Physics 221 Lecture 31 Line Radiation from Atoms and Molecules March 31, 1999

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1 Physics 221 Lecture 31 Line Radiation from Atoms and Molecules March 31, 1999 Reading Meyer-Arendt, Ch. 20; Möller, Ch. 15; Yariv, Ch.. Demonstrations Analyzing lineshapes from emission and absorption spectra 1. The end of the nineteenth century: line radiation. Besides blackbody radiation, the other ingredient of quantum optics is line radiation from atoms, ions and molecules. Like the case of blackbody radiation, the solution to the puzzle of line spectra lay in finding the right way to quantize the properties of atomic oscillators. a. Using the classical theory of the electron as an harmonic oscillator, Lorentz and others had formulated a theory of classical electrodynamics, which seemed to make perfect sense but which was seriously inconsistent with a number of experimental observations of radiation from atomic systems. In particular, it could not predict atomic line emission at all. b. The solution to this problem turned out to be the idea of quanta (Greek: a little bit ) introduced by Max Planck, coupled with the Bohr model of an atom with quantized electron orbits. The idea of quantization is central to our understanding of all phenomena at the atomic level. c. Early in the twentieth century, the key to the essential features of atomic radiation had been elaborated by Bohr, Schrödinger and others. 2. Early models of the atom and the puzzles of spectroscopy. Once the difficulties with the radiation from blackbodies appeared to be solved, there was another puzzle for the emerging quantum theory. This was a phenomenon again provided by the spectroscopists, who noticed that there were characteristic sharp emission (or absorption) lines which were characteristic of each element, but which seemed to have no known reason for being except for the mysterious numerical regularity in their sequence. a. Consider the spectra below and note the appearance of differing kinds of spectral lines from hydrogen, mercury and sodium lamps. Note also the dark absorption lines in the solar spectrum; here the line absorption is superimposed on the blackbody spectrum from the sun. 1

2 b. Even though Planck was not certain how seriously to take his quanta, others (notably Niels Bohr) quickly picked it up and applied it to the vexing problem of regularities in line emission from atoms. The initial idea was that electron waves had to be organized in space rather like standing acoustic waves on a string or a drumhead. Later this idea would be substantially refined by Schrödinger, Max Born and Werner Heisenberg into the concept of quantum states which satisfied the appropriate wave equation with symmetries which reflected the spatial distribution of electrons. 3. Spontaneous transitions and line broadening. To understand line radiation from atoms, we must first consider what happens to excited atoms which emit light spontaneously. We shall find that this is characterized not only by the energy difference between atomic or molecular energy states, but also by the lifetime of those states - which gives rise to the spectral lineshape. a. Imagine an atomic or molecular system with a set of energy levels {En} which are eigenstates {φn}. We want to look at the population of those levels and at the ways in which emission and absorption of photons affects the popula- 2

3 tion in the various energy levels. In thermal equilibrium, the population of the nth and the (n+1)st levels satisfy the Maxwell-Boltzmann statistical distribution condition: N n+1 = N n e -(E n+1 - E n )/kt This simply says that in thermal equilibrium, the relative populations of more energetic levels decreases exponentially with the energy difference from the ground state. b. Now we consider the behavior of atoms or molecules which make spontaneous transitions from higher to lower energy states. We shall find that this can be described with some simple intuitive concepts from quantum mechanics. i. Let us consider two levels among the entire manifold {En}, labelled E1 and E2. We assume that at some time t = 0, a large number of atoms N2 are in state which has an energy above that of atoms which are in state E1. The average number of these atoms per unit time which will make a spontaneous transition from the eigenstate φn+1 to the eigenstate φn is just given by, according to Einstein, - dn 2 N = A21 N dt 2 = 2 (τspont ) 21 The coefficient A12 has to vanish, because quantum states cannot make spontaneous transitions from states of lower to states of higher energy. It is possible to calculate the spontaneous emission rate from a knowledge of the eigenstates φn+1 and φn; however, for the present we simply accept this situation "on faith" and try to understand how the spontaneous emission lifetime can be characterized. ii. In characterizing the interaction of atoms or molecules with a surrounding radiation field, it is convenient to assume that τspont is a parameter of the system. From the quantum-mechanical uncertainty relationship between the energy and lifetime of a particular level, E t = (hν) t h ν t 1 Because there is an uncertainty in the lifetime of any individual atomic or molecular level, there will be a corresponding uncertainty in the frequency of the radiation emitted by a spontaneous transition. This uncertainty we shall characterize by defining a lineshape function g(ν), with the property that g(ν)dν is the probability that a given photon emitted 3

4 in a spontaneous transition from f to f will have a frequency between ν and ν+dν. Because the lineshape function has the character of a probability, it must also be true that: + g(ν) dν = 1 - iii. The detailed nature of the function g(ν) is a matter for quantum mechanics, but we can get some guidance by referring back to our electron oscillator model of the atom. Because atoms cannot stay in an excited state forever, we know there has to be some damping mechanism which dissipates energy. In elementary physics, you have already learned about damped oscillators under the rubric of RLC circuits; there it was shown that the amplitude of the oscillations in the electrical circuit took the form: g(ν) = 1 Γ 2π (ν - ν o ) 2 + (Γ/2) 2 Such a lineshape function is called a Lorentzian, and it is characteristic of the damped oscillator. [It turns out that there are still vigorous scientific investigations going on into the detailed form of the lineshape, and in many cases it appears not to be quite Lorentzian.] c. The quantity Γ/2 is the width at half maximum of the lineshape function, and as such is a direct measure of the uncertainty in the spontaneous lifetime. It also represents the damping factor for the oscillator, and is characteristic of the atom and of its environment. 4. Line broadening mechanisms. Now we explore the question of the mechanisms responsible for the observed linewidths G, which are evidently related to the lifetime uncertainty by Γ = 1/πτ. There are two possibilities for lifetime-changing mechanisms: those which are the same for all atoms or molecules in the ensemble, and those which depend on the specific state or environment of the atoms or molecules; these two categories are referred to as homogeneous and inhomogeneous lifetime broadening, respectively. a. Among the causes of homogeneous broadening are collisions, the natural lifetime, and the presence or absence of metallic boundaries. [Here one can mention the recent experiments showing that spontaneous emission can be inhibited by the presence of a boundary.] The total linewidth can thus be expressed as 4

5 Γ = 1 π τ τ1 + 1 τcoll 2 b. The most common causes of inhomogeneous broadening are crystal-field splitting in solids, reflecting spatial anisotropies in the random strains in the lattice of a solid, and Doppler broadening, which arises from the varying velocities of atoms in a gaseous ensemble. A similar equation can be written for this case. c. Example: Doppler broadening. If one calculates what the lineshape should be for the case of a gas at temperature T, it turns out to be (Yariv, Optical Electronics, Eq ) ν D = 2 ν o 2kT -7 ln 2 = νo Mc 2 T M o where T is in K, M is in atomic mass units (1.661 x kg), and νο is in Hz. For the Ne-Ne laser, for example, the laser transition is from the Ne atoms in a gas discharge; the Doppler width is of order 1.32 x 10 9 Hz, compared to the frequency of 4.74 x Hz. 5

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