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METHODS FOR OBTAINING ORIENTED NUCLEI
It is well known that many nuclei possess spins and magnetic moments. Under ordinary conditions the distribution of the nuclear spins of a macroscopic body over directions is completely chaotic. The obtaining of targets with oriented nuclei (nuclei are called oriented or polarized if, in an ensemble of nuclei, there is some preferential direction of the nuclear spin) is of considerable interest for nuclear physics, since by carrying out experiments with oriented nuclei and, for example, polarized neutrons, it will be possible to obtain much interesting information on the spin dependence of nuclear forces. However, as we shall see below, owing to the smallness of nuclear magnetic moments, the obtaining of oriented nuclei presents great difficulties.
It is clear that, in order to obtain a target with oriented nuclei, one must create such conditions under which states with different directions of the nuclear spin would have different energies. Then we obtain a Boltzmann distribution of the nuclei over these energy levels, and the direction of the nuclear spin which corresponds to the lowest energy state will be predominant. However, the energy differences of neighboring levels prove to be quantities of the order of \(0.01\)—\(0.02^\circ\mathrm{K}\) (in temperature units of energy), as a result of which the use of ultralow temperatures is required in order to obtain a significant degree of orientation of the nuclei.
Let the \(z\)-axis be the axis of quantization of the nuclear spin (this direction may be the direction of an external magnetic field or the direction of a crystal axis). Let \(m\) be the projection of the nuclear spin on the axis of quantization (in units of \(\hbar\)), and \(I\) the maximum projection of the spin (the magnitude of the spin is \(\sqrt{I(I+1)}\)). Then, as a measure of the degree of orientation of the nuclei, one may take the following quantity (called the nuclear polarization):
\[ f=\frac{\overline{m}}{I}, \]
where the average is taken over the ensemble of all nuclei of the given type in the sample. The quantity \(f\) is equal to zero in the absence of orientation and is equal to unity for complete polarization of the nuclei.
By some methods (see below) one obtains such samples in which the numbers of nuclei in states differing only in the sign of \(m\) are equal to one another. For describing the degree of orientation of such targets the quantity \(f\) is not suitable (it becomes zero), and it is convenient to introduce the following quantity (called the nuclear quadrupolarization):
\[ g=\frac{\overline{m^2}-\dfrac{I(I+1)}{3}}{I^2}. \]
For a chaotic distribution of nuclear spins over directions,
\[ \overline{m^2}=\frac{I(I+1)}{3} \]
and \(g=0\); for oriented nuclei, however, \(g\) takes values different from zero.
Let us now proceed to consider the methods of obtaining oriented nuclei. Up to the present time four methods have been proposed.
a) Direct method
A strong magnetic field is applied to the sample. It is clear that in this case
\[ f=\frac{M}{N\mu}, \]
where \(N\) is the number of nuclei in the sample, \(\mu\) is the maximum projection of the nuclear magnetic moment, and \(M\) is the total nuclear magnetic moment of the sample. Recalling the elementary formulas of the theory of paramagnetism, we obtain
\[ f = B_I\left(\frac{\mu H}{kT}\right), \]
where \(B_I\) is the Brillouin function. In order to obtain \(f\) close to unity, \(\frac{\mu H}{kT}\) must be greater than unity. But owing to the extreme smallness of \(\mu\), the method requires large values of \(\frac{H}{T}\).
Let us consider, for example, the case of \(\mathrm{Li}^7\) \(\left(I=\frac{3}{2}, \mu = 3.25\right.\) nuclear magnetons\(\left.\right)\). To obtain in this case \(f=0.75\), one requires
\[ \frac{H}{T} \simeq 2\cdot 10^7 \frac{\text{oersted}}{{}^{\circ}\mathrm{K}}, \]
i.e., for example, \(H=200\) kilooersteds at \(T=0.01^\circ\mathrm{K}\). In the case of protons, in order to obtain \(f=0.75\), at \(T=0.01^\circ\mathrm{K}\) one needs \(H=100\) kilooersteds. Thus, for a considerable polarization of nuclei, fields are required which are for the time being unattainable.
