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 pA > pB), named after Baldwin (2014). It is selected by setting the model to `B14 full'. The equation is

\mathrm{R}_{\textrm{2eff}}& = \frac{\mathrm{R}...
...}}(\nu_2 + 2\textrm{k}_{\textrm{AB}}p_D)\right)} ,

where Appendix 1 in Baldwin (2014) lists the recipe for the exact calculation of R2eff. 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

\Delta \mathrm{R}_2^0& = \mathrm{R}_{\mathrm{2...
...\sqrt{2}}\sqrt{-\Psi + \sqrt{\zeta^2 + \Psi^2} } .

The ground state ensemble evolution frequency f00 expressed in separated real and imaginary components in terms of definitions ζ, Ψ, and h4 is

f00 = $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \left(\vphantom{\mathrm{R}_{\mathrm{2A}}^0+ \mathrm{R}_{\mathrm{2B}}^0+ \textrm{k}_{\textrm{ex}}}\right.$R2A0 + R2B0 + kex$\displaystyle \left.\vphantom{\mathrm{R}_{\mathrm{2A}}^0+ \mathrm{R}_{\mathrm{2B}}^0+ \textrm{k}_{\textrm{ex}}}\right)$ + $\displaystyle {\frac{{\imath}}{{2}}}$$\displaystyle \left(\vphantom{\Delta\omega - h_4}\right.$Δω - h4$\displaystyle \left.\vphantom{\Delta\omega - h_4}\right)$. (11.38)

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

N & = h_3 + \imath h_4 , \\
NN^* & = h_3^2 + ...
...ft(2\Delta\omega ^2 + \imath \zeta\right) / NN^* .

The weighting factors for frequencies E0-2 emerging from a single CPMG block, F0-2, are

E_0 & = 2 \tau_{\textrm{CPMG}}\cdot h_3 , \\
...& = (h_3 - \imath h_4) \cdot \tau_{\textrm{CPMG}}.

Here τCPMG = 1/4νCPMG. The final result, with identities to assist efficient matrix exponentiation optimised for numerical calculation, is

\nu_{1c} & = F_0 \cosh(E_0) - F_2 \cos(E_2) , ...
...c{1}{T_{\textrm{relax}}} \log\left(\Re(T)\right) .

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 pD 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

$\displaystyle \sqrt{{\nu_{1c}^2-1}}$ $\displaystyle \approx$ ν2 +2kABpD. (11.42)

This occurs when kABpD tends to zero, and so ν2 = ν3. Setting kABpD 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 $\gg$ pB. In practise, significant deviations from the Carver and Richards equation can be incurred if pB > 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 user manual (PDF), created 2024-06-08.