PX391 Nonlinearity, Chaos, Complexity
Landau theory of Ferromagnets
Consider system which follows the minima of the free energy F(M):
F = F0 + F1M + F2 M 2 + F3 M 3 + F4 M 4
which can always be written as
F = F0 '+ F2 '( M − M 0 ) 2 + F3 '( M − M 0 )3 + F4 '( M − M 0 ) 4
since both are general polynomials up to degree 4 ( M ' = M − M 0 is the reqd.
transformation).
Case 1: no applied field. For symmetry F3 = 0
We then have (dropping primes) F ( M ) = F0 + α (T − Tc ) M 2 + β M 4
Extrema given by
dF
= 2α (T − Tc ) M + 4 β M 3 = 2 M (α (T − Tc ) + 2 β M 2 )
dM
i.e. at M = 0 or M 2 =
α (Tc − T )
2β
But M is real so M = ±
α (Tc − T )
2β
is an extremum for T < Tc
Look for minima
d 2F
= 2α (T − Tc ) + 12β M 2
dM 2
M =0:
M =±
Min for T > Tc ; Max for T < Tc
α (Tc − T )
2β
d 2F
dM 2
= 2α (T − Tc ) + 12 β .
α (Tc − T )
= −4α (T − Tc )
2β
Min for T < Tc , Max for T > Tc
We have a pitchfork bifurcation at T = Tc - see plot of M (T )
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PX391 Nonlinearity, Chaos, Complexity
As we go from T > Tc to T < Tc the system “falls” into one of the potential wells
- which one is determined by fluctuations at T = Tc .
Case 2: Now consider applied B field - asymmetric now F3 = γ ≠ 0
dF
= 2α (T − Tc ) M + 3γ M 2 + 4β M 3
dM
extrema now given by
that is, at M = 0 ,
dF
= 0 = M {2α (T − Tc ) + 3γ M + 4β M 2 }
dM
M=
−3γ ± (9γ 2 − 4.2α (T − Tc ).4 β )
2.4 β
There are 2 real values of M when 9γ 2 > 32αβ (T − Tc )
Now write M as
M=
−3γ ± 3 γ 2 − γ c 2
8β
Look for minima:
d 2F
= 2α (T − Tc ) + 6γ M + 12β M 2
dM 2
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PX391 Nonlinearity, Chaos, Complexity
M = 0 is a min at T > Tc
the M ≠ 0 extrema is given by:
which gives:
2α (T − Tc ) + 3γ M + 4β M 2 = 0
(
d 2F
d 2F
2
=
3
γ
M
+
8
β
M
,
or
= M ±3 γ 2 − γ c 2
dM 2
dM 2
)
Then in addition to the M = 0 extremum we have:
For γ 2 > γ c 2
For γ 2 = γ c 2
For γ 2 < γ c 2
2 real M ≠ 0 roots, one max, one min
32αβ (T − Tc )
−3γ
and γ c 2 =
; this is at T > Tc
8β
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- M is imaginary, no max/min
M=
also at γ c 2 = 0
T = Tc
(a) is (-)ve root hence
d 2F
>0
dM 2
M=
−3γ ± 3γ
8β
i.e.
M =0
or
M=
−6γ
8β
(a)
(b)
-this is a min.
(b) is an inflexion.
Finally, for γ c 2 < 0 - we have 2 real M ≠ 0 roots, both minima, and M = 0 which
is a maximum. Graphically:
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PX391 Nonlinearity, Chaos, Complexity
Now fluctuations are unimportant.
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