# Paramagnetism of Free Ions. Sign problem.

• LagrangeEuler
In summary, the energy of a system of magnetic dipoles is determined by the equation E_H=-\mu_0 \vec{\mu}\cdot \vec{H}=-\mu_0\mu_zH, where the first negative sign is due to the decrease in energy when a magnetic moment is parallel to the applied field. The second negative sign is unclear and may not be necessary. The direction of the magnetization, mu_z, is chosen to be parallel to the applied field, H, and is proportional to it. This direction is also dependent on the temperature of the material. In a paramagnet, the magnetization is parallel to the applied field, while in a diamagnet it is antiparallel.
LagrangeEuler
Energy of system of magnetic dipoles is
##E_H=-\mu_0 \vec{\mu}\cdot \vec{H}=-\mu_0\mu_zH##
why sign ''-''?
and ##\mu_z=-g_J\mu_Bm##
Again why ''-'' sign ?
##z## is choosen to be quantization direction, and along that direction is applied field ##H##.

E=m$\bullet$B but as the material has a magnetization M and B= H+M then H must face the opposite direction to B and that explains the first negative sign I think.

In a paramagnet, the magnetization of the material is parallel to the applied field. In a diamagnet it would be antiparallel (opposite).

The first "-" is there because the energy decreases when a magnetic moment is parallel to the applied field.

The second "-" does not make any sense to me.

H appears to be along the positive z direction. Then the magnetization mu_z should also be along the positive z direction, and it should be proportional to H. The magnitude of the magnetization also depends on the temperature.

## 1. What is paramagnetism?

Paramagnetism is a property of certain materials in which the atoms or molecules have unpaired electrons, causing them to be weakly attracted to an external magnetic field.

## 2. What are free ions?

Free ions are atoms or molecules that have lost or gained one or more electrons, resulting in a net electric charge. They are not bonded to other atoms and can move freely in a solution or gas.

## 3. How does paramagnetism of free ions differ from other types of magnetism?

Paramagnetism is different from other types of magnetism, such as ferromagnetism or diamagnetism, because it is not caused by the alignment of electron spins within a material. Instead, it is caused by the presence of unpaired electrons in the outermost electron orbitals of free ions.

## 4. What is the "sign problem" in relation to paramagnetism of free ions?

The "sign problem" refers to the difficulty in accurately predicting the magnetic behavior of free ions in a solution or gas. This is due to the fact that the magnetic moment of a free ion can be influenced by a variety of factors, such as the presence of other ions or the strength and direction of the external magnetic field.

## 5. How is the paramagnetism of free ions used in scientific research?

The paramagnetism of free ions is used in a variety of scientific research, such as in the study of chemical reactions, protein folding, and the behavior of materials. It can also be used in medical imaging techniques, such as magnetic resonance imaging (MRI), to analyze the distribution of free ions in the body.

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