Curie-Weiss Paramagnetism susceptibility

  • #1
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Homework Statement



(a) Show the curie-weiss behaviour.
(b) Estimate ##\lambda## and ##B_e## and exchange energy.[/B]
2011_B6_Q8.png


Homework Equations




The Attempt at a Solution



Part(a)
Since even when applied field is zero, ##B_{total} \neq 0## which gives rise to ##M\neq 0##. This is a fundamental property of ferromagnetism.
Consider the spin-1/2 ising model:
[tex]H = \sum\limits_{\langle i,j \rangle} J_{ij} \sigma_i \sigma_j - \sum\limits_i g \mu_B \sigma_i \cdot \vec B [/tex]
[tex]H_i = \left( g \mu_B B_{applied} - J \sum\limits_j \sigma_j \right)\sigma_i [/tex]
We represent the term in the brackets by an effective field ##B_e##:
[tex]H_i = \left( g \mu_B B_e \right)\sigma_i [/tex]
Average spin at site ##i## is given by
[tex]\langle \sigma \rangle = -\frac{1}{2} \tanh \left( \frac{\beta g \mu_B B_e}{2}\right) [/tex]
By the mean-field approach, we assume that ##\langle \sigma \rangle## is the same at all sites, giving:
[tex]g \mu_B B_{app} - J z \sigma_j = g \mu_B B_e [/tex]
where ##z## is the number of nearest neighbours, or coordination number.

This leads to a 'self-consistency' equation
[tex]\frac{1}{2}\tanh \left[ \frac{\beta}{2} \left( Jz\langle \sigma\rangle - g\mu_B B \right) \right] = \langle \sigma \rangle [/tex]
At zero appleid field, Curie temperature is given when gradient = 1, so
[tex] \frac{1}{2} \left( \frac{\beta}{2}Jz \right) = 1 [/tex]
[tex]T_c = \frac{Jz}{4 k_B} [/tex]

For the curie-weiss behaviour, we expand for small ##\langle \sigma \rangle##:
[tex] \langle \sigma\rangle = \frac{ \frac{\beta}{4}g \mu_B B_{app} }{frac{\beta}{4} Jz - 1} [/tex]
[tex] \langle \sigma\rangle = \frac{\frac{1}{4} \frac{g \mu_B B_{app}}{k_B}}{T-T_c}[/tex]
[tex]M = g n \mu_B \langle \sigma \rangle = \frac{\frac{1}{4} \frac{(g \mu_B)^2 n B_{app}}{k_B}}{T-T_c} [/tex]
[tex]\chi = \mu_0 \frac{\partial M}{\partial B_{app}} = \frac{1}{4} (g\mu_B)^2 \frac{n}{k_B} \frac{1}{T-T_C} [/tex]

Part(b)
I found the coordination number at ##T = 1024 K## and ##J=1## using
[tex]T_c = \frac{Jz}{4 k_B} [/tex]
It gave ##z = 5.7 \times 10^{-20}## which seems wrong as the number of neighbours should be a whole number..
 

Answers and Replies

  • #2
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bumpp on part (b) - non-integer number of neighbours!
 
  • #3
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bump on (b) - doesn't make sense to have non-integer number of neighbours
 
  • #4
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Bump on - (b) please
 
  • #5
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part (b) - number of neighbours?
 

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