Relativistic Behavior Research Articles

Linear prediction filter analysis of relativistic electron properties at 6.6 RE

Electrons with energies >500 keV in the Earth's outer magnetosphere exhibit very hard energy spectra and highly variable absolute intensities. The fluxes often show a strong 27‐day periodicity which is related to recurrent solar wind stream variations at 1 AU. We have used available solar wind speed data as well as continuous geomagnetic indices such as Kp and AE in order to characterize the relationship of relativistic electrons to these geophysical parameters. The present analysis emphasizes data taken in 1982–1985 and employs electron measurements from 3 to 40 MeV at geostationary orbit along with geomagnetic data from the National Geophysical Data Center CD ROM. The method of linear prediction filter (LPF) analysis is used to characterize and predict the general relationship between solar wind or geomagnetic indices as input time series and electron properties as the output time series. Filters are found that generally decrease strongly at zero lag time and then peak strongly at lags of 2–3 days. Cross‐covariance analyses show strong correlative peaks between electron fluxes and geomagnetic parameters at multiples of 13 and 27 days. The present analysis allows enhanced understanding of the relativistic electron behavior on both short and long time scales and permits improved prediction of both high‐altitude spacecraft operational environments and magnetosphere‐atmosphere coupling relationships.

Explicit illustration of causality violation: Noncausal relativistic wave-packet evolution.

A class of functions whose relativistic evolution is analytically solvable is found. These wave functions are initially localized (by a Newton-Wigner definition) and their spreading through time is investigated. Using these wave packets an explicit example of the violation of relativistic causality (in the spirit of Hegerfeldt's theorem) is presented. Some detailed qualitative features of relativistic quantum behavior are displayed.

The relativistic pair equation and related fundamental problems

By the use of projection operators, defined in Dirac space, it is shown how a relativistic pair equation, correct to order α2 Ry, can be constructed. It is pointed out how the Brown-Ravenhall disease (continuum dissolution) is rigorously avoided with this procedure. The pair equation treats correlation relativistically starting from a suitable choice of a fully relativistic one particle description of the atom. Second order contributions to the energy from the electrostatic correlation have been calculated for the 1s2 ground state of two-body systems with (Z = 2, 10). The starting point has been either hydrogen-like wavefunctions or Dirac-Fock orbitals. Excitations to s2, p2 and d2 have been treated and an unexpected behaviour of the p1/22 excitation relative to the p3/22 one is discussed. By iterative solutions of the exact pair equation a non-perturbative treatment of the two-body part of the static Coulomb interaction is obtained. This procedure has been used for s-excitation and a comparison of the relativistic and non-relativistic behaviour is made.

Correlation in a relativistic framework

By the use of projection operators, defined in Dirac space, it is shown how a relativistic pair equation, correct to order α 2Ry, can be constructed. It is pointed out how the Brown-Ravenhall disease (continuum dissolution) is rigorously avoided with this procedure. The pair equation treats correlation relativistically starting from a suitable choice of a fully relativistic one-particle description of the atom. Second order contributions to the energy from the electrostatic correlation have been calculated for the 1s 2 ground state of some two-body systems. The starting point has been either hydrogen-like wave functions ( Z = 2, 10) or Dirac-Fock orbitals ( Z 10−60). Excitations to s 2, p 2, d 2 and f 2 have been treated and an unexpected behaviour of the p 2 1 2 excita P 2 3 2 one is discussed. By iterative solutions of the exact pair equation a nonperturbative treatment of the two-body part of the s Coulomb interaction is obtained. This procedure has been used for s excitations ( Z = 2−20) and a comparison of the relativistic and nonrelativistic behaviour is made.

A dynamical interpretation of temperature

Quantum-mechanical interpretations are sought for tem-temperatureT and Boltzmann’s constantk. Jaynes’ information theory approach to statistical mechanics is used to include the uncertainties in particle energy levels caused by thermal interactions. Incorporating this minor correction reveals that the average energy spread per particle iskT. The energy spread is then related to the lifetime of a particle state with the result thatT can be interpreted as the average transition rate per particle. When this view is adopted, Boltzmann’s constant has units of action and its value is found to be 1/4h, whereh is Planck’s constant. The relativistic behavior of temperature is examined with the dynamical interpretation and it predicts that the temperature of a moving system is lower than that of the system at rest.

