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{{TAGDEF|SAXIS|[real array]| (0+, 0, 1)}}
{{TAGDEF|SAXIS|[real array]| (0, 0, 1)}}


Description: {{TAG|SAXIS}} specifies the quantisation axis for noncollinear spins.
Description: Set the global spin-quantization axis w.r.t. Cartesian coordinates.
----
----
The spin quantization axis may be specified by means of the {{TAG|SAXIS}}-tag,
{{TAG|SAXIS}} specifies the relative orientation of spinor space spanned by the Pauli matrices <math>\{\sigma_1</math>, <math>\sigma_2</math>, <math>\mathbf{\sigma}_3\}</math> with respect to Cartesian coordinates <math>\{\hat x, \hat y, \hat z\} </math>. The default is  <math>\sigma_1=\hat x</math>, <math>\sigma_2 =\hat y</math>, <math>\sigma_3 = \hat z</math>.
The direction of the spin-quantization axis <math>\sigma_3</math> with respect to Cartesian coordinates is set


   {{TAG|SAXIS}} =  s<sub>x</sub> s<sub>y</sub> s<sub>z</sub>    ! global spin quantisation axis
   {{TAG|SAXIS}} =  s<sub>x</sub> s<sub>y</sub> s<sub>z</sub>    ! global spin-quantization axis
such that <math>\sigma_3=\mathbf{s}/|\mathbf{s}|</math>, i.e., <math>\sigma_3</math> points along <math>\mathbf{s}=(s_x,s_y,s_z)^T</math>. The directions of <math>\sigma_1</math> and <math>\sigma_2</math> are a consequence of rotating <math>\sigma_3</math> to point along <math>\mathbf{s}</math> as described below.


where the default for {{TAG|SAXIS}} = (0+,0,1) (the notation 0+ implies an infinitesimal small positive number in ''x''-direction). All magnetic moments are now given with respect to the axis
The relative orientation of spinor space with respect to real space becomes important in case spin-orbit coupling is included ({{TAG|LSORBIT}}=True). All magnetic moments and spinor-like quantities written or read by VASP are given in the basis of the spinor space <math>\{\sigma_1</math>, <math>\sigma_2</math>, <math>\mathbf{\sigma}_3\}</math>. This includes the {{TAG|MAGMOM}} tag in the {{FILE|INCAR}} file, the total and local magnetizations in the {{FILE|OUTCAR}} and {{FILE|PROCAR}} file, the spinor-like orbitals in the {{TAG|WAVECAR}} file, and the magnetization density in the {{FILE|CHGCAR}} file.  
(s<sub>x</sub>,s<sub>y</sub>,s<sub>z</sub>), where we have adopted the convention '''that all magnetic moments and spinor-like quantities written or read by VASP are given with respect to this axis'''. This includes the {{TAG|MAGMOM}} line in the {{FILE|INCAR}} file, the total and local magnetizations in the {{FILE|OUTCAR}} and {{FILE|PROCAR}} file, the spinor-like orbitals in the {{TAG|WAVECAR}} file, and the magnetization density in the {{FILE|CHGCAR}} file. With respect to the Cartesian lattice vectors the components of the magnetization are (internally) given by


:<math>
== Coordinate system ==
\begin{align}
m_x & = & \cos(\beta) \cos(\alpha) m^{\rm axis}_x - \sin(\alpha) m^{\rm axis}_y + \sin(\beta) \cos(\alpha) m^{\rm axis}_z \\
m_y & = & \cos(\beta) \sin(\alpha) m^{\rm axis}_x + \cos(\alpha) m^{\rm axis}_y + \sin(\beta) \sin(\alpha) m^{\rm axis}_z \\
m_z & = & -\sin(\beta) m^{\rm axis}_x+ \cos(\beta) m^{\rm axis}_z
\end{align}
</math>


[[File:Saxis-angles.png|300px|thumb|Fig 1. Euler angles <math>\alpha</math> and  <math>\beta</math> defined by <math>\mathbf{s}=(s_x,s_y,s_z)^T</math>.]]


