Superphysics Superphysics
Part 4

Nuclear spin for the hyperfine-structure

December 13, 1946 6 minutes  • 1222 words

Which are the symmetry classes for other particles?

One example for particles with symmetrical wave functions only (assumption 2) was the photon.

This is:

  • an immediate consequence of Planck’s derivation of the spectral distribution of the radiation energy in the thermodynamical equilibrium.
  • necessary for the applicability of the classical field concepts to light waves in the limit where a large and not accurately fixed number of photons is present in a single quantum state.

The symmetrical class for photons occurs together with the integer value I for their spin.

  • The antisymmetrical class for the electron occurs together with the half-integer value 1⁄2 for the spin.

The important question of the symmetry classes for nuclei, however, had still to be investigated.

The symmetry class refers here also to the permutation of both the space coordinates and the spin indices of two like nuclei.

The spin index can assume 2 I + 1 values if I is the spin-quantum number of the nucleus which can be either an integer or a half-integer.

Before the electron spin was discovered, I proposed to use the assumption of a nuclear spin to interpret the hyperfine-structure of spectral lines.

This proposal had strong opposition from many sides.

But it influenced on the other hand Goudsmit and Uhlenbeck in their claim of an electron spin.

It was only some years later that my attempt to interpret the hyperfine-structure could be definitely confirmed experimentally by investigations in which also Zeeman himself participated and which showed the existence of a magneto-optic transformation of the hyperfine-structure as I had predicted it.

Since that time the hyperfine-structure of spectral lines became a general method of determining the nuclear spin.

In order to determine experimentally also the symmetry class of the nuclei, other methods were necessary.

The most convenient, although not the only one, consists in the investigation of band spectra due to a molecule with two like atoms16.

It could easily be derived that in the ground state of the electron configuration of such a molecule the states with even and odd values of the rotational quantum number are symmetric and antisymmetric respectively for a permutation of the space coordinates of the two nuclei.

There exist among the (2 I + 1) 2 spin states of the pair of nuclei, (2 I + 1) (I + 1) states symmetrical and (2 I + 1)I states antisymmetrical in the spins, since the (2 I+ 1) states with two spins in the same direction are necessarily symmetrical.

The conclusion was:

If the total wave function of space coordinates and spin indices of the nuclei is symmetrical, the ratio of the weight of states with an even rotational quantum number to the weight of states with an odd rotational quantum number is given by (I+ 1) : I.

In the reverse case of an antisymmetrical total wave function of the nuclei, the same ratio is I : (I + 1 ).

Transitions between one state with an even and another state with an odd rotational quantum number will be extremely rare as they can only be caused by an interaction between the orbital motions and the spins of the nuclei.

Therefore, the ratio of the weights of the rotational states with different parity will give rise to 2 different systems of band spectra with different intensities, the lines of which are alternating.

The first application of this method was the result that the protons have the spin 1⁄2 and fulfill the exclusion principle just as the electrons.

Dennison’s hypothesis was that, at this low temperature, the thermal equilibrium between the 2 modifications of the hydrogen molecule (ortho-H2: odd rotational quantum numbers, parallel proton spins; para-H2: even rotational quantum numbers, antiparallel spins) was not yet reached.

This hypothesis was later confirmed by the experiments of Bonhoeffer and Harteck and of Eucken.

It showed the theoretically predicted slow transformation of one modification into the other.

This removed the initial difficulties to understand quantitatively the specific heat of hydrogen molecules at low temperatures.

Among the symmetry classes for other nuclei, those with a different parity of their mass number M and their charge number Z are of a particular interest.

Consider a compound system consisting of numbers A1, A2, . . . of different constituents.

  • Each of these fulfills the exclusion principle.
  • A number S of constituents with symmetrical states, one has to expect symmetrical or antisymmetrical states if the sum AI + A 2 + . . . is even or odd.

This holds regardless of the parity of S.

Earlier one tried the assumption that nuclei consist of protons and electrons, so that M is the number of protons, M - Z is the number of electrons in the nucleus.

It had to be expected then that the parity of Z determines the symmetry class of the whole nucleus.

The counter-example of nitrogen has been known to have the spin I and symmetrical states.

After the discovery of the neutron, the nuclei have been considered, however, as composed of protons and neutrons.

  • A nucleus with mass number M and charge number Z should consist of Z protons and M - Z neutrons.

In case the neutrons would have symmetrical states, the parity of the charge number Z determines the symmetry class of the nuclei.

If, however, the neutrons fulfill the exclusion principle, the parity of M determines the symmetry class.

  • For an even M, one should always have symmetrical states, for an odd M, antisymmetrical ones.

The latter rule was confirmed by experiment without exception.

  • This proved that the neutrons fulfill the exclusion principle.

The most important and most simple crucial example for a nucleus with a different parity of M and Z is the heavy hydrogen or deuteron with M = 2 and Z = 1 which has symmetrical states and the spin I = 1, as could be proved by the investigation of the band spectra of a molecule with two deuterons.

From the spin value I of the deuteron can be concluded that the neutron must have a half-integer spin.

The simplest possible assumption that this spin of the neutron is equal to 1⁄2, just as the spin of the proton and of the electron, turned out to be correct.

There is hope, that further experiments with light nuclei, especially with protons, neutrons, and deuterons will give us further information about the nature of the forces between the constituents of the nuclei, which, at present, is not yet sufficiently clear.

These interactions are fundamentally different from electromagnetic interactions.

The comparison between neutron-proton scattering and proton-proton scattering showed that the forces between these particles are in good approximation the same, that means independent of their electric charge.

If one had only to take into account the magnitude of the interaction energy, one should therefore expect a stable di-proton or (M = 2, Z = 2) with nearly the same binding energy as the deuteron.

Such a state is, however, forbidden by the exclusion principle in accordance with experience, because this state would acquire a wave function symmetric with respect to the 2 protons.

This is only the simplest example of the application of the exclusion principle to the structure of compound nuclei, for the understanding of which this principle is indispensable, because the constituents of these heavier nuclei, the protons and the neutrons, fullfil it.

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