Z2 Structure Research Articles

Z2 structure of the stopping power for electron beams

The target atomic number (Z2) dependence of the collision stopping power for electron beams have been investigated for the purpose of determining primary mechanisms in stopping of electrons in elemental matter. The stopping powers of target elements with atomic numbers Z2=1–92 for the electrons have been evaluated on the basis of Gümüş method. The variation of the stopping power depending on energy of electrons and type of target has been studied to obtain the elements with the strongest stopping force property for electron beams. A strong Z2 oscillation in the collision stopping power has been observed whereas the mass collision stopping powers are found to be decreasing with increasing atomic number of the target. It is also found that the electron slowing down in matter mainly depends on the atomic density of target and initial energy of electrons.

Reciprocity in the electronic stopping of slow ions in matter

The principle of reciprocity, i.e., the invariance of the inelastic excitation in ion-atom collisions against interchange of projectile and target, has been applied to the electronic stopping cross section of low-velocity ions and tested empirically on ion-target combinations supported by a more or less adequate amount of experimental data. Reciprocity is well obeyed (within ~10%) for many systems studied, and deviations exceeding ~20% are exceptional. Systematic deviations such as gas-solid or metal-insulator differences have been looked for but not identified on the present basis. A direct consequence of reciprocity is the equivalence of Z1 with Z2 structure for random slowing down. This feature is reasonably well supported empirically for ion-target combinations involving carbon, nitrogen, aluminium and argon. Reciprocity may be utilized as a criterion to reject questionable experimental data. In cases where a certain stopping cross section has not been or cannot be measured, the stopping cross section for the inverted system may be available and serve as a first estimate. It is suggested to build in reciprocity as a fundamental requirement into empirical interpolation schemes directed at the stopping of low-velocity ions. Examination of the SRIM and MSTAR codes reveals cases where reciprocity is obeyed accurately, but deviations of up to a factor of two are common. In case of heavy ions such as gold, electronic stopping cross sections predicted by SRIM are asserted to be almost an order of magnitude too high.

Z2 structure and gas–solid effect in the stopping of slow ions

A recent claim by Paul of a systematic gas–solid difference in stopping cross sections for ions such as nitrogen and oxygen in the velocity range v ≃ v 0 is studied on the basis of existing experimental data. We find that all existing data support the commonly known Z 2 structure which, by and large, follows the valence structure of the target material. Existing experimental evidence is not found to support a specific gas–solid difference in the velocity range under consideration. The possibility of such an effect due to a gas–solid difference in charge state is rejected on theoretical grounds. Data for compound gases and solids are found to be well described by the Bragg additivity rule. We have also studied nitrogen/helium and oxygen/helium stopping ratios which determine the so-called effective-charge ratio. Taking into account the scatter of experimental data, we do not find clear evidence against Northcliffe’s assumption of a stopping ratio independent of Z 2 and common for gases and solids in the considered velocity range, although the absolute value appears too high.

Valence structure effects in the stopping of swift ions

The valence structure of a material may affect the stopping of swift charged particles primarily via Z2 structure, atom–molecule differences, gas–solid differences and metal–insulator differences. These material effects have a common physical origin and can therefore be considered from a unified point of view. Theoretical arguments focus on the effect of binding and orbital motion of the target electrons as well as projectile screening and Barkas–Andersen effect. Generally, valence effects depend on the atomic number, charge state and velocity of the projectile. Reference is made to recent calculations on the basis of binary stopping theory as well as experimental findings.

OPEN DESCENDANTS OF NAHE-BASED FREE FERMIONIC AND TYPE I ${\mathbb Z}_2^n$ MODELS

The NAHE set, that underlies the realistic free fermionic models, corresponds to ℤ2×ℤ2 orbifold at an enhanced symmetry point, with (h11,h21)=(27,3). Alternatively, a manifold with the same data is obtained by starting with a ℤ2×ℤ2 orbifold at a generic point on the lattice and adding a freely acting ℤ2 involution. In this paper we study type I orientifolds on the manifolds that underlie the NAHE-based models by incorporating such freely acting shifts. We present new models in the type I vacuum which are modulated by [Formula: see text] for n=2,3. In the case of n=2, the ℤ2×ℤ2 structure is a composite orbifold Kaluza–Klein shift arrangement. The partition function provides a simpler spectrum with chiral matter. For n=3, the case discussed is a ℤ2 modulation of the T6/ℤ2×ℤ2 spectrum. The additional projection shows an enhanced closed and open sector with chiral matter. The brane stacks are correspondingly altered from those which are present in the ℤ2×ℤ2 orbifold. In addition, we discuss the models arising from the open sector with and without discrete torsion.

SUPERSYMMETRY OF PARAFERMIONS

We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.

A Z2 structure in the configuration space of Yang–Mills theories

We argue for the presence of a Z2 topological structure in the space of static gauge–Higgs field configurations of SU(2n) and SO(2n) Yang–Mills theories. We rigorously prove the existence of a Z2 homotopy group of mappings from the two-dimensional projective sphere RP2 into SU(2n)/Z2 and SO(2n)/Z2 Lie groups, respectively. Consequently the symmetric phase of these theories admits infinite surfaces of odd-parity static and unstable gauge field configurations which divide into two disconnected sectors with integer Chern–Simons numbers n and n+1/2, respectively. Such a Z2 structure persists in the Higgs phase of the above theories and accounts for the existence of CS=1/2 odd-parity saddle point solutions to the field equations which correspond to spontaneous symmetry breaking mass scales.

TOPOLOGICAL STUDY OF GLOBAL QUANTIZATION IN NON-ABELIAN GAUGE THEORIES AND THE OPEN UNIVERSE

We study some topological aspects of non-Abelian gauge theories intimately connected with the Lie algebras of the gauge groups and the homotopy theory in the generalized gauge orbit space. The physics connection with the nonperturbative solution to the strong CP problem by magnetic monopoles and the quantization rule for the effective vacuum angle as originally proposed by the author are also discussed. In addition, some relevant topological formulas are given and discussed. We emphasize that a result7 from their physics application is that the usual gauge orbit space on the compactified space may contain at most a Z2 monopole structure in the SP (2N)(N= rank ) gauge theories; such a Z2 structure may induce spontaneous CP violations in SP (2N) gauge theories, as was noted. Some relevance to the open universe is discussed too. It is expected from our results that the open universe in the case of a nonvanishing θ can also be regarded as a quantum effect of the global quantization in the gauge orbit space. We expect that our results will be useful for other studies of non-Abelian gauge theories in general. Some remarks are also given related to our global quantization and the gauge theories in generalized gauge orbit space.