Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.
The second-order hyperfine energy in the 1s and 2s states of the hydrogen atom is calculated analytically using the Dirac equation for an electron in a Coulomb potential perturbed by a point nuclear magnetic moment. The (divergent) result is compared with the calculation using the non-relativistic form of the hamiltonian. It is emphasized that in this comparison all the contributions to the non-relativistic second-order energy must be included. It is shown that, at least for the Fermi contact interaction, the results are the same to order mc 2α6, so that for this case the reduction of the Dirac equation to non-relativistic form is not suspect, contrary to previous assertions.
The study of highly charged electronic and muonic hydrogen-like ions, provides an intriguing way to probe the internal structure of their atomic nuclei. In this work, we use nuclear structure calculations to accurately calculate the hyperfine splitting of electronic and muonic hydrogen-like ions, focusing in particular on the incorporation of finite-volume corrections, such as Bohr-Weisskopf and Breit-Rosenthal, due to the penetration of the electron and muon wavefunction into the nuclear electric charge and magnetic dipole densities. These corrections are essential for refining our understanding of the nuclear magnetic dipole and electric quadrupole moments. Our simulations use a Skyrme-Hartree-Fock-BCS model known for its effectiveness in modeling well-deformed nuclei such as
159Tb64+ and
165Ho66+, with particular emphasis on
161,163Dy65+ isotopes. It can also be generalised to multi-electron ions by studying the hyperfine anomaly between two isotopes.
Interactions between light and matter is one way to study both particles whose small spatial sizes render them irresolvable to our eyes and events where the characteristic time scale is too short for conventional recording and measurement techniques. The advent of free electron lasers allow laser pulses of high intensity and small temporal duration such that ultra-fast dynamics in matter systems and resulting non-linear effects can be resolved experimentally. This thesis discusses some theories to mathematically model such light-matter interactions and presents results based both on analytical expressions—where such expressions are available— from the Schrödinger and the Dirac equations as well as numerical calculations. Some models presented are a semi-classical model where light is treated classically and the atomic system quantum-mechanically and a statistical model which describes the light as quantized to feature effects like spontaneous emission and thermal excitation. The main theme of the thesis is relativistic effects and scalings of interaction effects with nuclear charge Z. The results discussed in the work shows that the equivalent characteristics of the dynamics of highly ionized hydrogen-like atoms predicted by the Schrödinger equation fail when relativistic effects are taken into account. This is shown to be a result of the quantities such as energy and mean radial distance from the nucleus not being exactly proportional to powers of Z in Dirac’s description of quantum mechanics. Further, deviations occur due to the necessary detuning of the driving laser field when spin-orbit interactions (a relativistic effect) split states of non-zero orbital momentum. Also, it is shown that as the characteristic time scales of interactions and inherent effects such as spontaneous emission scale differently with Z, the characteristic dynamics are not invariant to atomic number and it is not possible to rescale quantities to make them so. This work provides a guide for future Free-electron laser experiments on highly charged ions at X-ray wave- lengths.
In an isolated two-body composite system, the discussion of how the change of one body affects the state of the other will help to know the relation of a single particle's mixed and pure states. Given 5 one-dimensional hydrogen-like atoms models, each Coulomb interaction potential keeps invariant, while the masses of the nuclei are different. These two-body composite systems stay in their respective entangled state, each electron stays in a mixed state. If we suppose a one-dimensional hydrogen atom model having infinite nuclear mass, the electron stays in a pure state. We select position representation. The wave function of the ground state of the atom has the form of the square root of a δ function. To avoid calculation trouble of δ function, we select the first excited state and the superposed state of the first and the second excited states. We compare the two pure states, the first excited state and the superposed state, with those corresponding mixed states by fidelity and l1 norm coherence, and calculate the purities of the mixed states. The summations become integrations due to the position variable having a continuous eigenvalue spectrum. We investigate the changes in these quantities with the increase of the nuclear mass. The results show that the purities of the mixed states and the fidelities increase with the nuclear mass increasing. However, the coherences of the mixed states decrease with the nuclear mass increasing. This can be explained as that a mixed state with non-zero coherence may approach to a pure eigenstate, while the latter has zero coherence in the eigenspace. These mean that the greater a nuclear mass is, the closer the mixed state approaches to the corresponding pure state. Therefore, the two pure states are the approximations of the corresponding mixed states. The entangled state of the electron and the nucleus is related with the nuclear mass and the Coulomb interaction potential. So, each electron mixed state and its coherence are related with the nucleus and their Coulomb interaction potential. If the nuclear mass is great, the nucleus is approximately motionless or its state is approximately unchanged, and the Coulomb interaction potential approximates to the external Coulomb potential of the electron. The electron approximately stays in a pure state. The state and its coherence are related with the nucleus and the Coulomb interaction. From other point of view, the entangled state of the nucleus and the electron approximates to the product state of the unchanged nucleus state and the electron state. In this case, an electron mixed state approximates to its corresponding pure state, and then these states and their coherences are all related with the nucleus and the Coulomb interaction.
The eight-component relativistic wave equation for spin-
21 particles of Robson and Staudte (FV
21), a Feshbach--Villars linearization of the Feynman--Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the
n=1--3 levels of hydrogenic ions from H to U
91+ in the Schrödinger, Dirac and FV
21 theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV
21 and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin--field coupling the first-order roots become complex, and without the negative-norm states the
(Zα)2 coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the
2s1/2--
2p1/2 degeneracy needs its own spin--gradient term, and shifts placed by hand on the
κ-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular
(Zα)−2 second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the
(Zα)0 coefficients analytically and relates the
π2 terms of the
(Zα)2 coefficients to the singular ones. The Dirac weak/strong-field crossover for
n=2 of hydrogen lies near 2.8~kV cm
−1, and the perturbative window set by the fine structure grows as
Z5.
The hyperfine levels for the ground state of hydrogen are normally connected by magnetic dipole transitions. However, for astrophysical applications, it may be that two-photon electric dipole processes are also important over cosmological distances. This thesis sets up a general mathematical formalism for the calculation of two-photon processes within the hyperfine structure of an atom with spin
1/2, and applies it to the ground state of hydrogen. Rates are calculated for both spontaneous emission and absorption of radiation. Due to an extraordinary degree of cancellation in the matrix elements, the rates turn out to be too small to have a significant impact on astrophysical processes involving the absorption and emission of radiation. The results are nevertheless important in reducing the uncertainty due to possible contributions from these processes. Other two-photon processes involving Raman scattering are also considered and the relevant cross sections calculated. The calculations of the decay rate, absorption coefficient and Raman scattering cross section are extended to heavier hydrogenic ions as a function of the nuclear charge
Z, using an approximate interpolation formula for the hyperfine splitting in the ground state. Results are presented for the hydrogenlike ions from He
+ to Pb
81+. Modifications for the case of isotopes with a nuclear spin greater than
1/2 are briefly considered