ISOBARIC SPIN AND THE HYPOTHESIS OF CHARGE INDEPENDENCE OF NUCLEAR FORCES

G. I. Zel’dier

The present stage in the development of ideas about the structure of atomic nuclei begins with the establishment of the proton–neutron model of the nucleus. From that moment on, the study of the specific, i.e. non-electromagnetic, interaction between the particles composing the nucleus—protons and neutrons—has remained the central problem of nuclear physics.

From the standpoint of the proton–neutron model it has long been clear that both types of particles play approximately the same role in the structure of the nucleus. Not only the mass, but also the volume and binding energy of a nucleus are approximately proportional to the total number of particles in the nucleus, i.e. each of the nuclear particles, irrespective of whether it is a proton or a neutron, makes approximately the same contribution to these quantities. Proceeding from such considerations, as early as 1935 the idea was put forward¹ that forces very close in magnitude act between any two nuclear particles.

In a much more precise form, the hypothesis of the charge independence of nuclear forces was proposed and developed²,³,⁴,⁵,⁶ in connection with the results of experiments⁷ on the scattering of protons by protons. The interpretation of these experiments⁸ showed that, after subtracting the effect of Coulomb forces, the proton–proton interaction is almost indistinguishable from the proton–neutron interaction (at least in the \({}^{1}S\) state to which the experiments referred). When magnetic interactions⁹ were subsequently taken into account, the agreement proved still more exact. Although direct data on the neutron–neutron interaction were lacking, it was assumed that it too does not differ from the specifically nuclear neutron–proton or proton–proton interaction.

The latter assumption was supported by the results of measurements of the energies of mirror nuclei¹⁰. It turned out that the difference in the energies of the ground states of such nuclei can be attributed entirely to the difference in their Coulomb energy, and also to the difference

masses of the neutron and proton. This testifies in favor of the equality of the nuclear forces \(n—n\) and \(p—p\). (Such a statement is sometimes called “charge symmetry” of nuclear forces.)

The hypothesis of charge independence asserts the independence of nuclear forces from the charge of the nucleon, other conditions being equal, for any pair of particles, i.e. the equality of \(n—p\), \(p—p\), and \(n—n\) forces in identical space-spin states.

Of course, along with the nuclear forces one must also take into account the Coulomb forces between protons, as well as magnetic forces, which in any case violate the charge independence of the forces between nucleons. But Coulomb and other electromagnetic forces do not play a decisive role, at least in light nuclei. Moreover, they are known much better than nuclear forces, and their influence can usually be allowed for. However, it is meaningful, of course, to test the direct consequences of the hypothesis of charge independence only on light nuclei.

In addition to the effect of the Coulomb interaction, a certain difference in the behavior of neutrons and protons will be caused by the incomplete equality of their masses. However, the difference between the masses of \(n\) and \(p\) is small, and its influence is, as a rule, less than the influence of Coulomb forces. The principal manifestation of this difference—the different contribution of the rest masses of the neutron and proton to the rest mass of the nucleus—can be taken into account quite simply.

In studying the consequences of the hypothesis of charge independence of nuclear forces, it is expedient first to abstract from the two effects just mentioned.

Then the hypothesis of charge independence leads to the assertion that the Hamiltonian of a system of nucleons remains invariant when any proton is replaced by a neutron and any neutron by a proton. In other words, all isobaric nuclei will have a common Hamiltonian, symmetric with respect to all particles, i.e. invariant under the simultaneous interchange of the spatial and spin coordinates of any two particles.

Thus the hypothesis of charge independence can be expressed in the form of a symmetry principle for the nuclear Hamiltonian with respect to particle permutations.

The study of the properties of the permutation symmetry of states allowed by a symmetric Hamiltonian lies at the basis of most of the conclusions obtained from the hypothesis of charge independence. Here the essential point is the fact that, whatever the nuclear forces may be, protons and neutrons in any case are not identical particles in the quantum-mechanical sense. The strongest admissible assumption is that the proton and neutron are two states of one and the same particle, differing by the value of a certain internal coordinate on which the nuclear forces do not depend.

The idea that the proton and neutron should be regarded as two charge states of one particle—the nucl—

...nucleon—appeared immediately after the establishment of the proton–neutron model of the nucleus \(^{11}\). The fruitfulness of such a unification is suggested already by the closeness of the masses of the proton and the neutron, and also by the important role played, in the characterization of a nucleus, by the total number of particles in it. The immediate basis for such an interpretation was the attempt to interpret the forces between the neutron and the proton as forces of exchange character.

The unity of the proton and neutron states of the nucleon found its expression also in the recently clearly formulated concept of “nuclear charge” and in the law of its conservation \(^{12,13}\).

The possibility of treating the proton and neutron as two states of one and the same particle is not necessarily connected with the hypothesis of charge independence. But such an interpretation proves most fruitful precisely in connection with this hypothesis.

1. ISOBARIC SPIN

Usually a nucleon system—the atomic nucleus—is characterized by specifying the mass number \(A\), equal to the total number of particles in the system, and the atomic number \(Z\), equal to the number of protons in it. Such a characterization is, evidently, asymmetric with respect to neutrons and protons and is adapted rather to atomic than to nuclear problems.

In the unified treatment of the proton and neutron it is expedient to characterize a nucleon system by the number

\[ A = N + Z \tag{1,1} \]

and by the corresponding “antisymmetric” combination

\[ T_3 = \frac{N - Z}{2} = N - \frac{A}{2} = -\left(Z - \frac{A}{2}\right) \tag{1,2} \]

(\(N\) is the number of neutrons), indicating the excess of neutrons (deficiency of protons) over the average number \(A/2\) of particles of each type. The number \(T_3\), which will be integral or half-integral depending on the parity of \(A\) and, for a given \(A\), can take values from \(-A/2\) to \(+A/2\) in steps of one, is called the “third component” (or “\(z\)-component”) of the isobaric spin of a system of nucleons. For a given \(A\), different numbers \(T_3\) correspond to different isobaric nuclei*).

*) Until recently the term “isotopic spin,” going back to the work of \(^{5}\), was widespread. The term adopted in the text, adjoining \(^{32,33}\), is undoubtedly closer to the corresponding concept (different values of \(T_3\) are usually used to characterize different charge states of nuclei of the same mass number \(A\)).

If we are dealing with a single nucleon, then the third component of the isobaric spin will be

\[ \left. \begin{aligned} t_3 &= +\frac{1}{2}\quad &&(\text{for the neutron}),\\ t_3 &= -\frac{1}{2}\quad &&(\text{for the proton}). \end{aligned} \right\} \tag{1,3} \]

From the standpoint of a representation that distinguishes the neutron and the proton as different particles, the quantity \(t_3\) characterizes the type of particle. From the standpoint of a representation of the nucleon, however, \(t_3\) is a characteristic of its state, one of its coordinates, describing the “charge” degree of freedom. Corresponding to the two only possible charge states of the nucleon, the variable \(t_3\) takes only two values, i.e. it is, as one says, a dichotomic variable. In this, and only in this, consists its essential analogy with the variable of ordinary spin \(s_z\) (for a particle of total spin \(\frac{1}{2}\)),—an analogy to which the variable of isobaric spin owes its name. Starting for the moment only from the set of possible values of \(t_3\), one should assign to the nucleon the value of the “total isobaric spin” equal to \(\frac{1}{2}\). The choice of the values \(\pm \frac{1}{2}\) for the variable characterizing the charge state of the nucleon has, over any other possible choice, the advantage that it allows this analogy to be pursued far.

It is convenient, however, also to have a special notation for the quantity

\[ \tau_3 = 2t_3, \]

which takes for the neutron and the proton the values \(\pm 1\) (the analogue of the quantity \(\sigma_z\) in the theory of ordinary spin).

In order to obtain the complete set of coordinates of the nucleon, one must add the quantity \(t_3\) to its ordinary spatial and spin coordinates. Thus the complete set of coordinates of the nucleon will be

\[ \mathbf{r},\ s_z,\ t_3. \]

Accordingly, the wave function of a system of nucleons may be written in the form

\[ \Psi\bigl(\mathbf{r}^{(1)}, s_z^{(1)}, t_3^{(1)};\ \mathbf{r}^{(2)}, s_z^{(2)}, t_3^{(2)}, \ldots, \mathbf{r}^{(A)}, s_z^{(A)}, t_3^{(A)}\bigr) \tag{1,4} \]

(the superscripts number the particles). In what follows we shall sometimes denote the aggregate of spatial and (ordinary) spin coordinates by \(q^{(i)} = (\mathbf{r}^{(i)}, s_z^{(i)})\).

For a nucleus with mass number \(A\) and atomic number \(Z\), the wave function (1,4) must be different from zero only when

condition

\[ T_3=\sum_{i=1}^{A} t_3^{(i)}=\frac{N-Z}{2}. \]

Since we have reduced the difference between the neutron and the proton to the difference in the states of the nucleon, which are completely characterized by the set of its five coordinates, all nucleons appear in the theory as identical particles. Independently of assumptions about the character of the forces between nucleons, the Hamiltonian of the system will be a symmetric function of the variables belonging to the various nucleons. This means that the quantum-mechanical principle of identity of microparticles will be applicable to the nucleons, requiring symmetry or antisymmetry of the wave function with respect to a simultaneous interchange of all (in the present case, five) coordinates of any two identical particles. The factor \((+1\) or \(-1)\) acquired by the wave function under such an interchange cannot depend on the values of any arguments whatsoever of this function. Therefore we can determine it by considering the behavior of the function under interchange of two particles in the case where the values of \(t_3\) for them coincide (when both particles are either protons or neutrons). In this case the interchange of all five coordinates of both particles reduces to an interchange only of their spatial and spin coordinates. But for precisely such an interchange the Pauli principle, to which protons and neutrons are subject, requires antisymmetry. Hence, in the case we have chosen, the function (1.4) is antisymmetric. From what was said above it follows that it will also be antisymmetric for unequal values of \(t_3\) for the particles being interchanged, i.e. always.

At first sight it seems that the requirement of complete antisymmetry for the full wave function imposes additional physical restrictions on interchanges of neutrons and protons in comparison with the usual application of the Pauli principle to each type of particle separately. However, this is not so, because in passing from the separate treatment of the proton and neutron to the use, for distinguishing them, of the variable of isobaric spin, the increase in the number of symmetry conditions is accompanied by an increase in the number of functions differing by interchanges, which correspond to one and the same physical state. Let us also note that, for our argument, it was essential that protons and neutrons obey the same statistics—the Fermi–Dirac statistics.

Similarly to the representation of the wave function of a particle of spin \(\frac{1}{2}\) in the nonrelativistic theory, the wave function of a nucleon can be

to represent it in the form of a matrix of two rows (numbered by the values \(t_3\), for example in the order \(t_3=+\frac{1}{2}\), \(t_3=-\frac{1}{2}\))

\[ \psi=\left|\begin{array}{c} u\\ v \end{array}\right|, \tag{1.5} \]

where \(u\) and \(v\), which are functions of both the spatial and the ordinary spin coordinates, represent the probability amplitudes of the neutron and proton states of the nucleon, respectively. Then, evidently, the operator for the quantity \(t_3\) is represented in the form of the two-row matrix

\[ \hat t_3=\frac{1}{2}\hat\tau_3=\frac{1}{2}\left|\begin{array}{cc} 1&0\\ 0&-1 \end{array}\right|, \tag{1.6} \]

which coincides exactly with the third Pauli matrix, but acts not on the ordinary spin variables, but on the variables of isotopic spin. The eigenfunctions of this operator will be \(\left|\begin{array}{c}u\\0\end{array}\right|\) for the eigenvalue \(+\frac{1}{2}\) (neutron) and \(\left|\begin{array}{c}0\\v\end{array}\right|\) for the eigenvalue \(-\frac{1}{2}\) (proton), where \(u, v\) are arbitrary functions of the coordinates and of the ordinary spin.

Along with the matrix (1.6) we shall also introduce the analogues of the other two Pauli matrices:

\[ \frac{1}{2}\hat\tau_1=\frac{1}{2}\left|\begin{array}{cc} 0&1\\ 1&0 \end{array}\right| \quad \text{and} \quad \frac{1}{2}\hat\tau_2=\frac{1}{2}\left|\begin{array}{cc} 0&-i\\ i&0 \end{array}\right|. \tag{1.6′} \]

The purpose of introducing them is that, with their help, one can construct operators of transitions between the neutron and proton states of a nucleon, as well as the operator of interchange of the charge coordinates of two nucleons. These operators can be obtained as follows.

Introduce the combinations of the operators (1.6):

\[ \Pi=\frac{1}{2}\left(\hat\tau_1-i\hat\tau_2\right) = \left|\begin{array}{cc} 0&0\\ 1&0 \end{array}\right|; \qquad \Pi^{+}=\frac{1}{2}\left(\hat\tau_1+i\hat\tau_2\right) = \left|\begin{array}{cc} 0&1\\ 0&0 \end{array}\right|; \]

\[ T_{+}=\frac{1}{2}\left(1+\hat\tau_3\right) = \left|\begin{array}{cc} 1&0\\ 0&0 \end{array}\right|; \qquad T_{-}=\frac{1}{2}\left(1-\hat\tau_3\right) = \left|\begin{array}{cc} 0&0\\ 0&1 \end{array}\right|. \tag{1.7} \]

and consider their action on the wave function of a nucleon, written in the form (1.5). For example:

\[ \Pi\psi= \left|\begin{array}{cc} 0&0\\ 1&0 \end{array}\right| \left|\begin{array}{c} u\\ v \end{array}\right| = \left|\begin{array}{c} 0\\ u \end{array}\right|. \]

i.e., \(\Pi\), acting on a function describing a neutron state, transforms it into a function describing the same proton state; while acting on a function describing a proton state, it annihilates it. Briefly, the operator \(\Pi\) transforms a neutron into a proton (and annihilates a proton). Similarly, as is easy to see, the action of the remaining operators is as follows. \(\Pi^+\) transforms a proton into a neutron (and annihilates a neutron). The operator \(T_+\) acts on a neutron state as the identity operator, and on a proton state as an annihilating one. \(T_-\) acts as the identity on a proton state and as an annihilating one on a neutron state.

