Eine Integralgleichungsmethode zur Behandlung der Streuung an gebundenen Teilchen

First it is pointed out that various methods known for the treatment of multi-particle scattering problems such as the methods ofLax, Watson, andFaddeev are based on the same type ofT-operator equations with eliminated interactions. They only differ by the identification of the interactions. — Then an integral equation treatment for the scattering of a particle by a system ofn bound particles is developed. If the scattering occurs via local two-body forces, the interaction matrix element splits into that of the two-particle case and a momentum-dependent factor. This fact is used to simplify the scattering equations which then get a mathematical structure similar to that of theT-operator equations discussed at the beginning (however, involving sums of bound states rather than sums of interactions) and which, therefore, can be handled in a similar way. When the interactions are eliminated by means of the two-particle scattering amplitudes, the off-shell energy parameter of these amplitudes may be chosen to be dependent on quantum numbers of the bound system. Such a choice shows indeed to be favorable if one likes to keep only the lowest order approximation of the integral equations. The resulting approximate formula leads, after some further approximations, for resonant scattering to a formula ofLamb, and for weakly energy-dependent amplitudes to a formula ofFermi (being related to the impulse approximation). — The resonant scattering formula is applied to a quantum-mechanical derivation of a method for the determination of nuclear lifetimes which had been proposed on semiclassical arguments byCiocchetti et al. — Finally the method developed for the scattering of a particle by a system of bound particles is extended to collisions between composite particles.     

Delbrück-Streuung von 17 MeV-Photonen

Differential cross sections have been measured for the small angle scattering ofγ-rays by iron, silver, tantalum, lead and uranium with 17 MeV photons from Li⁷ (p, γ) Be⁸ at mean four momentum transfersq of 0·5mc and 1·3mc and with 7 MeV photons from F¹⁹(p, αγ)O¹⁶ at mean four momentum transfer of 0·5mc. Under these experimental conditions only Compton, Rayleigh and possibly Delbrück scattering are of importance. Extrapolation of known theoretical results to higher energies shows, that Rayleigh and Compton scattering from bound electrons should depend only onq for small angles, smallq and fixedZ. Using this it follows, that at 17 MeV andq≈0·5mc an additional scattering process must be present, which increases with growingZ and which is negligible in the measurements at 17 MeV withq≈1·3mc and at 7 MeV withq≈0·5mc. These results are in qualitative agreement with the approximate theory for Delbrück scattering ofBethe andRohrlich, however experimental cross sections at 17 MeV andq≈0·5mc are about a factor of 1·6 lower than those predicted by this theory. This discrepancy is not unexpected, since exact calculations of Delbrück scattering amplitude fromKessler andZernik at 2·62 MeV and 6·14 MeV show even greater deviations in the same sense.     

Mobility of electrons in nondegenerate semiconductors considering electron-electron scattering

Spitzer andHärm [1] have investigated the velocity distribution of electrons in the presence of a weak electric field in an ionized gas. Introducing the concept of time of relaxation τ′, due to electron-ion and electron-electron scattering, and using the results ofSpitzer andHärm [1], the authors have obtained an expression for τ′ which is applicable to semiconductors with little modification. In this paper the authors have used this expression for τ′ to obtain the mobility of electrons in nondegenerate semiconductors, taking into account the scattering by lattice vibrations, electron-ion interactions and electron-electron interactions.     

Einfluß der Pion-Pion-Wechselwirkung auf Pion-Nukleon-Streuphasen

The isospin dependent part of theS-wave pion-nucleon amplitude is calculated according to the CGLN approach, but recoil and projection on the partial waves are treated exactly. Then the difference against the experimental data is approximately a constant up to 100 MeV and amounts to 1/3 of the total value. It is shown, that at low energies the main corrections of the CGLN equations are taken into account by a subtraction of theπ+π→?+\(\overline \Re \),J=I=1 amplitudesf _± ¹ (t) as proposed byMandelstam andFrazer. Calculations have been done forS, P andD pion-nucleon amplitudes. The main result is a prediction of theS-wave amplitude obtained without adjusting free parameters up to 100 MeV. The relations of this method to the equations derived byBowcock, Cottingham andLurié according to the Cini-Fubini approach are discussed.     