Where ''m''<sup>axis</sup> is the externally visible magnetic moment. Here, <math>\alpha</math> is the angle between the {{TAG|SAXIS}} vector (s<sub>x</sub>,s<sub>y</sub>,s<sub>z</sub>) and the Cartesian vector <math>\hat x</math>, and <math>\beta</math> is the angle between the vector {{TAG|SAXIS}} and the Cartesian vector <math>\hat z</math>:
The default orientation is <math>\sigma_1=\hat x</math>, <math>\sigma_2 =\hat y</math>, <math>\sigma_3 = \hat z</math>.
To set <math>\hat{\sigma}_3=s/|s|</math>, VASP applies two rotations with Euler angles


:<math>
:<math>
\begin{align}
\begin{align}
\alpha &=& {\rm atan} \frac{s_y}{s_x} \\  
\alpha&=\arctan2\left(\frac{s_y}{s_x}\right) \in [-\pi,\pi]\\
\beta &=& {\rm atan} \frac{\sqrt{s_x^2+s_y^2}}{s_z}  
\beta&=\arctan2\left(\frac{\sqrt{s_x^2+s_y^2}}{s_z}\right) \in [0,\pi].
\end{align}
\end{align}
</math>
</math>
Here, <math>\alpha</math> is the angle between the projection of {{TAG|SAXIS}} onto the ''xy'' plane (s<sub>x</sub>,s<sub>y</sub>,0) and the Cartesian vector <math>\hat x</math>, and <math>\beta</math> is the angle between the vector {{TAG|SAXIS}} and the Cartesian vector <math>\hat z</math>, see Fig. 1. Search for `Euler angles` in the {{FILE|OUTCAR}} file to see what VASP uses. For the default <math>\mathbf{s}=(0,0,1)</math>, <math>\alpha=0</math> and <math>\beta=0</math>.


The inverse transformation is given by
The transformation of a vector <math>\mathbf{m}=(m_1,m_2,m_3)^T</math> given in the basis <math>\{\sigma_1</math>, <math>\sigma_2</math>, <math>\mathbf{\sigma}_3\}</math> into <math>\mathbf{m}'=(m_x,m_y,m_z)^T</math> in Cartesian coordinates and its inverse transformation read
 
:<math>
:<math>
\begin{align}
\begin{align}
m^{\rm axis}_x & = & \cos(\beta) \cos(\alpha) m_x + \cos(\beta) \sin(\alpha) m_y - \sin(\beta) m_z \\
\mathbf{m}&= m_1 \sigma_1 + m_2 \sigma_2 + m_3 \sigma_3 \\
m^{\rm axis}_y & = & -\sin(\alpha) m_z + \cos(\alpha) m_y \\
\mathbf{m}'&= m_x \hat x + m_y \hat y + m_z \hat z \\
m^{\rm axis}_z & = & \sin(\beta) \cos(\alpha) m_x + \sin(\beta) \sin(\alpha) m_y + \cos(\beta) m_z
\mathbf{m}'&= R_z^\alpha R_y^\beta \mathbf{m} \\
\mathbf{m} &= R_y^{-\beta} R_z^{-\alpha} \mathbf{m}' \\
\end{align}
\end{align}
</math>
</math>
where the rotation matrices are
:<math>
R_z^\alpha = \left(\begin{matrix}
  \cos(\alpha) & -\sin(\alpha) & 0 \\
  \sin(\alpha) & \cos(\alpha)  & 0 \\
        0    & 0            & 1 \\
\end{matrix}\right), \quad
R_y^\beta = \left(\begin{matrix}
  \cos(\beta)  &  0 & \sin(\beta) \\
      0      &  1 &    0        \\
  -\sin(\beta) &  0 & \cos(\beta) \\
\end{matrix}\right).
</math>
{{NB|mind|Apply the proper basis transformation when comparing vector-like quantities and spinor-like quantities.|}}
For instance, when {{TAG|LORBMOM}}=True the orbital angular momentum is written to the {{FILE|OUTCAR}} file in Cartesian coordinates. Thus, when comparing the orbital angular momentum (vector-like quantity) and the magnetization (spinor-like quantity), one has to perform a basis transformation on one of the quantities unless the bases agree (default).


It is easy to see that for the default (s<sub>x</sub>,s<sub>y</sub>,s<sub>z</sub>)=(0+,0,1), both angles are zero, i.e. <math>\beta=0</math> and <math>\alpha=0</math>. In this case, the internal representation is simply equivalent to the external representation:
== Example ==
 
* In case the bases have the same orientation, i.e., <math>\sigma_1=\hat x</math>, <math>\sigma_2 =\hat y</math>, <math>\sigma_3 = \hat z</math> (default)


:<math>
:<math>
\begin{align}
\begin{align}
m_x & = & m^{\rm axis}_x \\  
m_x & = & m_1, \\  
m_y & = & m^{\rm axis}_y \\  
m_y & = & m_2, \\  
m_z & = & m^{\rm axis}_z
m_z & = & m_3.
\end{align}
\end{align}
</math>
</math>
:For a single site this implies setting


The second important case, is ''m''<sup>axis</sup>=(0,0,''m''). In this case
{{TAG|MAGMOM}} = m<sub>x</sub> m<sub>y</sub> m<sub>z</sub> ! magnetic moment in Cartesian coordinates
{{TAG|SAXIS}} =  0 0 1  ! default
 
[[File:Spinor-space-example-saxis.png|300px|thumb|Fig 2. Example with <math>\mathbf{s}=(1,1,0)^T</math> and Euler angles <math>\alpha=\pi/4</math> and <math>\beta=\pi/2</math>.]]
 
