Magnetic susceptibility integral trouble

In summary, the conversation is about an algebra question related to the susceptibility equation. The speaker is trying to show that it can be expressed as ##V\chi_0=\mu_0\beta(\langle\vec{m_i}^2\rangle-\langle\vec{m_i}\rangle^2)## but they have made a mistake in their calculations. They provide an explanation of their calculations but ask for help in finding their mistake.
  • #1
maximus123
50
0
Hello,

This is really more of an algebra question. Here is the main integral I am using for the susceptibility

[itex] V\chi_0=\frac{d\langle \vec{m_i}\rangle}{d\vec{H_i}}=\frac{1}{Z}\int\vec{m_i}\frac{d}{d\vec{H_i}}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m-\frac{1}{Z^2}\frac{dZ}{d\vec{H_i}}\int\vec{m_i}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m[/itex]
Z is the partition function, m is the magnetic moment, H is the field, beta is just a constant in these calculations.

I have it that

[itex] \langle\vec{m_i}\rangle=\frac{1}{\mu_0\beta}\frac{1}{Z}\frac{dZ}{d\vec{H_i}}[/itex]

And ultimately I need to show that the susceptibility equation can be expressed as


[itex] V\chi_0=\mu_0\beta(\langle\vec{m_i}^2\rangle-\langle\vec{m_i}\rangle^2)[/itex]
I can get pretty close. If we look at the first term in the susceptibility equation at the top of the post


[itex]
\frac{1}{Z}\int\vec{m_i}\frac{d}{d\vec{H_i}}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m\\
=\frac{1}{Z}\int\mu_0\beta\vec{m_i^2}\,e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m\\
=\mu_0\beta\frac{1}{Z}\int\vec{m_i^2}\,e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m\\
=\mu_0^2\beta^2\langle\vec{m_i}^2\rangle\\

[/itex]
The last line simplification was achieved using the relation given at the top of the post (second equation down).
The second term then

[itex]
\frac{1}{Z^2}\frac{dZ}{d\vec{H_i}}\int\vec{m_i}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m\\
=\frac{1}{Z}\frac{1}{Z}\frac{dZ}{d\vec{H_i}}\int\vec{m_i}e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m\\
=\frac{1}{Z}\frac{1}{Z}\frac{dZ}{d\vec{H_i}}\frac{dZ}{d\vec{H_i}}\frac{1}{\mu_0\beta}\\
=\frac{1}{Z}\frac{dZ}{d\vec{H_i}}\langle\vec{m_i}\rangle\\
=\langle\vec{m_i}\rangle^2\mu_0\beta
[/itex]
Giving a final result for the susceptibility as

[itex]
V\chi_0=\mu_0^2\beta^2\langle\vec{m_i}^2\rangle-\langle\vec{m_i}\rangle^2\mu_0\beta
[/itex]

So basically I have an extra factor of [itex]\mu_0\beta[/itex] in the first term so I can't factorize it out to achieve the result I am supposed to. My apologies for the long winded nature of the post but if you could point out where I've made a mistake it would be greatly appreciated.
 
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  • #2
maximus123 said:
$$=\mu_0\beta\frac{1}{Z}\int\vec{m_i^2}\,e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m\\
=\mu_0^2\beta^2\langle\vec{m_i}^2\rangle\\
$$
The last line simplification was achieved using the relation given at the top of the post (second equation down).​

The last line line is incorrect. You should be able to interpret the expression ##\frac{1}{Z} \int\vec{m_i^2}\,e^{\mu_0\vec{m}\cdot\vec{H}\beta}d^2m##.
 

Related to Magnetic susceptibility integral trouble

1. What is magnetic susceptibility integral trouble?

Magnetic susceptibility integral trouble is a phenomenon in which the integration of the magnetic susceptibility over a certain region results in a value that is not physically meaningful. This can occur due to various factors such as incorrect assumptions, limitations of the measurement equipment, or errors in the measurement process.

2. How does magnetic susceptibility integral trouble affect scientific research?

Magnetic susceptibility integral trouble can significantly impact the accuracy and reliability of scientific research that involves the measurement of magnetic susceptibility. It can lead to erroneous conclusions and hinder the progress of understanding magnetic properties of materials.

3. What are some common causes of magnetic susceptibility integral trouble?

Some common causes of magnetic susceptibility integral trouble include instrumental drift, incorrect calibration, sample inhomogeneity, and improper sample preparation. These factors can introduce errors in the measurement process and result in inaccurate values for the magnetic susceptibility integral.

4. Can magnetic susceptibility integral trouble be avoided?

While it is not always possible to completely eliminate magnetic susceptibility integral trouble, there are steps that can be taken to minimize its occurrence. These include carefully calibrating the measurement equipment, properly preparing the sample, and conducting multiple measurements to reduce the effects of instrumental drift.

5. How can magnetic susceptibility integral trouble be corrected?

If magnetic susceptibility integral trouble is identified, it can be corrected by carefully reviewing and analyzing the measurement data, identifying the source of the error, and taking appropriate corrective measures. This may involve repeating the measurement, recalibrating the equipment, or adjusting the experimental parameters.

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