Okay, let's see. I need to find the partition function Z for a system of three spins S1, S2, and S3, each taking values +1 or -1. The energy is given by E = -J[S1S2 + S1S3 + S2S3]. The options are A to D. Hmm, partition function Z is the sum over all possible states of e^{-βE}, right? So first, I need to list all possible states of the three spins and calculate their respective energies, then compute the Boltzmann factors and sum them up.

Wait, each spin can be +1 or -1. So for three spins, there are 2^3 = 8 possible states. Let me list them out. Let's denote each state as (S1, S2, S3). The possible combinations are:

1. (+1, +1, +1)
2. (+1, +1, -1)
3. (+1, -1, +1)
4. (+1, -1, -1)
5. (-1, +1, +1)
6. (-1, +1, -1)
7. (-1, -1, +1)
8. (-1, -1, -1)

Now, for each state, I need to compute the energy E. The energy formula is E = -J(S1S2 + S1S3 + S2S3). Let's compute this for each state.

Starting with state 1: (+1, +1, +1)
E = -J[(+1)(+1) + (+1)(+1) + (+1)(+1)] = -J[1 + 1 + 1] = -3J
So the Boltzmann factor is e^{-βE} = e^{-β(-3J)} = e^{3Jβ}

State 2: (+1, +1, -1)
E = -J[(+1)(+1) + (+1)(-1) + (+1)(-1)] = -J[1 -1 -1] = -J(-1) = +J
So factor is e^{-βJ}

Wait, wait, let me check that again. S1S2 is 1*1=1, S1S3 is 1*(-1)=-1, S2S3 is 1*(-1)=-1. So sum is 1 -1 -1 = -1. Then E = -J*(-1) = +J. So e^{-βE} = e^{-βJ}

Similarly, state 3: (+1, -1, +1)
E = -J[(+1)(-1) + (+1)(+1) + (-1)(+1)] = -J[-1 +1 -1] = -J(-1) = +J. So same as state 2.

State 4: (+1, -1, -1)
E = -J[(+1)(-1) + (+1)(-1) + (-1)(-1)] = -J[-1 -1 +1] = -J(-1) = +J. So same as before.

Wait, but what about state 5? Let's check that as well, but maybe I can see a pattern. Let's do all states.

State 5: (-1, +1, +1)
E = -J[(-1)(+1) + (-1)(+1) + (+1)(+1)] = -J[-1 -1 +1] = -J(-1) = +J. Same as others.

State 6: (-1, +1, -1)
E = -J[(-1)(+1) + (-1)(-1) + (+1)(-1)] = -J[-1 +1 -1] = -J(-1) = +J.

State 7: (-1, -1, +1)
E = -J[(-1)(-1) + (-1)(+1) + (-1)(+1)] = -J[1 -1 -1] = -J(-1) = +J.

State 8: (-1, -1, -1)
E = -J[(-1)(-1) + (-1)(-1) + (-1)(-1)] = -J[1 +1 +1] = -3J. So factor e^{3Jβ}.

Wait, so looking at all the states, states 1 and 8 have E = -3J, so their factors are e^{3Jβ}. The other six states (2-7) each have E = +J, so their factors are e^{-βJ}.

So the partition function Z is the sum over all states: 2 states with e^{3Jβ} and 6 states with e^{-βJ}. Therefore, Z = 2e^{3Jβ} + 6e^{-βJ}. Wait, but looking at the options, option A is 2e^{3Jβ} + 6e^{-Jβ}, which matches. So the answer should be A.

Wait, but let me double-check. Let me recount the number of states. For E = -3J, only states where all spins are the same. There are two possibilities: all +1 or all -1. So that's two states. The rest six states have mixed spins. Each of those six has E = +J. So yes, the sum is 2e^{3Jβ} + 6e^{-βJ}, which is option A.

But wait, let me make sure I didn't make a mistake in calculating E for each state. Let's take state 2 again: (+1, +1, -1). S1S2 = 1, S1S3 = -1, S2S3 = -1. Sum is 1 -1 -1 = -1. So E = -J*(-1) = J. Correct. Similarly, state 3: (+1, -1, +1). S1S2 = -1, S1S3 = +1, S2S3 = -1. Sum is -1 +1 -1 = -1. E = J. Correct.

Another example: state 5: (-1, +1, +1). S1S2 = -1, S1S3 = -1, S2S3 = +1. Sum is -1 -1 +1 = -1. E = J. Correct.

So yes, all six mixed states have E = J. Therefore, their contributions are each e^{-βJ}, and there are six of them. The two homogeneous states (all +1 and all -1) have E = -3J, so their contributions are e^{3Jβ} each.

Therefore, Z = 2e^{3Jβ} + 6e^{-Jβ}, which is option A. But wait, looking at the options again:

A. Z= 2 e^(3Jβ)+6e^(-Jβ)

Yes, that matches. So the correct answer is A.
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