Let us now consider the operator

\[ P^{(\tau)}_{(i,k)}=\Pi^{(i)}\Pi^{(k)+}+\Pi^{(i)+}\Pi^{(k)}+T^{(i)}_{+}T^{(k)}_{+}+T^{(i)}_{-}T^{(k)}_{-}. \tag{1,8} \]

(the indices \(i\) and \(k\) denote the two particles to which the operators refer). Let us analyze the action of the operator (1,8) on all possible charge states of two particles. If the \(i\)-th particle is a neutron and the \(k\)-th is a proton, then only the first term will give a nonzero result, namely, the transformation of the \(i\)-th particle into a proton and the \(k\)-th into a neutron. If the \(i\)-th particle is a proton and the \(k\)-th is a neutron, then the action of the second term will lead to an analogous “exchange” of the proton and neutron. Finally, if both particles are neutrons (protons), then the third (fourth) term gives unity, while the others give zeros. In this case as well, the action of the operator \(P^{(\tau)}_{(i,k)}\) can be interpreted as an exchange of the (identical) charge coordinates.

Thus, the operator \(P^{(\tau)}_{(i,k)}\), defined by (1,8), is the operator of permutation of the charge coordinates, their exchange. Substituting in (1,8) the expressions for \(\Pi\) and \(T\) in terms of the operators \(\hat{\tau}\), we find:

\[ P^{(\tau)}_{(i,k)}=\frac{1+\left(\hat{\tau}^{(i)}\cdot\hat{\tau}^{(k)}\right)}{2}, \tag{1,9} \]

where, by definition,

\[ \left(\hat{\tau}^{(i)}\cdot\hat{\tau}^{(k)}\right)\equiv \hat{\tau}^{(i)}_{1}\hat{\tau}^{(k)}_{1}+ \hat{\tau}^{(i)}_{2}\hat{\tau}^{(k)}_{2}+ \hat{\tau}^{(i)}_{3}\hat{\tau}^{(k)}_{3} \]

is the scalar product of the operators \(\hat{\tau}^{(i)}\), \(\hat{\tau}^{(k)}\), whose components are given by relations (1,5) and (1,6).

Let us now introduce, for a system of \(A\) nucleons, the operators

\[ \hat{T}_{1}=\frac{1}{2}\sum_{i=1}^{A}\hat{\tau}^{(i)}_{1},\qquad \hat{T}_{2}=\frac{1}{2}\sum_{i=1}^{A}\hat{\tau}^{(i)}_{2},\qquad \hat{T}_{3}=\frac{1}{2}\sum_{i=1}^{A}\hat{\tau}^{(i)}_{3} \tag{1,10} \]

—three (noncommuting) components of the vector \(\hat{T}\) of isobaric

spin. They will, of course, possess all the mathematical properties of ordinary spin operators.

Let us find the eigenfunctions of the operator \(\hat T_3\).

The definition (1.6) of the operators \(\hat \tau_3\) can be rewritten in the form

\[ \hat{\tau}_3 \chi(\tau_3)=\tau_3\chi(\tau_3) \]

(the operator \(\hat{\tau}_3\) multiplies the function by \(+1\) when the variable \(\tau_3=1\), and by \(-1\) when \(\tau_3=-1\)). Therefore the action of the operator \(\hat T_3\) on the function \(\chi(\tau_3^{(1)},\tau_3^{(2)},\ldots,\tau_3^{(A)})\) will be

\[ \hat T_3\chi(\tau_3^{(1)},\tau_3^{(2)},\ldots,\tau_3^{(A)})= \]

\[ =\frac{1}{2}(\tau_3^{(1)}+\tau_3^{(2)}+\ldots+\tau_3^{(A)}) \chi(\tau_3^{(1)},\tau_3^{(2)},\ldots,\tau_3^{(A)}). \]

Here by \(\chi\) we may understand a function depending, in addition to the explicitly written isotopic variables, also on the remaining (spatial and spin) coordinates of the nucleons.

An eigenfunction of the operator \(\hat T_3\) is a solution of the equation

\[ \hat T_3\chi=T_3\chi, \]

where \(T_3\) is the eigenvalue. Therefore, for an eigenfunction it must be that

\[ \frac{1}{2}(\tau_3^{(1)}+\tau_3^{(2)}+\ldots+\tau_3^{(A)})\chi=T_3\chi \]

or, equivalently,

\[ \left\{\frac{1}{2}(\tau_3^{(1)}+\tau_3^{(2)}+\ldots+\tau_3^{(A)})-T_3\right\}\chi=0. \]

This means that either

\[ \frac{1}{2}(\tau_3^{(1)}+\tau_3^{(2)}+\ldots+\tau_3^{(A)})=T_3, \]

or \(\chi=0\). Hence it is seen that an eigenfunction of the operator \(\hat T_3\) is different from zero only for those values of the isotopic variables \(\tau_3^{(i)}\) for which \(N\) of them are equal to \(+1\), and the remaining \(Z\) are equal to \(-1\), with the numbers \(N\) and \(Z\) uniquely determined by the conditions

\[ \frac{N-Z}{2}=T_3,\qquad N+Z=A, \]

coinciding with (1.1) and (1.2).

In the sense of the variables \(\tau_3\), the eigenfunctions of the operator \(\hat T_3\) will thus describe states of the system of nucleons—

with a definite charge, i.e. with fixed numbers \(N\) of neutrons and \(Z\) of protons. In this case the eigenvalue \(T_3\) is interpreted as half the difference of the numbers of neutrons and protons, which also explains the notation adopted in (1.2) for this half-difference.

One can define the operator of the square of the total isobaric spin
\[ \hat T^2=\hat T_1^2+\hat T_2^2+\hat T_3^2=(\hat{\mathbf T},\hat{\mathbf T}). \tag{1.11} \]
Its eigenvalues can be written in the form \(T(T+1)\), where the numbers \(T\) are integers or half-integers depending on the parity of the number \(A\) of particles, and can take the values
\[ \frac{A}{2},\quad \frac{A}{2}-1,\ldots,\quad 0 \]
(or \(\frac{1}{2}\)).

A very important relation is that of the operator \(\hat T^2\) with the transposition operators \(P^{(\tau)}_{(i,k)}\). This relation is easily obtained from (1.11) and (1.9):
\[ \begin{aligned} \hat T^2 &=\left(\frac{1}{2}\sum_{i=1}^{A}\tau^{(i)},\ \frac{1}{2}\sum_{k=1}^{A}\tau^{(k)}\right) =\frac{1}{4}\left\{\sum_{i=1}^{A}\bigl(\hat{\tau}^{(i)}\bigr)^2 +2\sum_{i<k}^{A}\bigl(\hat{\tau}^{(i)}\hat{\tau}^{(k)}\bigr)\right\}\\ &=\frac{3}{4}A+\sum_{i<k}^{A}\left\{P^{(\tau)}_{(i,k)}-\frac{1}{2}\right\} =\frac{3}{4}A-\frac{A(A-1)}{4}+\sum_{i<k}^{A}P^{(\tau)}_{(i,k)}\\ &=A-\frac{A^2}{4}+\sum_{i<k}^{A}P^{(\tau)}_{(i,k)}, \tag{1.12} \end{aligned} \]
i.e. \(\hat T^2\) coincides, up to a constant term, with the sum of all pair transpositions of the charge coordinates of the nucleons. In particular, for a completely symmetric function all \(P^{(\tau)}_{(i,k)}=1\),
\[ \hat T^2=\frac{A}{2}\left(\frac{A}{2}+1\right),\qquad T=\frac{A}{2}. \]

In the simplest case of two particles
\[ \hat T^2=1+P^{(\tau)}_{(1,2)}, \tag{1.13} \]
so that the operator \(\hat T^2\) essentially coincides with the operator of the (single) transposition of two particles. Therefore, in this case the eigenfunctions of \(\hat T^2\) coincide with the eigenfunctions of the operator \(P^{(\tau)}_{(1,2)}\), i.e. are symmetric or antisymmetric functions of the variables \(t_3^{(1)}, t_3^{(2)}\). For a symmetric function \(P^{(\tau)}_{(1,2)}=1\), \(\hat T^2=T(T+1)=2,\ T=1\); for an antisymmetric function \(P^{(\tau)}_{(1,2)}=-1,\ T(T+1)=0,\ T=0\).

2. QUANTUM-MECHANICAL FORMULATION OF THE HYPOTHESIS OF CHARGE INDEPENDENCE

We shall now clarify how the hypothesis of charge independence can be introduced into the quantum-mechanical description of nucleons. In doing so we shall at first neglect Coulomb forces and the difference between the masses of the proton and the neutron. Then charge independence means, evidently, that the Hamiltonian \(H\) of the system of nucleons will not contain operators acting on the variables of isotopic spin\(^*\). In this case all operators of isotopic spin will commute with the Hamiltonian. In particular, the operators \(\hat T^2\) and \(\hat T_3\) will commute with the Hamiltonian. Since these latter, moreover, commute with one another, there will exist a complete system of common eigenfunctions of the operators \(\hat H\), \(\hat T^2\), and \(\hat T_3\). To these one may add the remaining operators that commute with the Hamiltonian and do not act on the variables of isotopic spin, such as the angular-momentum operators, \(\hat J^2\), the projection of angular momentum \(J_z\), and parity.

If the Hamiltonian contains no operators acting on the variables of isotopic spin, then the total wave function of the system of nucleons will be, in the absence of degeneracy of the corresponding Schrödinger equation, the product of a solution of this equation depending only on the spatial and ordinary spin variables and a function of the isotopic spin. In the presence of degeneracy the total wave function will be a linear combination of solutions belonging to the given eigenvalue, with coefficients depending on the variables of isotopic spin. In both cases the total wave function must be antisymmetric with respect to the simultaneous interchange of all five coordinates of any two nucleons.

\(^*\) More precisely, one should say that under the assumption of charge independence the Hamiltonian of the system can be written in a form containing no operators acting on the isotopic variables. For example, the presence in the Hamiltonian of operators of the form \((\tau^{(i)}\cdot\tau^{(k)})\) does not violate charge independence, because such operators can be expressed in terms of operators acting only on the coordinates \(q\). Indeed, according to Pauli’s principle, the operation of interchanging all coordinates of two nucleons \(P_{(i,k)} = P^{(\tau)}_{(i,k)} P^{(q)}_{(i,k)}\) reduces to multiplying the wave function by \(-1\). Therefore, when applied to admissible (antisymmetric) functions, \(P^{(\tau)}_{(i,k)} = -P^{(q)}_{(i,k)}\). From this relation and formula (1,9) we obtain
\((\tau^{(i)}\cdot\tau^{(k)}) = 2P^{(\tau)}_{(i,k)} - 1 = -(2P^{(q)}_{(i,k)} + 1)\), i.e. an expression of the operator \((\tau^{(i)}\cdot\tau^{(k)})\) in terms of spatial and spin operators.

We shall first analyze the consequences of the hypothesis of charge independence in the example of two nucleons, which will make it possible to clarify the fundamental side of the matter, and then indicate those complications which arise for an arbitrary number of particles.

For two nucleons the Schrödinger equation determining the space-spin part of the wave function of the stationary state will be

\[ \hat H(q^{(1)}, q^{(2)})\psi(q^{(1)}, q^{(2)})=E\psi(q^{(1)}, q^{(2)}). \tag{2,1} \]

The Hamiltonian of the system, containing, by the assumption of charge independence, only operators acting on the variables \(q\), will obviously be symmetric with respect to the simultaneous interchange of the spatial and spin coordinates of the two particles. Therefore, together with every eigenfunction belonging to a given eigenvalue \(E\), there will also belong to the same eigenvalue a function differing from the original function by the interchange of \(q^{(1)}\) and \(q^{(2)}\). In the absence of degeneracy this means that the eigenfunctions will be either symmetric or antisymmetric. If for some definite Hamiltonian a degeneracy is found, then even a small symmetric perturbation will remove it and will return us to the consideration of functions of definite symmetry as “proper linear combinations” of the zero approximation. Since there is always a sufficient variety of interactions, we may be sure that the state actually has a quite definite symmetry, if, of course, all the interactions under consideration are symmetric.

Thus, we can subdivide all eigenvalues and eigenfunctions into two systems:

\[ E_n^{(S)}, \qquad \psi_n^{(S)}(q^{(1)}, q^{(2)}) \tag{2,2} \]

and

\[ E_m^{(A)}, \qquad \psi_m^{(A)}(q^{(1)}, q^{(2)}) \tag{2,3} \]

corresponding respectively to symmetric and antisymmetric functions (\(n\) and \(m\) enumerate the different functions and energy levels of each of the systems).

To obtain the complete wave function it is still necessary to find functions depending on the isobaric spin. We shall require that they describe a state of the system of nucleons with a definite charge, i.e. that they be eigenfunctions of the operator

\[ \hat T_3=\frac{1}{2}\left(\hat\tau_3^{(1)}+\hat\tau_3^{(2)}\right); \]

\[ \hat T_3\chi=T_3\chi . \tag{2,4} \]

Then we can speak of a system of two neutrons \((T_3=1)\), or two protons \((T_3=-1)\), or one neutron and one proton \((T_3=0)\).

