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[http:The paper main equation 50 <br><math>R_{2,\textrm{eff}} = \frac{R_2^A+R_2^B+k_{\textrm{EX}}}{2}-\frac{N_{\textrm{CYC}}}{T_{\textrm{rel}}}\cosh{}^{-1}(v_{1c}) - \frac{1}{T_{\textrm{rel}}}\ln{\left( \frac{1+y}{2} + \frac{1-y}{2\sqrt{v_{1c}^2-1}}(v_2 + 2k_{\textrm{AB}}p_D )\right)} \\= R_{2,\textrm{eff}}^{\textrm{CR72}} - \frac{1}{T_{\textrm{rel}}}\ln{\left( \frac{1+y}{2} + \frac{1-y}{2\sqrt{v_{1c}^2-1}}(v_2 + 2k_{\textrm{AB}}p_D )\right)}</math> Which have these following definitions<br><math>v_{1c} = F_0\cosh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\cosh{\left(\tau_{\textrm{CP}}E_2\right)} \\v_{1s} = F_0\sinh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\sinh{\left(\tau_{\textrm{CP}}E_2\right)} \\v_{2}N = v_{1s}\left(O_B-O_A\right)+4O_B F_1^a \sinh{\left(\tau_{\textrm{CP}}E_1\right)} \\p_D N = v_{1s} + \left(F_1^a+F_1^b\right)\sinh{\left(\tau_{\textrm{CP}}E_1\right)}\\v_3 = \left( v_2^2 + 4 k_{\textrm{BA}} k_{\textrm{AB}} p_D^2 \right)^{1/www.nmr2} \\y = \left( \frac{v_{1c}-relax.com/manualv_3}{v_{1c}+v_3} \right)^{N_{\textrm{CYC}}}</Dispersion_model_summary.html Please see the summary of the model parameters here.]math>
B14
,The Baldwin 2014 2-site exact solution relaxation dispersion model for [[SQ CPMG-type data]]for all time scales whereby the simplification {{:R2Azero}} = {{:R2Bzero}} is assumed. This model is labelled as '''B14''' in [[Relaxation dispersion citation for relax|relax]]. The advantage of this code will be that you'll always get the right answer provided you've got 2-site exchange, in-phase magnetisation and on-resonance pulses. == Comments from the Author ==During the implementation of model B14, Andrew Baldwin wished to raise the attention to: '''1) Danger of errors of approximations'''.<br>'''2) That optimal accuracy analysis should include off resonance effects'''. Please read:* {{#lst:Citations|MyintIshima09}}* {{#lst:Citations|Bain11}} '''Differential relaxation effects''':<br> Please read:* {{#lst:Citations|Vallurupalli07}}* {{#lst:Citations|Vallurupalli08}}
== Equation ==
Note, <math> E_2 </math> is complex. If |x| denotes the complex modulus:<br>
<math>
\cosh{\left(\tau_{\textrm{CP}}E_2\right)} = \cos{\left(\tau_{\textrm{CP}}|E_2|\right)} \\\sinh{\left(\tau_{\textrm{CP}}E_2\right)} = i \sin{\left(\tau_{\textrm{CP}}|E_2|\right)}
</math>
The term <math>p_D</math> 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<br>
<math>
\sqrt{v_{1c}^2-1} \approx v_2 + 2k_{\textrm{AB}}p_D
</math><br>
This occurs when <math>k_{\textrm{AB}}p_D</math> tends to zero, and so <math>v_2=v_3</math>.<br>
Setting <math>k_{\textrm{AB}}p_D</math> to zero, amounts to neglecting magnetisation that starts on the ground state ensemble and end on the excited state ensemble and vice versa. <br>
This will be a good approximation when <math>p_A \gg p_B</math>.<br>
In practise, significant deviations from the Carver Richards equation can be incurred if <math>p_B > 1\%</math>.<br>
Incorporation of the correction term into equation (50), results in an improved description of the CPMG experiment over the Carver Richards equation.
=== Equation compared to Carver Richards 72 ===
[http://www.nmr-relax.com/manual/Dispersion_model_summary.html Please see the relax summary of the model parameters here.]
These definitions comes from the papers "Supplementary Section 4. Relation to Carver Richards equation".
