The Entropy of a Nuclear Spin J. Lee University of Manchester Monchester, England
System in a Magnetic Field An application of statistical mechanics
Entropy is popularly conceived as a measure of order in a system. This conception leads a t first sight to an erroneous qualitative impression of the entropy of a system of independent nuclear magnets. I n the absence of a magnetic field, we conceive a completely random set of orientations of the nuclei. I n the presence of a magnetic field, however small, the nuclear moments acquire a finite numher of angular orientations relative to the direction of the magnetic field. To take the simplest case, where I = on the basis of the most nafve point of view ( I ) , we conceive two nuclear orientations, one parallel and the other antiparallel to the field. On this same level of view, nuclei with I > are associated with 21 1 field-moment angles, 0, given by
+
cos
e
=
m/I
(1)
where I is the nuclear spin quantum numher, an integer or half-integer characteristic of the nucleus and
r n = I , ~ - I ,. . . , - I
Associated then with the 21 - 1 non-extreme angles is the idea that the nuclear moments precess (2) about the field direction with an angular velocity proportionate to the field magnitude. On a more refined basis, we invoke no parallel or antiparallel momentfield orientations, equation ( 1 ) being replaced (3) by COB
e
=
+
~ / [ I ( I 1)]%
(2)
and all orientations being subject to precession. Whether I = '/%orI > whether we adopt the nalve or the more refined viewpoint, there is no question about the fact that in comparison with the zero field situation, a magnetic field creates restricted orientation. Thus, if we proceed to considerations of entropy, we might conclude that the field reduces the entropy by increasing the order, and furthermore we would then encounter a discontinuity at zero field in the entropyfield relationship. This suspicious state of affairs leads one to doubt the basic premises of the argument and the purpose of this paper is to clarify the position. I n the presence of a magnetic field, we may distinguish the state of each nucleus by ( a ) its energy, this increasing with increasing deviation from field parallelity, or (b) its orientation as defined by the angle 0, hut not by the precessional angle, +, or (c) its nuclear spin wave function, each being nondegenerate. I n the zero field case, energies become equal and orientations random hut the wave functions, although now degenerate, persist and still serve as a means of distinction; in the absence of a reference direction, orientation in n o way plays a state-defining role. Essentially then, there is no gross distinction between zero and
non-zero field cases. A minor difference arises from a difference in the distribution between the one-nucleus quantum states, but this is a cmtinuous function of magnetic field intensity. The Spin-% Assembly
Let us illustrate the problem with the simplest situation of an assembly of equivalent nuclei of spin quantum numher '/I, for which there are two spin wave functions, a and 6. The algebraic expression for entropy as a function of magnetic field will he approached in two distinct ways, firstly from the point of view of the numher of a priori equally probable complexions, and secondly by use of the partition function concept. I n the zero-field situation, there will be an equal number of nuclei with the two wave functions, i.e., nuclei will he associated with a and with P, NObeing the Avogadro numher. Hence from the point of view of Maxwell-Boltzn~ann statistics, the number of complexions (4)
'/ao
'/ao
The question of whether the nuclei are localized or non-localized will not he considered here; it is assumed that an indistinguishability term, where appropriate, will be incorporated in a term associated with some other energy contribution, e.g., translation. Taking natural logarithms of equation (3) and then using Stirling's approximation, in x ! x In x - z, for large x (6):
-
Since the entropy S proximation S
=
R In 2
= =
k In W, we deduce the ap1.38 cill deg-' mole-'
(4)
The effect of an applied magnetic field of intensity, H, is to create two distinct magnetic interaction energies, + p a associated with function w and - p a associated with function (3, po being the maximum component of the magnetic moment in the field direction. From equation ( 2 ) , this component is related to the magnetic moment, p , by !a = A!
x
+
'/2/['/*('/% l ) l % .
If N, and N s are the numbers of nuclei in states w and 0, respectively, we have from the Boltzmann distribution law
Now a t normal temperatures and magnetic field intensities, d
