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Spectral profiles of certain atomic lines

Rosemary Beatrice Higgins

Published Jan 1, 1974• 0 citations
OpenAlex

Abstract

Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.

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Multiple-time-scale perturbation theory: Radiative decay of coupled atomic states

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The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.

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Allowed and Forbidden Transitions

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So far we have treated the energy levels of an atomic system as stable states with infinite lifetimes. The presence of an electromagnetic field changes this picture. Absorption of photons will excite atoms and ions to higher levels, while stimulated emission will make excited ones decay. Einstein (1917) showed, using purely statistical arguments, that an excited ion will decay through spontaneous emission in the absence of an electromagnetic field. This process can also be derived by using a rigorous quantum electrodynamic treatment, where the field is quantized. In this chapter we start by discussing transitions between atomic levels in a general sense, postulating the operators responsible for the decay. After the first few sections we will concentrate on the most important few multipoles or terms in the expansion of the electromagnetic field. We will introduce a number of different properties describing the transition, and eventually arrive at the measurable transition rate and, directly linked to it, the lifetime of an excited state. We will start, however, with the most basic property, the component strength. Our aim is to arrive at computationally useful expressions as soon as possible. For a more thorough treatment of the theory of radiative transitions we refer to the book by Sobelman (1972). For a brief review of the many factors affecting spectra of atoms, see Martin and Wiese (1996) .

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Rydberg atoms

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Introduction Series of atomic spectral lines had been studied by emission from excited vapors even before Balmer described the regularity of the visible lines of hydrogen in 1884. However, direct excitation had been restricted to the few lowest-lying levels, particularly those accessible by electric dipole transitions from the ground state using classical light sources. The advent of tunable lasers in the 1970s changed such spectroscopic limitations enormonsly, allowing excitation to a very much larger number of states. Atoms in highly excited states, typically with principal quantum numbers n ≥ 10, are called Rydberg atoms. They are oversize, fragile states of significant intrinsic interest. They are very rare in nature because of their size and fragility, but they are found in interstellar space (H I regions). One of the best sources for comprehensive descriptions of Rydberg atoms and references to the literature describing their exploration is Ref. [120]. It describes their characteristic properties and behavior both classically and quantum mechanically, and especially their strongly enhanced interactions with external fields. The first thing to note is that Rydberg atoms are usually characterized by a single, highly excited electron while all the other electrons remain in their original states. Thus their properties are very nearly classical, and the atoms are much closer to the ideal hydrogen-like model of Chap. 7, along with appropriate quantum defects (see Chap. 10). The spectroscopy of such states is especially interesting because the active electron is much less perturbed by the other electrons in the atom than when it is in lower states. Moreover, there are some special circumstances in which the energy of a Rydberg state is degenerate with an atomic state having two electrons excited to lower levels, and such configurations yield further new information. Second, Rydberg atoms do not usually decay back to the ground state by emission of characteristic radiation, so their study is mediated by other techniques that reveal new and interesting properties. Such properties arise because Rydberg atoms have a huge dipole moment and very low binding energy resulting from the large average distance between the excited electron and the nucleus-plus-core of the rest of the atom with net charge e . Third, transitions between Rydberg states are quite different from those connecting to the ground state for at least three reasons.

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The Structure of Emission Lines

Frank C. Hoyt

A method for obtaining the intensity distribution in a spectral line is derived from Dirac's quantum theory of the radiation field. The calculations are carried out for the case of emission from a first excited state (resonance line) and show that the intensity of frequency $\ensuremath{\nu}$ is, in this case, proportional to $\frac{1}{[4{\ensuremath{\pi}}^{2}{({\ensuremath{\nu}}_{0}\ensuremath{-}\ensuremath{\nu})}^{2}+{(A\ifmmode\cdot\else\textperiodcentered\fi{}2)}^{2}]}$ were $A$ is the rate at which atoms are leaving the upper state and ${\ensuremath{\nu}}_{0}$ is the frequency of the transition. The half-width is therefore $\frac{1}{2\ensuremath{\pi}}$ times the Einstein probability coefficient for spontaneous transition to the lower state. The more general case where the lower state has a finite life time is not treated.A general method for the treatment of radiation problems, which is essentially Weyl's modification of Dirac's method, is briefly described.

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The Decay of Resonance Radiation by Spontaneous Emission

Emil J. Hellund

A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.

Physical Review•Mar 1, 1953•45 citations

Frequency distribution of radiation emitted from excited atomic systems

Dwight D. Hearn

Abstract General expressions for the time‐dependent probability amplitudes of the quantum states of two arbitrary, interacting atoms are calculated when one atom is initially in an excited p state and the other atom is in an s ground state. The lifetimes of the excited states and the line shape of the emitted radiation are obtained as functions of both the atomic separation and the energy difference between the excited states of the two atoms. The emission line shape is shown to be doubly peaked and to agree with the line shape of the radiation scattered by a system of two interacting atoms. The expressions for the lifetimes of the excited states are found to be identical to those obtained for the radiation scattering situation.

