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Subsections
Energy of Ising system is
It can be written as
with molecular field
We substitute by its average value
and choose from consistency condition!
Let us write
The product:
Approximation: neglect . Result:
with
| |
(4) |
We separated energy into independent contributions from each spin!
Partition function:
Partition function for one spin
Free energy:
| |
(5) |
This depends on the magnetization M!
There are two ways to calculate M:
By definition, . Since only E_{i} depends on
, we have:
or, from (4)
| |
(6) |
This is a transcendent equation for M.
Free energy (5) depends on M. It should be minimal as
function of M:
or
This gives
| |
(7) |
The results (6) and (7) coincide! Our theory is
self-consistent.
We can rewrite equation (7) as
| |
(8) |
- 1.
- At large T, this is monotonic function:
This means one-phase regime
- 2.
- At low T, this is non-monotonic function:
From stability condition we see, that
this means phase separation
- 3.
- At critical temperature T_{c}, we have only one point, where
:
At this temperature
or, from (8)
kT=Jz
Phase diagram:
- Give same critical point
- Expansions near critical point coincide up to quadratic terms
- Give different results farther from critical point
Advantages of SCF:
- 1.
- We know how to extend it--just make a better approximation for
.
- 2.
- Works better farther from critical point
Next: Landau Theory
Up: Mean Field Approach
Previous: Flory Theory
© 1997
Boris Veytsman
and Michael Kotelyanskii
Tue Oct 14 22:58:59 EDT 1997