Paramagnetism of Free Ions. Sign problem.

[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 $H$.

 

[AlexCdeP]

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.

 

[M Quack]

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.