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Register a new userModification of Coulomb law and energy levels of the hydrogen atom in a superstrong magnetic field
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Bruno Machet
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We obtain the following analytical formula which describes the dependence of the electric potential of a point-like charge on the distance away from it in the direction of an external magnetic field B: Phi(z) = e/|z| [ 1- exp(-sqrt{6m_e^2}|z|) + exp(-sqrt{(2/pi) e^3 B + 6m_e^2} |z|) ]. The deviation from Coulombs law becomes essential for B > 3pi B_{cr}/alpha = 3 pi m_e^2/e^3 approx 6 10^{16} G. In such superstrong fields, electrons are ultra-relativistic except those which occupy the lowest Landau level (LLL) and which have the energy epsilon_0^2 = m_e^2 + p_z^2. The energy spectrum on which LLL splits in the presence of the atomic nucleus is found analytically. For B > 3 pi B_{cr}/alpha, it substantially differs from the one obtained without accounting for the modification of the atomic potential.
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The expanded variant of the lectures delivered at the 39th ITEP Winter School in 2011
The differential and partially integrated cross sections are considered for bremsstrahlung from high-energy electrons in atomic field with the exact account of this field. The consideration exploits the quasiclassical electron Greens function and wave functions in an external electric field. It is shown that the Coulomb corrections to the differential cross section are very susceptible to screening. Nevertheless, the Coulomb corrections to the cross section summed up over the final-electron states are independent of screening in the leading approximation over a small parameter $1/mr_{scr}$ ($r_{scr}$ is a screening radius, $m$ is the electron mass, $hbar=c=1$). Bremsstrahlung from an electron beam of the finite size on heavy nucleus is considered as well. Again, the Coulomb corrections to the differential probability are very susceptible to the beam shape, while those to the probability integrated over momentum transfer are independent of it, apart from the trivial factor, which is the electron-beam density at zero impact parameter. For the Coulomb corrections to the bremsstrahlung spectrum, the next-to-leading terms with respect to the parameters $m/epsilon$ ($epsilon$ is the electron energy) and $1/mr_{scr}$ are obtained.
We study electric potential of a charge placed in a strong magnetic field B>>4.4x10^{13}G, as modified by the vacuum polarization. In such field the electron Larmour radius is much less than its Compton length. At the Larmour distances a scaling law occurs, with the potential determined by a magnetic-field-independent function. The scaling regime implies short-range interaction, expressed by Yukawa law. The electromagnetic interaction regains its long-range character at distances larger than the Compton length, the potential decreasing across the magnetic field faster than along. Correction to the nonrelativistic ground-state energy of a hydrogenlike atom is found. In the infinite-magnetic-field limit the modified potential becomes the Dirac delta-function plus a regular background. With this potential the ground-state energy is finite - the best pronounced effect of the vacuum polarization.
A simple locally accurate uniform approximation for the nodeless wavefunction is constructed for a {it neutral} system of two Coulomb charges of different masses $(-q,m_1)$ and $(q,m_2)$ at rest in a constant uniform magnetic field for the states of positive and negative parity, ${(1s_0)}$ and ${(2p_0)}$, respectively. It is shown that by keeping the mass and charge of one of the bodies fixed, all systems with different second body masses are related. This allows one to consider the second body as infinitely-massive and to take such a system as basic. Three physical systems are considered in details: the Hydrogen atom with (in)-finitely massive proton (deuteron, triton) and the positronium atom $(-e,e)$. We derive the Riccati-Bloch and Generalized-Bloch equations, which describe the domains of small and large distances, respectively. Based on the interpolation of the small and large distance behavior of the logarithm of the wavefunction, a compact 10-parametric function is proposed. Taken as a variational trial function it provides accuracy of not less than 6 significant digits (s.d.) ($lesssim 10^{-6}$ in relative deviation) for the total energy in the whole domain of considered magnetic fields $[0,,,10^4]$ a.u. and not less than 3 s.d. for the quadrupole moment $Q_{zz}$. In order to get reference points the Lagrange Mesh Method with 16K mesh points was used to get from 10 to 6 s.d. in energy from small to large magnetic fields. Based on the Riccati-Bloch equation the first 100 perturbative coefficients for the energy, in the form of rational numbers, are calculated and, using the Pade-Borel re-summation procedure, the energy is found with not less than 10 s.d. at magnetic fields $leq 1$,a.u.
It is shown that hydrogen atom is a unique object in physics having negative energy of electric field, which is present in the atom. This refers also to some hydrogen-type atoms: hydrogen anti-atom, atom composed of proton and antiproton, and positronium.
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