K-shell ionisation by protons, deuterons and alpha particles at low projectile energies

The semiclassical approximation model (SCA) with hyperbolic projectile trajectories, including target recoil and symmetrisation, is used to study Coulomb ionisation in the incident energy range from 50 to 300 keV for a Ni target. Combined with corrections for the binding effect and the relativistic behaviour of the target electrons, the first-order SCA theory yields total cross sections consistent with recent measurements. No variation of parameters is performed.

Improvements to PARMILA

PARMILA is an internationally accepted standard for Monte Carlo simulation of linac and transport line performance. We discuss several modifications and improvements to this code. A generalized magnet routine allows the simulation of electrostatic and magnetic quadrupoles, solenoids, sextupoles, and octupoles. Optional inclusion of linear fringe fields and/or geometric aberrations is provided for in the quadrupole transformation. The dipole routine has been replaced with a more accurate algorithm. The accelerating gap transformation has been replaced by a set of implicit equations which accurately describe the relativistic particle behavior in the presence of longitudinal and transverse electric fields described by a set of 6 weighted Fourier moments (transit time factors). A simple model allows these moments, in turn, to be approximated from the cell geometry and the usual T and S functions. A number of added convenience features - interactive disk storage and retrieval of particle coordinates, individual particle input and observations, an interactively callable test for particle longitudinal stability, and an automated quadrupole tuning procedure - all add to the code's versatility, convenience, and strength as a design tool.

Undergraduate Relativity Experiment

The primary electrons in a γ-ray detector crystal are typically rather relativistic having kinetic energies of the order of their own rest mass. A technique is described which enables the student to observe the relativistic behavior of these electrons and compare his results with the classical and relativistic predictions. It is shown that it is possible to unambiguously determine the electron's momentum by knowing the Compton edge and photopeak energies of a γ-ray energy spectrum.

Relativistic Effects for Moving Terrestrial Clocks

In his communication1 on the relativistic behaviour of moving terrestrial clocks Hafele deduces that an airborne clock, after circumnavigating the Earth, could be either fast or slow when compared with a clock which remained stationary on the Earth's surface, depending on the direction of travel; and that in neither the east-bound nor the west-bound case would the generally accepted result be obtained—namely, that the “moving” clock would run slow by the factor (1−v2/c2)1/2. Hafele's apparently anomalous and non-relativistic conclusions have been criticized by Schlegel2, who objects to the background system of coordinates which Hafele employs to evaluate the clock rates and which, according to Schlegel, leads to erroneous results.

Relativistic Behaviour of Circumnavigating Clocks

CONSIDERABLE interest has been aroused by Hafele's suggestion1,2, subsequently supported by experiment3, that caesium beam clocks, borne by commercial jets circumnavigating the Earth, may provide supporting evidence for the famous prediction of Einstein's theory of relativity that if one member of a set of twins departs from the Earth in a spacecraft and later returns to reunite with his sibling, he will be the younger of the two. Strictly speaking, this statement is valid only if the frame of reference in which the home-bound sibling remains at rest is an inertial frame, which is not true of a frame of reference attached to the (rotating) Earth. This fact was ordinarily disregarded because the relative velocities involved were supposed to be much larger than the velocity difference caused by this circumstance. But in the case of clocks employed by Hafele and Keating3, this supposition is not true and the influence of Earth's rotation must be reckoned with.

Relativistic Behaviour of Moving Terrestrial Clocks

EINSTEIN concluded in his 1905 paper on relativity that “a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions”1. At that time, of course, he had not yet developed the general theory and did not realize that the difference in the gravitational potential at the equator and poles would just cancel the time dilatation from the difference in surface speeds, as has been pointed out recently by W. J. Cocke2. Cocke has also shown that, in so far as the Earth assumes the shape of the geoid (and weak tidal fields can be neglected), its entire surface is an “equal time” surface. There remains an interesting question, however: What would be the rate of a standard clock that is moving relative to stationary standard clocks on the geoid ? The standard answer that moving clocks run slow by the well known factor (1 - v2/c2)1/2 is almost certainly incorrect. The purpose of this note is to discuss this question of the rate of moving terrestrial clocks relative to stationary ones.