Untersuchung des ZerfallsBi^{212} (ThC)\xrightarrow{\beta }Po^{212} (ThC')\xrightarrow{\alpha }Pb^{208} (ThD) durch (γ, α)-Koinzidenzen

Theγ-spektrum of Po²¹² (ThC′) has been measured by (γ, α)-coincidences with a fast-slow coincidence apparatus. The results are in excellent agreement with the conversion lines found by theLatyshev group. The absoluteγ-intensities have also been determined in order to get spins and parities of the levels by calculating the absolute conversion coefficients. Theβ-intensity leading to the exited states of Po²¹²(ThC′) is estimated to be about one third of that measured byBurde andRozner by (β, α)-coincidences. Threeγ-lines measured byChinaglia andDemichelis by (γ, α)-coincidences do not agree with our results. The 2⁺ assignment of the first exited state is now well established, but for the other levels in the decay scheme there is still some uncertainty concerning spin and level assignment.     

Absorptionsquerschnitte und Transmissionskoeffizienten für Protonen und Neutronen nach dem optischen Modell

T _l -values and absorption cross sections for protons and neutrons are calculated with optical potentials, which have been fitted to experimental scattering data. Values are given forZ=10, 20, 30, 50, 70, 90, and energies between 0.1 und 25 MeV. For protons we used the local optical potential ofPerey and for neutrons the nonlocal optical potential ofPerey andBuck. The results are compared with those of other optical potentials. In addition it is shown, that the spin-orbit-interaction leads toT _{l j} -values which sometimes differ appreciably depending whetherj equalsl+1/2 orl?1/2.     

Test of a sum rule for the nucleon magnetic moments

It is shown that the original version of the sum rule relating the isovector part of the nucleon magnetic moment to aπ N scattering amplitude is not consistent with the experimental data. However there is good agreement for the corrected version ofMaiani andPreparata.     

Consistency of new π N phase-shift analyses with dispersion relations

New phase-shift analyses for pion-nucleon scattering are used to check the dispersion relations for the amplitudesB _±(v _L, 0), which are essentially determined by the pionnucleon spin-flip scattering extrapolated to forward direction. Within the experimental errors, the relations seem to be fulfilled. The pion-nucleon coupling constantf ² turns out to be consistent with the value found byHamilton andWoolcock, who obtainedf ²=0,081±0,003. However, in spite of the newer data and less sensitivity of our calculational procedure toward the high-energy behaviour of the amplitudes, the error off ² is still considered as somewhat larger. Our method also gives some information on the high-energy behavior of the spin-flip amplitude.     

Konversion der 238,6 keV-γ-Linie des ThB und Vergleich mit den Tabellen von Sliv und von Rose

The conversion ratiosK/L _I andK/L _II of the 238.6 keVγ transition of ThB have been measured with the new Heidelbergπ/2 √13Β-ray spectrometer. For this transition there are great discrepancies between the values as given bySliv andBand, and byRose. The experimental resultsK/L _I=5.96±0.26 andK/L _II=60.3±7.1 are in excellent agreement with the values ofSliv andBand.     

Über die Berechnung der Photoerzeugung von π0 aus Streudaten

The cross section forπ ⁰-production is calculated from the results of the dispersion relation approach ofChew, Goldberger, Low andNambu and compared with the experimental data. The predictions are made using the measured values of all scattering phase shifts. A fit of the theoretical result toσ(90?) between 260 and 370 MeV givesf ²=0,082. There is no systematic deviation from the predictions forσ (90?) up toE _γ =450 MeV. The general behavior of the asymmetry coefficientB is correct. A quantitative comparison which would show the contribution of the unknown electric dipole termN ⁽⁺⁾ ofChew et al. is only reasonable after the calculation ofB and the measurements have been improved.C/A depends strongly on the small phases up to 240 MeV. The effects of the small phases are pretty large forσ(l50?) but there is no discrepancy similar to the results forπ ⁺-production above 290 MeV. The formula for the cross section does not reduce to the results of the phenomenological theories ofBrueckner-Watson, Sachs et al. andFeld, if the simplifications made by these authors are taken into account. The origin of the differences is discussed.     

