Dispersion of the stress optic coefficient of the alkali halides

The dispersion of the stress optic coefficient C=n³/2 (q₁₁?q₁₂) of the alkali halides, NaCl, KCl, KBr and KI have been measured from the visible to the ultraviolet region. In general the value of “C” decreases with wavelength for all crystals. While the dispersion is only a few per cent in the visible region of wavelengths, it is enormous in the ultraviolet. NaCl shows a dispersion of about 100% from 5800 to 2400 Å; KCl about 200% from 5000 to 2400 Å; KBr about 300% from 5000 to 2400 Å; and KI about 400% from 5000 to 2800 Å. Also the potassium halides exhibit a change in sign of their “C” values in the ultraviolet. In KCl the sign reversal occurs at about 2550 Å; in KBr at 2760 Å and in KI at 3380 Å. Below these wavelengths, the potassium halides belong to the same class inMueller's classification as sodium chloride. The theory ofRamaseshan andSivaramakrishnan based on the assumption that a stress causes a change in the frequencies and oscillator strengths of atoms is unable to explain the observed behaviour of the alkali halides. On the other hand, the mere variation of the ionic refractivities with wavelength is also unable to explain the observed dispersion onMueller's theory. One is forced to assume that the strain polarisability constantK inMueller's theory varies with wavelength. When “K” is calculated from the experimentally observed values of “C”, it is found to increase with decreasing wavelength for all alkali halides. The variation with wavelength of “K” for all the alkali halides can be fitted up well by a formula of the type given byRamaseshan andSivaramakrishnan. Hence it appears that the total dispersion ofC can be explained only when we take into account the variation with wavelength of 1. theLorentz andCoulomb contributions fromMueller's theory and 2. the strain polarisability constant fromRamaseshan andSivaramakrishnan's theory.     

Meson baryon sum rules

The PCAC consistency condition, the Adler-Weisberger sum rule and the dispersion superconvergent sum rule have been saturated with resonance contributions only. Strangeness-changing currents and strangeness-conserving currents have been considered and compared. The results obtained also support PCAC for the strangeness conserving current. While in theπN case PCAC and the Adler-Weisberger sum rule are almost saturated with the famous N ₃₃ ^* resonance only, in the KN caseY ₁ ^* (1385),Y ₀ ^* (1405) andY ₀ ^* (1520) seem to be sufficient. The superconvergent sum rule works very well in the case ofπN scattering, including the contribution of twelve resonances. For KN scattering it can be neither established nor disapproved.     

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.     

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.     

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.     

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.     

On the combinatorics of Feynman graphs. I

As a preliminary step towards a rigorous proof ofDyson's conjectures on the combinatorics of Feynman graphs, the following lemma is proved rigorously in an explicit and elementary way: The generating functional of the time-orderedGreen's functions in perturbation theory is an exponential, the exponent of which is a sum of contributions from connected graphs only, with no contributions from vacuum graphs. The model used to demonstrate this is a scalar hermitean fieldA(x) with anA ⁴ selfcoupling.     

Vergleich der Photoerzeugung von π0 am Proton mit der Voraussage nach der Dispersionstheorie

The real parts of the photoamplitudesE 1S _1/2,M 1P _1/2,M 1P _3/2 have been calculated from the angular distribution of the reactionγ+p→p+π⁰ recently measured byGoldansky et al. at 160 to 240 MeV. One of the solutions fits pretty well to the theoretical prediction for theM 1P _3/2-amplitude according to the dispersion method ofChew, Goldberger, Low andNambu. There is a discrepancy forM 1P _1/2 ifα ₁₁ is taken from the effective range formula, but the positive values ofα ₁₁, necessary to give agreement, are not excluded by the results of the phase shift analysis, especially sincePontecorvo et al. have recently found positive values at higher energies. The prediction for the real part of theE 1 S_1/2-amplitude agrees with the experimental data, if pretty large recoil corrections are added which had been neglected byChew et al.     

