The full B14 2-site CPMG model

This is the model for 2-site exchange exact analytical derivation on all time scales (with the constraint that
*p*_{A} > *p*_{B}), named after Baldwin (2014).
It is selected by setting the model to `B14 full'.
The equation is

where Appendix 1 in Baldwin (2014) lists the recipe for the exact calculation of
R_{2eff}.
Note that the following definitions are different to those in the original publication, but match both the reference implementation and the relax implementation.
The definitions are functionally equivalent.
First establish the complex free precession eigenfrequency with

The ground state ensemble evolution frequency *f*_{00} expressed in separated real and imaginary components in terms of definitions *ζ*, *Ψ*, and *h*_{4} is

f_{00} = R_{2A}^{0} + R_{2B}^{0} + k_{ex} + Δω - h_{4}. |
(11.38) |

Define substitutions for `stay' and `swap' factors as

The weighting factors for frequencies *E*_{0-2} emerging from a single CPMG block, *F*_{0-2}, are

Here
*τ*_{CPMG} = 1/4*ν*_{CPMG}.
The final result, with identities to assist efficient matrix exponentiation optimised for numerical calculation, is

The advantage of these equations is that you will always obtain the correct answer provided you have 2-site exchange, in-phase magnetisation and on-resonance pulses.

The term *p*_{D} is based on product of the off diagonal elements in the CPMG propagator, see supplementary Section 3 (Baldwin, 2014).

It is interesting to consider the region of validity of the Carver and Richards result. The two results are equal when the correction is zero, which is true when

ν_{2} +2k_{AB}p_{D}. |
(11.42) |

This occurs when
k_{AB}*p*_{D} tends to zero, and so
*ν*_{2} = *ν*_{3}.
Setting
k_{AB}*p*_{D} 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
*p*_{A} *p*_{B}.
In practise, significant deviations from the Carver and Richards equation can be incurred if
*p*_{B} > 1%.
Incorporation of the correction term results in an improved description of the CPMG experiment over Carver and Richards (1972).

The reference for this equation is:

More information about the B14 full model is available from:

- the relax wiki at http://wiki.nmr-relax.com/B14_full,
- the API documentation at http://www.nmr-relax.com/api/3.2/lib.dispersion.b14-module.html,
- the relaxation dispersion page of the relax website at http://www.nmr-relax.com/analyses/relaxation_dispersion.html#B14_full.

The relax user manual (PDF), created 2024-06-08.