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Physical Review

The Structure of Emission Lines

Frank C. Hoyt

Published Sep 1, 1930• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.36.860

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Values of rate of decay and mean life of atoms and molecules in upper quantum states calculated from data on the intensity of absorption lines.---By combining F\"uchtbauer's method of determining from the intensity of absorption lines the probability that a molecule will absorb a quantum of energy, with Einstein's views as to the mechanism of light absorption and emission, the following equation is derived for calculating the rate at which molecules jump from upper to lower quantum states: ${A}_{21}=\left(\frac{8\ensuremath{\pi}{\ensuremath{\nu}}^{2}}{{c}^{2}{N}_{1}}\right)\left(\frac{{p}_{1}}{{p}_{2}}\right)\ensuremath{\int}{0}^{\ensuremath{\infty}}\ensuremath{\alpha}d\ensuremath{\nu}$ where ${A}_{21}$ is the chance per unit time that a molecule will jump spontaneously from quantum state 2 to quantum state 1, $\ensuremath{\nu}$ is the frequency of the light emitted in such a jump, ${p}_{1}$ and ${p}_{2}$ are the a priori probabilities of quantum states 1 and 2, and $\ensuremath{\alpha}$ is the absorption coefficient of the substance measured under conditions such that ${N}_{1}$ is the number of molecules per unit volume in the lower quantum state 1. The integral $\ensuremath{\int}\ensuremath{\alpha}d\ensuremath{\nu}$ is to be taken over the total effective width of the absorption line corresponding to the passage of molecules from quantum state 1 to quantum state 2. The mean life $\ensuremath{\tau}$ of molecules which decay from state 2 to state 1 is the reciprocal of ${A}_{21}$. Values of ${A}_{21}$ and $\ensuremath{\tau}$ are calculated from existing data for the mercury line $\ensuremath{\lambda}2537$, for a number of lines belonging to the alkali doublets, for the iodine line $\ensuremath{\lambda}5461$, and for a very considerable number of lines belonging to the rotation-oscillation spectra of the hydrogen halides. The values obtained agree with the meager data made available by other experimental methods. From these results the following conclusions are drawn.The mean life of molecules and atoms in upper quantum states may vary for different states at least over the range 1 to ${10}^{\ensuremath{-}8}$ seconds. The rate of decay is not a simple function of the frequency of the emitted light. The rate corresponding to the emission of a line of high frequency may be greater or less than that for a line of lower frequency. The data now available for the alkali doublets $1s\ensuremath{-}m{p}_{1}$ and $1s\ensuremath{-}m{p}_{2}$ indicate a higher rate of decay the smaller the change in total quantum number for the line under consideration. The rate of decay from a given $m{p}_{1}$ state is $m$ times as great as from the corresponding $m{p}_{2}$ state (already stated in another form by F\"uchtbauer and Hofmann). In the case of the rotation-oscillation spectra of the hydrogen halides the rate of decay is greater for quantum states with one unit of oscillation and many of rotation, than for those with one unit of oscillation and only a few of rotation; and in the case of different molecules but the same quantum numbers the rate of decay is greater for the molecule with the greater frequency of oscillation.Finally the possibility and method of calculating absolute values of ${A}_{21}$ from the Bohr correspondence principle is indicated.

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The Decay of Resonance Radiation by Spontaneous Emission

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Cambridge University Press eBooks•Feb 20, 2018•0 citations
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 \ensuremathν\ensuremath{\nu}\ensuremathν is, in this case, proportional to 1[4\ensuremathπ2(\ensuremathν0\ensuremath−\ensuremathν)2+(A\ifmmode⋅\else\textperiodcentered\fi2)2]\frac{1}{[4{\ensuremath{\pi}}^{2}{({\ensuremath{\nu}}_{0}\ensuremath{-}\ensuremath{\nu})}^{2}+{(A\ifmmode\cdot\else\textperiodcentered\fi{}2)}^{2}]}[4\ensuremathπ2(\ensuremathν0​\ensuremath−\ensuremathν)2+(A\ifmmode⋅\else\textperiodcentered\fi2)2]1​ were AAA is the rate at which atoms are leaving the upper state and \ensuremathν0{\ensuremath{\nu}}_{0}\ensuremathν0​ is the frequency of the transition. The half-width is therefore 12\ensuremathπ\frac{1}{2\ensuremath{\pi}}2\ensuremathπ1​ 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.
Abstract “Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.
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.
Values of rate of decay and mean life of atoms and molecules in upper quantum states calculated from data on the intensity of absorption lines.---By combining F\"uchtbauer's method of determining from the intensity of absorption lines the probability that a molecule will absorb a quantum of energy, with Einstein's views as to the mechanism of light absorption and emission, the following equation is derived for calculating the rate at which molecules jump from upper to lower quantum states: ${A}_{21}=\left(\frac{8\ensuremath{\pi}{\ensuremath{\nu}}^{2}}{{c}^{2}{N}_{1}}\right)\left(\frac{{p}_{1}}{{p}_{2}}\right)\ensuremath{\int}{0}^{\ensuremath{\infty}}\ensuremath{\alpha}d\ensuremath{\nu}$ where A21{A}_{21}A21​ is the chance per unit time that a molecule will jump spontaneously from quantum state 2 to quantum state 1, $\ensuremath{\nu}$ is the frequency of the light emitted in such a jump, p1{p}_{1}p1​ and p2{p}_{2}p2​ are the a priori probabilities of quantum states 1 and 2, and $\ensuremath{\alpha}$ is the absorption coefficient of the substance measured under conditions such that N1{N}_{1}N1​ is the number of molecules per unit volume in the lower quantum state 1. The integral $\ensuremath{\int}\ensuremath{\alpha}d\ensuremath{\nu}$ is to be taken over the total effective width of the absorption line corresponding to the passage of molecules from quantum state 1 to quantum state 2. The mean life $\ensuremath{\tau}$ of molecules which decay from state 2 to state 1 is the reciprocal of A21{A}_{21}A21​. Values of A21{A}_{21}A21​ and $\ensuremath{\tau}$ are calculated from existing data for the mercury line $\ensuremath{\lambda}2537$, for a number of lines belonging to the alkali doublets, for the iodine line $\ensuremath{\lambda}5461$, and for a very considerable number of lines belonging to the rotation-oscillation spectra of the hydrogen halides. The values obtained agree with the meager data made available by other experimental methods. From these results the following conclusions are drawn.The mean life of molecules and atoms in upper quantum states may vary for different states at least over the range 1 to ${10}^{\ensuremath{-}8}$ seconds. The rate of decay is not a simple function of the frequency of the emitted light. The rate corresponding to the emission of a line of high frequency may be greater or less than that for a line of lower frequency. The data now available for the alkali doublets $1s\ensuremath{-}m{p}_{1}$ and $1s\ensuremath{-}m{p}_{2}$ indicate a higher rate of decay the smaller the change in total quantum number for the line under consideration. The rate of decay from a given mp1m{p}_{1}mp1​ state is mmm times as great as from the corresponding mp2m{p}_{2}mp2​ state (already stated in another form by F\"uchtbauer and Hofmann). In the case of the rotation-oscillation spectra of the hydrogen halides the rate of decay is greater for quantum states with one unit of oscillation and many of rotation, than for those with one unit of oscillation and only a few of rotation; and in the case of different molecules but the same quantum numbers the rate of decay is greater for the molecule with the greater frequency of oscillation.Finally the possibility and method of calculating absolute values of A21{A}_{21}A21​ from the Bohr correspondence principle is indicated.
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.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.