Zum Zerfallπ→e+ν+γ

The radiative decay of the chargedπ-mesonπ→e+ν+γ is calculated using dispersion relation techniques. It is assumed that the weak interaction producing the decay has a V-A structure. The decay through the vector variant has been reinvestigated and found to be different from the result ofVaks andIoffe. The theoretical prediction of this paper is in agreement with a preliminary experiment recently performed at CERN.     

Comparison of theK ± p scattering data with the forward dispersion relations

The dispersion relations forK ^± p forward scattering were evaluated in order to test the compatibility with the experimental data and to obtain an estimation of the effective coupling constantG ^2/4π=(G _Λ ² +G _Σ ² )/4π.     

On the connection between the decaysπ 0→2γ andπ +→e++υ+γ

A result discovered byVaks andIoffe is newly derived using reduction technique and G-invariance only. It has some consequences concerning the decaysπ ⁰→2γ andπ ⁺→e⁺+υ+γ and various coupling constants of the? and ω particle.     

Zerfallsschema und β-Spektrum des MsTh1 (Ra228)

Theβ-spectrum of MsTh1 (Ra²²⁸) has been measured in a proportional counter in 2π — and in 4π-geometry. Comparison of these measurements shows that there are no conversion electrons. Theβ-decay of MsTh 1 therefore leads in all cases to the ground level of MsTh2. The Kurie plot is straight from 15 keV up to the upper energy limit ofE ₀=55±3 keV (logft=5·6). The result has been checked by a search for photons emitted by the MsTh1. Noγ-rays and no X-rays have been found. The measurements had to be made with a relatively large amount of natural Ra²²⁶ in the sources. — The results do not agree with those ofLecoin, Perey, Teillac andRiou, who proposed a complex decay sheme for MsTh1.     

Analysis of the photoproduction of pions by means of fixed t dispersion relations. I

An absolute prediction for the photoproduction of pions on protons is given in the region of the first pion nucleon resonance. The imaginary part of the amplitude was approximated by the resonant magnetic dipole amplitudeM ₁₊ ^3/2 taken from the theory of CGLN. With this approximation the real part of the amplitude was calculated by means of fixedt dispersion relations and discussed in terms of multipoles.π ⁺-andπ ⁰-excitation curves atΘ=90° for unpolarized as well as plane-polarizedγ's are compared with the experiments. In the appendix we give tables of the differential cross sections in the c.m. system.     

Physics results from HERMES

The Hermes experiment at the Desy laboratory in Hamburg, Germany, studies the spin structure of the nucleon. Its unique feature is the combination of a polarized internal gas target with the longitudinally polarized 27.5 GeV electron/positron beam of the Hera accelerator. Recent Hermes measurements include the proton spin structure function g ₁ ^p , the flavor decomposition of the polarized quark distributions in the nucleon, the DIS contribution to the generalized GDH sum rule, the first observation of a spin asymmetry in exclusive vector meson production and the first observation of a single-spin azimuthal asymmetry in semi-inclusive pion production. Additionally, the possibility of using various unpolarized gases as target material broadens the spectrum of physics measurements with Hermes.     

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.     