If we introduce the notation

\[ \delta(\tau,1)= \begin{cases} 1 & \text{for } \tau=1,\\ 0 & \text{for } \tau=-1 \end{cases} \quad \text{and} \quad \delta(\tau,-1)= \begin{cases} 0 & \text{for } \tau=1,\\ 1 & \text{for } \tau=-1 \end{cases} \tag{2,5} \]

for the functions of isotopic spin that describe, respectively, the neutron and proton states of a nucleon, then the function

\[ \delta\!\left(\tau_3^{(1)},1\right)\delta\!\left(\tau_3^{(2)},1\right) \tag{2,6a} \]

will refer to a system of two neutrons; the function

\[ \delta\!\left(\tau_3^{(1)},-1\right)\delta\!\left(\tau_3^{(2)},-1\right) \tag{2,6b} \]

to a system of two protons; and for the case of one proton and one neutron it will be possible to construct two functions:

\[ \delta\!\left(\tau_3^{(1)},1\right)\delta\!\left(\tau_3^{(2)},-1\right) \quad \text{and} \quad \delta\!\left(\tau_3^{(1)},-1\right)\delta\!\left(\tau_3^{(2)},1\right). \tag{2,6c} \]

Let us note that the choice of the isotopic-spin functions for a system of two particles in the form of products of functions describing the individual particles in no way restricts the generality of the argument. The four functions (2,6) form a complete orthonormal system in isotopic space, consisting of four points:

\[ \tau_3^{(1)}=1,\quad \tau_3^{(2)}=1; \qquad \tau_3^{(1)}=1,\quad \tau_3^{(2)}=-1; \]

\[ \tau_3^{(1)}=-1,\quad \tau_3^{(2)}=1; \qquad \tau_3^{(1)}=-1,\quad \tau_3^{(2)}=-1. \]

(Each of the functions (2,6) takes the value 1 at one of these four points and the value 0 at the other three.) Therefore the most general function of the isotopic variables is a linear combination of the functions (2,6). Substituting this combination into equation (2,4), it is easy to find that the eigenvalue \(T_3=1\) corresponds to the eigenfunction (2,6a), the value \(T_3=-1\) to the function (2,6b), while the eigenvalue \(T_3=0\) corresponds to an arbitrary linear combination of the two functions (2,6c).

The last requirement to which the isotopic-spin functions must be subject is that multiplying them by one of the coordinate-spin functions ((2,2) or (2,3)) must lead to a function antisymmetric in all five variables. For two particles this means that the isotopic-spin functions, like the coordinate-spin functions, must be symmetric or antisymmetric, i.e., must be eigenfunctions of the operator of permutation of the isotopic vari-

variables \(P^{(t)}(1,2)\). The eigenvalues of this operator are \(\pm 1\). From relation (1,13) between the permutation operator and the operator of total isobaric spin \(\hat{\mathbf T}^{\,2}\) we obtain that the isobaric-spin functions must be eigenfunctions of the operator \(\hat{\mathbf T}^{\,2}\) with eigenvalues 2 and 0. Since these eigenvalues are customarily expressed through the quantum number \(T\) by the formula \(T(T+1)\), the isobaric functions will be characterized by the quantum numbers \(T=1\) (symmetric function) and \(T=0\) (antisymmetric function).

The functions (2,6a) and (2,6b) directly satisfy the requirement of symmetry (and, consequently, belong to the eigenvalue \(T=1\) of the operator \(\hat{\mathbf T}^{\,2}\)). We denote them by

\[ \chi_1^1\left(\tau_3^{(1)}, \tau_3^{(2)}\right) = \delta\left(\tau_3^{(1)}, 1\right)\delta\left(\tau_3^{(2)}, 1\right) \tag{2,6a'} \]

and

\[ \chi_{-1}^1\left(\tau_3^{(1)}, \tau_3^{(2)}\right) = \delta\left(\tau_3^{(1)}, -1\right)\delta\left(\tau_3^{(2)}, -1\right), \tag{2,6b'} \]

where the upper index indicates the value of \(T\), and the lower one the value of \(T_3\) for the given function.

The functions (2,6c) are neither symmetric nor antisymmetric. But from them one can construct the symmetric combination

\[ \chi_0^1\left(\tau_3^{(1)}, \tau_3^{(2)}\right) = \frac{1}{\sqrt{2}} \left\{ \delta\left(\tau_3^{(1)}, 1\right)\delta\left(\tau_3^{(2)}, -1\right) + \delta\left(\tau_3^{(1)}, -1\right)\delta\left(\tau_3^{(2)}, 1\right) \right\} \tag{2,7} \]

and the antisymmetric combination

\[ \chi_0^0\left(\tau_3^{(1)}, \tau_3^{(2)}\right) = \frac{1}{\sqrt{2}} \left\{ \delta\left(\tau_3^{(1)}, 1\right)\delta\left(\tau_3^{(2)}, -1\right) - \delta\left(\tau_3^{(1)}, -1\right)\delta\left(\tau_3^{(2)}, 1\right) \right\}. \tag{2,8} \]

With the requirement of normalization, these combinations are, obviously, determined uniquely, up to an inessential phase factor.

The transition from the functions (2,6c) to the functions (2,7) and (2,8) has a clear physical meaning. The two functions (2,6c) differed in that the first of them described a state in which particle “1” was a neutron and particle “2” a proton, while the second described a state in which the neutron was particle “2” and the proton particle “1.” But such a distinction of particles by “numbers” is not physically possible, and in the functions (2,7), (2,8) it is erased. Both these functions describe a state with one neutron and one proton (eigenfunctions of the operator \(\hat T_3\) with \(T_3=0\)), but the charge state of a “particle of definite

“quantum number” is not defined (for the operators \(\hat{\tau}_3^{(1)}\) and \(\hat{\tau}_3^{(2)}\) the functions are not eigenfunctions).

Let us now write down the complete antisymmetric wave functions for a system of two nucleons.

The isobaric functions of two neutrons or two protons (2.6), being symmetric in their arguments, can be combined only with antisymmetric coordinate-spin functions (2.7). This gives:

\[ E_A,\qquad \Psi_{\mathrm{n},\mathrm{n}} = \psi^{(A)}\left(\mathbf r^{(1)}, s_z^{(1)};\mathbf r^{(2)}, s_z^{(2)}\right) \chi_1^1\left(\tau_3^{(1)}, \tau_3^{(2)}\right) \tag{2.9} \]

and

\[ E_A,\qquad \Psi_{\mathrm{p},\mathrm{p}} = \psi^{(A)}\left(\mathbf r^{(1)}, s_z^{(1)};\mathbf r^{(2)}, s_z^{(2)}\right) \chi_{-1}^1\left(\tau_3^{(1)}, \tau_3^{(2)}\right). \tag{2.10} \]

The presence here only of the antisymmetric function \(\psi^{(A)}\) is, of course, simply an expression of the Pauli principle, which for two neutrons or protons is applied to functions of the spatial and spin coordinates. The coincidence of the spatial-spin functions in (2.9) and (2.10) is a consequence of the assumed charge independence.

In the case of one neutron and one proton there arise two possibilities. Using the symmetric function (2.7), we obtain the functions

\[ E_A,\qquad \Psi_{\mathrm{n},\mathrm{p}} = \psi^{(A)}\left(\mathbf r^{(1)}, s_z^{(1)};\mathbf r^{(2)}, s_z^{(2)}\right) \chi_0^1\left(\tau_3^{(1)}, \tau_3^{(2)}\right), \tag{2.11} \]

which coincide in their spatial-spin part with the functions (2.9) and (2.10). In addition, a combination is possible of the antisymmetric function (2.8) with the symmetric function (2.2):

\[ E_S,\qquad \Psi_{\mathrm{n},\mathrm{p}} = \psi^{(S)}\left(\mathbf r^{(1)}, s_z^{(1)};\mathbf r^{(2)}, s_z^{(2)}\right) \chi_0^0\left(\tau_3^{(1)}, \tau_3^{(2)}\right). \tag{2.12} \]

These functions differ from the functions (2.9)—(2.11) not only in their charge part, but also in their dependence on the spatial and spin coordinates. The states of the proton–neutron system described by the functions (2.12) and belonging to the energy levels \(E_S\) have no analogues among the states of a system of two equally charged nucleons.

Let us note that the three functions (2.9)—(2.11) contain the isobaric functions \(\chi_1^1\), \(\chi_0^1\), \(\chi_{-1}^1\), all belonging to one and the same eigenvalue \(T=1\) of the total isobaric spin. On the other hand, the function (2.12) contains the function \(\chi_0^0\), the only one belonging to \(T=0\).

Thus, we see that the value of the total isobaric spin \(T\) characterizes the symmetry not only of the charge function, but also, through the Pauli principle, the symmetry of the spatial-spin function. Therefore different values of \(T\) correspond to

completely different spatial and spin properties of the states of the system of nucleons and, in particular, a different energy spectrum. On the other hand, formulas (2.9)—(2.11) show that, for a given value of \(T\), the difference in charge (i.e., the difference in the value of \(T_3\)) does not affect the spatial-spin properties of the system and its spectrum.

Let us recall that in our presentation we did not require from the outset that the wave functions be eigenfunctions of the operator \(\hat T^2\), but obtained this result as a consequence of the assumption that the spatial-spin functions possess a definite symmetry. Such symmetry is due, as was indicated above, to the presence of spatial-spin interactions. On the other hand, apart from the connection with the symmetry of the spatial-spin functions, which is based on the Pauli principle, there are no other grounds for expecting that a state with a definite energy can be characterized by a definite value of \(T\) (i.e., that there will be no splitting with respect to the values of \(T\)).

A set of states with one and the same value of \(T\) (and, of course, with the other quantum numbers coinciding), but differing in the values of \(T_3\), is called a charge multiplet, or \(T\)-multiplet. Under the assumption of charge independence (and neglecting the mass difference between the neutron and the proton), the energies and all other properties of the states belonging to the multiplet coincide. The concept of a charge multiplet and the quantum number \(T\), as applied to a system of any number of nucleons, was introduced by Wigner\(^5\) and Hund\(^6\).

The results obtained can now be formulated as follows. A system of two neutrons or two protons (\(T_3=+1\) or \(T_3=-1\)) can be only in a state belonging to a charge triplet (\(T=1\)), whereas for a proton–neutron system (\(T_3=0\)), in addition to the analogous charge-triplet state, states of a charge singlet (\(T=0\)) are also possible.

It is natural to apply the concept of \(T\)-multiplets to the question of the bound states of two nucleons. Only one such bound system is known—the deuteron, whereas neither the diproton nor the dineutron exists. If the absence of the diproton can be attributed to the influence of Coulomb forces, then the absence of the dineutron must find an explanation already within the framework of the hypothesis of charge independence. Such an explanation is indeed possible. As is known, the ground state of the deuteron is an even state with spin 1. In the case of two nucleons, the value of the spin and the parity also determine the value of the isobaric spin. This is connected with the fact that, for two particles, the operation of inversion of the system of coordi-

then whose origin is chosen at the midpoint of the straight line connecting the particles, is equivalent to an interchange of the spatial coordinates of the particles. Therefore, if the state of a system of two particles has a definite parity \(I(=\pm 1)\), then as a result of the interchange of the spatial coordinates the wave function acquires the factor \(I\). Further, for two particles an interchange of spins leads to the appearance of the factor \((-1)^{S+1}\) (\(S\) is the total spin), and an interchange of the isobaric variables gives the factor \((-1)^{T+1}\). The interchange of all five coordinates, however, must multiply the function by \(-1\). Hence

\[ I\cdot(-1)^{S+T}=-1,\qquad (-1)^T=I\cdot(-1)^{S+1}. \tag{2,13} \]

Thus, for \(S=0\), \(T=1\) in even states and \(T=0\) in odd states. Conversely, for \(S=1\), \(T=0\) in even states and \(T=1\) in odd states. Since the ground state of the deuteron is an even state with \(S=1\), for it

\[ T=0. \]

But a state with \(T=0\) is inadmissible for a system of two neutrons or two protons. Therefore for the dineutron or diproton there can be no state which, in its spatial and spin dependence, is identical to the ground state of the deuteron. The diproton or dineutron could exist in a state analogous to the virtual state of the deuteron \({}^{1}S\). Thus, the hypothesis of charge independence is in agreement with the existence of a single bound state of two nucleons with the spin and parity indicated above.

For two particles we have shown that the invariance of the space-spin properties of the system, even under the assumption of charge independence, holds not for arbitrary transformations of the functions of the isobaric spin \(\chi\), but only for transformations preserving the value of \(T\), or, equivalently, the value of the square of the length of the vector \(\mathbf T\). In geometrical language, one may say that charge independence leads to invariance of the space-spin properties with respect to rotations in “isobaric-spin space” (“charge space”). The conclusion obtained is in agreement with the fact, noted at the beginning of this section, that the presence in the Hamiltonian of operators of the form \((\boldsymbol{\tau}^{(i)},\boldsymbol{\tau}^{(k)})\), which are invariants of rotations in charge space, is compatible with charge independence.

Let us now turn to the case of an arbitrary number of particles. We shall still assume that the Hamiltonian contains no operators acting on the variables of the isobaric spin, so that the Hamiltonian is a symmetric function of spatial and ordinary spin operators alone. Hence, as before, together with each solution of the Schrödinger equation its

solutions for the same eigenvalue will include all functions obtained from the original solution by permutations of the particles. It follows that every time at least one not fully symmetric or not fully antisymmetric function belongs to a given eigenvalue, degeneracy will occur. (Let us emphasize that we are speaking of the degeneracy of solutions of the Schrödinger equation that determine only the space-spin functions, and not of the occurrence, for a given energy value, of several complete wave functions including dependence on the isobaric variables.)

In considering the case of two particles we assumed the absence of degeneracy and therefore restricted ourselves only to symmetric and antisymmetric solutions of the Schrödinger equation. When the number of nucleons is greater than two, one cannot restrict oneself to such solutions for all eigenvalues and, consequently, to the case of absence of degeneracy. This is evident at least from the fact that, as is known, in terms of the eigenfunctions of a Hermitian operator one can expand practically any function, while an arbitrary function with a number of variables greater than two obviously cannot be represented as a linear combination of a symmetric and an antisymmetric function.

We shall be interested only in such degeneracy as is necessarily caused by the symmetry of the Hamiltonian.

A permutation degeneracy of multiplicity \(n\) is regarded as necessary if from the \(n\) functions belonging to a given eigenvalue and transforming linearly into one another under permutations of the particles, it is impossible to construct \(m<n\) linear combinations which themselves would transform linearly into one another under permutations of the particles. To a necessary degeneracy there corresponds a “minimal” set of functions transforming into one another under the action of some group of transformations (in the present case, the group of particle permutations), which leaves the Hamiltonian of the system invariant. (In the terminology of group theory such a set forms a basis of a certain irreducible representation of the group.)

