## Vol. 14, No. 5 (1958)

##### 1958-03-05

###### CONTENT

1958, 44 (5): 369-375.
doi: 10.7498/aps.14.369

Abstract +

On the basis of the three premises of Stoner's collective electron theory of ferromagnetism, appropriate energy band diagrams are constructed. These diagrams vividly show the physical situation involved in Stoner's theory. With the help of diagrams, Stoner's criterion for occurrence of ferromagnetism, kθ′/ε0=ξ>2/3, and for complete magnetization, ξ≥2-1/3, can be deduced in a rather simple and straightforward way. With the help of diagrams, the transfer of electrons from d-band to s-band due to splitting of d-band in the ferromagnetic state is considered, and is estimated for nickel at 0°K to be about 1%.

1958, 44 (5): 376-392.
doi: 10.7498/aps.14.376

Abstract +

The focusing of high current electron beams by means of several different systems of periodic electric and magnetic fields has been studied theoretically. One of these periodic electron optical systems studied, i.e. a periodic electric field produced by a series of annular disks held at alternately higher and lower potentials, has also been investigated experimentally in an electrolytic tank.It is found that if the distance between the disks or half period of the field is long (i.e. two or three times longer than the diameter of the apertures at the center of the disks), the potential on the axis may be expressed by a saw-toothed function. Under this condition, the equation for describing the electron beam profile may be solved analytically. The conditions which must be satisfied to give periodic focusing have been obtained.However, if the distance between the disks is short and comparable with the diameter of the aperture, the potential on the axis should be expressed by a cosine function. Under such a condition, the equation for describing the electron beam profile becomes a complicated non-linear differential equation. This non-linear differential equation has been treated in earlier research and was able to obtain its first order approximate solution. In order to get higher order approximations, a perturbation method may be used.This perturbation method has also been applied to treat other periodic electric and magnetic fields, such as axially symmetric magnetic fields, quadrupole electric and magnetic fields. Higher order periodic approximate solutions, as well as the required conditions to give periodic focusing, have been obtained for these fields. It can be shown by numerical examples that for periodic focusing of high current electron beams higher order approximation should be included.

1958, 44 (5): 393-399.
doi: 10.7498/aps.14.393

Abstract +

A phase diagram of the alloys of the ternary system of Ag-Sb-Sn was constructed on the basis of the data obtained by X-ray analysis and metal-lographic examinations.The phase diagram obtained consists of six single-phase regions (namely a,β,γ,β′, Sn and Sb regions), seven two-phase regions(namely a+β,β+γ,γ+β′,γ+Sn,γ+Sb,β′+Sn,β′+Sb regions) and two three-phase regions(namely γ+β′+Sn,γ+β′+Sb regions). No new phase was observed in these alloys of the ternary system.

1958, 44 (5): 400-404.
doi: 10.7498/aps.14.400

Abstract +

, where α1 represents a set of commuting integrals of motion and α2 a set of observables not containing integrals of motion and introduce the assumption that for any integral of motion R, the matrix 1α′2Rα1"α2"> representing it is of the form φ(α′1α1")δ(α′2α2"). (3) Under such conditions, we prove that if the initial state of our system is an eigenstate of α1 corresponding to the eigenvalue α10, then the average of an observable F over time is given by ∑α210α2|F|α10α2>/∑α210α2|1|α10α2>, a result which is clearly to be expected. Extension to systems interacting with external surroundings is easily made.">In a paper by Klein, it is pointed out that for the quantum mechanical ergodic theorem (time average = average over states) to be valid, all integrals of motion R must satisfy the condition ∑ρ"(α′ρ"|R|β′ρ")=const δα′β′ (1) where α′, β′, γ, … refer to the assembly under question and α", β",…, ρ" ,… refer to the surrounding with which our assembly is in equilibrium. In this short note, it is pointed out that the argument of Klein is doubtful at one point and a slightly different approach is given. In this approach, it is shown that (1) is actually sufficient, but only after introducing an additional assumption that the different states of the surroundings have equal probabilities, and that in general, a stronger condition such as (α′ρ″|R|β′θ″)=const δα′β′δρ″θ″ (2) is needed. Cases with integrals of motion is also considered. We consider first an isolated system, write the wave function as (α1α2>, where α1 represents a set of commuting integrals of motion and α2 a set of observables not containing integrals of motion and introduce the assumption that for any integral of motion R, the matrix 1α′2Rα1"α2"> representing it is of the form φ(α′1α1")δ(α′2α2"). (3) Under such conditions, we prove that if the initial state of our system is an eigenstate of α1 corresponding to the eigenvalue α10, then the average of an observable F over time is given by ∑α210α2|F|α10α2>/∑α210α2|1|α10α2>, a result which is clearly to be expected. Extension to systems interacting with external surroundings is easily made.

1958, 44 (5): 405-422.
doi: 10.7498/aps.14.405

Abstract +

This paper is a brief discussion of the properties of expansors introduced by Dirac(1945). After transforming the expansors Anrst to ξ-representation defined by 〈ξ|〉=∑ξ0-n-1ξ1rξ2sξ3tAnrst (1) and showing that undergoes transformations identical with standardrepresentations in a Lorentz transformation, it is shown that of the two fundamental invariants J=-1/2IklIkl,I=-1/2εklmnIklImn (2) characterizing the different irreducible representations of the Lorentz group, the second one in the theory of expansors is always zero. It is also shown that the requirement of the different eigenfunctions of J in the space to behave regularly for all ratios of ξ leads to J1/2, (3) (ii)I′=±(1+J)1/2i,J′=1+J.(4) Explicit formula for such matrixes are also worked out.If we require expansors after operations by such operators remain as expansors, we must let -1≤J≤0 and confine ourselves to the selection rule I′=0,J′=1+J-2(1+J)1/2. (5) Since successive transformations of J by the above formula starting with an initial value of J, say J1, satisfying -1≤J1≤0 do not lead to values of J beyond J1 and J2 J2≡1+J1-2(1+J1)1/2 (both J1, J2 being negative), it is clear that in constructing a wave equation of the type (-irμpμ+k)ψ=0 (6) with ψ in expansor spaces, the simplest formulation is to let the space of ψ to consist of two such expansor spaces (J1 0), (J2,0). Of course, the matrixes γμ are so choosen that only the selection rule (5) is effective. It is shown that the operators ξvξv?/(?ξμ)-(1±(1+J)1/2)ξμ (7) transform the (J,0) space to the spaces (1+J±2(1+J)1/2,0) respectively. Thus γμ may be constructed easily in terms of the operator ξvξv?/(?ξμ)-(1-(1+J)1/2)ξμ. Investigations of such wave equations will be left later.