26.—The Strong Limit-2 Case of Fourth-order Differential Equations

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

23.—The Synthesis of Derivatives of Acenaphthenone

Acenaphthenequinone (I) condensed with ethyl cyanoacetate in ethanol to give the unsaturated ester (IIIb) while condensation with malonic acid in toluene in the presence of diethylamine gave the hydroxy acid (IIa). Esterification of this acid gave an ester (IIb) which could also be obtained by condensation of acenaphthenequinonewith ethyl hydrogen malonate. The acid (IIa) on dehydration gave 2-oxo-Δ1,α-acenaphtheneacetic acid (IIIa) of m.p. 230°C. This result seems to be in conflict with that of Rodionov and Federova (1950) who reported a m.p. of 160° for the acid (IIIa) which they obtained directly by condensation of acenaphthenequinone with malonic acid in ethanol in the presence of ammonia. Our efforts to repeat their result gave only impure polymeric material and it seems unlikely, therefore, that their product was simply the geometric isomer of our acid.

27.—The Quantum Theory of the Klystron and the Modulation of Electron Beams at Optical Frequencies

SynopsisThe relationship between the modulation of electron beams at optical frequencies (the Schwarz-Hora effect) and at microwave frequencies is discussed. At optical frequencies the interaction between the modulating field and the electron beam must be described quantum mechanically, although the field itself may be described classically; in the microwave case the process may be described entirely classically. The interaction modifies the state functions of the individual electrons, but the observable modulation of the total electron beam results from the coherence of the modulating field. The main features of the Schwarz-Hora effect result from ‘single photon’ processes, but the beam modulation in the klystron is a ‘multi-photon’ process.The exponential decay of the depth of modulation with distance from the interaction region, observable in the optical frequency case but not in the klystron, is not an inherently quantal effect. The periodic variation of the depth of modulation with distance along the beam, observed in the optical frequency case, is an essentially quantum mechanical effect, and is different, in its origins and in its dependence on the modulation frequency, from the space-charge waves which are observed on a klystron beam.

29.—Fine-structure in the Mass-yield Curve for Binary Fission—A Possible Secondary Effect

SynopsisThe recent suggestion (Feather 1974) that the short-range α-particles of fission are emitted predominantly from post-neutron-emission fragments in the mass-number range 140 to 145 is shown to have implications in relation to the fine-structure of the mass-yield curve. An examination of the mass-yield curves of Unik et al. (1973) discloses, in many of them, a fine-structure feature of the type predicted.

28.—Simultaneous Emission of two Light Charged Particles in Fission

SynopsisThe suggestion is made that in certain cases the light particles emitted in ternary fission may be released otherwise than in the ground state—and, in particular, that low-lying particle-unstable states may be involved. This suggestion is examined quantitatively in respect of the release of 8Be in the first excited short-lived state of 2·9 MeV excitation, and is shown to provide a satisfactory interpretation of the observation of quaternary fission in the α,α mode recently reported by Kataria et al. (1973). It appears probable that the other modes of quaternary fission observed by these authors should be similarly interpreted.

30.—The Chord Locus of a Certain Curve in [n]

SynopsisThe sharing of a common self-polar simplex by n−1 quadrics in [n] confers special features on their curve of intersection Γn. The three-dimensional locus Mn of chords of Γn. has certain singularities, some of which are described. In conclusion, a few comments refer to the case n = 4 when Mn is defined by a single equation.

24.—The Origin of the Short-range Alpha Particles of Fission

SynopsisThe suggestion is made, and rendered plausible by detailed calculation, that the short-range α-particles of fission, identified by Kugler and Clarke [1], are emitted, in competition with ‘prompt’ γ-rays, from highly excited post-neutron-emission fragments of even neutron number N and, predominantly, of mass number A in the range 140 ≦ A ≧ 145. A similar process is energetically forbidden in relation to triton emission.