Rotational relaxation: Difference between revisions
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however. Rather, relaxation times of the second | however. Rather, relaxation times of the second | ||
[[spherical harmonics|spherical harmonic]] are obtained: | [[spherical harmonics|spherical harmonic]] are obtained: | ||
:<math> | :<math>c_2(t)=\langle P_2( \mathbf{n}(0)\cdot\mathbf{n}(t) ) \rangle,</math> | ||
where <math>P_2(x)</math> is the second [[Legendre polynomials|Legendre polynomial]]. | where <math>P_2(x)</math> is the second [[Legendre polynomials|Legendre polynomial]]. | ||
According to simple [[rotational diffusion]] theory, the relaxation time | According to simple [[rotational diffusion]] theory, the relaxation time | ||
for <math>c_1(t)</math> would be given by | for <math>c_1(t)</math> would be given by | ||
<math>\tau_1 = 1 | <math>\tau_1 = \frac{1}{2D_\mathrm{rot}}</math>, and the relaxation time for | ||
<math>c_2(t)</math> would be <math>\tau_2 = 1 | <math>c_2(t)</math> would be <math>\tau_2 = \frac{1}{6D_\mathrm{rot}}</math>. | ||
Therefore, <math>\tau_1= 3 \tau_2</math>. This ratio is actually lower in simulations, | Therefore, <math>\tau_1= 3 \tau_2</math>. This ratio is actually lower in simulations, | ||
and closer to <math>2</math>; the departure from a value of 3 signals rotation | and closer to <math>2</math>; the departure from a value of 3 signals rotation | ||
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| HH || H-H axis || <math>\tau_2=2.0</math>ps (H-H dipolar relaxation NMR) | | HH || H-H axis || <math>\tau_2=2.0</math>ps (H-H dipolar relaxation NMR) | ||
|- | |- | ||
| OH || O-H axis || <math>\tau_2=1.95</math>ps (< | | OH || O-H axis || <math>\tau_2=1.95</math>ps (<sup>17</sup>O-H dipolar relaxation NMR) | ||
|- | |- | ||
| <math>\mu</math> || dipolar axis || not measurable, but related to bulk dielectric relaxation | | <math>\mu</math> || dipolar axis || not measurable, but related to bulk dielectric relaxation |
Latest revision as of 07:23, 21 October 2016
Rotational relaxation refers to the decay of certain autocorrelation magnitudes related to the orientation of molecules. If a molecule has an orientation along a unit vector , its autocorrelation will be given by
From the time decay, or relaxation, of this function, one may extract a characteristic relaxation time (either from the long-time exponential decay, or from its total integral, see autocorrelation). This magnitude, which is readily computed in a simulation is not directly accessible experimentally, however. Rather, relaxation times of the second spherical harmonic are obtained:
where is the second Legendre polynomial.
According to simple rotational diffusion theory, the relaxation time for would be given by , and the relaxation time for would be . Therefore, . This ratio is actually lower in simulations, and closer to ; the departure from a value of 3 signals rotation processes "rougher" than what is assumed in simple rotational diffusion (Ref 1).
Water[edit]
- Main article Rotational relaxation of water
Often, molecules are more complex geometrically and can not be described by a single orientation. In this case, several vectors should be considered, each with its own autocorrelation. E.g., typical choices for water molecules would be:
symbol | explanation | experimental value, and method |
HH | H-H axis | ps (H-H dipolar relaxation NMR) |
OH | O-H axis | ps (17O-H dipolar relaxation NMR) |
dipolar axis | not measurable, but related to bulk dielectric relaxation | |
normal to the molecule plane | not measurable |