* Another good choice is setting <math>\mathbf{s}</math> to point along the direction of the on-site magnetic moment such that


:<math>
:<math>
\begin{align}
\begin{align}
m_x & = & \sin(\beta)*\cos(\alpha) m = m s_x / \sqrt{s_x^2+s_y^2+s_z^2} \\
m_x & = & \sin(\beta)\cos(\alpha) m &= m\, s_x / \sqrt{s_x^2+s_y^2+s_z^2} \\
m_y & = & \sin(\beta)*\sin(\alpha) m = m s_y / \sqrt{s_x^2+s_y^2+s_z^2} \\
m_y & = & \sin(\beta)\sin(\alpha) m &= m\, s_y / \sqrt{s_x^2+s_y^2+s_z^2} \\
m_z & = & \cos(\beta) m = m s_z / \sqrt{s_x^2+s_y^2+s_z^2}
m_z & = & \cos(\beta) m &= m\, s_z / \sqrt{s_x^2+s_y^2+s_z^2},
\end{align}
\end{align}
</math>
</math>
:where <math>m</math> is the total on-site magnetic moment.
:For a single site, this case implies setting


{{TAG|MAGMOM}} = 0 0 m  ! magnetic moment along sigma3
{{TAG|SAXIS}} =  s<sub>x</sub> s<sub>y</sub> s<sub>z</sub> ! direction of sigma3
:Thus, there are two methods to rotate the initial magnetization in an arbitrary direction: either by changing the initial magnetic moments {{TAG|MAGMOM}} or by changing {{TAG|SAXIS}}. Both methods should, in principle, yield exactly the same energy, but for implementation reasons, the second method might be more precise.


Hence now the magnetic moment is parallel to the vector {{TAG|SAXIS}}. Thus there are two ways to rotate the spins in an arbitrary direction, either by changing the initial magnetic moments {{TAG|MAGMOM}} or by changing {{TAG|SAXIS}}.
* In case
 
To initialize calculations with the magnetic moment parallel to a chosen vector (''x'',''y'',''z''), it is therefore possible to either specify (assuming a single atom in the cell)


{{TAG|MAGMOM}} = x y z  ! local magnetic moment in x,y,z
  {{TAG|SAXIS}} =  1 1 0  ! alpha=pi/4, beta=pi/2
  {{TAG|SAXIS}} =  0 0 1   ! quantisation axis parallel to z
:the spinor space <math>\{\sigma_1</math>, <math>\sigma_2</math>, <math>\mathbf{\sigma}_3\}</math> will be rotated with respect to real space <math>\{\hat x, \hat y, \hat z\} </math> as shown in Fig. 2.
 
<!---
or
The second method also allows to read a preexisting {{FILE|WAVECAR}} file (from a collinear or non collinear run), and to continue the calculation with a different spin orientation. When a non collinear {{FILE|WAVECAR}} file is read, the spin is assumed to be parallel to {{TAG|SAXIS}} (hence VASP will initially report a magnetic moment in the ''z''-direction only).
 
{{TAG|MAGMOM}} = 0 0 total_magnetic_moment  ! local magnetic moment parallel to {{TAG|SAXIS}}
{{TAG|SAXIS}} =  x y z  ! quantization axis parallel to vector (''x'',''y'',''z'')
 
Both setups should in principle yield exactly the same energy, but for implementation reasons the second method is usually more precise. The second method also allows to read a preexisting {{FILE|WAVECAR}} file (from a collinear or non collinear run), and to continue the calculation with a different spin orientation. When a non collinear {{FILE|WAVECAR}} file is read, the spin is assumed to be parallel to {{TAG|SAXIS}} (hence VASP will initially report a magnetic moment in the ''z''-direction only).


