s = +1 s = -1

V

Fp ,

N

Fp

0en 1log ,1log 3id3id

N

FNkTF

0

0 dp

N

F

N

F

exid FFF

exexid pkTppp

N

F

N

Fp exidex

N

Fp exex

Phase diagrams of the 1/2-spin Ising model

H=0

similar to phase diagrams of fluids

EJERCICIO 9:

except for T < Tc

j

jj

jij

jiji mssssssN 1

21

si

(1944)

average field due to neighbouring spins

external field

effective field

averaged magnetisation independent of

dimensionality!

HmJzmsHJzme BiBi 2)(Energy of i-th spin:

Total energy: HmNNJzmE B 2

2

1 (factor 1/2 to avoid counting spin pairs twice)

(quasi) ideal:

rama metaestable

rama metaestable

región inestable

región inestable

DOMINIOS MAGNÉTICOS

JzmmHB arctanh

spinodal region

nucleation region

nucleation region

nucleation kinetics:

nucleation barrier, critical domain size, growth by motion of

grain boundaries

spinodal decomposition:

instability at q=0 (long-wavelength fluctuation)

ordered

(J>0)

)()( aai Hs

)()( aai Hs

)()( bbi Hs

)()( bbi Hs

EJERCICIO 10

study the nature of the transition for |H| small

B

B

EJERCICIO 11a

EJERCICIO 11b

Stirling's approx. xxxx log!log

Types of boundary conditions:

11 ssN

diverges but only at T=0

...1 1

l

N

lN JZ cosh22

1

1

A B A A B

B A A B A

B B A A B

A A B A B

B A B B A

NaF/AlF3 mixture

structure of -brass, CuZn

number of bonds probability of

one neighbour being A

probability of the other neighbour

being B

Now we make a mean-field approximation.

the Bragg-Williams approximation:

Entropic contribution. Favours mixing (disorder). Has minimum at x=1/2 (complete mixing) and maxima at x=0 and 1 (pure phases)

Energetic contribution. Favours segregation (order). Has minima at x=0 and 1 (pure phases) and maximum at x=1/2

AAN BB N

CONSTANTS HAVE BEEN DROPPED!