Exploring Spin Magnetization in Non-Ideal Systems

In summary, the Hamiltonian of a system with N spins arranged in one dimension can be described by the energy function E(m), where m is the spin magnetization. The minimum energy configurations are determined by the direction of the spins and the external magnetic field.
  • #1
Marioweee
18
5
Homework Statement
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Relevant Equations
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We have a set of N spins arranged in one dimension that can take the values $$s_i=\pm 1$$. The Hamiltonian of the system is:
$$H=-\frac{J}{2N}\sum_{i \neq j}^{N} s_i s_j -B\sum_{i=1}^{N}s_i.$$
where $$J>0$$, B is an external magnetic field, and the first sum runs through all the values of i and j between 1 and N different from each other.
To analyze the behavior of the system, the most interisting observable is spin magnetization, whose definition is:
$$m\equiv \frac {1}{N}\sum_{i}^{N} si, \; \; m\in [-1,1].$$
Deduces the E(m) energy of a configuration $${s_i}_{i=1}^N$$ as a function of its magnetization by
spin. Find the minimum energy configurations.
Well, the truth is that I do not know very well how to answer the question that is asked. I have already solved the problem in the case that there is no interaction between spins. The problem is that I am not familiar with non-ideal Hamiltonians, that is, that there is interaction between the particles that make up the system. In addition, I am puzzled by the fact that I am asked to obtain the energy as a function of spin magnetization, since this leads me to think that it is not necessary to determine the partition function to solve this question. Thank you very much for reading and for the help.
 
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  • #2
The energy of a configuration $\{s_i\}_{i=1}^N$ as a function of its magnetization can be written as:$$E(m)=-\frac{J}{2N}\sum_{i \neq j}^{N} s_i s_j -B N m.$$The minimum energy configurations are those which maximize the spin magnetization $m$. This means that for $B>0$, the minimum energy configurations are those with all spins pointing in the same direction (all positive or all negative). For $B<0$, the minimum energy configurations are those with an equal number of positive and negative spins.
 

Related to Exploring Spin Magnetization in Non-Ideal Systems

What is spin magnetization?

Spin magnetization is a measure of the alignment of the spin of particles in a material. It is a vector quantity that describes the strength and direction of the magnetic moment in a material.

Why is it important to explore spin magnetization in non-ideal systems?

Non-ideal systems refer to materials that do not have a perfect crystal structure, such as amorphous or disordered materials. Exploring spin magnetization in these systems can provide insights into the effects of disorder on magnetic properties and can help improve our understanding of how these materials behave in real-world applications.

How is spin magnetization measured?

Spin magnetization can be measured using various techniques, such as magnetometry, nuclear magnetic resonance (NMR), and electron spin resonance (ESR). These methods involve applying a magnetic field to the material and measuring the response of the spin of particles.

What factors can affect spin magnetization in non-ideal systems?

In non-ideal systems, factors such as defects, impurities, and disorder can affect the spin magnetization. These can alter the magnetic interactions between particles and lead to changes in the overall magnetic properties of the material.

How can exploring spin magnetization in non-ideal systems benefit technology?

Understanding spin magnetization in non-ideal systems can help improve the design and performance of magnetic materials used in technology. This knowledge can also aid in the development of new materials with desired magnetic properties for specific applications, such as data storage, sensors, and medical devices.

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