# Spins in an external magnetic field

• Jolb
In summary, an external magnetic field is a magnetic field created outside of a material or object, typically by a magnet or electric current. Spin, a quantum property of a particle, causes it to behave like a tiny magnet and align with or against an external magnetic field. This alignment can result in different energy levels for particles, leading to phenomena like magnetic resonance and spin precession. In an external magnetic field, particles can have a spin up or spin down state, which can be used to manipulate and control their behavior. The behavior of spins in an external magnetic field is crucial in technologies like MRI machines, magnetic storage devices, and quantum computing, and is studied extensively in research.
Jolb
This is from Kardar's Statistical Physics of Particles, p.123, question 8.

## Homework Statement

Curie suceptibility: consider N non-interacting quantized spins in a magnetic field $$\vec{B}=B\hat{z}$$ and at temperature T. The work done by the field is given by BMz, with a magnetization
$$M_z=\mu\sum_{i=1}^{N}m_i$$. For each spin, mi takes only the 2s+1 values -s, -s+1, ..., s-1, s.

[Parts a-c omitted.]

d) Show that
$$C_B -C_M = c \frac{B^2}{T^2}$$, where CB and CM are heat capacities at constant B and M, respectively.

## Homework 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 a-c 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:
$$G=G=-k_B T\mathrm{ln}Z$$(Kardar 4.88)
$$M = -\frac{\partial G}{\partial B}$$ (Kardar 4.97)
$$H = -\frac{\partial \mathrm{ln}Z}{\partial \beta }$$ where β=kBT (Kardar 4.89)
$$C_M=\frac{\partial H}{\partial T}$$ (given in Kardar just below 4.89)
$$C_B=-B\frac{\partial M}{\partial T}$$ (given in Kardar just below 4.98)

My equations:
$$Z=\left (\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}} \right )^N$$
$$G = -Nk_B T\mathrm{ln}\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}}$$

## The Attempt at a Solution

I've written down CB and CM 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 a-c, which are all in "my equations."Thanks.

Dear forum post author,

Thank you for your question. I have taken a look at your attempt and I believe I have found the issue. In your equations for Z and G, you are using the partition function and the Gibbs free energy for a single spin, rather than for N spins. This is why you are getting exponentials that do not cancel out.

To fix this, you need to use the partition function for N spins and the Gibbs free energy for N spins. These can be found by raising the partition function and Gibbs free energy for a single spin to the power of N. This will give you the correct expressions for CB and CM.

I hope this helps. Let me know if you have any further questions.

## What is an external magnetic field?

An external magnetic field is a magnetic field that is created outside of a material or object. It can be created by a magnet or by an electric current.

## What is spin in relation to an external magnetic field?

Spin is a quantum property of a particle that describes its intrinsic angular momentum. In the presence of an external magnetic field, the spin of a particle causes it to behave like a tiny magnet, aligning with the field or against it.

## How does an external magnetic field affect spins?

An external magnetic field exerts a force on the spins of particles, causing them to align with the field. This alignment can result in different energy levels for the particles, leading to various phenomena such as magnetic resonance and spin precession.

## What is the difference between spin up and spin down in an external magnetic field?

In an external magnetic field, the spin of a particle can either align with the field (spin up) or against the field (spin down). This results in different energy levels for the particle and can be used to manipulate and control the spin state of the particle.

## How is the behavior of spins in an external magnetic field used in research and technology?

The behavior of spins in an external magnetic field is the basis for many important technologies such as MRI machines, magnetic storage devices, and quantum computing. It is also studied extensively in research to understand the fundamental properties of particles and to develop new technologies.

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