Str1k3
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Temperature in a nuclear spin 1/2 system??
A solid at temperature T contains 10^20 protons which have a spin I = 1/2 and a nuclear g-factor of 5.59. Calculate the temperature such that 75% of the protons have their magnetic dipole moment aligned parallel to the applied magnetic field that has a magnitude of 1.0T. The nuclear magneton is 5.05X10^-27 Am^2.
Single particle partition function Z = [tex]\sum e^{-\beta\epsilon}[/tex]
[tex]\epsilon = -mg\mu B[/tex] where m = -I, -I+1,..., I-1, I
[tex]\beta = \frac{1}{k_{B}T}[/tex]
I have tried using the partition function by setting the probability of 0.75 = e^(-beta*epsilon)/Z and then just solving for the temperature, but this gives a very small negative answer. Also, using this way doesn't take the number of protons into account which surely isn't right? I'm not sure if i should be worrying about indistinguishability either?
Homework Statement
A solid at temperature T contains 10^20 protons which have a spin I = 1/2 and a nuclear g-factor of 5.59. Calculate the temperature such that 75% of the protons have their magnetic dipole moment aligned parallel to the applied magnetic field that has a magnitude of 1.0T. The nuclear magneton is 5.05X10^-27 Am^2.
Homework Equations
Single particle partition function Z = [tex]\sum e^{-\beta\epsilon}[/tex]
[tex]\epsilon = -mg\mu B[/tex] where m = -I, -I+1,..., I-1, I
[tex]\beta = \frac{1}{k_{B}T}[/tex]
The Attempt at a Solution
I have tried using the partition function by setting the probability of 0.75 = e^(-beta*epsilon)/Z and then just solving for the temperature, but this gives a very small negative answer. Also, using this way doesn't take the number of protons into account which surely isn't right? I'm not sure if i should be worrying about indistinguishability either?