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) .
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) .
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.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.
Radiative transitions are quantum-state transitions in matter quantum systems (such as atoms， ions， and molecules) caused by their interaction with the electromagnetic field， and accompanied by the emission or absorption of photons.Their theoretical description involves physical theories including quantum mechanics and quantum electrodynamics， as well as mathematical tools such as spherical tensors and group theory.Focusing on radiative transitions of doped ions in solids， this paper summarizes earlier related theories and aims to concisely introduce fundamental theories and mathematical expressions while maintaining rigor， and to clearly describe the underlying physical pictures and mechanisms.The topics covered include atomic quantum states， the quantized electromagnetic field， multipole radiative transitions and selection rules， stimulated absorption， stimulated emission and spontaneous emission， transition probability and excited-state lifetimes.The article also discusses f-f and 4f-5d transitions of rare-earth ions and dd transitions of transition-metal ions， involving Judd -Ofelt theory， crystal-field theory， electronvibration coupling， and the Dorenbos model， as well as statistical analysis， machine learning， and firstprinciples methods.This work may provide a theoretical reference for the study of rare-earth-and transition-metaldoped luminescent materials.Keywords： radiative transition； theory； physical picture； f-f transition； d-d transition； 4f-5d transition 文章编号： 1000-7032(XXXX)XX
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
Many processes, including absorption and stimulated emission occurring in lasers, can be handled using a semi-classical model for the atom–radiation interaction, in which the matter is given a quantum description, but the radiation is represented as a classical electromagnetic field (see Chapter 2). There are other phenomena that cannot be adequately described without quantizing the radiation. For example, it has been known since the 1930s that spontaneous emission can only be treated correctly using a fully quantum framework for the interaction, in which both the matter and the radiation are quantized, as we shall see in Chapter 6. However, it was not until the 1970s that situations were found in which a free electromagnetic field , far from sources, exhibited properties and behaviour that could not be described by a classical field, but which could be perfectly well interpreted in terms of a quantized field. This chapter is devoted to the quantization of the free electromagnetic field, far from the charges and currents sourcing it. This free electromagnetic field will be called radiation , and in Chapter 6 we shall specify exactly what is meant by radiation when sources are present. The canonical quantization procedure used here starts from a description of the classical dynamics of the field in the framework of the Hamiltonian formalism, the basic features of which are discussed in Section 4.1.
In Section 5.4.2 where we treated time-dependent perturbation theory, we took the perturbation to be of the general form Hˆ(1)(r, t) = 2Hˆ(1)(r) cosωt (9.1) but for absorption and emission of photons, we want the perturbation to be an electromagnetic field of the form E = E0 cosωt eˆz (9.2) so that the perturbing potential is V (r, t) = − ∫ E dz = −qE0z cosωt (9.3) and because of the factor of 2 in Equation (9.1), the matrix element is Hˆ(1)kj = − 12E0〈k|℘z|j〉 (9.4) where ℘z ≡ qz is the z-component of the dipole moment. In the derivation of Equation (5.48) from Equation (5.47), we integrated over the energy E = ~ω, whereas if we instead integrate over ω, the result is Rj→k = 2pi ~2 |Hˆ(1)kj |2 , (9.5) where the δ-function in Equation (5.48) tells us to evaluate the expression at ω = ωkj . The energy density in an electric field is given by and we let u→ ρ(ω)dω so that Rj→k = pi ²0~2 |〈k|℘z|j〉|2ρ(ω) , (9.6) and There is nothing special about the z-direction in evaluating the matrix element, so it is common to use the magnitude instead of the single component, where by symmetry, |℘|2 = |℘x|2 + |℘y|2 + |℘z|2 = 3|℘z|2 so the final result is expressed as Rj→k = pi 3²0~2 |〈k|℘|j〉|2ρ(ω) . (9.7) If we imagine a closed system in equilibrium, where there are atoms in both the lower state (ψk) and in the upper state (ψj), we can be assured that there is a balance between the rates of emission and absorption. If we define the spontaneous emission rate to be A∗ so that if there are Nj atoms in the upper state, there will be NjA transitions per second from state j to state k, as illustrated in Figure 5.2c. There are also transitions from state j to state k by stimulated emission as illustrated in Figure 5.2a. This rate is proportional to the energy density in the electromagnetic field of the wave and is given by Bjkρ(ω), where Bjk is proportional to Rk from Equation (5.48) and ρ(ω) is the energy density in the electric field of the wave at frequency ω. The total number of transitions per second from state j to state k due to stimulated emission is therefore NjBjkρ(ω). If there is an equilibrium, there must be an equal number of transitions from state k to state j by absorbing photons as illustrated in Figure 5.2b. The total number of transitions per second from state k to state j is then given by NkBkjρ(ω) where Nk is the number of atoms in the lower state (ψk) and Bkj is the transition rate for the absorption process. The rate equation is therefore dNj dt = −NjA−NjBjkρ(ω) +NkBkjρ(ω) . (9.8) In equilibrium, there is a steady state, so dNj/dt = 0, so that the energy density in the wave must be given by ρ(ω) = NjA NkBkj −NjBjk = A (Nk/Nj)Bkj −Bjk . (9.9) Equilibrium also implies, however, that there is a temperature T and that the relative probability of finding an atom in one state compared to another is given by the Boltzmann factor of Equation (7.26), which depends on the energies of the two states such that Nk Nj = e(Ej−Ek)/KT , (9.10) but with Ej − Ek = ~ω, this becomes ρ(ω) = A e~ω/KTBkj −Bjk . (9.11) We wish to relate this energy density expression to the black-body distribution since both are equilibrium expressions, but Equation (9.11) is in terms of the energy density of the electromagnetic field, while our black-body expression of Equation (1.3), which is the same as Equation (7.31) without the zero-point term, is given in terms of the intensity, I(λ, T ). In order to relate these two expressions, we revisit the expression for the density of states given by Equation (7.4) where dxdy dz = V , dpxdpydpz = 4pip2 dp, and p = ~k. Then we have N dxdy dz dpxdpydpz = 4piV h3 ~3k2 dk = V 2pi2 k2 dk . (9.12) Then, since k = ω/c, and since there are two independent photon states for each energy (spin 1 and spin −1), the density of photons is Nω V = ω2 pi2c3 dω e~ω/KT − 1 (9.13) where we have included the Bose-Einstein factor since photons have integral spin, and the energy density is therefore ρ(ω)dω = ~ω Nω V = ~ω3 pi2c3 dω e~ω/KT − 1 . (9.14) Comparing this result with Equation (9.11), we first conclude that Bkj = Bjk, which means that the transition rates are the same for the upward and downward transitions. Solving for A, we find for the spontaneous emission rate, A = ~ω3 pi2c3 Bkj , (9.15) and since, from Equation (9.7) Bkj = pi|〈k|℘|j〉|2 3²0~2 , (9.16) the spontaneous rate is given by A = ω3|〈k|℘|j〉|2 3pi²0~c3 . (9.17) and The lifetime in an excited state is the reciprocal of the spontaneous rate, although if there is more than one final state, the lifetime is the reciprocal of the sum of the various rates.