<math>
\tau_{\textrm{CP}} = \frac{1}{4\nu_\textrm{CPMG}} \\
\alpha_- = k_{\textrm{AB}} - k_{\textrm{BA}} \\
\zeta = 2 \Delta \omega \, \alpha_- = h_1\\
\Psi = \alpha_-^2 + 4 k_{\textrm{BA}} k_{\textrm{AB}} - \Delta \omega^2 = h_2\\
\xi = \frac{2\tau_{\textrm{CP}}}{\sqrt{2}}\sqrt{\Psi + \sqrt{\Psi^2 + \zeta^2}} = 2h_3 \tau_{\textrm{CP}} = \tau_{\textrm{CP}}E_0\\
\eta = \frac{2\tau_{\textrm{CP}}}{\sqrt{2}}\sqrt{-\Psi + \sqrt{\Psi^2 + \zeta^2}} = 2h_4 \tau_{\textrm{CP}} = \tau_{\textrm{CP}}E_2\\
D_+=\frac{1}{2}\left(1+\frac{\Psi+2\Delta \omega^2}{\sqrt{\Psi^2+\zeta^2}} \right) = F_0 \\
D_-=\frac{1}{2}\left(-1+\frac{\Psi+2\Delta \omega^2}{\sqrt{\Psi^2+\zeta^2}} \right) = F_2
</math>
Physical meanings:<br>
<math>\xi</math> and <math>\eta</math> are differences of the real and imaginary components of the free precession frequencies <math>f_{00}</math> and <math>f_{11}</math> (equation (41)).<br>
<math>D_+</math> and <math>D_-</math> are the stay/stay (<math>F_{0}</math>), and swap/swap (<math>F_{2}</math>) coefficients (equation (36)).<br>
<math>\zeta</math> and <math>\Psi</math> are parameters that enable the free precession frequencies to be written in a more concise form (equation (12)).<br>
Note that in reference 2 ([http://dx.doi.org/10.1016/S0076-6879(01)39315-1 10.1016/S0076-6879(01)39315-1]), in equation 25, the definition used for <math>\tau_{\textrm{CP}}</math> is twice that used in this work but is otherwise identical.
== Parameters ==
The B14 model has the parameters {$R_2^0${{:R2zero}}, $...$, $p_A${{:pA}}, $\Delta\omega${{:Deltaomega}}, $k_{ex{:kex}}$}. == Code ==[http://search.gmane.org/?query=https%3A%2F%2Fgna.org%2Fsupport%2F%3F3154&author=&group=gmane.science.nmr.relax.scm&sort=revdate&DEFAULTOP=and&TOPDOC=10&xP=https%09gna%09org%09support%093154&xFILTERS=Gscience.nmr.relax.scm---A Commits that created the code, approx 20-30 commits, with corrections. ] [https://gna.org/support/?3154 Here is gna.org tracker that followed this code. It includes valuable comments.] The library code: {{relax url|path=lib/dispersion/b14.py}} The target function func_B14(): {{relax url|path=target_functions/relax_disp.py}} === Example graph of test data ===[[File:CPMG Model B14 Example data disp 1 H.png|thumb|center|upright=2|Graph from systemtest "relax -s Relax_disp.test_baldwin_synthetic"]]
== Reference ==
The reference for the B14 model is:
* Carver, J. and Richards, R. (1972). General 2-site solution for chemical exchange produced dependence of T2 upon Carr-Purcell pulse separation. ''J. Magn. Reson.'', '''6'''(1), 89-105. ([http{{#lst://dx.doi.org/10.1016/0022-2364(72)90090-X 10.1016/0022-2364(72)90090-X]).Citations|Baldwin14}}
== Related models ==
The B14 model is a linear correction to the CR72 [[CR72]] model, and algorithms based on this have significant advantages in both precision and speed over existing formulaic approaches.
== Links ==
The [[Relaxation dispersion citation for relax|implementation of the CR72 B14 model in relax]] can be seen in the:* [http://www.nmr-relax.com/manual/The_reduced_B14_2_site_CPMG_model.html relax manual],* [http://www.nmr-relax.com/api/3.1/lib.dispersion.b14-module.html API documentation],* [http://www.nmr-relax.com/analyses/relaxation_dispersion.html#B14 relaxation dispersion page of the relax website].
== See also ==
[[Category:Models]][[Category:Dispersion models]][[Category:Relaxation_dispersionanalysis]]