Relativistic Slater's integrals for theK andL shells

TheZ-dependent form of theSlater's integrals for electrons in theK andL shells is studied by expanding the relativistic hydrogenic radial wave functions in the manner explained byLayzer andBahcall. When screening is not taken into consideration, theseSlater's integrals can be put in the form AZ(1+a Z²) wherea is positive for all integrals considered except G¹(1s, 2p) and G²(2¯p, 2p). Values ofA anda are given.     

The Adler-Weisberger sum rule

Starting from a weak form of the axial current algebra and usingFubini's covariant dispersion method we give a fairly rigorous derivation of the Adler-Weisberger sum rule. Special attention is given to the necessary continuation in the “pion mass”. Combining the sum rule with the forwardπ ^± p dispersion relation we obtain a sum rule containing essentially low energy data. A numerical evaluation is given for two sets of existing s-waveπN scattering lengths. The one reproducesAdler's andWeisberger's results, the other improves the agreement with experiment giving r_A=1.20.     

Notes on number density fluctuations and the scattering cross section for electromagnetic waves scattered from a thermal nonequilibrium plasma

The scattering cross section and the Doppler spectrum for electromagnetic waves scattered by the electron density fluctuations of a plasma, where the mean kinetic temperature of the electronsT _e may differ from that of the ionsT _i , has been obtained among others byFejer, Buneman, Renau, Camnitz andFlood, andSalpeter. These authors use different methods of approach to calculate the autocovariance of the electron number-density fluctuations (from the mean) and then obtain the scattering cross section. Because of the differing results, the methods, concepts, and derivations of the scattering cross section are carefully examined in this paper. It is shown that the short-time dynamical considerations incorporated in the formulation of the statistical theory of the electron number-density fluctuations of the plasma as used by several authors (for instanceFejer, Buneman, Salpeter,) leads to results of limited validity. In addition, a fundamental error in calculating the electron density fluctuations leads these latter authors to an incorrect scattering cross section. The theory of scattering of electromagnetic waves from a plasma, where the electrons arenot in thermal equilibrium with the ions but statistical equilibrium exists, is developed in a general way. The covariance of the number-density fluctuations from the mean of the charged species of the plasma and the scattering cross sectionσ(q) are obtained. In particular it is shown that for a wavelength λ much greater than the effective Debye lengthd, the backscattering cross section increases and approaches complete incoherent scattering asT _e /T _i increases. This result is explained by noting that in the case of thermal equilibrium, the predicted value of the back-scattering cross section is smaller than that of the backscattering cross section from completely uncorrelated electron density fluctuations because the electrostatic interaction between the charged particles of the plasma, which is a function ofT _e andT _i , introduces a certain amount of organization in otherwise completely uncorrelated electron density fluctuations. When the mean temperature of the electrons increases relative to that of the ions, the organization introduced in the fluctuations diminishes because of the increasing thermal agitation of the electrons relative to that of the ions, and the backscattering process approaches that of incoherent backscattering (Thomson-type scattering). The spectrum function of incoherent scattering of electromagnetic waves from a nonequilibrium plasma is obtained and some cases of current interest are plotted.     

A new approach to potential scattering

An approximate integral-representation of theS-matrix in partial-wave expansion is derived for a scalar Schrödinger particle in a central field. The method consists of linearizingCalogero's Riccati equation for the interpolatingS-matrix in such a way that the solution of the linearized equation deviates as little as possible from the exact one. TheS-matrix thus obtained exhibits exact crossing-symmetry and uniform convergence independent of the coupling constant of the scattering potential. In the weak coupling limit it is especially shown thatour method is more accurate than the second Born approximation. In the second part of the paper we specialize ourS-matrix to low and large energies. At low energies, a general integral for the scattering length is obtained and at large energies the summation over all angular momenta is carried out yielding an expression for the scattering amplitude.     

