# deer_analyt.m

Analytical expression for a DEER trace for two spins in the presence of dipolar and exchange (EPR) or J (NMR) coupling. This function returns the powder averaged dipolar modulation signal in DEER, PELDOR and in-phase ESEEM type experiments, as well as in solid state NMR spectroscopy where dipolar modulation has been isolated by selective recoupling.

## Syntax

    deer=deer_analyt(D,J,t)


## Arguments

    D - dipolar coupling, angular frequency units, the
coefficient in front of (1-3*cos(theta)^2)*Lz*Sz
in the spin Hamiltonian

J - exchange coupling, angular frequency units, NMR
convention (no factor of 2 in front), the coef-
ficient in front of Lz*Sz in the spin Hamiltonian

t - array of time points, seconds


## Outputs

    deer - an array of DEER form factor values of the same
dimension as t


## Examples

See examples/esr_solids directory for multiple examples from EPR spectroscopy.

An example DEER modulation trace: numerical simulation vs. analytical expression used by this function.

## Notes

1. Distance distributions in real molecules are rarely delta functions. If the distribution has a width, it would impose and additional exponential or Gaussian decay on the modulation produced by this function.
2. The presence of observer spins can generate a background signal that may be approximately described by a stretched exponential.