Spin Dynamics PhD Level

• 6 points: Gone in Spring 1997
• # Compendium

Quantum Dynamics in Simple Spin Systems. A Density-Matrix Treatment within the Liouville Formalism.
by Nikolas P. Benetis

A functional formalism to calculate experimental measureables using Quantum mechanics, Mathematical Statistics and Group Theory.

Autumn 1987 Linkoping University, IFM Revised January 1994. Currently under revision, 1997.

• 0 Spin Dynamics
• Spin dynamics, studies the forces which affect the spins and the motions that the above forces cause on the spins. As a "by-product" one gains information about the motions and the organization in the lattice, i.e. the environment of the spins.
Magnetic resonance experiments like for example NMR Relaxation and ESR Lineshapes provide one with dynamics information about the molecular system under investigation.
Relaxation depends on the interaction of the spins with the lattice.
Except from the dc magnetic field, each spin in the sample feels also a local field, which is time dependent due to the motions of the lattice such as rotation, vibration, translation, chemical reorganization or exchange etc.
In a magnetic resonance experiment we observe the total magnetic moment of a large collection, ensemble, of particles or in other words the macroscopic magnetization. The number of particles in a sample is of the order of the Avogadro number 10
23 units. It would be unpractical to give a detailed microscopic description of the motion for each one of these particles, even if we could obtain such an information.
We must be satisfied with a statistical description.
We can for example describe reorientation in the following way. We affirm that the average molecule keeps its orientation, relative to a laboratory coordinate system, a certain time interval in average. Such a time interval which is called in this case the rotational correlation time tau-R, is a parameter which describes the reorientation of an almost spherical molecule in a macroscopic sample. If the molecule has a more complicated shape several parameters are needed.
The rest of the possible motions can be described similarly.
Within this statistical description of motion, the interaction between the spins and the environment becomes random. Take for example the dipole-dipole interaction DD between a spin I which we directly observe, and another spin S in the same molecule, interacting with I. The strength of the DD interaction depends on the instantaneous orientation of the position vector between I and S with respect to the applied field. Because the molecule is reorienting in a random manner (tumbles) the strength of the DD interaction changes also in a similar fashion, it fluctuates. Such an interaction leads to relaxation. Relaxation broadens the lines of the NMR and the ESR signals.

Selected References

A. Abragam, The Principles of Nuclear Magnetism, Clarendon Press, Oxford,1961
R.R Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Clarendon Press, Oxford, 1987
Maria Engström,Theoretical Treatment of Pulses in Magnetic Resonance, Diploma work, Linköping University, 1995
F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill Int. Book Comp 1965 Donald A. McQuarrie, Statistical Mechanics, Harper & Row, NY 1976
R.G. Gordon, Adv. Magn. Res. 3 (1968) 1

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