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FINE STRUCTURE OF NUCLEAR MAGNETIC RESONANCE ABSORPTION
In several papers that appeared during the last two years,[^1-5] the fine structure of the nuclear magnetic absorption line of protons in various substances has been studied.
The experimental arrangement in these works was the same as in earlier studies of nuclear magnetic resonance absorption.[^6] A large homogeneous magnetic field \(H_0\) (of the order of several thousand oersteds) is applied to the system; this removes the spatial degeneracy of the nuclear spins and gives \(2I+1\) equally spaced Zeeman levels (\(I\) is the nuclear spin, equal to \(1/2\)). In the papers reviewed, substances containing hydrogen were studied; in this case application of the field \(H_0\) gives 2 spin states of the proton \(\left(I=\frac{1}{2}\right)\), the energy separation between which is:
\[ \hbar \omega_0 = 2\mu H_0, \]
where \(\mu\) is the magnetic moment of the proton. Introducing the gyromagnetic ratio
\[ \gamma = \frac{\mu}{I} = \frac{2\mu}{\hbar} \]
(equal for the proton to \(2.7 \cdot 10^4\ \text{oersted}^{-1}\ \text{sec}^{-1}\)), we obtain:
\[ \omega_0 = \gamma H_0. \]
Perpendicular to the field \(H_0\), an alternating field of small amplitude \(H_1 \cos \omega t\) \((H_1 \ll H_0)\) is applied. If the frequency of this field \(\omega\) is close
to \(\omega_0\), then the alternating field will induce transitions of protons from one spin state to another. As a result of the greater “population” of the spin level with lower energy, absorption of the energy of the radio-frequency field by the system of nuclear spins will occur.
The frequency \(\omega\) is kept constant and the field \(H_0\) is modulated about the value \(H^*=\dfrac{\omega}{\gamma}\). In this case, the absorption coefficient is obtained on the oscilloscope as a function of \(H_0\).
Usually the absorption line has one maximum at exact resonance, i.e. at \(H_0=H^*=\dfrac{\omega}{\gamma}\).
On each proton, in addition to the field \(H_0\), there acts a field from neighboring protons. This local field is of the order \(\dfrac{\mu}{a^3}\), where \(a\) is the distance between the two nearest protons. If numerical values are substituted, it is found that the local field is of the order of 10 oersteds. It is clear that this local field will broaden the absorption line, and the width of the line will be of the order of the local field. A detailed theory of the width of the magnetic absorption line was developed by Van Vleck\(^7\).
In the works reviewed, substances were studied in which, owing to certain features, the line of magnetic absorption of protons possesses not one but several maxima.
In Pake’s work\(^1\), the resonance of protons in gypsum \((\mathrm{CaSO}_4\cdot 2\mathrm{H}_2\mathrm{O})\) was studied. Both single crystals and gypsum powders were investigated; we shall confine ourselves only to presenting the results obtained for single crystals.
It turned out that the observed pattern depends on the orientation of the crystal relative to the field \(H_0\). In the general case the absorption line has four maxima; however, for some orientations of the crystal the number of maxima is equal to one or two.
Gypsum is a monoclinic crystal. In Pake’s experiments the magnetic field \(H_0\) was rotated in the plane \((001)\) (\([010]\) is the direction of the monoclinic axis). Let us denote by \(\varphi\) the angle between \(H_0\) and the direction \([100]\).
The qualitative explanation of the fine structure is that the strongest influence on each proton is exerted by the local field from the proton located in the same water molecule. Therefore, in a first approximation, for the resonant value of the field \(H_0\) only the projection onto \(H_0\) of the local field from the nearest proton plays a role. This projection depends on the direction of the spin of the nearest proton and on the angle \(\vartheta\) between the field \(H_0\) and the line joining the two nearest protons. A detailed calculation, carried out by Pake, shows that the resonant value of the field is given by the following formula:
\[ H_0 = H^* \pm \frac{3}{2}\frac{\mu}{a^3}(\cos^3 \vartheta - 1), \]
in which the presence of two signs is due to the two possible directions of the spin of the nearest neighboring proton.