The recommended procedure for the calculation of magnetic anisotropies is therefore (please check the section on {{TAG|LMAXMIX}}):
The recommended procedure for the calculation of magnetic anisotropies is therefore (please check the section on {{TAG|LMAXMIX}}):
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VASP reads in the {{FILE|WAVECAR}} and {{FILE|CHGCAR}} files, aligns the spin quantization axis parallel to {{TAG|SAXIS}}, which implies that the magnetic field is now parallel to {{TAG|SAXIS}}, and performs a non selfconsistent calculation. By comparing the energies for different orientations the magnetic anisotropy can be determined. Please mind, that a completely selfconsistent calculation ({{TAG|ICHARG}} = 1) is in principle also possible with VASP, but this would allow the spinor wavefunctions to rotate from their initial orientation parallel to {{TAG|SAXIS}} until the correct groundstate is obtained, i.e. until the magnetic moment is parallel to the easy axis. In practice this rotation will be slow, however, since reorientation of the spin gains little energy. Therefore if the convergence criterion is not too tight, sensible results might be obtained even for fully selfconsistent calculations (in the few cases we have tried this worked beautifully).
VASP reads in the {{FILE|WAVECAR}} and {{FILE|CHGCAR}} files, aligns the spin quantization axis parallel to {{TAG|SAXIS}}, which implies that the magnetic field is now parallel to {{TAG|SAXIS}}, and performs a non selfconsistent calculation. By comparing the energies for different orientations the magnetic anisotropy can be determined. Please mind, that a completely selfconsistent calculation ({{TAG|ICHARG}} = 1) is in principle also possible with VASP, but this would allow the spinor wavefunctions to rotate from their initial orientation parallel to {{TAG|SAXIS}} until the correct groundstate is obtained, i.e. until the magnetic moment is parallel to the easy axis. In practice this rotation will be slow, however, since reorientation of the spin gains little energy. Therefore if the convergence criterion is not too tight, sensible results might be obtained even for fully selfconsistent calculations (in the few cases we have tried this worked beautifully).
 
--->
== Related Tags and Sections ==
== Related tags and articles ==
{{TAG|LNONCOLLINEAR}},
{{TAG|LNONCOLLINEAR}},
{{TAG|MAGMOM}},
{{TAG|MAGMOM}},
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----
----


[[Category:INCAR]][[Category:Magnetism]][[Category:Noncollinear magnetism]]
[[Category:INCAR tag]][[Category:Magnetism]][[Category:Noncollinear magnetism]]

Latest revision as of 09:55, 21 September 2023

SAXIS = [real array]
Default: SAXIS = (0, 0, 1) 

Description: Set the global spin-quantization axis w.r.t. Cartesian coordinates.


SAXIS specifies the relative orientation of spinor space spanned by the Pauli matrices , , with respect to Cartesian coordinates . The default is , , . The direction of the spin-quantization axis with respect to Cartesian coordinates is set

 SAXIS =   sx sy sz    ! global spin-quantization axis

such that , i.e., points along . The directions of and are a consequence of rotating to point along as described below.

The relative orientation of spinor space with respect to real space becomes important in case spin-orbit coupling is included (LSORBIT=True). All magnetic moments and spinor-like quantities written or read by VASP are given in the basis of the spinor space , , . This includes the MAGMOM tag in the INCAR file, the total and local magnetizations in the OUTCAR and PROCAR file, the spinor-like orbitals in the WAVECAR file, and the magnetization density in the CHGCAR file.

Coordinate system

Fig 1. Euler angles and defined by .

The default orientation is , , . To set , VASP applies two rotations with Euler angles

Here, is the angle between the projection of SAXIS onto the xy plane (sx,sy,0) and the Cartesian vector , and is the angle between the vector SAXIS and the Cartesian vector , see Fig. 1. Search for `Euler angles` in the OUTCAR file to see what VASP uses. For the default , and .

The transformation of a vector given in the basis , , into in Cartesian coordinates and its inverse transformation read

where the rotation matrices are

Mind: Apply the proper basis transformation when comparing vector-like quantities and spinor-like quantities.

For instance, when LORBMOM=True the orbital angular momentum is written to the OUTCAR file in Cartesian coordinates. Thus, when comparing the orbital angular momentum (vector-like quantity) and the magnetization (spinor-like quantity), one has to perform a basis transformation on one of the quantities unless the bases agree (default).

Example

  • In case the bases have the same orientation, i.e., , , (default)
For a single site this implies setting
MAGMOM = mx my mz ! magnetic moment in Cartesian coordinates
SAXIS =  0 0 1   ! default
Fig 2. Example with and Euler angles and .
  • Another good choice is setting to point along the direction of the on-site magnetic moment such that
where is the total on-site magnetic moment.
For a single site, this case implies setting
MAGMOM = 0 0 m   ! magnetic moment along sigma3
SAXIS =  sx sy sz ! direction of sigma3
Thus, there are two methods to rotate the initial magnetization in an arbitrary direction: either by changing the initial magnetic moments MAGMOM or by changing SAXIS. Both methods should, in principle, yield exactly the same energy, but for implementation reasons, the second method might be more precise.
  • In case
SAXIS =  1 1 0   ! alpha=pi/4, beta=pi/2
the spinor space , , will be rotated with respect to real space as shown in Fig. 2.

Related tags and articles

LNONCOLLINEAR, MAGMOM, LSORBIT

Examples that use this tag