In addition to such necessary degeneracy, a stronger degeneracy may also occur (especially under an artificial mathematical simplification of the problem, neglect of small perturbations, etc.), in which several such “irreducible” (“minimal”) sets of functions belong to a given eigenvalue. Such degeneracy is called accidental, bearing in mind that it depends strongly on the concrete form of the Hamiltonian and can easily be removed when some perturbation is taken into account, even one that does not change the symmetry of the system.

The degeneracy that was discussed in considering two particles, when we regarded symmetric and antisymmetric

coordinate-spin functions belonging to different energy values is precisely such an accidental degeneracy.

As in the case of two particles, for any number of particles the absence of accidental degeneracy will be connected with the existence of a definite permutation symmetry of the functions belonging to the given eigenvalue of the Hamiltonian. Indeed, from \(A!\) completely nonsymmetric functions, differing by permutations of \(A\) particles, one can construct, by linear combination, both a completely symmetric and a completely antisymmetric function, each of which separately forms a “complete set” of functions transforming into one another under permutations. The impossibility of obtaining these two linear combinations means that the function which we would like to symmetrize (antisymmetrize) is antisymmetric (symmetric) in some pairs of particles, i.e. in fact possesses a certain definite symmetry. The presence of such symmetry or antisymmetry conditions in some pairs of particles reduces the multiplicity of the degeneracy from \(A!\) for a completely nonsymmetric function to, generally speaking, considerably smaller multiplicities corresponding to the required degeneracy.

In order for the degeneracy to be necessary, the symmetry conditions for the functions belonging to the given eigenvalue must be sufficiently restrictive. It is necessary that the symmetry could not be increased by passing to linear combinations of these functions. A description of the types of such “maximal” symmetry of functions can be found, for example, in \(^{14}\)*).

But if the assumption of the absence of accidental degeneracy leads to the requirement of a definite symmetry of the space-spin functions, then it turns out that, for the possibility of forming an antisymmetric total wave function, the isobaric-spin functions must also possess a quite definite symmetry, in a certain sense opposite to it. Thus the Pauli principle extends the requirement for the existence of a definite permutation symmetry also to the isobaric-spin functions.

Between the symmetry types of the space-spin and isobaric functions that are necessary for the possibility of obtaining an antisymmetric total wave function there exists a one-to-one correspondence. However, not every type of symmetry is possible for functions of isobaric spin (as also for ordinary spin). Isobaric-spin functions can be antisymmetric only in pairs of particles, but not in groups of three or a larger number of particles. Since the isobaric-spin variable

* ) An elementary proof of the theorems on the types (“characters”) of the permutation symmetry of functions is contained in the original paper \(^{15}\). For the application of the methods of this paper to the theory of isobaric spin, see \(^{6}\).

Since the spin for each particle can take only two different values \(\left(\pm \frac{1}{2}\right)\), in any group of three or more particles the values of at least two isobaric variables will coincide, and a function antisymmetric with respect to the particles of this group will vanish. This leads to the fact that, for some types of symmetry of the space-spin functions, the construction of the complete wave function is altogether impossible, and the corresponding solutions of the Schrödinger equation for a system of nucleons have no physical meaning. For example, one cannot obtain a completely antisymmetric total wave function from a coordinate-spin function symmetric in all particles if the number of particles is greater than two.

The origin of this restriction is clear. The types of symmetry of the space-spin wave function admissible in the theory of isobaric spin must not, it would seem, differ from those which would be obtained by a direct application of the Pauli principle to a system of neutrons and protons as different particles. But in that case only such solutions of the Schrödinger equation have meaning as can be antisymmetrized with respect to permutations of any two neutrons, and also of any two protons. “Too symmetric” functions, like the one mentioned in the example given, do not permit such antisymmetrization for any choice of the numbers of neutrons and protons out of the given total number of particles, and in the general case the restrictions on the types of symmetry of the space-spin wave functions that follow from the theory of isobaric spin have the same meaning.

In the case of two nucleons we saw that to each type of symmetry of the isobaric-spin functions there corresponds a definite value of the total isobaric spin (the quantum number \(T\)). This is also true for an arbitrary number of nucleons.

For a number of nucleons \(A\), the most general type of “maximal” symmetry\({}^{14}\) for the functions of the isobaric-spin variables may be denoted by specifying some partition of the number \(A\) into two nonnegative integer summands:

\[ A=n_1+n_2 \qquad (n_1 \geq n_2). \tag{2,14} \]

The function described by such a partition is antisymmetric in \(n_2\) pairs of particles and symmetric in the remaining \(n_1-n_2\) particles.

It can be shown that the function defined by the partition (2,14) is an eigenfunction of the operator \(\hat T^{\,2}\), belonging to the eigenvalue \(T(T+1)\), where the quantum number \(T\) is connected with the partition numbers \(n_1,n_2\) by the formula

\[ T=\frac{n_1-n_2}{2}. \tag{2,15} \]

The proof of this formula is given in the Appendix. It is based on the connection between the operator \(\hat{\Gamma}^{2}\) and the operators of permutations of charge variables, established by formula (1,12).

It is clear that, for a given number of nucleons \(A\), the symmetry of the isobaric-spin function (i.e., the numbers \(n_1\) and \(n_2\)) is completely determined by the quantum number \(T\). Since the symmetry of the isobaric-spin function in turn determines, by the Pauli principle, the symmetry of the space-spin function, we again, as in the case of two particles, become convinced that the general space-spin properties will be possessed by those different charge states of the system of nucleons which have one and the same total isobaric spin \(T\).

From the interpretation of the numbers \(n_1, n_2\) and formula (2,15) it is seen that the maximum value \(T=\dfrac{A}{2}\) (attained for \(n_1=A,\ n_2=0\)) corresponds to complete symmetry of the isobaric-spin function and, hence, to complete antisymmetry of the space-spin function. A decrease of \(T\) corresponds to an increase of antisymmetry in the isobaric-spin function and an increase of symmetry in the space-spin function. The minimum value of \(T\) will be equal to zero if \(A\) is even (then the equality \(n_1=n_2\) is possible), and equal to \(\dfrac{1}{2}\) if \(A\) is odd (then the minimum value is \(n_1-n_2=1\)).

Let us now determine what different charge states may possess a symmetry characterized by the quantum number \(T\).

The isobaric-spin function of symmetry \(T\) is nonzero only in the case when no two of its antisymmetrically related variables have identical values. Therefore, in order for it to be nonzero, the number of coincident values \(\left(+\dfrac{1}{2}\ \text{or}\ -\dfrac{1}{2}\right)\) of its variables must not exceed \(n_1\). (The values of all symmetrically related variables may coincide and, in addition, one value from each antisymmetric pair.) If, moreover, the function under consideration describes a state with a definite charge, then it is nonzero only on the condition that exactly \(N\) of its variables are equal to \(+\dfrac{1}{2}\), while the remaining \(Z\) are equal to \(-1\) \((N+Z=A,\ N-Z=2T_3)\). In order that the function not vanish identically, both conditions must be compatible. For this purpose the larger of the numbers \(N\) and \(Z\) must not exceed \(n_1\). This larger number may be represented in the form

\[ \frac{N+Z}{2}+\frac{|N-Z|}{2}, \]

so that we obtain the condition

\[ \frac{N+Z}{2}+\left|\frac{N-Z}{2}\right|\leqslant n_1. \]

Taking into account that \(N+Z=n_1+n_2\), we find:

\[ \left|\frac{N-Z}{2}\right|\leqslant \frac{n_1-n_2}{2}, \]

and, using the definition of \(T_3\) and formula (2.15), we finally obtain:

\[ |T_3|\leqslant T. \tag{2.16} \]

Hence it is seen that the possible values of \(T_3\) for a given \(T\) will be

\[ -T,\ -T+1,\ldots,\ T-1,\ T \tag{2.17} \]

—a total of \(2T+1\) states of the charge multiplet.

Thus, the hypothesis of charge independence leads to the result that a system of nucleons may be assigned, in addition to the usual quantum numbers such as spin and parity, a quantum number \(T\)—the “total isotopic spin,” characterizing the permutation symmetry of the charges and, consequently, also of the coordinate-spin functions.

Under the condition of charge independence, the total isotopic spin is a constant of the motion; the operator \(\hat T^2\) commutes with the Hamiltonian. For a given number of nucleons \(A\), the space-spin properties and, in particular, the energy values coincide for those isobaric states which correspond to a definite value of \(T\). To each value of \(T\) there belong \(2T+1\) isobaric states forming a charge multiplet, for which

\[ |T_3|=\left|\frac{N-Z}{2}\right|\leqslant T. \]

It follows from this that, for a system of nucleons with close numbers of neutrons and protons, there exist states which systems with a large difference in the numbers of these and the other particles cannot possess. On the other hand, for a prescribed number of protons and neutrons the possible values of \(T\) are bounded from below by the inequality just given.

Of course, these results, analogous to the well-known results of the theory of ordinary spin, could have been seen directly from the fact that, in a formal respect, the properties of the isotopic-spin operators coincide with the properties of the operators of ordinary spin.

However, at the basis of spin theory (for example, in deriving permutation relations for the operators) there is usually placed the considera-

the isotropy of physical space and, accordingly, the invariance of the description of phenomena with respect to rotations of the coordinate system. Therefore, in the case of ordinary spin, for example, the circumstance that the operators \(\hat{s}_x, \hat{s}_y, \hat{s}_z\) form a three-dimensional vector is quite clear in its physical content. At the same time, for isotopic spin the “rotational” aspect and, in particular, the three-dimensionality of “charge space” are far less obvious.

The possibility of speaking about rotation in a three-dimensional symbolic charge space is a consequence of the dichotomy of the variable of isotopic spin for each nucleon. The wave function of a nucleon at each point of real space and of ordinary spin space is represented by two complex numbers entering into a column matrix. Such a most general charge state of the nucleon may be obtained, for example, from any state with definite charge by means of some unitary transformation, and even a unitary transformation with determinant equal to unity. But to every two-dimensional transformation of this type one can unambiguously assign a transformation of rotation in three-dimensional real space, and essentially different unitary transformations (differing not only by a sign) will be assigned different rotations \(^{4,17}\). In this way the group of transformations of the charge states of a single nucleon is connected with the group of rotations in symbolic three-dimensional space.

In the case of several nucleons, rotation in charge space is, of course, not the most general transformation allowing one to transform a given charge state into any other charge state. But the hypothesis of charge independence also does not assert the equivalence of all charge states of a system of nucleons. It is asserted only that, for properties of the system which do not depend explicitly on charge, no charge state of a single nucleon—neutron, proton, or any superposition of these two states—is singled out. From the point of view of charge properties, on the other hand, it is natural, of course, to single out the “pure” neutron and “pure” proton states. This singling out is effected by choosing the axis “3” in the symbolic charge space in such a way that states with a definite projection \(\tau_3\) of the vector \(\boldsymbol{\tau}\) correspond to the neutron and the proton. But the theory must be invariant with respect to the choice of that nucleon state to which a definite value of the isotopic variable \(\tau_3\) corresponds. Passing to another choice also means a “rotation of the coordinate system in the charge space of the nucleon” (according to the connection noted above between rotations and unitary transformations of the charge functions of the nucleon). Analogously, one may consider unitary transformations

of the system of nucleons itself, corresponding to its own “rotation” in charge space. The state arising as a result of the transformation differs from the initial one in that the role which, in the latter, was played, for example, by the neutron state of each nucleon passes to that superposition of neutron and proton states which is obtained under the same “rotation” of the single neutron in charge space. Under the assumption of charge independence, the state of the system of nucleons obtained in this way will coincide with the initial one in its space-spin properties.

Let us emphasize once again that charge independence is equivalent to invariance with respect to rotation in the space defined by the manifold of charge states of a single nucleon.

III. ENERGY LEVELS OF ISOBARIC NUCLEI

Simultaneously with the establishment of the concept of total isobaric spin, attempts were made to calculate energy levels for light nuclei4, 6, 16, for which the violation of charge independence by Coulomb forces is insignificant. In doing so, a model of individual particles with \(L\)-\(S\) coupling was used, and the filling of spatial shells by nucleons was considered. Under the simplest assumptions about the type of forces, data were obtained on the sequence of levels as a function of the quantum numbers \(L, S, T\).

Recently, calculations have been carried out under the assumption of \(j\)-\(j\) coupling18, and an intermediate coupling has also been considered33.

The most important result is that the low energy levels and, in particular, the ground states of nuclei must correspond to the smallest possible value, for a given nucleus, of the quantum number \(T\), i.e. to the case \(T = T_3\).

This is evidently connected with the fact that a decrease in the value of \(T\), i.e. an increase of antisymmetry in the isobaric-spin function, leads to an increase of symmetry in the space-spin function and, as a result, makes possible a higher (permutation) symmetry of the spatial dependence of the complete wave function. It is precisely this highest possible permutation symmetry of the spatial function that corresponds to the minimum value of the energy (if the forces between particles have the character of attraction), since any increase of antisymmetry means a decrease in the probability of finding particles at very small distances from one another. (In view of the opposite sign of the forces, the situation in nuclei is the reverse of that which takes place for the electrostatic interaction of atomic electrons.) One may expect that the conclusion drawn has general significance and is not connected with detailed assumptions about nuclear forces.

States with high spatial permutation symmetry, i.e., with small values of \(T\), prove to be still more preferable energetically if one takes into account the exchange character of the nuclear forces. It is known that among the exchange forces the greatest role is played by Majorana-type forces. These forces have the character of attraction between a pair of particles if the wave function is symmetric in the spatial coordinates of these particles, and the character of repulsion if spatial antisymmetry is present. As a result, the dependence of the binding energy on the symmetry of the wave function turns out to be very sharp. Only states with high spatial symmetry, i.e., with small values of \(T\), are realized in light nuclei. This explains, without abandoning charge independence, such an important fact as

Fig. 1.

the absence of light nuclei with a considerable difference in the number of neutrons and protons. For such nuclei small values of \(T\) would be impossible \(\left(T \geq \dfrac{|N-Z|}{2}\right)\); effective use of the exchange forces for obtaining binding energy is impossible.

