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Physical Review A

Time-dependent perturbation of Mössbauer spectra

M. I. Salkola, Stig Stenholm

Published Apr 1, 1990• 23 citations
Publisher

Abstract

In this paper we derive a complete semiclassical description of M\"ossbauer spectroscopy with arbitrary external or internal perturbations of the active levels. The \ensuremath{\gamma} rays are taken to be weak enough to allow us to evaluate the linear M\"ossbauer response to lowest order in the intensity, in which case the \ensuremath{\gamma}-ray field can be treated classically. It induces transitions between level manifolds that can have arbitrary interactions inside themselves. The nuclear system is described by a density matrix. In principle, time-dependent perturbations can be handled by the formalism, but the equations are easier to solve for systems that display steady-state behavior. We consider, in particular, the case of magnetic radio-frequency modulation of the M\"ossbauer levels. Utilizing a matrix-continued-fraction technique, we solve for the absorption spectrum, and we choose numerical illustrations with parameters such that they describe the nucleus $^{57}\mathrm{Fe}$. In this paper we treat mainly the case of magnetic modulation, but other experimental arrangements are discussed too. The calculations are able to treat both saturation and interference effects, which become important when the radio-frequency modulation is very strong. The new feature that emerges is the occurrence of resonances arising from the interference of different physical processes. The calculational method and its results are discussed and compared with other works.

DOI
10.1103/physreva.41.3838

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Theoretical description of quantum beats of recoil-free γ radiation

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A classical optical theory is given for the phenomenon of $\ensuremath{\gamma}$-ray quantum beats observed by Perlow in a study of the M\"ossbauer effect of a frequency-modulated source. The intensity $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of the radiation from such a source transmitted through a resonant absorber is obtained as a function of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ and $t$, where ${\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ is the frequency of an absorber resonance, ${\ensuremath{\omega}}_{0}$ is the central frequency of the frequency-modulated source, and $t$ is the laboratory time. An average is taken over the unobserved initial formation time of the excited nuclear state in the source. When viewed at fixed ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, the calculated intensity displays beats in the time spectrum, and when viewed at fixed $t$, the intensity shows dispersion at appropriate values of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, responsible for the observed enhancement of intensity above background. The harmonic content of the quantum beats is calculated explicitly in the thin-absorber limit, and the observed linear variation about ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}=0$ of the ratio of Fourier components $\frac{{D}_{1}}{{D}_{2}}$ is explained. The use of $\frac{{D}_{1}}{{D}_{2}}$ to measure small frequency shifts is analyzed by a statistical comparison with the method of switching between steepest points of the line, as used in gravitational-red-shift measurements. The variances are comparable. The effect on $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of line broadening due to sample inhomogeneities is calculated for a Lorentz distribution of center frequencies in source and absorber, and a prescription is given for modifying the various terms in $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ accordingly. Finally, the effect of a distribution of phase and amplitude of the motion of the vibrating source is discussed.

