Paramagnetism of Free Ions. Sign problem.

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SUMMARY

The discussion centers on the energy equation for a system of magnetic dipoles, specifically the expression ##E_H=-\mu_0 \vec{\mu}\cdot \vec{H}##. The negative sign in the equation indicates that the energy decreases when the magnetic moment is aligned parallel to the applied magnetic field, ##H##. The second negative sign associated with ##\mu_z=-g_J\mu_Bm## raises questions, as the magnetization, ##\mu_z##, is expected to be proportional to the applied field, ##H##, and is influenced by temperature. The relationship between magnetization and the applied field direction is clarified, distinguishing between paramagnetic and diamagnetic materials.

PREREQUISITES
  • Understanding of magnetic dipole moments
  • Familiarity with the concepts of magnetization and magnetic fields
  • Knowledge of the Landé g-factor, ##g_J##
  • Basic principles of thermodynamics as they relate to magnetization
NEXT STEPS
  • Study the implications of the Landé g-factor in magnetic systems
  • Explore the relationship between temperature and magnetization in paramagnetic materials
  • Investigate the differences between paramagnetism and diamagnetism
  • Learn about the mathematical derivation of magnetic energy equations
USEFUL FOR

Physicists, materials scientists, and students studying magnetism and magnetic materials will benefit from this discussion, particularly those focusing on the behavior of magnetic dipoles in external fields.

LagrangeEuler
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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##.
 
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E=m\bulletB 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.
 

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