Under the conditions of exact charge independence and in the absence of Coulomb forces and of the difference between the proton and neutron masses, the schemes of the energy levels of light nuclei would have the form shown in Fig. 1, \(a\) and \(b\), respectively for nuclei of even and odd mass. For simplicity the figure shows only one charge multiplet, having a definite value of \(T\) (indicated as a subscript to \(E\)). As was clarified above, each multiplet extends over all isobaric nuclei having \(T_3\) values from \(-T\) to \(T\).

The levels of the various isobaric nuclei forming a multiplet not only coincide energetically, but also have identical other quantum characteristics (angular momenta, parities). This is connected with the fact that the entire theory set forth above, as applied

Isobaric Spin

...to the eigenfunctions of the Hamiltonian of the system, can be repeated with respect to the common eigenfunctions of the complete set of operators commuting with the Hamiltonian and not depending explicitly on the charge.

Let us now consider what changes must be introduced into the scheme of nuclear energy levels in order to take account of two obvious deviations from the exact charge independence of the Hamiltonian—the difference between the masses of the neutron and the proton, and also the Coulomb interaction.

Strictly speaking, the very existence of a difference between the masses of the neutron and the proton leads to the appearance in the Hamiltonian of terms containing the operators \(\hat{\tau}_3^{(i)}\) in such a combination that it does not commute with the operator \(\hat{T}^2\). For example, if one uses the nonrelativistic approximation, then the terms in the Hamiltonian corresponding to the rest energy and kinetic energy will be

\[ \sum_{i=1}^{A}\left\{ m_n c^2 \frac{1+\hat{\tau}_3^{(i)}}{2} + m_p c^2 \frac{1-\hat{\tau}_3^{(i)}}{2} \right\} - \sum_{i=1}^{A}\left\{ \frac{\hbar^2}{2m_n}\frac{1+\hat{\tau}_3^{(i)}}{2} + \frac{\hbar^2}{2m_p}\frac{1-\hat{\tau}_3^{(i)}}{2} \right\}\Delta_i . \tag{3,1} \]

The first sum, representing the rest energy of the nucleons, can be written in the form

\[ A\frac{(m_n+m_p)}{2}c^2+(m_n-m_p)c^2T_3, \tag{3,2} \]

showing that it commutes with \(\hat{T}^2\). Thus, the contribution to the Hamiltonian from the rest energy leads only to a splitting of the levels (determined by the quantity \((m_n-m_p)c^2T_3\)), but the levels can still be characterized by the quantum number \(T\).

The second sum in (3,1), giving the kinetic energy, is equal to

\[ -\frac{\hbar^2}{2m}\sum_{i=1}^{A}\Delta_i + \frac{\hbar^2(m_n-m_p)}{4m_n m_p} \sum_{i=1}^{A}\hat{\tau}_3^{(i)}\Delta_i . \tag{3,3} \]

Here the first term corresponds to the kinetic energy of a system of particles, each of which has mass

\[ m=\frac{2m_n m_p}{m_n+m_p}\simeq m_{\text{nucleon}} . \]

This term is the main one in the kinetic energy and, obviously, commutes with \(\mathbf{T}^2\). The second term in (3.3) does not commute with \(\hat{\mathbf{T}}^2\), but is a small correction. The results obtained from the hypothesis of charge independence are valid only when this term is neglected. If it is regarded as a perturbation, it will lead to transitions between states with different values of \(T\). In what follows we shall take the correction for the mass difference into account only in the rest energy (3.2).

The operator of the Coulomb interaction

\[ H_c=\frac{e^2}{4}\sum_{i<j}\frac{\left(1-\hat{\tau}_3^{(i)}\right)\left(1-\hat{\tau}_3^{(j)}\right)}{r_{ij}} \tag{3.4} \]

also violates the charge independence of the Hamiltonian and does not commute with \(\mathbf{T}^2\). The Coulomb interaction will lead both to a displacement of the energy levels and to the appearance of a mixture of states with different values of \(T\). To estimate the latter effect, the method of stationary perturbation theory was applied in work \(^{19}\), using the shell model. If \(\psi(T)\) is the wave function of a nucleus in a state with isotopic spin \(T\), obtained without taking Coulomb forces into account, then in the first approximation of perturbation theory the wave function will be a superposition of states with different \(T\):

\[ \Phi_T=\psi(T)+\sum_{T'}\frac{H^c_{T'T}}{E_T-E_{T'}}\psi(T')=\psi(T)+\sum_{T'}\alpha_T(T')\psi(T'). \tag{3.5} \]

The quantity \(\left|\alpha_T(T')\right|^2\) may be taken as a measure of the fraction of the state \(\psi(T')\) in the superposition, if in the zeroth approximation the state was \(\psi(T)\).

In the cited work \(^{19}\), an estimate was made of the quantities \(a_T^2(T')\) at \(T=0\) for the case of nuclei with two or four nucleons outside closed shells. The possibility of excitation of the core by Coulomb forces was ignored. For two nucleons outside a closed shell (the nucleus \(\mathrm{Li}^6\) or \(\mathrm{N}^{14}\)), the probability of adding to the basic state \(T=0,\ S=1,\ L=0,\ J=1\), configuration \((1p)^2\), the state \(T=1,\ S=1,\ L=0,\ J=1,\ (1p2p)\), proves to be of the order

\[ a_0^2(1)\simeq 2.5\cdot 10^{-3}, \tag{3.6} \]

while the probabilities of adding other states are still smaller than the indicated value or even equal to zero. For the case of four nucleons the quantities \(a_0^2(1)\) also turn out to be very small \((10^{-5}—10^{-6})\). Apparently, admixtures of states should be more appreciable if the initial \(T\ne 0\).

The general conclusion is that Coulomb forces lead to mixing of states with different values only to a very weak degree. Therefore, if substantial mixing is discovered, it will indicate a deviation from charge independence of the specific nuclear forces.

Thus, while preserving to a considerable degree the “purity” of states with respect to isobaric spin, the Coulomb interaction, on the other hand, leads to a quite noticeable splitting of the energy levels of the components of a \(T\)-multiplet.

For calculating the magnitude of this splitting, the elementary approach which leads to the well-known semiempirical formulas for nuclear masses proves satisfactory. The contribution to the mass from the Coulomb energy can be written in the form

\[ \Delta_c M(Z)=\frac{3e^2}{5c^2R}Z(Z-1) \tag{3,7} \]

or, equivalently,

\[ \Delta_c M(T_3)=\frac{3e^2}{5c^2R}\left(\frac{A}{2}-T_3\right)\left(\frac{A}{2}-T_3-1\right), \tag{3,8} \]

where for the nuclear radius the expression \(^{22}\)

\[ R=r_0 A^{\frac13}=1.45\cdot 10^{-13} A^{\frac13}\ \text{cm}. \tag{3,9} \]

is adopted.

From (3,2) and (3,8) we obtain that the splitting of the levels of a \(T\)-multiplet caused by the mass difference of the neutron and the proton, as well as by the Coulomb forces, will be determined by the expression

\[ M(T_3)=A\frac{(m_n+m_p)}{2}+(m_n-m_p)T_3+ \]

\[ +\frac{3e^2}{5c^2r_0}\left(\frac{A}{2}-T_3\right)\left(\frac{A}{2}-T_3-1\right)A^{-\frac13}, \]

which can also be written in the form

\[ M(T_3)-M(0)=(m_n-m_p)T_3-\frac{3e^2}{5c^2r_0}T_3(A-1-T_3)A^{-\frac13}. \tag{3,10} \]

Here nuclear masses stand on the left. If, as is customary, the calculation is made for atomic masses, then in the right-hand side of the equality one must replace the mass difference of the neutron and proton by the mass difference of the neutron and the hydrogen atom:

\[ M_a(T_3)-M_a(0)=(m_n-m_H)T_3-\frac{3e^2}{5c^2r_0}T_3(A-1-T_3)A^{-\frac13}. \tag{3,11} \]

Substituting the values of the constants \(^{22}\), we obtain:

\[ M_a(T_3)-M_a(0)= \]

\[ =\left\{0.781T_3-0.598T_3(A-1-T_3)A^{-\frac13}\right\}\ \text{MeV}. \tag{3,12} \]

The first term, taking into account the difference between the masses of n and p, leads to an increase in the energy of the level with increasing \(T_3\); the second—the Coulomb term—has the opposite effect. Except in the case of the very lightest nuclei, the second term predominates, so that the splitting of the levels of a \(T\)-multiplet for the various mirror nuclei entering it takes the form shown in Fig. 2, \(a\) and \(b\).

If the energy-level differences of mirror nuclei can be interpreted by a formula of type (3.11), then this speaks in favor of the fact that, apart from the Coulomb interaction and the effect of the difference between the masses of n and p, the forces between nucleons may be regarded as charge-independent.

Fig. 2.

The mass differences of mirror pairs of nuclei, i.e. mirror pairs of the type \(N_0+\mathrm{n}\) and \(N_0+\mathrm{p}\), \(N_0+2\mathrm{n}\) and \(N_0+2\mathrm{p}\), where \(N_0\) is the “residual nucleus,” consisting of equal numbers of neutrons and protons, have been studied in detail \(^{20,21,22}\).

The “mirror” transformation of a system of nucleons consists in replacing all its neutrons by protons and all its protons by neutrons. In this process, evidently, only the number of pairs n—n and p—p changes, but not the number of pairs n—p. Therefore comparison of the characteristics of mirror nuclei gives information only about the relative magnitude of the neutron—neutron and proton—proton forces (but does not permit comparison of these forces with the neutron—proton forces). In particular, the coincidence of the energy levels of mirror nuclei up to the Coulomb displacement (and the difference between the masses of n—p) testifies only to the equality of the n—n and p—p forces to each other.

In the notation of isobaric-spin theory, mirror nuclei differ by the replacement of \(T_3\) by \(-T_3\). For a nucleus of the type \(N_0+k\mathrm{n}\)

\[ T_3 = +\frac{k}{2}, \quad \text{for a nucleus } N_0 + kp \quad T_3 = -\frac{k}{2}. \]
According to the number \(k=1, 2, \ldots\), mirror pairs of the first, second, etc. orders are distinguished.

From formula (3.12) we have:

\[ M_a\left(-\frac{k}{2}\right) - M_a\left(\frac{k}{2}\right) = k\left\{0.598(A-1)A^{-\frac{1}{3}} - 0.781\right\} \, \text{Mev}. \tag{3.13} \]

Light odd nuclei are grouped into mirror pairs of the first order. In this case, one of the nuclei of the pair is \(\beta\)-radioactive.

Table I

Mirror nuclei of the first order.
Mass differences

No.	Mirror pair of nuclei	Values of \(\Delta M_a\) in Mev, experiment	Values of \(\Delta M_a\) in Mev, by formula (3.13), \(k=1\)
1	\(\mathrm{n}^1 \to \mathrm{H}^1\)	0.781	0.781
2	\(\mathrm{H}^3 \to \mathrm{He}^3\)	0.0185	0.048
3	\(\mathrm{Be}^7 \to \mathrm{Li}^7\)	0.864	1.095
4	\(\mathrm{C}^{11} \to \mathrm{B}^{11}\)	1.990	1.908
5	\(\mathrm{N}^{13} \to \mathrm{C}^{13}\)	2.227	2.270
6	\(\mathrm{O}^{15} \to \mathrm{N}^{15}\)	2.705	2.614
7	\(\mathrm{F}^{17} \to \mathrm{O}^{17}\)	2.754	2.941
8	\(\mathrm{Ne}^{19} \to \mathrm{F}^{19}\)	3.254	3.253
9	\(\mathrm{Na}^{21} \to \mathrm{Ne}^{21}\)	3.52	3.554
10	\(\mathrm{Mg}^{23} \to \mathrm{Na}^{23}\)	3.92	3.845
11	\(\mathrm{Si}^{27} \to \mathrm{Al}^{27}\)	4.61	4.402
12	\(\mathrm{P}^{29} \to \mathrm{Si}^{29}\)	4.65	4.67
13	\(\mathrm{S}^{31} \to \mathrm{P}^{31}\)	4.92	4.931
14	\(\mathrm{Cl}^{33} \to \mathrm{S}^{33}\)	5.22	5.184
15	\(\mathrm{Ar}^{35} \to \mathrm{Cl}^{35}\)	5.42	5.435
16	\(\mathrm{K}^{37} \to \mathrm{Ar}^{37}\)	5.59	5.680
17	\(\mathrm{Ca}^{39} \to \mathrm{K}^{39}\)	6.15	5.920
18	\(\mathrm{Sc}^{41} \to \mathrm{Ca}^{41}\)	5.96	6.156

Starting with \(A=7\), the nuclei with one extra neutron are stable; the other nucleus of the pair is \(\beta^+\)-active or undergoes \(K\)-capture (\(\mathrm{Be}^7\)). The mass differences of the nuclei of a mirror pair may be pos-

obtained experimentally from measurements of the limiting energy of \(\beta\)-decay, and also from data on the thresholds of the reactions \((p,n)\), \((d,n)\), \((\gamma,n)\).

A comparison of the experimental values of mass differences with the differences calculated by formula (3.13) (with \(k=1\)) is given in Table I (p. 479)\(^{22}\). This comparison shows that the displacement of the ground states of the nuclei of a mirror pair in energy can indeed be attributed to Coulomb forces. Evidently, these states are components of an isobaric doublet

\[ \left(T=|T_3|=\frac{1}{2}\right). \]

Not only the ground states, but also other energy levels of mirror nuclei must coincide after subtraction of the effect of the Coulomb forces. Such subtraction is to a considerable extent achieved automatically if the ground states of the nuclei of a mirror pair are made to coincide on the energy scale, as was done in Fig. 3\(^{22}\). Even if coincidence of individual levels is not thereby achieved, the similarity of the whole system of levels is demonstrated very convincingly. It is difficult to expect more, if one takes into account that the assumption of an identical influence of Coulomb forces on the displacement of different excited levels is rather crude.

