Nuclear Overhauser effect/Advanced: Difference between revisions

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{Def|Nuclear Overhauser effect}
Nuclear Overhauser effect: Noe(Definition) : {{Def|Nuclear overhauser effect}}


The following discussion is relevant for studies in solution/liquid where the molecules are undergoing rapid isotropic rotational motion.
<br/>
The Noe enhancement is quantitatively defined as  
The Noe enhancement is quantitatively defined as  
: <math> \eta = \frac{S_z - S_{z,equil}}{S_{z,equil}}            \qquad Eq. 1 </math>  
: <math> \eta = \frac{<S_z> - <S_{z,equil}>}{<S_{z,equil}>}            \qquad Eq. 1 </math>  


For a pair of nonidentical spins I and S, :  
For a pair of nonidentical spins I and S with dipolar interactions, subject to random perturbation from the environment (lattice), the expressions for the time dependence of the the expectation values of the magnetization calculated by using time dependent perturbation theory are the Solomon equations<ref>Quantum description of high resolution NMR in liquids. M.Goldman. Oxford.</ref>:
: <math> \frac{d<I_z>}{dt} = -\rho_I (<I_z> - <I_{z,equil}>) - \sigma (<S_z> - <S_{z,equil}>)  \qquad Eq. 2</math>
: <math> \frac{d<I_z>}{dt} = -\rho_I (<I_z> - <I_{z,equil}>) - \sigma (<S_z> - <S_{z,equil}>)  \qquad Eq. 2</math>
: <math> \frac{d<S_z>}{dt} = -\rho_S (<S_z> - <S_{z,equil}>) - \sigma (<I_z> - <I_{z,equil}>) \qquad Eq. 3 </math>
: <math> \frac{d<S_z>}{dt} = -\rho_S (<S_z> - <S_{z,equil}>) - \sigma (<I_z> - <I_{z,equil}>) \qquad Eq. 3 </math>
: <math> \sigma </math> is called the cross relaxation rate and is responsible for the Nuclear overhauser effect.
: <math> \sigma </math> is called the cross relaxation rate and is responsible for the Nuclear overhauser effect.
: <math> \rho_I = \frac{\gamma_I^2\gamma_S^2\hbar^2}{10 r^6 } ( J(w_I-w_S) + 3J(w_I) + 6 J(w_I + w_S) ) \qquad Eq. 4 </math>
: <math> \rho_I = \frac{\gamma_I^2\gamma_S^2\hbar^2}{10 r^6 } \big( J(w_I-w_S) + 3J(w_I) + 6 J(w_I + w_S) \big) \qquad Eq. 4 </math>
: <math> \sigma = \frac{\gamma_I^2\gamma_S^2\hbar^2}{10 r^6 } ( -J(w_I-w_S) +  6 J(w_I + w_S) )) \qquad Eq. 5 </math>
: <math> \sigma = \frac{\gamma_I^2\gamma_S^2\hbar^2}{10 r^6 } \big( -J(w_I-w_S) +  6 J(w_I + w_S) ) \big) \qquad Eq. 5 </math>


: <math> \frac{1}{T_2} = \frac{\gamma^2\gamma_S^2\hbar^2}{20 r^6 } ( 4J(0) + J(w_I - w_S) + 3J(w_I) + 6 J(w_I + w_S) + 6 J(w_S) ) \qquad Eq. 6 </math>


In the steady state <math> \frac{d<S_z>}{dt} = 0 </math>, when the resonance frequency of spin I is irradiated , <math> <I_z> = 0</math>, therefore:
In the steady state <math> \frac{d<S_z>}{dt} = 0 </math>; when the resonance frequency of spin I is irradiated , <math> <I_z> = 0</math>, therefore:
: <math> (<S_z> - <S_{z,equil}>)=  \frac{\sigma}{\rho_S} (<I_{z,equil}>)  \qquad (from Eq. 3) </math>
: <math> (<S_z> - <S_{z,equil}>)=  \frac{\sigma}{\rho_S} (<I_{z,equil}>)  \qquad (from \quad Eq. 3) </math>


Assuming that the expectation values of magnetization are proportional to the magnetogyric ratios:
Assuming that the expectation values of magnetization are proportional to the magnetogyric ratios we obtain:
: <math>\eta = \frac{<S_z> - <S_{z,equil}>}{<S_{z,equil}>} = \frac{\sigma}{\rho_S} \frac{\gamma_I}{\gamma_S} \qquad Eq. 7 </math>
: <math>\eta = \frac{<S_z> - <S_{z,equil}>}{<S_{z,equil}>} = \frac{\sigma}{\rho_S} \frac{\gamma_I}{\gamma_S} \qquad Eq. 6 </math>
This indicates that considerable enhancement in the intensity of the S signal can be obtained by irradiation at the frequency of the I spin, provided that <math> \frac{\gamma_I}{\gamma_S} > 1 </math>, because <math> \frac{\sigma}{\rho_S} \rightarrow 1/2 </math> when <math> w\tau_c << 1 </math>.  
This indicates that considerable enhancement in the intensity of the S signal can be obtained by irradiation at the frequency of the I spin, provided that <math> \frac{\gamma_I}{\gamma_S} > 1 </math>, because <math> \frac{\sigma}{\rho_S} \rightarrow 1/2 </math> when <math> w\tau_c << 1 </math>.  
However, when <math> w\tau_c >> 1 </math>, <math> \frac{\sigma}{\rho_S} \rightarrow -1 </math> and negative Noe enhancements are obtained.   
However, when <math> w\tau_c >> 1 </math>, <math> \frac{\sigma}{\rho_S} \rightarrow -1 </math> and negative Noe enhancements are obtained.   
<br/>
<br/>
The sign of <math> \eta </math> changes from positive to negative when <math> w\tau_c </math> is close to one and under such conditions the Noe effect may not be observable.  This happens for rigid molecules with relative molecular mass about 500 at room temperature e.g. many hexapeptides.
The sign of <math> \eta </math> changes from positive to negative when <math> w\tau_c </math> is close to one and under such conditions the Noe effect may not be observable.  This happens for rigid molecules with relative molecular mass about 500 at room temperature e.g. many hexapeptides.
== References ==
<references/>
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An advanced level version of Nuclear Overhauser effect.

Nuclear Overhauser effect: Noe(Definition) : Change in intensity of a signal when irradiation is carried out at the resonance frequency of a spatially proximal nucleus.

The following discussion is relevant for studies in solution/liquid where the molecules are undergoing rapid isotropic rotational motion.


The Noe enhancement is quantitatively defined as

For a pair of nonidentical spins I and S with dipolar interactions, subject to random perturbation from the environment (lattice), the expressions for the time dependence of the the expectation values of the magnetization calculated by using time dependent perturbation theory are the Solomon equations[1]:

is called the cross relaxation rate and is responsible for the Nuclear overhauser effect.


In the steady state ; when the resonance frequency of spin I is irradiated , , therefore:

Assuming that the expectation values of magnetization are proportional to the magnetogyric ratios we obtain:

This indicates that considerable enhancement in the intensity of the S signal can be obtained by irradiation at the frequency of the I spin, provided that , because when . However, when , and negative Noe enhancements are obtained.
The sign of changes from positive to negative when is close to one and under such conditions the Noe effect may not be observable. This happens for rigid molecules with relative molecular mass about 500 at room temperature e.g. many hexapeptides.

References

  1. Quantum description of high resolution NMR in liquids. M.Goldman. Oxford.