In a gypsum single crystal there are two orientations of the line joining the nearest protons\(^8\). Therefore, in the general case, an absorption line with four maxima is obtained. Interaction with the remaining protons broadens each of these maxima.
In Blumberg’s work\(^5\), the resonance of protons in copper vitriol \((\mathrm{CuSO}_4\cdot 5\mathrm{H}_2\mathrm{O})\) was studied. In this case the picture proved to be more complicated and more interesting than in the case of gypsum.
A frequency \(\nu = 30.5\) megacycles was used, and the field \(H_0\) was modulated around 7000 oersteds. At room temperature an absorption line with one maximum and a width of about 15 oersteds was obtained. When the temperature is lowered the line splits; at liquid-helium temperatures the absorption line consists of 10 components. The maximum splitting at a temperature of \(1.2^\circ\) K is about 600 oersteds. Let us note that the position and width of each maximum depend on the orientation of the crystal relative to the field \(H_0\). The \(\mathrm{Cu}^{++}\) ion is a paramagnetic ion. Therefore, on each proton, in addition to the field \(H_0\) and the local field from neighboring protons, there also acts the field from the nearest \(\mathrm{Cu}^{++}\) ions.
The average magnetic moment of a copper ion in the field \(H_0\) is, in order of magnitude, equal to:
\[ \overline{\mu}_1 = \frac{\mu_1^{2}}{3kT} H_0, \]
where \(\overline{\mu}_1\) is the magnetic moment of the copper ion.
At the point where the proton is located, the copper ion creates an average field of the order
\[ \frac{\overline{\mu}_1}{r^3}, \]
where \(r\) is the distance from the copper ion to the proton. Substituting: \(r = 2.5\ \text{\AA}\), \(H_0 = 7000\) oersteds, we obtain for the field acting on the proton from the copper ion a value of the order of 1 oersted at \(T = 300^\circ\) K, 15 oersteds at \(T = 20^\circ\) K, and 300 oersteds at \(T = 1^\circ\) K.
In view of the fact that the line width produced by the magnetic interaction of the protons is of the order of 10–15 oersteds, at room temperature the field of the copper ions, owing to its relatively small magnitude, does not affect the width of the absorption line, which is also confirmed experimentally. At liquid-hydrogen temperatures the copper field is of the order of the line width; as a result, an incompletely resolved fine structure is obtained. At helium temperatures such a structure of the absorption line will be fully resolved.
A detailed calculation carried out by Blumberg shows that for each pair of protons there should be four lines. The unit cell of copper sulfate \(2\mathrm{CuSO}_4 \cdot 10\mathrm{H}_2\mathrm{O}\) contains 20 protons; however, owing to the presence of a center of symmetry, for each proton one can indicate another, located in the same unit cell, where the copper ions create the same field. In view of this there should be only 20 lines, and the calculation shows that these 20 lines must constitute 10 pairs. Each pair, according to Blumberg’s calculations, contains 2 very close lines, and therefore in experiment an absorption line with 10 maxima is obtained.
Let us also note a number of works \(^{2—4}\) in which an attempt was made to study the structure of solids and hindered rotation in them by investigating the fine structure of the nuclear magnetic-resonance absorption line.
G. Khutsishvili
CITED LITERATURE
- G. E. Pake, J. Chem. Phys. 16, 327 (1948).
- H. S. Gutowsky a. G. E. Pake, J. Chem. Phys. 16, 1164 (1948).
- H. S. Gutowsky, G. B. Kistiakowsky, G. E. Pake a. E. M. Purcell, J. Chem. Phys. 17, 973 (1949).
- H. S. Gutowsky and G. E. Pake, J. Chem. Phys. 18, 162 (1950).
- N. Bloembergen, Physica 16, 95 (1950).
- N. Bloembergen, E. M. Purcell and R. V. Pound, Phys. Rev. 73, 679 (1948).
- J. H. Van-Vleck, Phys. Rev. 74, 1168 (1948).
- W. A. Wooster, Zs. f. Krist. 94, 375 (1936).