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Multiplicity function
Adrian Down
August 31, 2005
1 Review: system of N spins of
We assume that N is even. Usually it is so large that it doesnโt matter.
2 n^ possible arrangements
(N + 1) possible values of M : N m, (N โ 2)m,... , โN m
g = number of ways of choosing M
n โ=
N 2
โ, n โ=
N 2
โ s
Spin excess is defined n โ โn โ= 2s. Note that we put the factor of 2 in for
convenience.
Magnetic potential energy: U = โM B = โ 2 smB. Energy is greatest for
magnetic moments anti-parallel to the field.
2 Probability of โ or โ
2.1 Probability function
Let Pโ be the probability of spin โ.
Likewise for Pโ
Must have
Pโ + Pโ = 1
For B = 0, there is no difference between the states, so
Pโ = Pโ =
For B > 0, energy is lower for spins pointing up.
Pโ > Pโ
Probability of finding a particular configuration with 2s excess spins is
P
nโ โ P^
nโ โ =^ P^
(N 2 +s) โ P^
(N 2 โs) โ
Probability of finding the value M = (2s)m is
P (N, s) = g(N, s)P
(N 2 +s) โ P^
(N 2 โs) โ
g(N, s) is the multiplicity function, the number of arrangements of N spins
to achieve spin excess of 2S. The other part is the probability of any one
configuration.
2.2 Example: N = 6, 2 s = 4
Six ways of achieving this value of s:
Thus g(6, 2) = 6
2.3 Evaluating g(N, s)
In how many ways can we distribute nโ = N 2 + s spins over N sites? Just
pick out the spins.
1 st^ can be put on any of the N sites.
2
nd can only be on one of the remaining (N โ 1) sites.
3 rd^ can only be on one of the remaining (N โ 2) sites.
nth^ can only be on one of the remaining (N โ nโ + 1) sites.
Want to simplify
g(N, s) =
N!
(N
2 +^ s
(N
2 โ^ s
Best way is to take the natural log of both sides.
ln g(N, s) = ln N! โ ln
N
! โ ln
N
โ s
Now we massage the factors on the right.
( N
2
N
N
N
N
N
N
ln
N
! = ln
N
โ^ s
k=
ln
N
Similarly,
ln
N
โ s
! = ln
N
โ^ s
k=
ln
N
โ k + 1
Hence,
ln
N
! + ln
N
โ s
! = 2 ln
N
โ^ s
k=
ln
N 2 +^ k N 2 โ^ k^ + 1
Divide by N 2
in the last summation.
Use the assumption that |k| โค |s| N to expand the logarithms.
โ^ s
k=
ln
2 k N 1 โ (^2) Nk + (^) N^2
โ^ s
k=
ln
2 k
N
โ^ s
k=
ln
2 k
N
N
โ^ s
k=
2 k
N
2 k
N
N
N
โ^ s
k=
k โ
N
s^2
2
2 s^2
N
ln g(N, s) = ln N! โ 2 ln
N
2 s^2
N
Take exponentials of both sides.
g(N, s) =
N!
N 2!
e
โ 2 s
2 N
Another very important result is
g(N, s) = g(N, 0)e
โ 2 s
2 N
where
g(N, 0) =
N!
N 2!
g(N, s) falls to
1 e of its peak value when^ s^ =^
N 2
2
Note that we assumed |s| N. This means that the Gaussian fails out in
the wings. The approximation is only good near the peak. For most systems
we care about, this is an amazingly good approximation.
For N โผ 1022 (โผ 1 cm^3 of atoms in a solid) width of the peak โผ 1011 ,
but the fractional width โผ 10 โ^11.