To obtain \(f=0.2\) at \(T=0.01^\circ\mathrm{K}\), considerably smaller fields are required, for example: about 30 kilooersteds in the case of \(\mathrm{Li}^7\), and about 20 kilooersteds in the case of protons.
It is known that ultralow temperatures are obtained by adiabatic demagnetization of paramagnetic salts (for example, chromium or iron alums). Therefore, when applying the direct method, the experiment will have the following form: the sample (diamagnetic or metallic) must be brought into good thermal contact with the cooled paramagnetic salt, and then a strong magnetic field must be applied to the sample (but not to the salt, otherwise the latter will be heated).
In view of the technical difficulties of applying the direct method, in recent years three other methods have been proposed for obtaining oriented nuclei.
b) Quadrupole method\(^{1}\) (Pound’s method).
It is known that nuclei (with spin \(I>1\)) possess quadrupole moments. In the place where some nucleus (possessing a quadrupole moment) is situated, there is an inhomogeneous electric field produced both by the electron shell of the ion itself of which the given nucleus is a part, and by neighboring ions. It is known from theory that, when an electric quadrupole is placed in an inhomogeneous electric field which does not possess spherical symmetry, splitting of the energy levels occurs. In particular, an electric field with axial symmetry produces a splitting in which states with projections of the nuclear spin \(+m\) and \(-m\) on the axis of symmetry of the electric field coincide. For example, for spin \(I=\frac{3}{2}\) we obtain two levels corresponding to \(m=\pm\frac{3}{2}\) and \(m=\pm\frac{1}{2}\).
These splittings, owing to the smallness of nuclear quadrupole moments, are usually extremely small; however, in some molecular crystals (namely in those crystals in whose molecules there are covalent bonds of type \(p\), for example \(\mathrm{CH}_3\mathrm{Br}\); let us note that in such cases an electric field with axial symmetry, produced by the remaining charges of the molecule, is present at the halogen nucleus) these splittings reach \(0.01\text{–}0.02^\circ\mathrm{K}\). At temperatures of the same order we obtain a fairly considerable degree of orientation of the nuclei (with \(f=0,\ g\ne 0\)).
For carrying out the experiment, a single crystal of such a substance must be brought into thermal contact with a cooled paramagnetic salt.
The other two methods of obtaining oriented nuclei are based on the interaction of the nuclear spin with the spin of the electron shell of a paramagnetic ion. Let us consider a paramagnetic ion containing a nucleus with spin (for example, the odd isotopes \( \mathrm{Gd}^{+++}, \mathrm{Fe}^{+++}, \mathrm{Co}^{++}, \mathrm{Cu}^{++} \)). It is clear that there will be a magnetic interaction between the nuclear spin and the spin of the electron shell. This interaction was evaluated by studying the hyperfine structure of paramagnetic resonance. It turned out that the splittings caused by this interaction are of the order of \(0.01\)—\(0.03^\circ\mathrm{K}\). Let us note that it follows from this that an unclosed \(3d\) or \(4f\) shell creates at the nucleus a field of the order of several hundred kilooersteds.
c) Orientation by a weak field\(^2\) (Gorter–Rose method).
In view of the comparatively large magnitude of the magnetic moment of an unfilled \(3d\) or \(4f\) shell, orientation of the spins of the electron shells is much easier than orientation of nuclear spins (for orientation of the shell spins it is sufficient that
\[ \frac{H}{T} \simeq 2 \cdot 10^4 \frac{\text{oersted}}{^\circ\mathrm{K}} \]
). If, however, the shell spins are oriented, then all nuclei will be acted upon by equally directed internal fields (as noted above, they are of the order of several hundred kilooersteds), which at a sufficiently low temperature will produce considerable nuclear polarization.
