FUN_PARAMETERS_RELAX module

t1t2ne.scripts.f_ParaMeters_relax.OuterSphere(B, c=1, d=3.6e-10, D_target=1e-10, D_cosolute=2.6e-10, f=0.5, taue=1e-09, tauv=2.6e-11, deltat=0.014, AMe=None, I=None, g=None, S=3.5, nuc='1H')[source]

Calculate the outer sphere relaxation rates R1 and R2 using the Freed model:

\[\begin{split}k_{outer} = \frac{16\pi}{81} \left( \frac{\mu_0}{4\pi} \right)^2 h^2 \gamma_n^2 \gamma_e^2 S(S+1) N_A f \frac{c}{d(D_{target}+D_{cosolute})}\\ R_1 = k_{outer} [7 J_{outer} (\omega_S) + 3 J_{outer} (\omega_I)]\\ R_2 = \frac{1}{2} k_{outer} [13 J_{outer} (\omega_S) + 3 J_{outer} (\omega_I) + 4 J_{outer} (0)]\end{split}\]

Optionally, the transient zero-field splitting (ZFS) contribution to electron relaxation is included with the Bloembergen-Morgan model:

\[\begin{split}R_{1e} = \frac{2}{15} \Delta_t^2 \left(4S(S+1) - 3\right) \left(J(\omega_e) + 2J(2\omega_e)\right)\\ R_{2e} = \frac{1}{15} \Delta_t^2 \left(4S(S+1) - 3\right) \left(3J(0) + 5J(\omega_e) + J(2\omega_e)\right)\end{split}\]

Default values are for 1 mM Gd-DOTA and a protein of 10 kDa at room temperature. Values for Gd-DOTA are taken from Li et al. 2002.

Parameters:

B (float) – magnetic field strength in Tesla
c (float) – concentration of the paramagnetic cosolute in mM (default 1 mM)
d (float) – distance of closest approach between the nucleus and the paramagnetic center in Angstroms
D_target (float) – diffusion coefficient of the target molecule in m^2/s (default 1e-10 m^2/s, corresponding to a protein of ~10 kDa at room temperature)
D_cosolute (float) – diffusion coefficient of the paramagnetic cosolute in m^2/s (default 2.6e-10 m^2/s, corresponding to DOTA at room temperature)
f (float) – fraction of the sphere of accessibility for the cosolute (default 0.5)
taue (float) – electron relaxation time in seconds (default 1e-9 s)
tauv (float or None) – correlation time for transient ZFS fluctuations in seconds (default 2.6e-11 s, for Gd^3+ complexes)
deltat (float or None) – transient ZFS parameter in cm^-1 (default 0.014 cm^-1 for Gd^3+ complexes, None for no transient ZFS)
AMe (float or None) – Hyperfine coupling constant to the metal center in Hz (default None, not used for Gd-DOTA)
I (float or None) – nuclear spin quantum number (default None, not used for Gd-DOTA)
g (float or None) – electron g-factor (default None, uses free electron g-factor, not used for Gd-DOTA)
S (float) – electron spin quantum number (default 3.5, for Gd^3+)
nuc (str) – nuclear spin quantum number (default ‘1H’)

Returns:

R1_outer (float) – longitudinal relaxation rate in s^-1
R2_outer (float) – transverse relaxation rate in s^-1

t1t2ne.scripts.f_ParaMeters_relax.SBM(r, B, taue=1e-09, taur=1e-09, tau_M=inf, S=0.5, tauv=None, deltat=None, AMe=None, I=None, g=None, nuc='1H')[source]

Calculate the dipolar relaxation rates R1 and R2 using the Solomon-Bloembergen-Morgan (SBM) equations, with optional transient zero-field splitting (ZFS) contributions.

Parameters:

r (float) – distance of between the nucleus and the paramagnetic center in Angstroms
B (float) – magnetic field strength in Tesla
g (float, or list of two floats, or 3x3 numpy array) – electron g-factor (default is the free electron g-factor)
taue (float) – electron relaxation time in seconds (default 1e-9 s)
taur (float) – rotational correlation time in seconds (default 1e-9 s)
tau_M (float) – exchange correlation time in seconds (default np.inf)
S (float) – electron spin quantum number (default 0.5, one unpaired electron)
deltat (float or None) – transient ZFS parameter in cm^-1 (default None, no transient ZFS)
tauv (float or None) – correlation time for transient ZFS fluctuations in seconds (default None, no transient ZFS)

Returns:

R1_SBM (float) – longitudinal relaxation rate in s^-1
R2_SBM (float) – transverse relaxation rate in s^-1

t1t2ne.scripts.f_ParaMeters_relax.compute_taue(B, tauv, S=0.5, deltat=None, AMe=None, I=None, g=None)[source]

Compute the electron relaxation time taue using the Bloembergen-Morgan model for transient zero-field splitting (ZFS) contributions.

