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The Baldwin (2014) 2site exact solution model for all time scales B14.
This module is for the function, gradient and Hessian of the B14 model.
The model is named after the reference:
The equation used is:
R2A0 + R2B0 + kex Ncyc 1 ( 1+y 1y ) R2eff =    * cosh^1 * v1c   ln(  +  * (v2 + 2*kAB*pD ) ) , 2 Trel Trel ( 2 2*sqrt(v1c^2 1 ) ) 1 ( 1+y 1y ) = R2eff(CR72)   ln(  +  * (v2 + 2*kAB*pD ) ) , Trel ( 2 2*sqrt(v1c^2 1 ) )
Which have these following definitions:
v1c = F0 * cosh(tauCP * E0) F2 * cosh(tauCP * E2) , v1s = F0 * sinh(tauCP * E0) F2 * sinh(tauCP * E2) , v2*N = v1s * (OBOA) + 4OB * F1^a * sinh(tauCP * E1) , pD N = v1s + (F1^a + F1^b) * sinh(tauCP * E1) , v3 = ( v2^2 + 4 * kBA * kAB * pD^2 )^1/2 , y = ( (v1cv3)/(v1c+v3) )^NCYC ,
Note, E2 is complex. If x denotes the complex modulus:
cosh(tauCP * E2) = cos(tauCP * E2) , sinh(tauCP * E2) = i sin(tauCP * E2) ,
The term pD is based on product of the off diagonal elements in the CPMG propagator (Supplementary Section 3).
It is interesting to consider the region of validity of the Carver Richards result. The two results are equal when the correction is zero, which is true when:
sqrt(v1c^21) ~ v2 + 2*kAB * pD ,
This occurs when 2*kAB * pD tends to zero, and so v2=v3. Setting "kAB * pD" to zero, amounts to neglecting magnetisation that starts on the ground state ensemble and end on the excited state ensemble and vice versa. This will be a good approximation when pA >> p_B.
In practise, significant deviations from the Carver Richards equation can be incurred if pB > 1 %. Incorporation of the correction term into equation (50), results in an improved description of the CPMG experiment over the Carver Richards equation.
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.
More information on the B14 model can be found in the:
More information on the B14 full model can be found in the:
Comparison to CR72 model can be found in the:
Comparison to CR72 full model can be found in the:





g_fact = 0.70710678118654746


__package__ =

Imports: arccosh, arctan2, array, cos, cosh, isfinite, log, max, power, sin, sinh, sqrt, sum

Calculate the R2eff values for the CR72 model. See the module docstring for details.

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