Book review: Handbook of early childhood literacy

Joanne Larson and Shira May Peterson highlight the paucity of research on how talk is used in pre-school settings compared to school settings. They locate the wealth of research studies on talk and discourse in formal learning settings in terms of the different ideological positions adopted. They use two continua; Streets (1995) distinction between ideological and autonomous conceptions of literacy and whether the function of literacy is seen as fixed or fluid to separate the studies into four quadrants, each with a distinctive ideological base. This means that their analysis does not highlight the groundbreaking impact of individual studies, but it does illustrate very effectively how ideological assumptions shape both research design and the resulting recommendations for practice. It also highlights some key research problems: that ideological rigidity means researchers can miss opportunities to build on each others' findings

Supercriticality and Transmission Resonances in the Dirac equation

It is shown that a Dirac particle of mass $m$ and arbitrarily small momentum will tunnel without reflection through a potential barrier $V=U_c(x)$ of finite range provided that the potential well $V=-U_c(x)$ supports a bound state of energy $E=-m.$ This is called a supercritical potential well.Comment: Submitted to Physical Review Letter

Solutions to the 1d Klein-Gordon equation with cutoff Coulomb potentials

In a recent paper by Barton (J. Phys. A40, 1011 (2007)), the 1-dimensional Klein-Gordon equation was solved analytically for the non-singular Coulomb-like potential V_1(|x|) = -\alpha/(|x|+a). In the present paper, these results are completely confirmed by a numerical formulation that also allows a solution for an alternative cutoff Coulomb potential V_2(|x|) = -\alpha/|x|, ~|x| > a, and otherwise V_2(|x|) = -\alpha/a.Comment: 8 pages, 4 figure

Positron Tunnelling through the Coulomb Barrier of Superheavy Nuclei

We study beams of medium-energy electrons and positrons which obey the Dirac equation and scatter from nuclei with $Z > 100.$ At small distances the potential is modelled to be that of a charged sphere. A large peak is found in the probability of positron penetration to the origin for $Z \approx 184.$ This may be understood as an example of Klein tunnelling through the Coulomb barrier: it is the analogue of the Klein Paradox for the Coulomb potential.Comment: 3 figures, to be published in Physics Letters

The Strong Levinson Theorem for the Dirac Equation

We consider the Dirac equation in one space dimension in the presence of a symmetric potential well. We connect the scattering phase shifts at E=+m and E=-m to the number of states that have left the positive energy continuum or joined the negative energy continuum respectively as the potential is turned on from zero.Comment: Submitted to Physical Review Letter

Klein Tunnelling and the Klein Paradox

The Klein paradox is reassessed by considering the properties of a finite square well or barrier in the Dirac equation. It is shown that spontaneous positron emission occurs for a well if the potential is strong enough. The vacuum charge and lifetime of the well are estimated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Klein tunnelling is a property of relativistic wave equations, not necessarily connected to particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of sufficiently large $Z$ will bind positrons. Correspondingly, it is expected that as $Z$ increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling.Comment: 17 page

Electron transmission through step- and barrier-like potentials in graphene ribbons

The list of textbook tunneling formulas is extended by deriving exact expressions for the transmission coefficient in graphene ribbons with armchair edges and the step-like and barrier-like profiles of site energies along the ribbon. These expressions are obtained by matching wave functions at the interfaces between the regions, where quasiparticles have constant but different potential energies. It is shown that for an $U_0$ high barrier and low-energy electrons and holes, the mode transmission of charge carriers in this type of ribbons is described by the textbook formula, where the constant barrier is replaced by an effective, energy-dependent barrier, $U_0\to U(E)$. For the lowest/highest electron/hole mode, $U(E)$ goes, respectively, to zero and nonzero value in metallic and semiconducting ribbons. This and other peculiarities of through-barrier/step transmission in graphene are discussed and compared with related earlier results.Comment: Edited, misprints correcte

The hydrino and other unlikely states

We discuss the tightly bound (hydrino) solution of the Klein-Gordon equation for the Coulomb potential in 3 dimensions. We show that a similarly tightly bound state occurs for the Dirac equation in 2 dimensions. These states are unphysical since they disappear if the nuclear charge distribution is taken to have an arbitrarily small but non-zero radius.Comment: Submitted to Physics Letters