International Journal of Quantum Chemistry•Sep 1, 1971•2 citations
Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.
The spontaneous radiative decay to the ground state of an atomic system initially in the higher of two excited states, coupled in a radiationless fashion by an external field, is investigated by the method of multiple-time-scale perturbation theory. The coupled differential equations of motion for the probability amplitudes are solved to second order, while the usual secular behavior of conventional time-dependent perturbation theory is eliminated in terms of two natural time scales: a fast time scale corresponding to the reciprocal of the level transition frequency and a slower time scale associated with the inverse of the radiation linewidth. Expressions are obtained to this order for the energy shifts and decay characteristics of the excited states and the energy distribution of the final-state photons, quantities previously determined by Fourier analysis and contour integration and by the phenomenological approach of Weisskopf and Wigner. A detailed comparison of these various methods of solution includes conditions under which the present results agree to second order with the previous calculations.
So far we have treated the energy levels of an atomic system as stable states with infinite lifetimes. The presence of an electromagnetic field changes this picture. Absorption of photons will excite atoms and ions to higher levels, while stimulated emission will make excited ones decay. Einstein (1917) showed, using purely statistical arguments, that an excited ion will decay through spontaneous emission in the absence of an electromagnetic field. This process can also be derived by using a rigorous quantum electrodynamic treatment, where the field is quantized. In this chapter we start by discussing transitions between atomic levels in a general sense, postulating the operators responsible for the decay. After the first few sections we will concentrate on the most important few multipoles or terms in the expansion of the electromagnetic field. We will introduce a number of different properties describing the transition, and eventually arrive at the measurable transition rate and, directly linked to it, the lifetime of an excited state. We will start, however, with the most basic property, the component strength. Our aim is to arrive at computationally useful expressions as soon as possible. For a more thorough treatment of the theory of radiative transitions we refer to the book by Sobelman (1972). For a brief review of the many factors affecting spectra of atoms, see Martin and Wiese (1996) .
Introduction Series of atomic spectral lines had been studied by emission from excited vapors even before Balmer described the regularity of the visible lines of hydrogen in 1884. However, direct excitation had been restricted to the few lowest-lying levels, particularly those accessible by electric dipole transitions from the ground state using classical light sources. The advent of tunable lasers in the 1970s changed such spectroscopic limitations enormonsly, allowing excitation to a very much larger number of states. Atoms in highly excited states, typically with principal quantum numbers n ≥ 10, are called Rydberg atoms. They are oversize, fragile states of significant intrinsic interest. They are very rare in nature because of their size and fragility, but they are found in interstellar space (H I regions). One of the best sources for comprehensive descriptions of Rydberg atoms and references to the literature describing their exploration is Ref. [120]. It describes their characteristic properties and behavior both classically and quantum mechanically, and especially their strongly enhanced interactions with external fields. The first thing to note is that Rydberg atoms are usually characterized by a single, highly excited electron while all the other electrons remain in their original states. Thus their properties are very nearly classical, and the atoms are much closer to the ideal hydrogen-like model of Chap. 7, along with appropriate quantum defects (see Chap. 10). The spectroscopy of such states is especially interesting because the active electron is much less perturbed by the other electrons in the atom than when it is in lower states. Moreover, there are some special circumstances in which the energy of a Rydberg state is degenerate with an atomic state having two electrons excited to lower levels, and such configurations yield further new information. Second, Rydberg atoms do not usually decay back to the ground state by emission of characteristic radiation, so their study is mediated by other techniques that reveal new and interesting properties. Such properties arise because Rydberg atoms have a huge dipole moment and very low binding energy resulting from the large average distance between the excited electron and the nucleus-plus-core of the rest of the atom with net charge e . Third, transitions between Rydberg states are quite different from those connecting to the ground state for at least three reasons.
A method for obtaining the intensity distribution in a spectral line is derived from Dirac's quantum theory of the radiation field. The calculations are carried out for the case of emission from a first excited state (resonance line) and show that the intensity of frequency $\ensuremath{\nu}$ is, in this case, proportional to $\frac{1}{[4{\ensuremath{\pi}}^{2}{({\ensuremath{\nu}}_{0}\ensuremath{-}\ensuremath{\nu})}^{2}+{(A\ifmmode\cdot\else\textperiodcentered\fi{}2)}^{2}]}$ were AAA is the rate at which atoms are leaving the upper state and ${\ensuremath{\nu}}_{0}$ is the frequency of the transition. The half-width is therefore $\frac{1}{2\ensuremath{\pi}}$ times the Einstein probability coefficient for spontaneous transition to the lower state. The more general case where the lower state has a finite life time is not treated.A general method for the treatment of radiation problems, which is essentially Weyl's modification of Dirac's method, is briefly described.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
Abstract General expressions for the time‐dependent probability amplitudes of the quantum states of two arbitrary, interacting atoms are calculated when one atom is initially in an excited p state and the other atom is in an s ground state. The lifetimes of the excited states and the line shape of the emitted radiation are obtained as functions of both the atomic separation and the energy difference between the excited states of the two atoms. The emission line shape is shown to be doubly peaked and to agree with the line shape of the radiation scattered by a system of two interacting atoms. The expressions for the lifetimes of the excited states are found to be identical to those obtained for the radiation scattering situation.