Physical Review A•Oct 1, 1979•64 citations
In this paper we derive a complete semiclassical description of M"ossbauer spectroscopy with arbitrary external or internal perturbations of the active levels. The \ensuremath{\gamma} rays are taken to be weak enough to allow us to evaluate the linear M"ossbauer response to lowest order in the intensity, in which case the \ensuremath{\gamma}-ray field can be treated classically. It induces transitions between level manifolds that can have arbitrary interactions inside themselves. The nuclear system is described by a density matrix. In principle, time-dependent perturbations can be handled by the formalism, but the equations are easier to solve for systems that display steady-state behavior. We consider, in particular, the case of magnetic radio-frequency modulation of the M"ossbauer levels. Utilizing a matrix-continued-fraction technique, we solve for the absorption spectrum, and we choose numerical illustrations with parameters such that they describe the nucleus 57Fe^{57}\mathrm{Fe}57Fe. In this paper we treat mainly the case of magnetic modulation, but other experimental arrangements are discussed too. The calculations are able to treat both saturation and interference effects, which become important when the radio-frequency modulation is very strong. The new feature that emerges is the occurrence of resonances arising from the interference of different physical processes. The calculational method and its results are discussed and compared with other works.
We study theoretically resonance absorption of \ensuremath{\gamma} rays by nuclei exposed to an external radio-frequency (rf) magnetic field. The external field is taken to couple directly to the nuclear magnetic moments. Perturbation theory is developed for M\"ossbauer spectroscopy in terms of Floquet eigenstates that incorporate the rf field exactly. The present treatment allows for calculation of the conventional time-averaged absorption spectrum, as well as analysis of the time dependence of absorption, resolved on the scale of a period of the rf field. An efficient numerical implementation of the theory is described and comparisons with the equally general approach of Salkola and Stenholm [M. Salkola and S. Stenholm, Phys. Rev. A 41, 3838 (1990)] are made. We demonstrate that even at off-resonance frequencies a strong modulation leads to shifts of the original static hyperfine lines. Examples of line splittings of NMR's associated with Rabi flopping are also shown. \textcopyright{} 1996 The American Physical Society.
As part of our general programme to develop the field of quantum nucleonics, we have studied Mössbauer spectroscopy when there is Zeeman splitting of the nuclear levels and a further interaction due to an applied rf-radiation field. We have applied the `dressed' state concept, developed in quantum electronics, to this situation. In particular, we have studied the case when the rf frequency is in the neighbourhood of the ground state (spin =1/2) splitting. The dressed-state approach in this case treats the coupling of nuclear Zeeman levels, due to an rf field, by considering the total system made up of nucleus, static magnetic field and rf field as one global quantum system. This allows the time evolution of the system to be handled straightforwardly. The energy levels and corresponding eigenstates of the system are calculated as a function of the rf frequency and the magnitude of the rf magnetic flux density. Mössbauer spectra are calculated for the case in which the source is subjected to both the static and radiation fields while the absorber nuclear levels are unsplit. Spectra are given as a function of the rf frequency, the magnitude of rf-field magnetic flux density, as well as the photon direction.
In recent years there became actual studies of quantum interference on gamma spectroscopic transitions induced by coherent perturbations.A special attention is paid to gamma transitions in compound systems. In this work it is shown that quantum interference effects (QIE) occur in gamma quantum spontaneous emission spectrum of an electron-nuclear paramagnetic system when electron spin resonance is induced. Occurrence of QIE is connected to preparation of the quantum system initial state. The role of electron relaxation is defined: increase of relaxation rate leads to extinguishing of QIE. The cases of stationary excitation of Mossbauer levels by wide and narrow spectra are considered.
A classical optical theory is given for the phenomenon of $\ensuremath{\gamma}$-ray quantum beats observed by Perlow in a study of the M\"ossbauer effect of a frequency-modulated source. The intensity $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of the radiation from such a source transmitted through a resonant absorber is obtained as a function of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ and ttt, where ${\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ is the frequency of an absorber resonance, ${\ensuremath{\omega}}_{0}$ is the central frequency of the frequency-modulated source, and ttt is the laboratory time. An average is taken over the unobserved initial formation time of the excited nuclear state in the source. When viewed at fixed ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, the calculated intensity displays beats in the time spectrum, and when viewed at fixed ttt, the intensity shows dispersion at appropriate values of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, responsible for the observed enhancement of intensity above background. The harmonic content of the quantum beats is calculated explicitly in the thin-absorber limit, and the observed linear variation about ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}=0$ of the ratio of Fourier components D1D2\frac{{D}_{1}}{{D}_{2}}D2​D1​​ is explained. The use of D1D2\frac{{D}_{1}}{{D}_{2}}D2​D1​​ to measure small frequency shifts is analyzed by a statistical comparison with the method of switching between steepest points of the line, as used in gravitational-red-shift measurements. The variances are comparable. The effect on $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of line broadening due to sample inhomogeneities is calculated for a Lorentz distribution of center frequencies in source and absorber, and a prescription is given for modifying the various terms in $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ accordingly. Finally, the effect of a distribution of phase and amplitude of the motion of the vibrating source is discussed.