epyr.relaxation.models

Time-domain relaxation models for EPR T1/T2 measurements.

Mathematical Background:

Mono-exponential decay/recovery, stretched exponential, bi-exponential, and a combined homogeneous/spectral-diffusion decay model used to extract spin relaxation times (T1, T2) from pulsed EPR time traces.

References

Eaton, S.S. and Eaton, G.R., “Relaxation Times of Organic Radicals and Transition Metal Ions”, Biol. Magn. Reson., 19, 2000.

Functions

biexponential(t, amplitude1, tau1, ...[, offset])

Compute a two-component exponential decay.

gamma_gaussian_decay(t, amplitude, Gamma0, ...)

Compute an echo decay with homogeneous and spectral-diffusion terms.

inversion_recovery(t, amplitude, T1[, offset])

Compute an inversion-recovery T1 curve.

mono_exponential(t, amplitude, T[, offset])

Compute a single-rate exponential decay.

saturation_recovery(t, amplitude, T1[, offset])

Compute a saturation-recovery T1 curve.

stretched_exponential(t, amplitude, T, beta)

Compute a stretched-exponential decay.
epyr.relaxation.models.mono_exponential(t, amplitude, T, offset=0.0)[source]

Compute a single-rate exponential decay.

V(t) = amplitude * exp(-t / T) + offset

Parameters:

  • t (np.ndarray) – Time axis, in the unit chosen by the caller (e.g. ns or us).

  • amplitude (float) – Signal amplitude at t = 0, relative to offset.

  • T (float) – Decay (or recovery) time constant, in the same unit as t.

  • offset (float, optional) – Asymptotic signal level as t approaches infinity (default: 0.0).

Returns:

Signal values, same shape as t.

Return type:

np.ndarray

epyr.relaxation.models.stretched_exponential(t, amplitude, T, beta, offset=0.0)[source]

Compute a stretched-exponential decay.

V(t) = amplitude * exp(-(t / T) ** beta) + offset

beta = 1 reduces to a mono-exponential decay. beta < 1 describes a distribution of relaxation rates, common in pulsed EPR echo decays.

Parameters:

  • t (np.ndarray) – Time axis, in the unit chosen by the caller.

  • amplitude (float) – Signal amplitude at t = 0, relative to offset.

  • T (float) – Characteristic decay time constant, in the same unit as t.

  • beta (float) – Stretching exponent (dimensionless).

  • offset (float, optional) – Asymptotic signal level as t approaches infinity (default: 0.0).

Returns:

Signal values, same shape as t.

Return type:

np.ndarray

epyr.relaxation.models.biexponential(t, amplitude1, tau1, amplitude2, tau2, offset=0.0)[source]

Compute a two-component exponential decay.

V(t) = amplitude1 * exp(-t / tau1) + amplitude2 * exp(-t / tau2) + offset

Parameters:

  • t (np.ndarray) – Time axis, in the unit chosen by the caller.

  • amplitude1 (float) – Amplitude of the first component at t = 0.

  • tau1 (float) – Decay time constant of the first component, same unit as t.

  • amplitude2 (float) – Amplitude of the second component at t = 0.

  • tau2 (float) – Decay time constant of the second component, same unit as t.

  • offset (float, optional) – Asymptotic signal level as t approaches infinity (default: 0.0).

Returns:

Signal values, same shape as t.

Return type:

np.ndarray

Notes

tau1 and tau2 are not necessarily T1 or T2: either component can represent either relaxation process, depending on the experiment.

epyr.relaxation.models.inversion_recovery(t, amplitude, T1, offset=0.0)[source]

Compute an inversion-recovery T1 curve.

V(t) = amplitude * (1 - 2 * exp(-t / T1)) + offset

Parameters:

  • t (np.ndarray) – Delay time after inversion, in the unit chosen by the caller.

  • amplitude (float) – Fully-recovered signal amplitude relative to offset.

  • T1 (float) – Spin-lattice relaxation time, same unit as t.

  • offset (float, optional) – Baseline signal level (default: 0.0).

Returns:

Signal values, same shape as t.

Return type:

np.ndarray

epyr.relaxation.models.saturation_recovery(t, amplitude, T1, offset=0.0)[source]

Compute a saturation-recovery T1 curve.

V(t) = amplitude * (1 - exp(-t / T1)) + offset

Parameters:

  • t (np.ndarray) – Delay time after saturation, in the unit chosen by the caller.

  • amplitude (float) – Fully-recovered signal amplitude relative to offset.

  • T1 (float) – Spin-lattice relaxation time, same unit as t.

  • offset (float, optional) – Baseline signal level (default: 0.0).

Returns:

Signal values, same shape as t.

Return type:

np.ndarray

epyr.relaxation.models.gamma_gaussian_decay(t, amplitude, Gamma0, GammaG, offset=0.0)[source]

Compute an echo decay with homogeneous and spectral-diffusion terms.

V(t) = amplitude * exp(-Gamma0 * t) * exp(-(GammaG * t) ** 2) + offset

Gamma0 captures the homogeneous (exponential) relaxation rate. GammaG captures an additional Gaussian-shaped decay from spectral diffusion.

Parameters:

  • t (np.ndarray) – Time axis, in the unit chosen by the caller.

  • amplitude (float) – Signal amplitude at t = 0, relative to offset.

  • Gamma0 (float) – Homogeneous decay rate, in inverse time units (1 / t unit).

  • GammaG (float) – Gaussian (spectral-diffusion) decay rate, in inverse time units.

  • offset (float, optional) – Asymptotic signal level as t approaches infinity (default: 0.0).

Returns:

Signal values, same shape as t.

Return type:

np.ndarray