critical exponents for
different models and selected experimental values [92, 93].
β γ δ
Ising .33 1.24 4.8
Heisenberg .36 1.39 4.8
Mean Field .5 1 3
ZrZn 2 .50(3) 1.02(5) 3.1(3)
Fe, Ni, YIG .37(2) 1.2(2) 4(1)
3.2.2.2.5 Landau mean field theory
Near the critical temperature T C the molecular field, or mean field model
(section 3.2.2.2.5) predicts mean field critical exponents (Table 3-1). The
Landau theory of continuous, second order phase transitions (excluding
fluctuations) arrives at the same mean field result. Here the free energy is
expanded in a Taylor series of the order parameter (M in the case of
ferromagnetism). Due to the symmetry of the order parameter, only even
powers of M are nonzero. The first few terms are [94]: G = G 0 + a(T - T C )M 2 +
bM 4 - HM. At a given H and T, M can be found by minimizing the free energy
G. The general solution is H/M = 2a(T - T C ) + 4bM 2 . Below T C the saturation
magnetization (H = 0) is found to be M 2 = (T C - T)a/2b, giving the critical
exponent β = 1/2. Above T C in a field, M is small so the bM 4 term can be
ignored. This gives a susceptibility χ = M/H = (T - T C ) -1 /2a, and a critical