Temperature in a nuclear spin 1/2 system?

In summary, the problem involves calculating the temperature at which 75% of the protons in a solid, with a spin of 1/2 and a nuclear g-factor of 5.59, have their magnetic dipole moment aligned parallel to a 1.0T magnetic field. Using the single particle partition function and setting the probability of 0.75 equal to e^(-beta*epsilon)/Z, the resulting temperature is a small negative value. However, this method does not take into account the number of protons and the concept of indistinguishability.
  • #1
Str1k3
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Temperature in a nuclear spin 1/2 system??

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?
 
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  • #2


can you show in detail how you did?
 

1. What is the temperature in a nuclear spin 1/2 system?

The temperature in a nuclear spin 1/2 system refers to the average kinetic energy of the particles within the system. It is measured in units of Kelvin (K) and is directly related to the random motion of the particles.

2. How does temperature affect a nuclear spin 1/2 system?

Temperature can affect a nuclear spin 1/2 system in several ways. It can increase the energy of the particles, causing them to move faster and generate more heat. It can also affect the alignment of the nuclear spins and the strength of the magnetic field within the system.

3. What is the relationship between temperature and entropy in a nuclear spin 1/2 system?

Entropy is a measure of the disorder or randomness within a system. In a nuclear spin 1/2 system, higher temperatures lead to increased entropy as the particles become more disordered and have higher levels of random motion.

4. How is temperature controlled in a nuclear spin 1/2 system?

In a nuclear spin 1/2 system, temperature can be controlled through various methods such as using external cooling sources like liquid nitrogen or by adjusting the magnetic field to manipulate the energy levels of the particles.

5. What is the significance of temperature in nuclear magnetic resonance (NMR) experiments?

Temperature plays a crucial role in NMR experiments as it affects the alignment and behavior of nuclear spins, which is the basis for NMR spectroscopy. Accurate temperature control is necessary for obtaining reliable and reproducible results in NMR experiments.

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