In cases where the angular momenta and parities of similar states are known, they turn out to coincide, as also follows from charge independence. These cases are marked by dots in Fig. 3.

Among even nuclei we encounter mirror pairs of the second order (\(k=2\)). The experimental data are compared with the results of calculation by formula (3.13) in Table II\(^{22}\).

Table II

Mirror nuclei of the second order.
Mass differences

No.	Mirror pair of nuclei	\(\Delta M_a\) values in \(10^{-6}\,M_{\mathrm{ev}}\), experiment	\(\Delta M_a\) values in \(10^{-6}\,M_{\mathrm{ev}}\), by formula (3.13), \(k=2\)
1	\(B^8 - Li^8\)	\(200 \pm 30\)	282
2	\(C^{10} - Be^{10}\)	\(380 \pm 11\)	369
3	\(N^{12} - B^{12}\)	\(463 \pm 9\)	450
4	\(O^{14} - C^{14}\)	\(535 \pm 11\)	525
5	\(Na^{20} - F^{20}\)	\(\sim 890\)	732
6	\(Al^{24} - Na^{24}\)	\(910 \pm 30\)	857

Fig. 3.

It may be considered that the ground states of second-order mirror pairs are two states of the charge triplet \((T=1,\ T_3=\pm 1)\). Then, according to the hypothesis of charge independence, there must exist a state for a nucleus with equal numbers of neutrons and protons \((T_3=0;\) such nuclei are called charge-symmetric), belonging to the same triplet, analogous to the ground states of the mirror pair in its quantum characteristics and differing in energy only by the “Coulomb displacement” \((3,11)\). However, this state of the charge-symmetric nucleus need not necessarily be the ground state, since for a nucleus with \(T_3=0\), besides states with \(T=1\), states with \(T=0\) are possible, probably lower. The component of the triplet \(T=1,\ T_3=0\), analogous to the ground states of the mirror pair, should appear as an excited state of the charge-symmetric nucleus.

Fig. 4.

As an example, let us consider the nucleus \(N^{14}\) \((T_3=0)\), for which a state is expected that, together with the ground states of \(O^{14}\) and \(C^{14}\), belongs to the charge triplet \((T=1)\). In Fig. 4, borrowed from work 2, a scheme is given of the levels of the nuclei \(C^{14}\), \(N^{14}\), and \(O^{14}\). The energy difference of the ground states of the nuclei \(C^{14}\) and \(N^{14}\) is determined from the boundary of the \(\beta\)-decay of \(C^{14}\). The ground level of \(N^{14}\) lies lower than the ground level of \(C^{14}\), although in the transition from \(C^{14}\) to \(N^{14}\) the Coulomb energy increases. This lower ground level of the charge-symmetric nucleus \(N^{14}\) evidently belongs to \(T=0\). But if we add to the energy of the ground state of \(C^{14}\) \((0.15\ \text{MeV}\) above the ground state of \(N^{14})\) the Coulomb displacement \([M_a(0)-M_a(1)]\) of the levels in \(N^{14}\) relative to the levels in \(C^{14}\), calculated by formula \((3,12)\) and equal to \(2.20\ \text{MeV}\), then we arrive at the value \(2.35\ \text{MeV}\) for the expected level with \(T=1\) in the nucleus \(N^{14}\). This agrees well with the known level \(2.31\ \text{MeV}\) in the nucleus \(N^{14}\), which is thus identified as the lowest level in this nucleus having \(T=1\). Subtraction of the Coulomb displacement for the ground state of \(O^{14}\) \((T_3=-1)\) likewise leads to an almost exact coincidence of this state with the level \(2.31\ \text{MeV}\) in \(N^{14}\). Thus, all three components of the charge triplet have been identified.

It is known that the nucleus \(C^{14}\) in the ground state has \(J=0^+\); the same characteristics should be assigned also to the two other components of the triplet. It is interesting that there exists a \(\beta\)-transition from the ground state of \(O^{14}\) to the level \(2.31\ \text{MeV}\) of the nucleus \(N^{14}\). The hypothesis

charge independence leads to the conclusion that this will be a \(0 \to 0\) transition, forbidden by Teller’s selection rules.

Fig. 4 also demonstrates the similarity of the excited states of \(\mathrm{C}^{14}\) and \(\mathrm{N}^{14}\) and the presence, in the charge-symmetric nucleus, of levels that have no analogues in the neighboring isobar (obviously, these are levels with \(T = 0\)). This figure is, as it were, a realization of the general scheme of Fig. 2, \(a\).

For a more convenient survey of the energy-level schemes of even isobaric nuclei, it is expedient to reduce the schemes to a form similar to Fig. 1, \(a\). This can be achieved if, in a scheme of the type of Fig. 4, the positions of the ground states of nuclei with \(T_3 = \pm 1\) are shifted by the amount of the Coulomb displacement calculated from formula (3,11). Then we obtain such an arrangement of levels as should be expected in the absence of Coulomb effects. Since formula (3,11), of course, is not exact, it will be more reliable, having used it to estimate the magnitude of the Coulomb displacement, to align the ground states of nuclei with \(T_3 = \pm 1\) with the nearest level of the nucleus with \(T_3 = 0\). Then one may expect that this level is the lowest level of the nucleus with \(T_3 = 0\) having \(T = 1\). In some cases such an identification is aided by consideration of data obtained from nuclear reactions, as will be discussed in more detail in the next section.

As a result of the transformation described above, which has the meaning of a semi-empirical subtraction of Coulomb effects, one obtains the schemes shown in Fig. 5, \(a\), \(b\), and \(c\) for isobars with mass numbers \(A = 6, 8, 10, 12, 14, 16\) and \(T_3 = 0, \pm 1\), borrowed from work \(^{33}\)*). In the schemes the energies are given in MeV; the ground state of the nucleus with \(T_3 = 0\) is chosen as the origin. Levels established unreliably are marked with a dotted line; broad levels are hatched. The hatching at the right edge of the scheme denotes unstudied energy regions. Above the levels marked by the letters \(p\), \(n\), \(\alpha\), the nuclei are unstable with respect to emission of the corresponding particles.

The schemes also give some data on the quantum characteristics of the levels.

Data on the excited states of nuclei with an excess of protons \((T_3 = -1)\) are absent. The ground states could, after subtraction of the Coulomb displacement, be aligned with the lowest state of the nuclei with \(T_3 = 0\) having \(T = 1\), as we saw above in the example of the isobaric triad \(\mathrm{C}^{14}\), \(\mathrm{N}^{14}\), \(\mathrm{O}^{14}\).

*) Instead of calculating the “theoretical” Coulomb displacement by a formula of type (3,11), work \(^{33}\) used a somewhat different method. However, since alignment with experimentally known levels was then carried out, the results do not differ from those given by the method described in the text. See also the comparison of levels of isobaric nuclei performed in \(^{34,35}\).

Fig. 5.

Visible annotations in the figure:

a)
\(H^{6}\); \(5.3\); \(3.58\); \((0)\); \(2.19\); \(n\); \(\rho\); \(\alpha\); \(1\); \(L^{6}\).
\(\alpha\); \(n\); \(19.1\); \(16.8\); \(L^{8}\); \(19.9\); \(19.0\); \(18.4\); \(18.8\); \(16.7\); \(17.1\); \(15.0\); \(14.7\); \(21\); \(3\); \(11.1\); \(9.8\); \(1.5\); \(4.9\); \(5.0\); \(3.0\); \((1)\); \((2^{+})\); \(Be^{8}\).
\(3\); \(12.57\); \(13.77\); \(14.87\); \(17.21\); \(18.46\); \(18.49\); \(16.79\); \(17.10\); \(17.44\); \(15.14\); \(n\); \(p\); \(d\); \(t\); \(\alpha\); \(B^{12}\); \(9.62\); \(7.3\); \(4.44\); \(2^{+}\); \(0\); \(C^{12}\); \(1+(0)\).

b)
\(9.28\); \(9.78\); \(8.0\); \(\alpha\); \(n\); \(8.68\); \(5.11\); \(1.74\); \(Be^{10}\); \(0\); \((2)\); \((0)\); \(2.68\); \(7.58\); \(4.79\); \(2.15\); \(0.714\); \(0.29\); \(3\); \(8.0\); \(p\); \(d\); \(\gamma\); \(\alpha\); \((1)\); \((2^{+})\); \(0^{+}\).

c)
\(n\); \(8.42\); \((2.32)\); \(0^{+}\); \(C^{14}\); \(5.46\); \(5.10\); \(4.8\); \(2.32\); \(p\); \(d\); \(\alpha\); \((1^{-})\); \((-)\); \((+)\); \(1^{+}\); \(N^{14}\).
\(14.1\); \(12.5\); \(13.6\); \(11.6\); \(11.2\); \(10.8\); \(9.5\); \(8.2\); \(7.0\); \(6.01\); \(5.11\); \(6.05\); \(0\); \(0\); \(N^{16}\); \(O^{16}\); \(p\); \(d\); \(^{3}\!He\); \(\alpha\); \((2)\); \(2^{+}\); \(0^{+}\).

An important difference is revealed when comparing the schemes of isobars of the types \(A=4n\) and \(A=4n+2\). If, for isobars of the type \(A=4n\), the lowest levels with \(T=1\) lie in the region \(12\text{--}17\) MeV, then isobars of the type \(A=4n+2\) have excitation energies of such levels only of the order of \(1.7\text{--}3.6\) MeV.

The relative proximity of the levels with \(T=1\) in isobars \(A=4n+2\) to the ground state is apparently explained by the competition between the types of permutation symmetry of the charge and spin functions, leading to maximal symmetry of the spatial dependence of the wave function. (In the zero approximation of \(L\)-\(S\) coupling\(^{6,16}\), the maximum possible symmetry of the spatial dependence is realized for such isobars not only for \(T=0\), \(S=1\), but also for \(T=1\), \(S=0\).)

Examination of the schemes of Fig. 5, \(a\), \(b\), and \(c\) shows that, even after a rough subtraction of the effects associated with Coulomb forces (and with the neutron–proton mass difference), the arrangement of the energy levels of even isobaric nuclei proves to be close to the arrangement required by charge independence of nuclear forces. For odd nuclei an analogous result was obtained above from consideration of mirror pairs of first order. Thus one may conclude that charge independence of nuclear forces does indeed regulate the most general regularities in the arrangement of the levels of light isobaric nuclei.

IV. ISOBARIC SPIN AND NUCLEAR REACTIONS. SELECTION RULES

Under the assumption of charge independence of the interaction, the total isobaric spin is a constant of the motion. Expressed in this form, the hypothesis of charge independence of nuclear forces can be applied directly to the consideration of nuclear reactions involving only nucleons\(^{23}\). For such reactions, in addition to the requirement

\[ \Delta T_3 = 0, \tag{4,1} \]

which expresses conservation of electric charge, the requirement

\[ \Delta T = 0, \tag{4,2} \]

must also be satisfied, i.e. the law of conservation of total isobaric spin.

The concept of total isobaric spin proves useful not only in considering charge-independent interactions between nucleons, but also in application to photonuclear reactions, and also to \(\beta\)-decay. The interaction of a system of nucleons with the photon field or with the field of \(\beta\)-particles may be regarded as weak in the sense of perturbation theory. Then in the unperturbed system charge-independent forces will act, and its states may

will be characterized by the quantum number \(T\). A perturbation—for example, interaction with a photon field—will lead to transitions between different states of the unperturbed system, and (in view of the charge dependence of the perturbation) one can no longer expect the total isotopic spin necessarily to be conserved. It turns out, however, that in this case too selection rules can be established for the quantum number \(T\).

In order to obtain selection rules for \(\gamma\)-transitions \(^{24}\), let us consider the Hamiltonian of the interaction of a system of nucleons with an electromagnetic field. In the nonrelativistic approximation the Hamiltonian can be written in the form

\[ H'=-\sum_{i=1}^{A}\left\{ \frac{e}{mc}\,\mathbf p_i\mathbf A(\mathbf r_i)\frac{1-\hat{\tau}^{(i)}_3}{2} + \left[ \mu_n\frac{1+\hat{\tau}^{(i)}_3}{2} + \mu_p\frac{1-\hat{\tau}^{(i)}_3}{2} \right] \hat{\boldsymbol\sigma}^{(i)}\operatorname{rot}\mathbf A(\mathbf r_i) \right\}. \tag{4,3} \]

Here the first term describes the interaction of the orbital motion of protons with the field, and the second—the interaction with the field of the intrinsic magnetic moments of the nucleons. The factors \(\dfrac{1\pm\hat{\tau}^{(i)}_3}{2}\) take into account the difference between the neutron and proton electromagnetic characteristics (charges and magnetic moments), \(\sigma^{(i)}\) is the usual spin operator, \(m\) is the nucleon mass, \(\mu_n,\ \mu_p\) are the magnetic moments of the neutron and proton, and \(\mathbf A\) is the vector potential of the electromagnetic field.

The probability of an electromagnetic transition is determined by the square of the matrix element

\[ (\alpha'T'\mid H'\mid \alpha T), \tag{4,4} \]

where \(\alpha,\ \alpha'\) denote all the remaining quantum numbers, apart from \(T\), by which the initial and final states are characterized, respectively.

