The excitation of nuclei by electrons By I. N. Sneddon and B. F. Touschek Department of Natural Philosophy, the University of Glasgow {Communicated by P. I. Dee, F.R.S.—Received 10 1947) The cross-section for the inelastic scattering of electrons by atomic nuclei is calculated assuming the process to arise from the electromagnetic interaction between electrons and protons. For fast electrons scattered by cadmium the cross-section so obtained is in fair agreement with recent experimental estimates. Difficulties, due to the failure of Born’s approximation, arise when an attempt is made to apply the theory to the explanation of the electron excitation curves recently observed by Wiedenbeck.
The excitation of nuclei by electrons 345 however, is based on Born’s approximation method and thus restricts the applic ability of its results to large energies of both incoming and outgoing electron and small atomic numbers.
346 I. N. Sneddon and B. F. Touschek This breakdown of the method of approximation is well known to be responsible for the interesting anomalies in the electron excitation curves of optical transitions (Massey & Mohr 1931). Its effect in this analogy is a very rapid increase in the excita tion curve near the threshold leading to a maximum, the width of which is roughly determined by the wave-length of the scattered electron. In the optical case, how ever, the effect is complicated by exchange phenomena.
The excitation of nuclei by electrons 347 scattered electrons and kmc is the momentum of the recoiling nucleus we obtain from the conservation of momentum the relation p = p +k. (3) Now if MA is the mass of the nucleus the relation between the energy and momentum glVeS T = mk2/2MA^A so that we may omit T in equation (2) as long as | k | is not too large. Physically this infers that the nucleus is capable of absorbing any amount of momentum without affecting the energy balance.
348 I. N. Sneddon and B. F. Touschek we may make use of the continuity equation d lv j+ | = ° to express ^pe^-^drni terms of J. Considering that p has the time dependent factor eiAt, integrating the continuity equation throughout space after multiplication by ei(k-r), and making use of Gauss’s theorem we obtain jpe^^dr = i(k.J). (8) Substituting from equation (8) into equations (6) and (5) we obtain AP = {(a0k —Aa) J}, (9) where the product involved is a scalar product. Writing for simplicity s {<T,cr') = a0k — Aa0 (10) s, and putting 2 £s*s (11) cr cr' where now the product on the left-hand side of the equation is dyadic, we obtain from equation (9) A2S S |F |2 = (J*.£.J). (12) <r or' It follows from its definition (11) that the quantity is a Hermitian tensor in three- dimensional space, i.e. S*j = S^.The evaluation of the tensor S is straightforward and will be carried out in the next section. The averaging over the initial orientations of the nucleus is a much more difficult problem; consequently we do not attempt to develop the general theory but instead restrict ourselves to certain simplified cases.
The excitation of nuclei by electrons 349 and in the cases of the electric quadrupole and magnetic dipole it is defined by J.= iATk, (14) where the tensor T is given by the formula iAT7 Jjrdr, (15) where the product of the vectors j and r is dyadic.
350 L N. Sneddon and B. F. Touschek where the operators K and K' are defined by tf = (<x.p)+/?+e, = (a.p') + /?+e' (20) and give 0 when applied to the amplitudes of states with negative energy. The summations can then be carried out over all states, positive and negative, of the electron, and use can be made of the orthogonality relation S*.(<r)u?(<r) = <T in which a and /? are tha spinor indices. Carrying out this summation we obtain ee'S = £ sp (k - Aa) K’(k — Aa) K = A - B (21) where sp denotes the spur with respect to the spin indices and A, B and C are defined by the relations A =isp(k^'kZk -I £ = iA sp ( kK'aK + aK'kK), 1 (22) C — £A2sp .)Ka'Ka(J Making use of the relations a(a. p) + (a. p) a = 2p„ sp a* = sp = 0 we finally obtain A = kkfee' + (p.p') + 1], B = A{e'(kp+pk) + e(kp'+p'k)}, (23) C = A2[/{ee'-(p.p')-l}+pp'+p'pL so that the tensor S is determined.
The excitation of nuclei by electrons 351 It follows from equation (24) that for the dipole case we have to calculate (S). This quantity is obtained from the expression for S by writing scalar products instead of dyadic products and using the fact that (/) = 3. This gives (4) = &2{ee' + (p.p') + l}, (B) = 2A{e'(k.p) + e(k.p')}, (25) (C) = A2{3[ee' - (p. p') - 1] + 2(p. p')}v The occurrence of ( k 2— A2)2 in the denominator of the expression on the right-hand side of equation (4) makes an expansion of — + in powers of 2 — A2 desirable.
352 I. N. Sneddon and B. F. Touschek In the limiting case ^->0 we have e = 1 + A and e' = 1, so that equation (27) reduces to ,r _ 0 = 47rr2|x|2^-|^ + Aj (28) near the threshold of the reaction. On the other hand in the limiting case oo, p' = p, we have 2 0 = Snrl | x2|[loge^— lj • (29) The close resemblance of equation (29) to Bohr’s formula (Bohr 1915) for the ionization losses of a charged particle should be observed.
The excitation of nuclei by electrons 353 where c = a x b is a unit vector. Inserting this expression into equation (30) we obtain therefore 1 -2S E | P |2 = ((cxk)^(cxk)).