Heisenburg Ferromagnetic Model

In summary, the conversation discusses a question involving an equation and a substitution. The goal is to show that the substitution leads to a dispersion relation, which is expressed as an equation involving various variables. The attempt at a solution involves substituting the given equation into the original equation and solving, but this raises some confusion about the definition of certain values and the presence of an 'i' factor. The marking scheme is discussed as well.
  • #1
shedrick94
30
0

Homework Statement


There are several parts to this question, however I could complete these parts. It is just an equation used in the prior part to the question that is need to solve this:

If we define \begin{equation} \sigma_{n}^{-}=\sigma_{n}^{x}+i\sigma_{n}^{y} \end{equation} and with the wavelike substitution \begin{equation} \sigma_{n}^{-}=Ae^{i(kna-wt)} \end{equation} show that one obtains the dispersion relation: \begin{equation} \hbar\omega=2JS[1-cos(ka)] \end{equation}

Homework Equations


The substitution referred to is into this formula:

\begin{equation} \hbar \frac{d\mathbf{\sigma}_{n}}{dt}=JS\hat{z}\times (\sigma_{n-1}-2\sigma_{n}+\sigma_{n+1}) \end{equation} and we also know: \begin{equation} \sigma_{n}=(\sigma_{n}^{x},\sigma_{n}^{y},0) \end{equation}

The Attempt at a Solution



I thought it would simply be that we substitute in the \begin{equation} \sigma_{n}^{-}=Ae^{i(kna-wt)} \end{equation} into the equation and solve. However, that wouldn't make sense to me as the \begin{equation} \sigma_{n}^{-} \end{equation} value does not seem to be defined as a vector but the right hand side of the equation is. Even so when we do this the differential on the LHS would return a factor of 'i' and I'm not sure where that would disappear to either. It seems to me that in the marking scheme it has been fudged to make the answer correct. If not then I think I just cannot follow what has been done. I have attached the mark scheme bellow as an image.

XgdVTB7.png
 
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  • #2
A is a vector in the substitution. You are multiplying the derivative by i so when you take the derivative of the complex exponential the i(s) cancel.
 
Last edited:

1. What is the Heisenburg Ferromagnetic Model?

The Heisenburg Ferromagnetic Model is a mathematical model used to describe the behavior of magnetic materials at the atomic level. It was proposed by physicist Werner Heisenburg in 1928 and is based on the principles of quantum mechanics.

2. How does the Heisenburg Ferromagnetic Model work?

The model works by considering the interactions between individual magnetic moments, or spins, within a material. These spins can align in parallel or anti-parallel directions, resulting in either a ferromagnetic or anti-ferromagnetic state. The model takes into account the energy associated with these interactions to determine the magnetic properties of a material.

3. What are some applications of the Heisenburg Ferromagnetic Model?

The model has been used to study and predict the behavior of various magnetic materials, such as iron, nickel, and cobalt. It has also been applied in the field of spintronics, which explores the use of electron spin as a means of storing and transmitting information in electronic devices.

4. What are the limitations of the Heisenburg Ferromagnetic Model?

One of the main limitations of the model is that it does not take into account the effects of temperature and thermal fluctuations on the magnetic properties of a material. This can make it less accurate in predicting the behavior of materials at higher temperatures. Additionally, the model assumes that all spins within a material are interacting with each other, which may not always be the case in real materials.

5. How does the Heisenburg Ferromagnetic Model relate to other models of magnetism?

The Heisenburg Ferromagnetic Model is one of the most well-known and widely used models in the field of magnetism. It is often compared and contrasted with other models, such as the Ising model and the Stoner model, to better understand the behavior of magnetic materials. Each model has its own strengths and limitations, and they are often used in conjunction with each other to provide a more comprehensive understanding of magnetism.

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