Module cr72

The Carver and Richards (1972) 2-site all time scale exchange CR72 and CR72 full models.

Description

This module is for the function, gradient and Hessian of the CR72 and CR72 full models.

References

The model is named after the reference:

• Carver, J. P. and Richards, R. E. (1972). General 2-site solution for chemical exchange produced dependence of T2 upon Carr-Purcell pulse separation. J. Magn. Reson., 6, 89-105. (DOI: 10.1016/0022-2364(72)90090-X).

Equations

The equation used is:

   R2eff = 1/2 [ R2A0 + R2B0 + kex - 2.nu_cpmg.cosh^-1 (D+.cosh(eta+) - D-.cos(eta-)) ] ,

where:

          1 /        Psi + 2delta_omega^2 \ 
   D+/- = - | +/-1 + -------------------- | ,
          2 \        sqrt(Psi^2 + zeta^2) /

            2^(2/3)
   eta+/- = ------- sqrt(+/-Psi + sqrt(Psi^2 + zeta^2)) ,
            nu_cpmg

   Psi = (R2A0 - R2B0 - pA.kex + pB.kex)^2 - delta_omega^2 + 4pA.pB.kex^2 ,

   zeta = 2delta_omega (R2A0 - R2B0 - pA.kex + pB.kex).

kex is the chemical exchange rate constant, pA and pB are the populations of states A and B, and delta_omega is the chemical shift difference between the two states in ppm.

CR72 model

Importantly for the implementation of this model, it is assumed that R2A0 and R2B0 are identical. This simplifies some of the equations to:

   R2eff = R20 + kex/2 - nu_cpmg.cosh^-1 (D+.cosh(eta+) - D-.cos(eta-) ,

where:

   Psi = kex^2 - delta_omega^2 ,

   zeta = -2delta_omega (pA.kex - pB.kex).

Links

More information on the CR72 model can be found in the:

• relax wiki,
• relax manual,
• relaxation dispersion page of the relax website.

More information on the CR72 full model can be found in the:

• relax wiki,
• relax manual,
• relaxation dispersion page of the relax website.

Functions

r2eff_CR72(r20a=None, r20b=None, pA=None, dw=None, kex=None, cpmg_frqs=None, back_calc=None, num_points=None)
Calculate the R2eff values for the CR72 model.

Variables

__package__ = 'lib.dispersion'

Imports: arccosh, cos, cosh, sqrt

Function Details

r2eff_CR72(r20a=None, r20b=None, pA=None, dw=None, kex=None, cpmg_frqs=None, back_calc=None, num_points=None)

Calculate the R2eff values for the CR72 model.

See the module docstring for details.

Parameters:

• r20a (float) - The R20 parameter value of state A (R2 with no exchange).
• r20b (float) - The R20 parameter value of state B (R2 with no exchange).
• pA (float) - The population of state A.
• dw (float) - The chemical exchange difference between states A and B in rad/s.
• kex (float) - The kex parameter value (the exchange rate in rad/s).
• cpmg_frqs (numpy rank-1 float array) - The CPMG nu1 frequencies.
• back_calc (numpy rank-1 float array) - The array for holding the back calculated R2eff values. Each element corresponds to one of the CPMG nu1 frequencies.
• num_points (int) - The number of points on the dispersion curve, equal to the length of the cpmg_frqs and back_calc arguments.