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Solving non-linear frequency problem, NMR

  1. Jan 12, 2007 #1
    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?
     
  2. jcsd
  3. Jan 12, 2007 #2

    marcusl

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    Since A, B and C enter the same way in every equation, they cannot be separated.
     
  4. Jan 12, 2007 #3
    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
     
  5. Jan 12, 2007 #4
    ps if someone can think of a transform or method for solution, then you will at the very least get an acknowledgement on a paper
     
  6. Jan 12, 2007 #5

    marcusl

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    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.
     
  7. Jan 12, 2007 #6
    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
     
    Last edited: Jan 12, 2007
  8. Jan 12, 2007 #7

    marcusl

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    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 embarass myself with this question, but how about double-quantum coherence?
     
  9. Jan 12, 2007 #8
    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!
     
  10. Jan 12, 2007 #9

    marcusl

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    That's usually the way it goes. If it were easy someone would already have done it...
     
  11. Jan 12, 2007 #10
    yeah, ain't physics grand :)
     
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