Possible violation of the δS=δQ selection rule in leptonic decays of σ+-hyperons

We have found a possible example of the rare decayσ ⁺ →nΜ ⁺ Ν, which violatesδS=δQ. The positive decay track of theσ ⁺ comes to rest in the hydrogen bubble chamber and decays into ane ⁺. This track has all the characteristics of a stoppingΜ ⁺. The decay neutron fortuitously scatters twice, producing two recoil protons. The only other possible interpretation of the event isσ ⁺→nγ(π ⁺ →Μ ⁺ Ν), where theπ ⁺ →Μ ⁺ Ν decay produces no deflection (θ<0.1 rad) and no significant change in curvature. Using thep-wave radiative decay predictions ofBarshay et al. we calculate that the integrated branching ratio for such “accidental” events isγ(σ ⁺ →nγ(π ⁺ →Μ _stop ⁺ Ν))/γ(σ ⁺ →nπ ⁺)=1.6×10^?6. Most of the contribution to this “accidental” branching ratio comes from radiative decays where theπ ⁺ mesons have ranges less than 1 mm (p _π<20 MeV/c). If one excludes thoseΜ's with ranges less than 1.2 cm the above “accidental” branching ratio becomes 5.5×10^?7. With this figure we estimate that we should have seen 6.5×10^?2 events of this type thusfar in our experiment. The neutron momentum does not help in deciding between the two hypotheses. We therefore assign a confidence level of 7% for the radiative hypothesis. For the leptonic hypothesis we obtain an estimate of the branching ratio,γ(σ ⁺ →nΜ ⁺ Ν)/γ(σ ⁺ →nπ ⁺)=5×10^?5. If one further accepts theσ ⁺ →nπ ⁺ Ν event reported byBarbaro-Galtieri et al. and theσ ⁺ →ne ⁺ Ν event reported byNauenberg et al., one obtains theδS=?δQ leptonic branching ratio [γ(σ ⁺ →nΜ ⁺ Ν)+γ(σ ⁺ →ne ⁺ Ν)]/γ(σ ⁺ →nπ ⁺)=(4±3)×10^?5.     

Pion-pion interaction andK 2x03C0;x03C0;x03C0;-decay

The enhancement of theK _2π ⁺ -decay compared to theK _2π ⁰ -decay is discussed on the basis of the\(|\mathop {\Delta {\rm I}}\limits^ \to | = \tfrac{1}{2}\)-rule. The enhancement factor is calculated by dispersion methods which yield an expression depending only on the phase shift of the two pion system in theJ=0,I=0,2 state. This expression has been studied in the framework of simple models for the two-pion interaction in order to obtain a survey of the possibilities for the cause of the anomalous large ratioK _2π ⁺ /K _2π ⁰ . Only characteristic cases have been considered and, as far as possible, experimental results of theπ-π-interaction are taken into account.     

Bemerkungen über relativistische Stoßinvarianten

The relativistic analogue of the theorem ofGrad is proven stating that the general additive collision invariant is given by the expressionβ _a p ^a +c. It is used to determine the equilibrium distribution of a 1-component general relativistic quantum gas. The proof is not analogous to the one initially given byGrad, but covers the non-relativistic case.     

Theory of the density of states of pure type-II superconductors in high magnetic fields

This paper deals with the theory of the density of states of pure type-II superconductors in high magnetic fields. An approximate expression of theGreen's function is derived by using the fact that it has the periodicity of the lattice of flux lines in the center-of-mass coordinates. In comparison to the normal stateGreen's function a correction term proportional to the absolute square of the order parameter arises. This is seen to act as a self-energy part depending strongly on the polar angleΘ of the “quasiparticle” momentum relative to the direction of the field. The corresponding density of states, as a function of the excitation energy, is found to vary from the gapless to the BCS type as the parameterΘ changes fromπ/2 to zero (orπ). The averaged state density varies gradually from a uniform to a BCS-like state density as the external field decreases sufficiently far below the upper critical field. The results of the present theory for the averaged density of states agree fairly well with the results due to a conjecture byMaki if the field is very close to the upper critical field.     

Contributions to the theory of alpha-decay

The theory of α-decay is reformulated in order to point out the necessary approximations in a rigorous way. The connection between the formulas given byThomas and byMang is easily established.Mang's explicit expressions of reduced widths, valid for shell model wave functions, are brought into a general form. The theory is applied to BCS wave functions and the decay constants of several Po-isotopes are calculated as an example. Furthermore results are presented which favour a spin assignmentI=18 and a predominating (h9/2) ₈ ² (i11/2) ₁₀ ² configuration for the Po²¹² isomeric state. The penetrabilities are calculated numerically, since the WKB-method is found to be unsatisfactory.