PHY3703
ASSIGNMENT 3
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA
2026
Page 1 of 12
, SOLUTION:
We start from the mean-field theory result for the energy per spin at 𝐻 = 0, given by
equation (5.120):
𝐸 1 𝑞𝐽𝑚 2
= − 𝐽𝑞 [tanh ( )] .
𝑁 2 𝑘𝑇
However, the mean-field equation for the magnetization is
𝑞𝐽𝑚
𝑚 = tanh ( ),
𝑘𝑇
which holds for all 𝑇 at 𝐻 = 0. Substituting this directly into the energy expression
simplifies it to
𝐸 1
= − 𝐽𝑞 𝑚2 .
𝑁 2
This is equation (5.119) in the text, and it is valid for all 𝑇 when 𝐻 = 0. For 𝑇 > 𝑇𝑐 , we
have 𝑚 = 0, so 𝐸 = 0 and thus 𝐶 = 0. For 𝑇 < 𝑇𝑐 , 𝑚 is nonzero, and we need its behavior
near 𝑇𝑐 .
We are told that for 𝑇 ≲ 𝑇𝑐 ,
𝑇𝑐 − 𝑇
𝑚2 ≈ 3 .
𝑇𝑐
This result comes from expanding the mean-field equation near 𝑇𝑐 . Let us verify it
explicitly. The mean-field equation is
𝑇𝑐
𝑚 = tanh ( 𝑚) ,
𝑇
Page 2 of 12
ASSIGNMENT 3
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA
2026
Page 1 of 12
, SOLUTION:
We start from the mean-field theory result for the energy per spin at 𝐻 = 0, given by
equation (5.120):
𝐸 1 𝑞𝐽𝑚 2
= − 𝐽𝑞 [tanh ( )] .
𝑁 2 𝑘𝑇
However, the mean-field equation for the magnetization is
𝑞𝐽𝑚
𝑚 = tanh ( ),
𝑘𝑇
which holds for all 𝑇 at 𝐻 = 0. Substituting this directly into the energy expression
simplifies it to
𝐸 1
= − 𝐽𝑞 𝑚2 .
𝑁 2
This is equation (5.119) in the text, and it is valid for all 𝑇 when 𝐻 = 0. For 𝑇 > 𝑇𝑐 , we
have 𝑚 = 0, so 𝐸 = 0 and thus 𝐶 = 0. For 𝑇 < 𝑇𝑐 , 𝑚 is nonzero, and we need its behavior
near 𝑇𝑐 .
We are told that for 𝑇 ≲ 𝑇𝑐 ,
𝑇𝑐 − 𝑇
𝑚2 ≈ 3 .
𝑇𝑐
This result comes from expanding the mean-field equation near 𝑇𝑐 . Let us verify it
explicitly. The mean-field equation is
𝑇𝑐
𝑚 = tanh ( 𝑚) ,
𝑇
Page 2 of 12