Schrödinger equation in multiply connected spaces and phase optics

The magnetic vector potential\(\vec A\) in a field free spaceR ₀ cannot be removed by gauge transformations in general, ifR ₀ is multiply connected.Aharonov andBohm ¹ have noticed recently that\(\vec A\) therefore should have more physical meaning than only to give the magnetic field by differentiation. They could show that\(\vec A\) inR ₀ may influence the phase ofSchrödinger'sψ-function in an observable manner. We want to point out here that this influence can be expressed in a simple, general form: “A closed magnetic field line operates uponψ like ae Φ/?-phase-shifter placed on any area bounded by the field line.” Surface like phase shifters are familiar in phase optics. There exists a narrow relationship between electron scattering at magnetic fields and some special problems of phase optics. An electron phase contrast microscope is discussed.     

Untersuchungen zur Mehrfach-Emission von Sekundärelektronen

CuBe- and NaCl-targets are bombarded by single electrons (100–600 eV). The secondary electrons accelerated by 40 kV strike the crystal of a scintillation counter, backed by a multichannel analyser. The probabilityP _n of emission ofn=0, 1, 2, 3, ... secondaries can be found from the pulse height distribution. The probability distributionP _n =f(n) shows a characteristic deviation from aPoisson's distribution. There was no evidence that there is a preference for even numbers ofn as found byBarrington andAnderson.     

Quasielastic kaon knockout from a proton and momentum distribution of virtual kaons in the proton

The momentum distribution of kaons in the proton and the coupling-constant ratio fKΛN/fKΣN are determined on the basis of the ³P₀ quark model of meson-baryon coupling. The longitudinal cross section for kaon electroproduction is calculated. The results of the present study are compatible with available experimental data, but more detailed data on the longitudinal cross section would make it possible to refine the absolute values of the coupling constants fKΛN and fKΣN.     

Paramagnetische Resonanz von Zentren mit den Symmetrienn,\bar n undn/mundn/m

The consequences of applying the spin-Hamiltonians ofAbragam andPryce to paramagnetic centers with the low symmetriesn,\(\bar n\) andn/m(C _n ,C _nh ,S _n ) are examined. The asymmetry of theg- and the hfs-tensors, described in an earlier paper, is considered in this context. The point is that for the symmetries under consideration the coordinate systems are not determined by the symmetry elements. For this reason it is possible to introduce separate coordinate systems for the magnetic field, the electron spin and the spins of the nuclei and this allows the symmetrisation of the tensors under certain conditions. This procedure also leeds to an understanding of the independent parameters found in the Hamiltonians. Ambiguities between the spin-Hamiltonians and the esr-spectra indicate the limits of theAbragam andPryce formalism. The application of the theory to paramagnetic centers under the influence of external, electrical fields is discussed and reveals interesting aspects.     

Zur Theorie der infinitesimalen Holonomiegruppe in der Allgemeinen Relativitätstheorie

Starting from the infinitesimal holonomy groupH _i of aV ⁴, (+++?) the spinholonomy group\(\tilde H_i \equiv \bar \sigma ^1 (H_i )\) defined by the covering isomorphism\(\sigma :G \to L_ + ^ \uparrow \) is introduced. In Einstein-spaces we may replace its real Lie-algebra by a complex one. With the complex calculus we may reproduce the results ofSchell, Goldberg andKerr with very much simplified proofs. A theorem on non-empty Einstein-spaces is given. In part 4 we prove a theorem on the connection between theH _i -behaviour of a vector (spinor) and its covariant derivative in aV ⁴. With its help we get in a simple manner the metiics of aV ⁴ with givenH _i and Dim (H _i ) <6; our results agree with those given byGoldberg andKerr, Cahen andDebever. Finally we make some new statements on imperfect holonomy groups.