
#1
Mar312, 03:25 AM

P: 419

This is from Kardar's Statistical Physics of Particles, p.123, question 8.
1. The problem statement, all variables and given/known data Curie suceptibility: consider N noninteracting quantized spins in a magnetic field [tex]\vec{B}=B\hat{z}[/tex] and at temperature T. The work done by the field is given by BM_{z}, with a magnetization [tex]M_z=\mu\sum_{i=1}^{N}m_i[/tex]. For each spin, m_{i} takes only the 2s+1 values s, s+1, ..., s1, s. [Parts ac omitted.] d) Show that [tex]C_B C_M = c \frac{B^2}{T^2}[/tex], where C_{B} and C_{M} are heat capacities at constant B and M, respectively. 2. Relevant equations I'm not totally sure the following equations are absolutely correct, since I've tried using them to no avail. Some of them are the ones I derived in parts ac and others are ones from Kardar. So I'll separate the ones I found from the ones in Kardar, and for the ones in Kardar, I'll include where Kardar puts them if you want to check if they apply. Kardar equations: [tex]G=G=k_B T\mathrm{ln}Z [/tex](Kardar 4.88) [tex]M = \frac{\partial G}{\partial B}[/tex] (Kardar 4.97) [tex]H = \frac{\partial \mathrm{ln}Z}{\partial \beta }[/tex] where β=k_{B}T (Kardar 4.89) [tex]C_M=\frac{\partial H}{\partial T}[/tex] (given in Kardar just below 4.89) [tex]C_B=B\frac{\partial M}{\partial T}[/tex] (given in Kardar just below 4.98) My equations: [tex]Z=\left (\frac{1e^{\frac{(2s+1)B\mu}{k_B T}}}{1e^{\frac{B\mu}{k_B T}}} \right )^N[/tex] [tex]G = Nk_B T\mathrm{ln}\frac{1e^{\frac{(2s+1)B\mu}{k_B T}}}{1e^{\frac{B\mu}{k_B T}}}[/tex] 3. The attempt at a solution I've written down C_{B} and C_{M} but I have exponentials hanging around which don't cancel and therefore don't leave me with B^2/T^2 proportionality. I suspect there's something wrong with the results of my ac, which are all in "my equations." Thanks. 


Register to reply 
Related Discussions  
Alignment of micro dipoles in magnetic material in external magnetic field.  Classical Physics  5  
Torque and Force due to an external magnetic field  Classical Physics  2  
Magnetic dipole in an external magnetic field  General Physics  3  
Magnetic Field inside and external to a wire  Introductory Physics Homework  1  
how does the electron spins affects the magnetic field??  Atomic, Solid State, Comp. Physics  3 