# pair_style spin/exchange command

## Syntax

pair_style spin/exchange cutoff

• cutoff = global cutoff pair (distance in metal units)

## Examples

pair_style spin/exchange 4.0
pair_coeff * * exchange 4.0 0.0446928 0.003496 1.4885
pair_coeff 1 2 exchange 6.0 -0.01575 0.0 1.965


## Description

Style spin/exchange computes the exchange interaction between pairs of magnetic spins:

$H_{ex} = -\sum_{i,j}^N J_{ij} (r_{ij}) \,\vec{s}_i \cdot \vec{s}_j$

where $$\vec{s}_i$$ and $$\vec{s}_j$$ are two neighboring magnetic spins of two particles, $$r_{ij} = \vert \vec{r}_i - \vec{r}_j \vert$$ is the inter-atomic distance between the two particles. The summation is over pairs of nearest neighbors. $$J(r_{ij})$$ is a function defining the intensity and the sign of the exchange interaction for different neighboring shells. This function is defined as:

${J}\left( r_{ij} \right) = 4 a \left( \frac{r_{ij}}{d} \right)^2 \left( 1 - b \left( \frac{r_{ij}}{d} \right)^2 \right) e^{-\left( \frac{r_{ij}}{d} \right)^2 }\Theta (R_c - r_{ij})$

where $$a$$, $$b$$ and $$d$$ are the three constant coefficients defined in the associated “pair_coeff” command, and $$R_c$$ is the radius cutoff associated to the pair interaction (see below for more explanations).

The coefficients $$a$$, $$b$$, and $$d$$ need to be fitted so that the function above matches with the value of the exchange interaction for the $$N$$ neighbor shells taken into account. Examples and more explanations about this function and its parameterization are reported in (Tranchida).

From this exchange interaction, each spin $$i$$ will be submitted to a magnetic torque $$\vec{\omega}$$, and its associated atom can be submitted to a force $$\vec{F}$$ for spin-lattice calculations (see fix_nve_spin), such as:

$\vec{\omega}_{i} = \frac{1}{\hbar} \sum_{j}^{Neighb} {J} \left(r_{ij} \right)\,\vec{s}_{j} ~~{\rm and}~~ \vec{F}_{i} = \sum_{j}^{Neighb} \frac{\partial {J} \left(r_{ij} \right)}{ \partial r_{ij}} \left( \vec{s}_{i}\cdot \vec{s}_{j} \right) \vec{e}_{ij}$

with $$\hbar$$ the Planck constant (in metal units), and $$\vec{e}_{ij} = \frac{\vec{r}_i - \vec{r}_j}{\vert \vec{r}_i-\vec{r}_j \vert}$$ the unit vector between sites $$i$$ and $$j$$.

More details about the derivation of these torques/forces are reported in (Tranchida).

For the spin/exchange pair style, the following coefficients must be defined for each pair of atoms types via the pair_coeff command as in the examples above, or in the data file or restart files read by the read_data or read_restart commands, and set in the following order:

• $$R_c$$ (distance units)

• $$a$$ (energy units)

• $$b$$ (adim parameter)

• $$d$$ (distance units)

Note that $$R_c$$ is the radius cutoff of the considered exchange interaction, and $$a$$, $$b$$ and $$d$$ are the three coefficients performing the parameterization of the function $$J(r_{ij})$$ defined above.

None of those coefficients is optional. If not specified, the spin/exchange pair style cannot be used.

## Restrictions

All the pair/spin styles are part of the SPIN package. These styles are only enabled if LAMMPS was built with this package, and if the atom_style “spin” was declared. See the Build package doc page for more info.