**4. Conclusion**

466 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

2 2 2 2 4 2 0 00 3 2 0 44 22

*m ae ae <sup>e</sup> H HH*

<sup>R</sup> R RR

. 2 12 <sup>864</sup> <sup>4</sup>

Here the first term is the energy of the electron gas in the absence of a magnetic field. The second term that makes the main addend in the magnetic moment is the product of the

magnetic field (formula (56)). The third term in the formula (60) is created by the addends *Emb* , the squared additional term of the Fermi energy, and the negative addend in the result

the spin polarization. It always is smaller than the second term. The magnetic moment is:

 

 <sup>2</sup> 2 4 <sup>2</sup> 0 0 0 3 0 43 2

where <sup>2</sup> *E eH m <sup>p</sup>* R 4 . The first term is the diamagnetic moment of the orbital motion in the states of the conduction band. It is proportional to the electron density and third power of the radius as distinct from the result of the Landau theory (see to formula (5)). This formula would be obtained by other way. Let us consider the density of electric current

*<sup>P</sup> <sup>P</sup> E E E ae ae e H HH HH m m m*

0 2 2

*m me E E <sup>H</sup>*

 

R R

 **b**

d

*mb*

 

Fermi energy of the electron gas

of integration 22 2 2 *m <sup>b</sup>* R

2 2

(formula (32)) in the ground state with zero angular momentum:

1

*N*

The first term equals to zero because <sup>2</sup>

change in the second addend <sup>2</sup>

The magnetic moment is:

will cancel.

to <sup>2</sup> *N* R  magnetic moment will start is:

2 2 2

*m mm*

2

*m*

. The last term in this formula describes the energy lowering by

<sup>2</sup> 6 216 4

M R RR (61)

 

*nl nl*

*r nl r* n . Then the integration gives both the

and , 0 *nl* <sup>n</sup> *nll* . Let us

*E H mb* . When *Nb* will be equal

ˆ ˆ 2 2 2 **rr rr**

 <sup>2</sup> 2 2 <sup>2</sup> <sup>3</sup> <sup>0</sup> 0 0 <sup>d</sup> , <sup>d</sup> , d <sup>2</sup> n n *r r*

<sup>0</sup> 2 d1 *nl*

0 the diamagnetic susceptibility will be zero. When the magnetic field is

 

*nl nl <sup>e</sup> e H j rr r nll r r r r nl r r*

M (63)

*r i ri i ri i i*

*i i i e l e Hr l e Hr <sup>j</sup> m r r*

*m m*

 *r rr* <sup>R</sup>

first and the second terms of the formula (61). If the sum , *nl* <sup>n</sup> *nll* will be computed in the common theory, the linear with respect to the magnetic field terms in the formula (63)

The orbital diamagnetic susceptibility decreases with increasing of the magnetic field. This

stronger, the Fermi energy remains on the level *n N <sup>F</sup>* of the magnetic band and increases proportionally to magnetic field. The magnetic moment remains constant. The paramagnetic moment also does not increase because the density of states on the Fermi level decreases linearly. The value of the magnetic induction whereby the saturation of the

 

, *nl nl* 

function also has a spikes that is caused by the addend 2 2

<sup>R</sup> R R

<sup>0</sup> and the additional term in it, which depends on the

(60)

(62)

The fundamental theory of statistical mechanics requires taking into account the law of the angular momentum conservation. The fulfilment of this requirement does not introduce any essential alterations into statistical thermodynamics, when the angular momentum of the system equal zero and the system Hamiltonian is a positive definite quadric form of all momenta. An equilibrium isolated system would have nonzero angular momentum only if an attraction of particles can resist centrifugal forces, as it is in nebulas. A gas can be in equilibrium with a rotating envelope that is a termospinstat. The condition of this equilibrium is the equality of the average value of sum of particles angular velocities to the angular velocity of the envelope. The Gibbs density of distribution and the thermodynamical functions are generalized for this case. If a system has the angular momentum equal to zero, the conservation of this value is important only when the Hamiltonian or/and an averaged quantity depend on the angular momentum. The problem of an electron gas in a uniform magnetic field is considered with taking into account the conservation of the zero value of the angular momentum. This consideration eliminates the paradoxical statement of the conventional theory that diamagnetic moment of the gas equals zero in classical as well as quantum physics (the Bohr – van Leeuwen theorem). The new formulae for the magnetic moment of the electron gas are obtained. It also leads to the effect of confinement of two-dimensional gas of charged particles by magnetic field. This results in effect of a non-uniform density of a gas, which decreases with distance from a center according both to classical as well as quantum theory. Then the model of noninteracting charged particles does not have areas of application. Many theories should be reconsidered, if they are founded on this model and on the statistical mechanics which does not take into account the angular momentum conservation law.
