BEXT: Difference between revisions

Revision as of 19:15, 8 February 2024

BEXT = [real array]

Default: BEXT	= 0.0	if ISPIN=2
	= 3*0.0	if LNONCOLLINEAR=.TRUE.
	= N/A	else

Description: BEXT specifies an external magnetic field.

By means of the BEXT one may specify an external magnetic field that acts on the electrons in a Zeeman-like manner. This interaction is carried by an additional potential of the following form:

For ISPIN = 2:

{\displaystyle V^{\uparrow} = V^{\uparrow} + B_{\rm ext} }

{\displaystyle V^{\downarrow} = V^{\downarrow} - B_{\rm ext} }

and

B_{\rm ext}

= BEXT (in eV).

For LNONCOLLINEAR = .TRUE.:

{\displaystyle V_{\alpha\beta} = V_{\alpha\beta} + \vec{B}_{\rm ext} \cdot \vec{\sigma}_{\alpha \beta} }

where

({B}^x_{\rm ext}, {B}^y_{\rm ext}, {B}^z_{\rm ext})

= BEXT (in eV), and

\vec{\sigma}

is the vector of Pauli matrices.

Heuristically, the effect of the above is most easily understood for the collinear spinpolarized case (ISPIN=2):

The eigenenergies of spin-up states are raised by $B_{\rm ext}$ eV, whereas the eigenenergies of spin-down states are lowered by the same amount.

The total energy changes by:

{\displaystyle \Delta E = (n^{\uparrow} - n^{\downarrow}) B_{\rm ext} }

eV

where

n^{\uparrow}

and

n^{\downarrow}

are the number of up- and down-spin electrons in the system.

Shifting the eigenenergies of the spin-up and spin-down states w.r.t. each other may lead to a redistribution of the electrons over these states (changes in the occupancies) and hence to changes in the density with all subsequent consequences.

The energy difference between two Zeeman-splitted electronic states is given by:

{\displaystyle \hbar \omega = g_e \mu_B B_0 }

where $\mu_B$ is the Bohr magneton and $g_e$ is the electron g-factor.

For ISPIN=2, for purely Zeeman splitted states, we have:

{\displaystyle V^{\uparrow} - V^{\downarrow} = 2 B_{\rm ext} }

This leads to the following relationship between our definition of $B_{\rm ext}$ (in eV) and the magnetic field $B_0$ (in T):

{\displaystyle B_0 = \frac{2 B_{\rm ext}}{g_e \mu_B} }

where $\mu_B$ = 5.788 381 8060 x 10^-5 eV T^-1, and $g_e$ = 2.002 319 304 362 56.

@@ Line 52: / Line 52: @@
 </math>
 where <math>\mu_B</math>= 5.788 381 8060 x 10<sup>-5</sup> eV T<sup>-1</sup>, and <math>g_e</math>= 2.002 319 304 362 56.
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