Spin Temperature in Magnetism and in Magnetic Resonance
In Chapter 5 we employed the concept of spin temperature to discuss relaxation. The idea of spin temperature was introduced by Casimir and du Pre [6.1] to give a thermodynamic treatment of the experiments of Gorter and his students on paramagnetic relaxat
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6.1 Introduction In Chapter 5 we employed the concept of spin temperature to discuss relaxation. The idea of spin temperature was introduced by Casimir and du Pre [6.1] to give a thermodynamic treatment of the experiments of Gorter and his students on paramagnetic relaxation. It was Van Vleck [6.2] who first employed the concept for a detailed statistical mechanical calculation of the relaxation times of paramagnetic ions. Both in this case, and also in his general statistical mechanical treatment of static properties of paramagnetic atoms [6.3], he recognized and emphasized the fact that expansion of the partition function Z in powers of liT enabled one to calculate Z without the necessity of solving for the energies and eigenfunctions of the Hamiltonian. Waller evidently was the first person to use this property [6.4]. From the partition function, one can compute all the static properties of the system, such as the specific heat, the entropy, the magnetization, and the energy. For example, the average energy of a system, E, at a temperature T is given by -
E
a InZ = kT 2 aT
.
(6.1)
In 1955, Redfield [6.5] showed that the conventional theory of saturation did not properly account for the experimental facts of nuclear resonance in solids. In one of the most important papers ever written on magnetic resonance, he showed that the conventional approach in essence violated the second law of thermodynamics. He went on to show that saturation in solids can be described simply by applying the concept of spin temperature to the reference frame that rotates in step with the alternating field HI. To understand his ideas, one needs to understand certain concepts which predate the discovery ,of magnetic resonance - ideas such as adiabatic demagnetization. ' In this chapter we begin by describing a simple experiment which displays the failing of the pre-Redfield theory of magnetic ~sonance. Then we turn to a discussion of the use of spin temperature in nonresonance cases to build background for the application of these same ideas in the rotating reference frame. We then discuss the Redfield theory of saturation in solids.
219 C. P. Slichter, Principles of Magnetic Resonance © Springer-Verlag Berlin Heidelberg 1990
6.2 A Prediction from the Bloch Equations Let us consider a simple resonance experiment with a rotating magnetic field of angular frequency w, transverse to the static field Ho, tuned exactly to resonance w=,Ho
.
(6.2)
We discuss it by means of the Bloch equations. It is convenient to transform to a reference frame rotating at w with HI. along the x-axis as is done in Sect. 2.8. Exactly at resonance, the Bloch equations become dMz dt
= -,MyHI + Mo -
TI
Mz
(6.3a) (6.3b) (6.3c)
Suppose we now orient M along HI so that at t =0 Mz = Mo, My = Mz =O. From (6.3b) we see that Mz will decay to zero in a time T2 • The low HI steadystate solution of the Bloch equations shows that they describe a Lorentzian line with a frequency width I
Llw = T2
.
(6.4)
For solids typically Llw 'a!' ,Hneighbor 'a!' '~ 'a!' a
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