Thermodynamics with use of Zusammenstand and probability

[Zinggy]

Homework Statement

Three-state system. The nucleus of the nitrogen isotope 14N acts, in some ways, like a spinning, oblate sphere of positive charge. The nucleus has a spin of lft and an equatorial bulge; the latter produces an electric quadrupole moment. Consider such a nucleus to be spatially fixed but free to take on various orientations relative to an external inhomogenous electric field (whose direction at the nucleus we take to be the z-axis). The nucleus has three energy eigenstates, each with a definite value for the projection sz of the spin along the field direction. The spin orientations and the associated energies are the following: spin up (s_z = 1h), energy = £o; spin "sideways" (s_z = 0), energy = 0; spin down (s_z = -1h), energy = £o (again). Here £o denotes a small positive energy

h=Planks constant

a.)In thermal equilibrium at temperature T , what is the probability of finding the nucleus with spin up? In what limit would this be 1/3?
b.)Calculate the energy estimate (e) in terms of εo, T et cetera. Sketch (e) as a function of T
c.)What value does the estimate (s_z) have? Give a qualitative reason for your numerical result.

Homework Equations

KT^2 δ/δT ln(z) Where z=Zusmenmenstand = e^s(E_tot)

The Attempt at a Solution

a.) We attempted to solve the probability problem by using 1=P_up+P_side+P_down=1/z(e^s_zB/kt+e^-s_zb/kt+e^0b/kt
But we've only used this for magnetic moments in class before so we don't know how to translate to spin up probability.

b.)We're assuming energy estimate = expected energy, ∴ <E>=KT²∂/∂T ln(z)
Substituting in we get, <E>=KT²1/KT ⇒ <E> = T Meaning for T it is a linear relationship? We also don't know where ε₀ is coming from.
c.) <s_z> = 1 because the sum of all probabilities must equal 1?

 

[DrClaude]

I'm confused by what you wrote. Using +,0,- to designate spin-up, spin-"side", and spin-down, respectively, is the energy
$$
E_+ = \epsilon_0, E_0 = 0, E_- = \epsilon_0
$$
(as in the problem statement) or
$$
E_+ = -\epsilon_0, E_0 = 0, E_- = \epsilon_0
$$
(as in your attempt)?

Also, it would help if you wrote down explicitly the partition function (which is how we call in English the Zusmenmenstand).

Edit: What is also needed is the formula for expected (average) values.

 

[Zinggy]

I'm confused by what you wrote. Using +,0,- to designate spin-up, spin-"side", and spin-down, respectively, is the energy
$$
E_+ = \epsilon_0, E_0 = 0, E_- = \epsilon_0
$$
(as in the problem statement) or
$$
E_+ = -\epsilon_0, E_0 = 0, E_- = \epsilon_0
$$
(as in your attempt)?

Also, it would help if you wrote down explicitly the partition function (which is how we call in English the Zusmenmenstand).

Edit: What is also needed is the formula for expected (average) values.

Sorry if anything is confusing, I'm not use to posting in this format. Here is the question verbatim

My group and I are only assuming you have to use Zusmenmenstand because the lecture notes we have. Unfortunately this is all we're given.

 

[DrClaude]

Ok, that clears up my first question.

My group and I are only assuming you have to use Zusmenmenstand because the lecture notes we have. Unfortunately this is all we're given.

Yes, it is correct to use Z. But you must have seen the formulas for Z, for the probability of being in a given state, and for expected values?