Remark to the interpretation of the Kennedy-Thorndike experiment

It is shown that the result of the Kennedy—Thorndike experiment can be understood in terms of the Lorentz contraction only and it is not necessary — as it is generally supposed — to make use of the relativistic behaviour of clocks for the interpretation of the result.

Spin-Orbit Coupling and the Knight Shift in Nontransition-Metal Superconductors

The Knight shift ${K}_{s}$ of nontransition-metal superconductors is discussed in terms of three contributions, namely: (a) The Van Vleck part of the contact shift attributed to the spin-orbit coupling force of the periodic potential which, in the presence of an external magnetic field, causes virtual high-energy rearrangements of conduction electrons near and inside the Fermi surface; (b) that part of the contact shift which is attributed to low-energy rearrangements of conduction electrons and affected by spin-reversing scattering; and (c) the diamagnetic orbital shift. For the calculation of (a), Wannier's theory of a Bloch electron in a magnetic field is generalized to the original Pauli Hamiltonian ${\mathcal{H}}_{0}$, describing the relativistic dynamical behavior of a conduction electron in the effective periodic potential of the lattice. This leads to an effective Hamiltonian which couples only Bloch-type spinors of the same band index, but of the same and of different spin indices. With the help of the eigenfunctions of ${\mathcal{H}}_{0}$, the hyperfine contact interaction is treated by perturbation theory. To arrive at simple expressions for the corresponding Knight shift and nuclear spin-relaxation time, valid for arbitrary strength of spin-orbit coupling, the energy-band function in the absence of the field is approximated by a parabola. Formulas for (b) and (c) are taken from the literature. The relative importance of the three contributions to ${K}_{s}$ is discussed for Al, Sn, and Hg, where the Knight shift has been observed in the normal and in the superconducting states. If one assumes that spin-reversing scattering is caused merely by spin-orbit interactions at atomic imperfections such as displaced surface atoms of small particles, and not by paramagnetic impurities, one finds that in Al neither of the two spin-orbit coupling effects is sufficiently strong to account for more than \ensuremath{\sim}2% of the residual shift ${K}_{s}(0)$. For the experimental Sn sample, (a) and (b) have the ratio 1:3 and, together with the orbital shift, can account for $\frac{2}{3}$ of the observed ${K}_{s}(0)$. For Hg, (a) and (b) are of comparable magnitude at $T=0$ and together account for more than \textonehalf{} of the observed shift ${K}_{s}(0)$.

Lecture Demonstration of Relativistic Behavior of Electrons

A lecture demonstration is described, which allows verification of the relativistic relation between kinetic energy and momentum, (mc2+E)2 = m2c4+c2p2. Electrons are accelerated by a small high-voltage generator (van de Graaff, of approximately 250 kv) and deflected in a magnetic field of approximately 100 gauss. The measured values of high-voltage and deflecting magnetic field are plotted on a straight line which represents the relation given above.

Uniformly Moving Dislocations

An expression is derived for the displacements in an isotropic elastic medium which contains an edge dislocation moving with uniform velocity c. When c=0 the solution reduces to that given by Burgers for a stationary edge dislocation. The energy density in the medium becomes infinite as c approaches c2, the velocity of shear waves in the medium; this velocity therefore sets a limit beyond which the dislocation cannot be accelerated by applied stresses. The atomic structure of the medium is next partly taken into account, following the method already used by Peierls and Nabarro for the stationary dislocation. The solution found in this way differs from the one in which the atomic structure is neglected only within a region of width ζ which extends not more than a few atomic distances from the centre. ζ varies with c and vanishes when c=cr, the velocity of Rayleigh waves. It becomes negative when cr< c<c2. Thus cr rather than c2 appears to be the limiting velocity when the atomic nature of the medium is taken into account. Since cr similar, equals 0.9c2 the difference is not of much importance. The same method applied to a screw dislocation gives, in the purely elastic case, the expression already derived by Frank. The corresponding Peierls-Nabarro calculation shows that the width ζ is proportional to (1 - c2/c22)½. This relativistic behaviour is analogous to Frenkel and Kontorowa's results for their one-dimensional dislocation model.