Chemistry - Nonlinear magnetic field

From Maxwell's nonlinear equations follow equations for the nonlinear magnetic field. In dimensionless variables they are

$\nabla \times \vec H=-\vec u H^2$

$k^2\nabla^2\vec u=\vec A H^2$

where k² is interaction constant of free w-field. A and H are the vector potential and the strength of magnetic field respectively. The vector u is potential of w-field. It is the analogue of the velocity of charged particles in an electrically neutral system.

Consider the magnetic field with simplest geometry. It is

$\vec u =u(x) e_z$

$\vec A=A(x) e_z$

$\vec H=-\dot Ae_y$

In this case fields equations are the same as for nonlinear Coulomb field

$\dot H=uH^2$

$k^2\ddot u=AH^2$

And they are valid for axial symmetrical distributions of jets if take variables

x = ln r

r 2 + z 2 = R 2

On infinity we require $H = 0$ .

Because space variable now run all axis magnetic field may have additional symmetry. For paramagnetic field $H (x) = - H ( - x)$ for diamagnetic states $H (x) = H ( - x)$

From the coherence condition exist no one type solutions of field equations.

In vacuum state $u = c o n s t a n t$ . Then for diamagnetic state $u = 0$ . From condition on infinity magnetic field also is zero.

Paramagnetic field is not zero if on dividing surface is flat current of charged particles. In this case in area $x > 0$ for flat symmetry

$H=-\frac{H_0}{1-uH_0 x}$

for axial symmetry

$H=-\frac{H_0}{r(1-H_0 u \ln r)}$

In area $x < 0$ need replace $H 0$ $\rightarrow$ $- H 0$ . Always $u < 0$ . These are usual Biot –Savart-Laplace field. With more complicated geometry such states are observable in plasma as neutral current sheet .

In coherences states solutions field equations are

$u=-\frac{A}{k}+2\sum_{0}^N\frac{1}{A-A_n}$

$H\sim \prod (A-A_n)^2 exp\left(-\frac{A^2}{2k}\right)$

where the Hermite numbers are determined by equations

$A_n=2k\sum_{s\ne n}$ $\frac{1}{A_n-A_s}$

In coherence states self energy of free w-field have singularities in points $A = A n$ . For paramagnetic field these singularities are cancel if field is close between

A n - 1 < A < A n

Then the potential is periodic function and field have a stratum structure in all space. From the requirement of finite energy these solutions need restrict or taking solutions with special symmetry or closing this field between two sheets with parallel flat jets of charge particles. This look out as magneticcondenser or coaxial guide . Because field equations have scale symmetry small thickness of layers is not necessary require big strength of field.

Thus nonlinear magnetic field have more reach structure compare with linear field.