Solving non-linear frequency problem, NMR

[zeta]

Hi All;
Long time reader, first time poster.
Here's a tricky one. I have a solid state NMR set of data in 3D. Let's say the signal phases
in the three dimensions are:
exp(i*(A+B*C+C)t_1)
exp(i*(A+B*C+C)t_2)
exp(i*(A+B*C+C)t_3)

given the information, one should be able to (in theory) determine A,B,C... however given the fact that there is a patently non-linear term this is non-trivial. Any ideas?

 

[marcusl]

Since A, B and C enter the same way in every equation, they cannot be separated.

 

[zeta]

sorry, I should have included the fact that there are coefficients in front of the unknowns ie.,
one can write the coefficients as a 3*3 matrix which is singular.
The frequencies in the phase are derived from Average Hamiltonian theory
thanks for the interest!

ED: error

 

[zeta]

ps if someone can think of a transform or method for solution, then you will at the very least get an acknowledgment on a paper

 

[marcusl]

Still don't get it. Do the coefficients differ in the 3 eqns? Are they known?

It would help if you can make a more detailed and exact statement of the problem.

 

[zeta]

Here's what the phases amount to in terms of equations:

a*A+b*B*C+c*C
d*A+e*B*C+f*C
g*A+h*B*C+i*C

I would like to solve for A,B,C given the coefficients, which depend on the spin angular momentum quantum numbers of the transitions (r <-> c), as well as Clebsch-Gordon coefficients.
In the mid nineties frydman introduced the shearing transformation whereby in 2d NMR one may remove the quadrupolar anisotropy ( term C) by essentially eliminating this from line 2 (indirect detected dimension, delta m \neq +/- 1) by multiplying line two by a multiple of line one (direct detected dimension, delta m = +/- 1).
In the indirect dimension the frequency is thus a linear sum of the isotropic contributions, terms A, B.

ED: errors

hence the MO for doing 3d is to completely isolate the isotropic frequencies

 

[marcusl]

I'm not worried about the non-linearity, but if your coefficient matrix is truly singular then i think you are hosed. Can you vary your experiment to get a matrix that's non-singular? (Well-conditioned is better yet!)

Can you get additional info some other way? My NMR knowledge is poor so I'll probably embarrass myself with this question, but how about double-quantum coherence?

 

[zeta]

yes, 'hosed' is the right word for it. The underlying problem is the linear dependence of the C-G coefficients unfortunately. However I think you hit the nail viz more information and this is the tack I was taking. The second line is in fact double quantum (the third triple) and my thought was to lift the linear dependence by including third order effects on this non-symmetric transition (3/2<->-1/2), but then my elegant paper/experiment goes to hell. such is life I guess...

thanks for the advice!

 

[marcusl]

That's usually the way it goes. If it were easy someone would already have done it...

 

[zeta]

yeah, ain't physics grand :)