It is proposed to demagnetize a paramagnetic salt, whose ions contain nuclei with spin, from large fields (and temperatures of the order of \(1^\circ\mathrm{K}\)) down to fields of the order of several hundred oersteds. If in this process a temperature of the order of several hundredths of a degree is obtained (the final temperature will be the lower, the higher the initial field), then considerable nuclear polarization will be obtained.
d) Orientation by anisotropic spin–spin interaction\(^3\) (Blin method).
Experiments on the hyperfine structure of paramagnetic resonance show that in paramagnetic salts the interaction of the nuclear spin with the spin of the electron shell is, generally speaking, anisotropic, namely, this interaction is described by the following Hamiltonian:
\[ AS_z I_z + B(S_x I_x + S_y I_y) \]
(where \(S\) is the spin of the electron shell, \(z\) is the axis of symmetry of the crystalline field), and, generally speaking, \(A \ne B\). In addition, experiments show that in Tutton salts of Co and Cu (Tutton salts are compounds of the type \(M_2^{I}M^{II}(\mathrm{SO}_4)_2 \times 6\mathrm{H}_2\mathrm{O}\), where \(M^{I}\) is a monovalent and \(M^{II}\) a divalent metal, for example Co or Cu) \(A \gg B\). For example, in the case of the salt \((\mathrm{NH}_4)_2\mathrm{Co}(\mathrm{SO}_4)_2 \cdot 6\mathrm{H}_2\mathrm{O}\), \(A = 0.035^\circ\mathrm{K}\), \(B = 0.001^\circ\mathrm{K}\). Neglecting in this case the small second term, we obtain that the energy of interaction of the nuclear spin with the shell spin has the form \(AS_z I_z\). This interaction will cause a splitting of the energy levels; in particular, for \(S = \frac{1}{2}\), \(S_z\) can take the values \(\pm \frac{1}{2}\), and we obtain \(2I+1\) doubly degenerate energy levels. For example, for \(I = \frac{3}{2}\) we obtain four levels with the following values of the projection of the nuclear spin on the crystalline axis
\[ m = \pm \frac{3}{2}, \ \pm \frac{1}{2}, \ \mp \frac{1}{2} \ \text{and} \ \mp \frac{3}{2} \]
and with an energy difference between neighboring
levels equal to \(\dfrac{A}{2}\). If the temperature is of the order of, or less than, \(A\), the lower level will be populated more than the others, and we shall obtain a considerable degree of orientation of the nuclei (with \(f = 0,\ g \ne 0\)).
It is proposed to demagnetize a paramagnetic salt with anisotropic hyperfine structure from large fields and helium temperatures down to zero field. In this way a rather low temperature is obtained and, consequently, a fairly strong degree of nuclear orientation.
So far we have not touched upon the following circumstance: when the specimen is brought to some temperature \(T\), a certain time is required (the so-called relaxation time of the nuclear spin with the lattice) for the nuclear spins to come into thermal equilibrium with the lattice; only after this equilibrium has been established shall we obtain a Boltzmann distribution of the nuclear spins over their spin levels. For the experiments to be successful, this relaxation time must not be too long.
In the case of methods b and c relaxation will apparently be sufficiently rapid as a consequence of the interaction of the nuclear spins with paramagnetic spins (electron shells), which are rather strongly coupled to the lattice. In the case of method b, however, in order that the relaxation time not be too long, paramagnetic atoms will have to be added to the specimen.
In conclusion, let us note that, of all the methods considered, given today’s technical possibilities, the Bloch method gives the strongest degree of orientation, since the final temperature of the specimen in this case is lower than when other methods are applied.
G. Kh.
References
- R. V. Pound, Phys. Rev. 76, 1410 (1949).
- C. G. Gorter, Physica 14, 504 (1948); M. E. Rose, Nucleonics 3, 6, 23 (1948).
- B. Bleaney, Proc. Phys. Soc. A 64, 315 (1951); Phil. Mag. 42, 442 (1951).