Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 239
– J – J + J + J
Figure 5.3: The four possible microstates of the N = 2 Ising chain.
and one spin down with energy +J (see Figure 5.3). Thus Z2 is given by
Z2 = 2e βJ +2e −βJ = 4coshβJ. (5.37)
In the same way we can enumerate the eight microstates for N = 3. We find that
Z3 = 2e 2βJ +4+2e −2βJ = 2(e βJ +e −βJ ) 2
(5.38a)
= (e βJ +e −βJ )Z2 = (2coshβJ)Z2. (5.38b)
The relation (5.38b) between Z3 and Z2 suggests a general relation between ZN and ZN−1:
ZN = (2coshβJ)ZN−1 = 2 2coshβJ N−1 . (5.39)
We can derive the recursion relation (5.39) directly by writing ZN for the Ising chain in the
form
ZN =
···
e βJ N−1 i=1 sisi+1 . (5.40)
s1=±1
sN=±1
The sum over the two possible states for each spin yields 2N microstates. To understand the
meaning of the sums in (5.40), we write (5.40) for N = 3:
Z3 =
e βJs1s2+βJs2s3 . (5.41)
s1=±1 s2=±1 s3=±1
The sum over s3 can be done independently of s1 and s2, and we have
Z3 =
s1=±1 s2=±1
=
s1=±1 s2=±1
e βJs1s2 e βJs2 +e −βJs2
e βJs1s2 2coshβJs2 = 2coshβJ
s1=±1 s2=±1
(5.42a)
e βJs1s2 . (5.42b)
We haveused the fact that the cosh function is evenand hence coshβJs2 = coshβJ, independently
of the sign of s2. The sum over s1 and s2 in (5.42b) is Z2. Thus Z3 is given by
in agreement with (5.38b).
Z3 = (2coshβJ)Z2, (5.43)
The analysis of (5.40) for ZN proceeds similarly. We note that spin sN occurs only once in
the exponential, and we have, independently of the value of sN−1,
sN=±1
e βJsN−1sN = 2coshβJ. (5.44)