For selection rules associated with the quantum number \(T\), only the structure of the dependence of the operator (4,3) on the isotopic-spin operators is essential. Therefore we shall rewrite (4,3) schematically in the form

\[ H'=H_0+H_1=H_0+\sum_{i=1}^{A} f_i\hat{\tau}^{(i)}_3, \tag{4,5} \]

where \(H_0\) and \(f_i\) do not depend on \(\hat{\tau}^{(k)}_3\). In other words, \(H_0\) is a scalar in isotopic space, whereas \(H_1\) is the “3”-component of a vector in this space. The conditions for the nonvanishing of a matrix element of type (4,4), when \(H'\) is a scalar or a compo-

vector are well known for the ordinary angular momentum (see, for example, \(^{14}\), § 27). They carry over directly to the case of isobaric spin, since in the formal relation (the commutation rules) it does not differ from ordinary spin. The matrix element of the scalar \(H_0\) is different from zero only under the conditions

\[ \Delta T=0,\qquad \Delta T_3=0, \tag{4,6} \]

and the matrix element of the “3”-component of a vector—under the conditions

\[ \Delta T=0,\ \pm 1,\qquad \Delta T_3=0, \tag{4,7} \]

with the transitions

\[ \Delta T=0\quad \text{for}\quad T_3=0 \tag{4,8} \]

excluded, and, in particular, the transition \((T=0)\to(T=0)\). As a result we obtain the selection rules for photodisintegration reactions:

\[ \Delta T=0,\ \pm 1,\qquad \Delta T_3=0. \tag{4,9} \]

Directly, these rules are not strong restrictions, since in light nuclei there are no known states with high values of \(T\). Therefore the rules (4,9) have not yet been subjected to experimental verification.

However, rules that are more valuable in practical terms can be obtained by considering the multipolarity of electromagnetic radiation \(^{24,25}\). Electric dipole radiation is associated only with the first term in the interaction Hamiltonian (4,3). The contribution from this term to \(H_0\) will be

\[ -\frac{e}{2mc}\sum_{i=1}^{A}\mathbf p_i\mathbf A(\mathbf r_i). \]

This expression corresponds to the Hamiltonian of interaction with the radiation field for \(A\) particles having one and the same charge-to-mass ratio, equal to \(\frac{e}{2m}\). But when the charge-to-mass ratio is equal, the time derivative of the dipole moment, which determines the principal term of electric dipole radiation, will be equal to zero. In this case dipole radiation can nevertheless occur owing to correction terms arising from higher powers of \((kr_i)\) in the coefficient of the zero term of the expansion of the exponential entering the vector potential in Legendre polynomials. These correction terms will give an intensity of dipole radiation reduced by \((kR)^4\) relative to the ordinary intensity, i.e. comparable with electric octupole or magnetic quadrupole radiation. Therefore, if the principal term of dipole radiation does not vanish, then the correction terms can usually be neglected.

This is also what happens if there are no other selection rules except the rules for angular momentum and parity, which affect all terms of the dipole radiation simultaneously. In our case, however, it should be borne in mind that the radiation arising from the term \(H_0\) in the Hamiltonian (4.5) is not strictly forbidden as electric dipole radiation. Attention was drawn to the role of this circumstance in connection with the selection rules for isobaric spin in work \(^{25}\). However, since the dipole transitions determined by the correction terms are sharply weakened, we shall speak of them as “forbidden” and even “impossible.”

In view of the reservation made, we arrive at the conclusion that electric dipole radiation can occur only through the term \(H_1\) in (4.5), and for it, in addition to the general selection rules (4.7), the exclusion of transitions (4.8) must hold. In other words, an electric dipole transition in charge-symmetric nuclei (\(T_3=0\)) is allowed only with a change of the total isobaric spin \(T\) by one unit, and, in particular, the transition \((T=0)\to(T=0)\) is forbidden as an electric dipole transition.

As for the selection rules for \(\beta\)-decay, they can be obtained analogously to the rules (4.7). The Hamiltonian of the interaction of nucleons with the field of \(\beta\)-particles will contain linearly the operators \(\hat{\tau}\) only in the combinations

\[ \frac{\hat{\tau}_1 \pm i\hat{\tau}_2}{2}, \]

corresponding to the transformation of a neutron into a proton and conversely, i.e. it has the form of combinations

\[ B_1 \pm iB_2 \]

of the components of a vector in isobaric space. Consideration of the transition matrix element for an operator of this form leads to the selection rules \(^{26}\)

\[ \Delta T = 0,\ \pm 1,\qquad \Delta T_3 = \pm 1, \tag{4.10} \]

which, with respect to the possible change of the total isobaric spin, coincide with the rules (4.9) for \(\gamma\)-radiation.

The selection rules listed may be applied both to determining the isobaric spin of certain nuclear states, if the hypothesis of charge independence is assumed to be valid, and to testing the hypothesis itself. To determine \(T\) for excited states, one can use the fact that, by the law of conservation of isobaric spin, the collision of a nucleus with a deuteron or an alpha particle, for which always \(T=0\), leads to a state with the same value of \(T\) as the original nucleus. Likewise, the decay of a state with emission of \(d\) or \(\alpha\) does not change \(T\). Therefore, for example, the levels \(4.47\) Мэв

and 9.7 MeV in C\(^{12}\), arising in the reaction N\(^{14}(d,\alpha)\)C\(^{12*}\), must, like the ground state of N\(^{14}\), have \(T=0\). Conversely, the absence of excitation of the 1.74 MeV level in B\(^{10}\) in inelastic deuteron scattering gives an additional argument in favor of the fact that this level differs in the value of \(T\) from the ground state (with \(T=0\)) and probably has \(T=1\), as follows from the coincidence of its energy (to within the Coulomb correction) with the energy of the ground states of Be\(^{10}\) and C\(^{10}\). In general, the requirement of conservation of isobaric spin can affect inelastic scattering of deuterons and alpha particles by nuclei of the type \(A=4n+2\) (Li\(^{6}\), B\(^{10}\), N\(^{14}\)), where low-lying levels with \(T=1\) are expected. These levels should not be appreciably excited in such reactions. Since the reaction cross section is proportional to the square of the transition matrix element, even a substantial, in amplitude, admixture of an allowed state may lead to a small effective cross section.

As a further example of the application of the law of conservation of isobaric spin, one may consider\(^{23}\) the reaction O\(^{16}(d,\alpha)\)N\(^{14}\). Since the nucleus O\(^{16}\), the deuteron, and the alpha particle all have isobaric spin zero, the final nucleus N\(^{14}\) must also be in a state with \(T=0\). This means that the law of conservation of isobaric spin forbids the reaction to the excited 2.3 MeV state of N\(^{14*}\), which, as mentioned above, probably belongs to the charge triplet (C\(^{14}\), N\(^{14*}\), O\(^{14}\)) and has \(T=1\). Indeed, despite searches for this reaction,\(^{36}\) it has not been found*).

Conservation of isobaric spin may also manifest itself in the small width of certain excited levels of nuclei. Thus, in resonant reactions \((p,\alpha)^{28}\) on a nucleus with \(T_3=\frac{1}{2}\), \(T=\frac{1}{2}\), the intermediate nucleus must have \(T=0\) or \(T=1\). But the final states in such reactions usually contain charge-symmetric nuclei in low-lying energy states, for which \(T=0\). Then the partial width for \(\alpha\)-decay at the second stage of the reaction will, for some resonant levels (having \(T=1\)), be unusually small, since the corresponding transition must proceed only at the expense of violation of the purity of the states in isobaric spin by Coulomb forces. This may explain why, in the reaction B\(^{11}+p\to\)C\(^{12*}\to\)Be\(^{8}+\alpha\), the resonance at a proton energy of 165 keV has a width of 0.1 eV for the most energetic group of alpha particles and a total

*) It has been noted, however, that the prohibition of reactions like the one indicated can be explained not only on the basis of considerations of charge independence, but also from the less stringent requirement of charge symmetry, i.e. invariance of the interaction with respect to the replacement of neutrons by protons and protons by neutrons\(^{27}\).

the $\alpha$-width by several volts, although the energy available for alpha decay is about $9$ MeV. Analogous levels, for which alpha decay was not observed, also exist in $\mathrm{B}^{10}$^{23}.

Relations may be obtained between the probabilities of transitions to states of one and the same $T$-multiplet. Let us consider, for example, the reactions $\mathrm{Be}^9(d,p)$ and $\mathrm{Be}^9(d,n)$, the first to the ground state of $\mathrm{Be}^{10}$ $(T=1,\ T_3=1)$, the second to the excited state of $\mathrm{B}^{10*}$ with energy $1.74$ MeV, belonging to the same multiplet. Denote the charge wave functions of these nuclei by $\varphi_1^1$ and $\varphi_0^1$, and the charge functions of the neutron and proton by $\tau_{1/2}^{1/2}$ and $\tau_{-1/2}^{1/2}$, respectively. Since the initial state had

\[ T=\frac{1}{2}, \qquad T_3=\frac{1}{2}, \]

then, under the condition of charge independence of the interaction, the final state must have the same quantum numbers. Therefore the charge wave function of the final state cannot be simply $\varphi_1^1 \tau_{-1/2}^{1/2}(\mathrm{Be}^{10}+p)$ or $\varphi_0^1 \tau_{1/2}^{1/2}(\mathrm{B}^{10*}+n)$, but must be such a superposition of these functions that has

\[ T=\frac{1}{2}, \qquad T_3=\frac{1}{2} \]

and is equal to the charge wave function of the initial state $X_{1/2}^{1/2}$. This gives

\[ X_{1/2}^{1/2} = \frac{1}{\sqrt{3}} \left( \sqrt{2}\,\varphi_1^1 \tau_{-1/2}^{1/2} - \varphi_0^1 \tau_{1/2}^{1/2} \right), \]

where the coefficients are determined by the well-known formulas for addition of angular momenta (see, for example, ^{29}, p. 78). The relative probability of the transitions $\mathrm{Be}^9(d,p)\mathrm{Be}^{10}$ and $\mathrm{Be}^9(d,n)\mathrm{B}^{10*}$ will be equal to the ratio of the squares of these coefficients, multiplied by the ratio of the phase volumes for the corresponding final states.

Let us now turn to consideration of the influence of selection rules with respect to isotopic spin on $\gamma$-radiation and photodisintegration reactions ^{25}. This influence should be manifested mainly in nuclei with $T_3=0$, for which, according to (4,8), electric dipole transitions without a change of isotopic spin are forbidden. Such transitions can occur only owing to correction terms of the dipole radiation and then have an intensity of the order of that of magnetic quadrupole radiation, and also owing to the impurity of the states involved with respect to isotopic spin. This means that, in charge-symmetric nuclei, electric dipole

emission with a transition between states having \(T=1\) must be much weaker than emission between similar states in neighboring isobars. It is possible that such a character is possessed by the radiation corresponding to the transition between the levels \(8.05\) and \(2.3\) MeV of the nucleus \(\mathrm{N}^{14}\), which are analogous respectively to the \(6.1\) MeV level and the ground state of \(\mathrm{C}^{14}\). Then the width of the transition to the \(2.3\) MeV level in \(\mathrm{N}^{14}\) must be much smaller than the width of the transition in \(\mathrm{C}^{14}\), although with respect to angular momentum and parity both transitions are allowed.

States with \(T=0\), between which, as was established above, electric dipole emission is forbidden, are of special interest from the point of view of the selection rules for isobaric spin.

The question of \(\gamma\)-transitions in \(\mathrm{O}^{16}\) was considered in works \(^{30,25}\). \(\mathrm{O}^{16}\) has a state \((1-)\) at \(7.12\) MeV, \((2+)\) at \(6.91\) MeV, \((3-)\) at \(6.14\) MeV, and \((0+)\) at \(6.05\) MeV; the ground state is \(0+\). Apparently all these states have \(T=0\) (the lowest state with \(T=1\), analogous to the ground state of \(\mathrm{N}^{16}\), should be near \(13\) MeV). Thus, the decay of the \(7.12\) MeV level to the ground state is forbidden by the selection rules for isobaric spin. On the other hand, this decay is the most favorable energetically. According to the data of work \(^{30}\), it proceeds 120 times faster than decay to the \((3-)\) state, which is not forbidden by isobaric spin. This result can be explained \(^{25}\) on the basis of an estimate of the correction terms of dipole emission, which are of order \(M2\). Then, for the ratio of the probability of radiation with transition to the ground state to the probability of the \(E2\) transition to the \((3-)\) state, one obtains the value \(7^5:100 \simeq 170\).

Further, in the same nucleus \(\mathrm{O}^{16}\), in the absence of special selection rules one would expect for the \((2+)\) state a predominant \(E1\) decay to the \((3-)\) level. However, the \(E2\) decay to the ground state \((0+)\) occurs at least 200 times faster. This can be interpreted as the result of the action of the selection rule for \(E1\) radiation with respect to isobaric spin.

The ground state of \(\mathrm{B}^{10}\) has \(T=0\); the lowest of the states with \(T=1\) (an analogue of the ground state of \(\mathrm{Be}^{10}\)) lies at \(1.74\) MeV. Another state with \(T=1\) in \(\mathrm{B}^{10}\), analogous to the first excited state of \(\mathrm{Be}^{10}\) (\(3.37\) MeV), should be expected near \(5\) MeV. In this region there is a doublet at \(5.11\) and \(5.16\) MeV. Investigation of the reaction \(\mathrm{Li}^{6}(\alpha,\gamma)\mathrm{B}^{10}\) showed \(^{30}\) that the \(5.16\) MeV level has a small width (\(\sim 0.2\) eV), while the \(5.11\) MeV level was not observed at all in this reaction. One could assign to the latter level the value \(T=1\) and consider that its excitation would violate the law of conservation of isobaric

spin in the reaction \(\mathrm{Li}^6+\alpha\). However, using the estimate of the expected admixture of the \(T=1\) state to the ground state of \(\mathrm{Li}^6\), caused by the influence of Coulomb forces, the authors\(^{30}\) find that the level width due to such an admixture should be much greater than the upper limit they obtained for the level width \(5.11\) MeV and close to the observed level width \(5.16\) MeV. Therefore they assign the value \(T=1\) to the \(5.16\)-MeV state. As for the \(5.11\)-MeV level, assuming for it \(T=0\) and \(J=2^-\), one may think that its decay to the ground state of \(\mathrm{B}^{10}\) (\(J=3^+\)) is forbidden as a dipole transition between states with \(T=0\).