Parameters:

B (float) – magnetic field strength in Tesla
S (float) – electron spin quantum number (default 0.5, one unpaired electron)
tauv (float) – correlation time for transient coordination sphere fluctuations in seconds (default None, no transient ZFS)
deltat (float or None) – transient ZFS parameter in cm^-1 (default None, no transient ZFS)
AMe (float or None) – molecular alignment parameter (default None)
I (float or None) – nuclear spin quantum number (default None)
g (float or None) – electron g-factor (default None, uses free electron g-factor)

Returns:

T1e (float) – longitudinal electron relaxation time in seconds
T2e (float) – transverse electron relaxation time in seconds

t1t2ne.scripts.f_ParaMeters_relax.contactrelax(A, B, taue=1e-09, tau_M=inf, S=0.5, tauv=None, deltat=None, AMe=None, I=None, g=None)[source]

Calculate the contact relaxation rates R1 and R2 using a simple contact relaxation model, with optional transient zero-field splitting (ZFS) contributions.

Parameters:

A (float) – contact coupling constant in Hz
B (float) – magnetic field strength in Tesla
taue (float) – electron relaxation time in seconds (default 1e-9 s)
tau_M (float) – exchange correlation time in seconds (default np.inf)
deltat (float or None) – transient ZFS parameter in cm^-1 (default None, no transient ZFS)
tauv (float or None) – correlation time for transient ZFS fluctuations in seconds (default None, no transient ZFS)
S (float) – electron spin quantum number (default 0.5, one unpaired electron)

Returns:

R1_con (float) – longitudinal relaxation rate in s^-1
R2_con (float) – transverse relaxation rate in s^-1

t1t2ne.scripts.f_ParaMeters_relax.curie(B, r, S=0.5, nuc='1H', T=298, tau_r=1e-09, tau_M=inf, chi=None, sigma=None)[source]

Curie-spin relaxation is implemented with two set of equations: - If the \(\chi\) tensor is not provided the Curie-spin relaxation is computed using equations 4.30 and 4.31 in Bertini et al. 2016, which assume an isotropic susceptibility. - If the \(\chi\) tensor is provided, the Curie-spin relaxation is computed using equations 17-19 in Suturina et al. 2018, which account for anisotropy in the susceptibility.

Parameters:

B (float) – magnetic field strength in Tesla
r (float or np.ndarray) – distance between the nucleus and the paramagnetic center in Angstroms as norm or as vector
S (float) – electron spin quantum number (default 0.5, one unpaired electron)
nuc (str) – nucleus type (default ‘1H’)
T (float) – temperature in Kelvin (default 298 K)
chi (np.ndarray, optional) – susceptibility tensor (default None, isotropic)
sigma (float, optional) – chemical shielding anisotropy (default None, not used)

Returns:

R1_curie (float) – longitudinal relaxation rate in s^-1
R2_curie (float) – transverse relaxation rate in s^-1

t1t2ne.scripts.f_ParaMeters_relax.rotational_taue(g, B, tauv, A=None, I=None)[source]

Calculate the rotational contribution to the electron relaxation times.

Parameters:

g (float or array-like) – Electron g-factor or g-tensor
B0 (float) – Magnetic field strength in Tesla
tauv (float) – Correlation time for rotational fluctuations in seconds
A (float or array-like, optional) – Hyperfine coupling constant or tensor
I (float or array-like, optional) – Nuclear spin quantum number or tensor

Returns:

R1e (float) – Electron longitudinal relaxation rate in s^-1
R2e (float) – Electron transverse relaxation rate in s^-1

t1t2ne.scripts.f_ParaMeters_relax.transient_zfs(deltat, B, tauv, S)[source]

Calculate the transient zero-field splitting (ZFS) contributions to the electron relaxation rates T1e and T2e using the Bloembergen-Morgan model:

\[\begin{split}R_{1e} = \frac{2}{15} \Delta_t^2 \left(4S(S+1) - 3\right) \left(J(\omega_e) + 2J(2\omega_e)\right) \\ R_{2e} = \frac{1}{15} \Delta_t^2 \left(4S(S+1) - 3\right) \left(3J(0) + 5J(\omega_e) + J(2\omega_e)\right)\end{split}\]

Parameters:

deltat (float) – transient ZFS parameter in cm^-1
B (float) – magnetic field strength in Tesla
tauv (float) – correlation time for transient ZFS fluctuations in seconds
S (float) – electron spin quantum number

Returns:

R1e (float) – electron longitudinal relaxation rate in s^-1
R2e (float) – electron transverse relaxation rate in s^-1