Grumpy, on 20 January 2013 - 08:48 AM, said:
This sounds like a physics question.
I was sleeping until now. West coast time.
ThirtyEight, on 20 January 2013 - 07:02 AM, said:
Thus a shorter correlation time means the molecule is moving faster. But why does this generate a wide range of frequencies with relatively low amplitude, while slow motion generates a much more limited range of frequencies, but these have higher amplitude? And how does this explain the T1, T2 vs Tc graph (the one that has a straight like for T2 and a v-shaped line for T1)?
The correlation times are dependent upon spectral density functions that are Fourier transforms of the detected signals. In other words, you do an FT to get the signal in frequency space (a spectrum) from the signal that's measured in the time domain. In any FT, if the length of a time-dependent signal is short (ie, pulsed quickly), the frequency spectrum of that signal is broad, contains high frequency components, and is low amplitude for any given part of the spectrum (because it's broad). If the signal is long and coherent, the FT gives a spectrum in frequency space that is sharp with high amplitude, highly peaked, and not broad at all. FT fun fact: If two signals have the same total energy, but one is a fast pulse and the other is a slow pulse, the integral of the signal in the frequency domain will be equal for both- that's why the amplitude is lower for the broader signal.
Now, the plots you describe for the dependencies of T1 (spin-lattice) and T2 (spin-spin) sound like the could be vs temperature or Tc (they're inversely related). Tc can, in simplest terms, be considered to be opposite of temperature: high temp=low Tc, low temp=high Tc. In either case, it's all about how easy it is for the group of particles to fall from the excited state to the lower state. For a given T or Tc, the magnetized and unmagnetized energy states have a given energy gap. When a particle crosses that energy gap from high to low, it gives off energy. To cross that gap from low to high, it requires energy.
In the spin-spin case, it's pretty simple: the magnet's switched off, and particles begin to fall from the high energy state to the low energy state. If the group of particles is cold, many particles will jump down and stay down, not many particles at all will be able to jump back up to the higher state randomly. If the group of particles is hot, particles will jump down, but many will have enough energy to randomly jump back up- so it takes a longer time to relax.
The spin-lattice case is more complex. Here, we have to think about how easy it is for the relaxing particles to give their energy to the surrounding particles. At low temperature, there's not many vibrational and rotational energy states that the lattice particles can access, so the probability that the relaxing particles can put energy into one of those states is low. That makes for a long T1 time. As temperature increases, more rotational and vibrational states become available, making it more likely that the energy can be transferred. T1 time is decreased. But if the relaxing particles go to extremely high temperatures, the energy gap of the relaxation is too big for most of the vibrational and rotational states in the lattice, and the probability of energy transfer is decreased. That makes T1 large again.
ThirtyEight, on 20 January 2013 - 07:02 AM, said:
And finally, one explanation (i think) for this was that basically dipole-dipole relaxation is short range and if tc is small, then the molecule is moving too fast to interact with neighboring molecules. However, I thought interactions were controlled by the frequency of the two molecules, not by proximity?
Dipole-dipole interactions are short range, but energy transfer is also dependent upon the temp/Tc of the of the particles, so both proximity and frequency are important. Imagine you were looking at a stoplight through a green piece of glass, and you wanted to see a lot of light- you'd have to be close enough to the light to see it (not miles away) AND the light would have to be green for you to see it (not red). In this analogy, how far you are from the light is the proximity of particles, what piece of glass (filter) you use represents the energy states available in the other particle or lattice, and the color the of the light represents the frequency emitted by the relaxation. To get low T1 times, you need to be close, and you need to emit energy at a frequency that can be absorbed by the other particle.
Hope this helps, I haven't had breakfast yet.
SDS, on 20 January 2013 - 10:33 AM, said:
This sounds like straight Fourier theory when moving from the time domain to the frequency domain. A delta function (instantaneous event) transforms to the frequency domain as a constant (ie - it contains equal amounts of all frequencies). So, the faster the event, the more frequencies that need to be involved. Likewise, an event that is more spread out doesn't need as many higher frequencies. I assume the amplitudes issue is a conservation of energy thing...
Yep, that's right.
Edited by IKnowPhysics, 20 January 2013 - 01:26 PM.