Selection rules in the quantum number \(T\) for radiative transitions have also been applied to the consideration of photodisintegration reactions with absorption of \(\gamma\)-rays and emission of particles\(^{28}\). The influence of the selection rules will be most noticeable for even-even nuclei with \(T_3=0\) (\(A=4n\)), in which the ground state has \(T=0\) and \(J=0^+\), while the first excited state with \(T=1\) lies at a high energy \(W_1\). We shall restrict ourselves to considering only electric dipole (\(E1\)), electric quadrupole (\(E2\)), and magnetic dipole (\(M1\)) transitions.

For photon energy \(W<W_1\), all absorption leads to an intermediate nucleus with \(T=0\). In this case electric dipole absorption will be forbidden. Absorption may be either \(M1\), leading to an intermediate nucleus with \(J=1^+\), or \(E2\), leading to an intermediate nucleus with \(J=2^+\). Absorption at any energy in the reactions \((\gamma,d)\) and \((\gamma,\alpha)\) will have the same character if the final nucleus remains in a state with \(T=0\). If the final state in the \((\gamma,\alpha)\) reaction is the ground state of the nucleus, then the absorption must be electric quadrupole, and the intermediate nucleus must have \(J=2^+\).

Absorption leading to an intermediate nucleus with \(T=1\) is possible only for \(W>W_1\), and then it may be of any multipolarity. But the reaction can end with emission of a deuteron or an alpha particle only if the energy is sufficient for the final nucleus also to remain in a state with \(T=1\).

The considerations presented would be strictly valid under the condition of exact charge independence of the interaction between nucleons. It is necessary, however, to take into account the corrections introduced by Coulomb forces. As a result, there will be some probability of such processes as are forbidden by the selection rules in isobaric spin. If there are no effectively competing processes, such as emission of a neutron or proton, then following absorption into a state with \(T=1\) (for example, dipole absorption) a certain number of alpha particles may be observed, leaving the nucleus in a state with \(T=0\).

The arguments set forth have been applied to the nucleus \(\mathrm{C}^{12}\). At low photon energies the absorption will have the character \(E2\)

or \(M1\). The first level with \(T=1\)—an analogue of the ground states \(B^{12}\) and \(N^{12}\)—must lie near \(15\ \mathrm{Mev}\) and have \(J=1+\). It is possible that this level is \(15.09\ \mathrm{Mev}\). The analogue of the first excited state of \(B^{12}\) (\(0.95\ \mathrm{Mev}\)) is the level \(16.07\ \mathrm{Mev}\) with \(J=2+\). These data determine the absorption thresholds, respectively, \(M1\) and \(E2\), with transition to the intermediate nucleus with \(T=1\). The absorption threshold \(E1\) must, obviously, lie higher, at some energy \(W_1(E1)\), where the lowest level with \(T=1,\ J=1-\) is located. When the photon energy reaches this value, \(E1\) absorption will begin to increase rapidly and will soon become dominant. This should manifest itself in the processes \((\gamma,p)\) and \((\gamma,n)\), whereas the processes \((\gamma,\alpha)\) can at first proceed only as forbidden by isobaric spin, i.e. at the expense of impurity of the states. The threshold of the allowed reaction \((\gamma,\alpha)\), leading to the lowest state with \(T=1\) in \(Be^8\) (\(16.8\ \mathrm{Mev}\)), is estimated at \(26\ \mathrm{Mev}\).

The experimental data on the reactions \((\gamma,n)\) and \((\gamma,p)\) in \(C^{12}\) indicate a steep rise of the effective absorption cross section near \(20\ \mathrm{Mev}\). This is interpreted as the onset of \(E1\) absorption, i.e. as the threshold \(W_1(E1)\). The reactions \((\gamma,\alpha)\) give peaks at \(18\ \mathrm{Mev}\) and \(29\ \mathrm{Mev}\). The first peak must be due to \(M1\) and \(E2\) absorption, which is confirmed by the experimental data. In the region between \(20\) and \(26\ \mathrm{Mev}\) there is a large peak in the reactions \((\gamma,n)\) and \((\gamma,p)\), whereas the effective cross section \((\gamma,\alpha)\) in this region is small. Such a result was to be expected on the basis of the preceding arguments, since the processes \((\gamma,\alpha)\) can proceed here only as allowed \(E2\) and \(M1\) or as forbidden \(E1\). The above-mentioned peak near \(29\ \mathrm{Mev}\) is evidently due to allowed \(E1\) processes. This is confirmed by the fact that in the region above \(26\ \mathrm{Mev}\) the reaction \(C^{12}(\gamma,\alpha)Be^8\) goes in 88% of cases to levels of \(Be^8\) near \(17\ \mathrm{Mev}\), and among them mainly to the \(16.8\ \mathrm{Mev}\) level[^31]. From the energy point of view the \(16.8\ \mathrm{Mev}\) level in \(Be^8\) proves to be an analogue of the ground states of \(Li^8\) and \(B^8\), and therefore has essentially \(T=1\). Although it also decays with emission of alpha particles, its width is small (\(<0.3\ \mathrm{Mev}\)). The data on the angular distribution and angular correlation of alpha particles make it possible to assert that absorption in the reaction \(C^{12}(\gamma,\alpha)\), leading to this level, is indeed electric dipole absorption, and that the level itself has \(J=2+\). The transition to the \(3\ \mathrm{Mev}\) level, which also has \(J=2+\) and is energetically preferable, occurs 6 times less often, apparently precisely because of the isobaric-spin selection rule for dipole absorption.

If, in the region above \(26\ \mathrm{Mev}\), despite the inclusion of allowed \(E1\) absorption, the effective cross section of the reaction \((\gamma,\alpha)\) nevertheless does not increase very significantly in comparison with the region of lower energies, then this may be explained by a strong decrease of the total absorption above \(26\ \mathrm{Mev}\), as indicated by—

affects the behavior of the effective cross sections of the reactions \((\gamma,n)\) and \((\gamma,p)\). An analogous consideration was also carried out for reactions induced by \(\gamma\)-rays in the \(O^{16}\) nucleus.

An interesting prediction has been made\(^{25}\) concerning the competition of the processes \((\gamma,d)\) and \((\gamma,np)\) for odd-odd nuclei with \(T_z=0\), in the case of an electric dipole character of the absorption. In such reactions the final state must have \(T=1\) and contain an even-even nucleus. Therefore the emission of a deuteron will be forbidden if the energy is insufficient for the final nucleus to be left in one of its (high-lying) states with \(T=1\). On the other hand, the simultaneous emission of a neutron and a proton in a state with isobaric spin \(1\) (for example, in a virtual \({}^{1}S_0\) state) will be allowed. Thus, under the indicated conditions, an excess of \((\gamma,np)\) reactions over \((\gamma,d)\) reactions should be observed.

The study of the influence of charge independence on nuclear reactions has begun only recently. But it is already clear that such an approach is of considerable interest for the interpretation of processes in light nuclei. A systematic consideration of reactions “from the point of view of the quantum number \(T\)” is apparently a matter for the near future.

APPENDIX

Let us show that if the function \(\chi\) of the isobaric-spin variables for a system of \(A\) particles has the symmetry defined by the partition

\[ A=n_1+n_2 \qquad (n_1\geq n_2), \tag{2,14} \]

i.e. is antisymmetric in \(n_2\) pairs of particles and symmetric in the remaining \(n_1-n_2\) particles, then it is an eigenfunction of the operator \(\hat T^2\) with eigenvalue \(T(T+1)\), where

\[ T=\frac{n_1-n_2}{2}. \]

According to (1,12), the operator \(\hat T^2\) can be expressed in terms of the operators of pair interchanges of the isobaric variables:

\[ \hat T^2=A-\frac{A^2}{4}+\sum_{i<k}^{A} P^{(\tau)}_{(i,k)} . \tag{1,12} \]

Let us consider the action of the sum of pair interchanges on the function \(\chi\). Since the function is symmetric in \(n_1-n_2\) particles, their interchanges give in this sum

\[ \frac{(n_1-n_2)(n_1-n_2-1)}{2} \]

terms.

of which are equal to 1. Permutations of particles within antisymmetric pairs will give in the sum \(n_2\) terms equal to \(-1\). It remains to consider permutations of each of the particles belonging to antisymmetric pairs with all particles not belonging to the given pair. We shall consider jointly the permutations of each of two antisymmetrically connected particles with a particle not belonging to the given pair. Let, for example,

\[ \chi(\widehat{1\,2}\,3\ldots) \]

be a function of the isobaric-spin variables in which particles “1” and “2” are connected into an antisymmetric pair. Then

\[ \left(P^{(\tau)}_{(1,3)}+P^{(\tau)}_{(2,3)}\right)\chi(\widehat{1\,2}\,3\ldots) = \chi(\widehat{3\,2}\,1\ldots)+\chi(\widehat{1\,3}\,2\ldots). \]

It is easy to verify that the function

\[ \chi(\widehat{1\,2}\,3\ldots)-\chi(\widehat{3\,2}\,1\ldots)-\chi(\widehat{1\,3}\,2\ldots) \]

is antisymmetric in all three particles, and therefore, like any such function of isobaric-spin variables, is equal to zero. Therefore

\[ \left(P^{(\tau)}_{(1,3)}+P^{(\tau)}_{(2,3)}\right)\chi(\widehat{1\,2}\,3\ldots) = \chi(\widehat{1\,2}\,3\ldots). \]

Consequently, the permutations under consideration can be grouped pairwise in such a way that each pair contributes to the sum occurring in (1,12) a term equal to 1. But the number of such permutations is equal to

\[ \frac{(n_1-n_2)(n_1+n_2-1)}{2} - \left[\frac{(n_1-n_2)(n_1-n_2-1)}{2}+n_2\right] = 2(n_1-1)n_2. \]

Combining the results obtained, we find that the action of the operator which is the sum of all pair permutations on the function \(\chi\) reduces to multiplication of it by the factor

\[ \frac{(n_1-n_2)(n_1-n_2-1)}{2} - n_2 + (n_1-1)n_2. \]

Therefore

\[ \hat T^{\,2}\chi = \left\{ A-\frac{A^2}{4} + \sum_{i<k}^{A} P^{(\tau)}_{(i,k)} \right\}\chi = \]

\[ = \left\{ n_1+n_2-\frac{(n_1+n_2)^2}{4} + \frac{(n_1-n_2)(n_1-n_2-1)}{2} - n_2 + \right. \]

\[ \left. + (n_1-1)n_2 \right\}\chi = \frac{n_1-n_2}{2} \left( \frac{n_1-n_2}{2}+1 \right)\chi. \]

Thus, under the assumed properties of the permutation symmetry of the function \(\chi\), we indeed obtain:

\[ \hat T^{\,2}\chi=T(T+1)\chi,\qquad \text{where } T=\frac{n_1-n_2}{2}. \]

References

1. L. A. Young, Phys. Rev. 47, 972 (1935); 48, 913 (1935).
2. B. Cassel and E. U. Condon, Phys. Rev. 50, 846 (1936).
3. G. Breit and E. Feenberg, Phys. Rev. 50, 850 (1936).
4. E. Feenberg and E. Wigner, Phys. Rev. 51, 95 (1937).
5. E. Wigner, Phys. Rev. 51, 106 (1937).
6. F. Hund, Zeits. f. Phys. 105, 202 (1937).
7. Tuve, Heydenburg and Hafstad, Phys. Rev. 50, 806 (1936).
8. Breit, Condon and Present, Phys. Rev. 50, 825 (1936).
9. J. Schwinger, Phys. Rev. 78, 135 (1950).
10. Fowler, Delsasso and Lauritsen, Phys. Rev. 49, 561 (1936).
11. W. Heisenberg, Zeits. f. Phys. 77, 1 (1932).
12. Ya. B. Zel’dovich, DAN 86, 505 (1952).
13. E. Wigner, Proc. Nat. Acad. Sc. USA 38, 449 (1952).
14. L. Landau and E. Lifshitz, Quantum Mechanics, Part I, Gostekhizdat, 1948.
15. F. Hund, Zeits. f. Phys. 43, 788 (1927).
16. E. Feenberg and Phillips, Phys. Rev. 51, 597 (1937).
17. V. I. Smirnov, A Course of Higher Mathematics, Vol. III, Part I, Gostekhizdat, 1951.
18. D. Kurath, Phys. Rev. 88, 804 (1952).
19. L. A. Radicati, Proc. Phys. Soc. A 66, 139 (1953).
20. B. S. Dzhelepov, ZhETF 19, 360 (1949).
21. B. S. Dzhelepov, Izv. AN, Ser. Phys. 15, 496 (1951).
22. B. S. Dzhelepov, Izv. AN, Ser. Phys. 17, 391 (1953).
23. R. K. Adair, Phys. Rev. 87, 1041 (1952).
24. L. A. Radicati, Phys. Rev. 87, 521 (1952).
25. M. Gell-Mann and Telegdi, Phys. Rev. 91, 169 (1953).
26. E. Wigner, Phys. Rev. 56, 519 (1939).
27. N. M. Kroll and L. L. Foldy, Phys. Rev. 88, 1177 (1952).
28. S. Devons, Proc. Phys. Soc. A 66, 665 (1953).
29. Condon and Shortley, The Theory of Atomic Spectra, IL, Moscow, 1949.
30. G. A. Jones and D. H. Wilkinson, Phys. Rev. 90, 722 (1953).
31. J. J. Wilkin and F. K. Goward, Proc. Phys. Soc. A 66, 661 (1953).
32. F. Ajzenberg and T. Lauritsen, Rev. Mod. Phys. 24, 321 (1952).
33. D. R. Inglis, Rev. Mod. Phys. 25, 390 (1953).
34. T. Lauritsen, Ann. Rev. of Nuclear Sci. 1, 67 (1952).
35. W. E. Burcham, Progr. Nucl. Phys. 2, 174 (1952).
36. Ashmore and Raffle, Proc. Phys. Soc. A 64, 754 (1950); Burrows et al., Proc. Roy. Soc. A 209, 478 (1951).