Atomic Energy Levels and Spontaneous Emission

This is a continuation from the previous tutorial - working definitions of cutoff wavelength.

Our objective in this tutorial is to give a very brief introduction to the concepts of atomic energy levels and of spontaneous emission between those levels. We attempt to demonstrate heuristically that atoms (or ions, or molecules) have quantum-mechanical energy levels; that atoms can be pumped or excited up into higher energy levels by various methods; and that these atoms then make spontaneous downward transitions to lower levels, emitting radiation at characteristic
transition frequencies in the process.

The Helium Spectrum

Figure 1.5 illustrates a simple experiment in which a small helium discharge lamp (or lacking that, a neon sign) is viewed through an inexpensive transmission diffraction grating of the type available at scientific hobby stores.

When viewed directly the discharge helium lamp appears to emit pinkish-white light. When viewed through the diffraction grating, however, each wavelength in the light is diffracted at a different angle. Upon looking through the grating, you therefore observe multiple images of the lamp, each displaced to a different discrete angle, and each made up of a different discrete wavelength or color emitted by the helium discharge.

A strong yellow line at 5876 Å (or 588 nm) is particularly evident, but violet, green, blue, red, and deep red lines are also readily seen. These visible wavelengths are plotted in Figure 1.6, along with (as a matter of curiosity) the relative response of the human eye, and the wavelengths of four of the more common visible lasers.

These different wavelengths are, of course, only a few of the discrete components in the fluorescence spectrum of the helium atoms. In the helium discharge tube a large number of neutral helium atoms are present, along with a small number of free electrons and a matching number of ionized helium atoms to conduct electrical current.

The free electrons are accelerated along the tube by the applied electric field, and collide after some distance with the neutral helium atoms. The helium atoms are thereby excited into various higher quantum energy levels characteristic of the helium atoms.

A small fraction are also ionized by the electron collisions, thereby maintaining the electron and ion densities against recombination losses, which occur mostly at the tube walls.

After being excited into upper energy levels, the helium atoms soon give up their excess energy by dropping down to lower energy levels, emitting spontaneous electromagnetic radiation in the process. This spontaneous emission or fluorescence is the mechanism that produces the discrete spectral lines.

Quantum Energy Levels

Figure 1.7 shows the rather complex set of quantum energy levels possessed by even so simple an atom as the He atom. The solid arrows in this diagram designate some of the spontaneous-emission transitions that are responsible for the stronger lines in the visible spectrum of helium.

The dashed arrows indicate a few of the many additional transitions that produce spontaneous emission at longer or shorter wavelengths in the infrared or ultraviolet portions of the spectrum, lines which we can "see" only with the aid of suitable instruments.

Every atom in the periodic table, as well as every molecule or ion, has its own similar characteristic set of quantum energy levels, and its own characteristic spectrum of fluorescent emission lines, just as does the helium atom.

Understanding and explaining the exact values of these quantum energy levels for different atoms and molecules, through experiment or through complex quantum analyses, is the task of the spectroscopist.

The complex labels given to each energy level in Figure 1.7 are part of the working jargon of the spectroscopist or atomic physicist. In this text we will not be concerned with predicting the quantum energy levels of laser atoms, or even with understanding their complex labeling schemes, except in a few simple cases.

Rather, we will accept the positions and properties of these levels as part of the data given us by spectroscopists, and will concentrate on understanding the dynamics and the interactions through which laser action is obtained on these transitions.

Planck's Law

The relationship between the frequency \(\omega_{21}\) emitted on any of these transitions and the energies \(E_2\) and \(E_1\) of the upper and lower atomic levels is given by Planck's Law

\[\tag{1}\omega_{21}=\frac{E_2-E_1}{\hbar}\]

where \(\hbar\equiv{h}/2\pi\), and Planck's constant \(h=6.626\times10^{-34}\) Joule-second.

In this tutorial, as in real life, optical and infrared radiation will sometimes be characterized by its frequency \(\omega\), and sometimes by its wavelength \(\lambda_0\) expressed in units such as Angstroms (Å), nanometers (nm), or microns (μm).

Quantum transitions and the associated transition frequencies are also very often characterized by their transition energy or photon energy, measured in units of electron volts (eV), or their inverse wavelength \(1/\lambda_0\) measured in units of "wavenumbers" or \(\text{cm}^{-1}\).

Since we will be jumping back and forth between these units, it will be worthwhile to gain some familiarity with their magnitudes. Some useful rules of thumb to remember are that

\[\tag{2}1\text{ }\mu\text{m}(\text{"one micron"})\equiv1,000\text{nm}\equiv10000\text{ Å}\]

and that, in suitable energy units,

\[\tag{3}[\text{transition energy }E_2-E_1\text{ in eV}]\approx\frac{1.24}{[\text{wavelength }\lambda_0\text{ in microns}]}\]

Hence 10000 Å or 1 μm matches up with 10,000 \(\text{cm}^{-1}\) or ~ 1.24 eV. A visible wavelength of 500 nm or 5000 Å or 0.5 μm thus corresponds to a photon energy of 20,000 \(\text{cm}^{-1}\) or ~ 2.5 eV. Note that this also corresponds to a transition frequency of \(\omega_{21}/2\pi=6\times10^{14}\) Hz, expressed in the conventional units of cycles per second, or Hertz.

Energy Levels in Solids: Ruby or Pink Sapphire 

As another simple illustration of energy levels, try shining a small ultraviolet lamp (sometimes called a "mineral light") on any kind of fluorescent mineral, such as a piece of pink ruby or a sample of glass doped with a rare-earth ion, or on a fluorescent dye such as Rhodamine 6G.

These and many other materials will then glow or fluoresce brightly at certain discrete wavelengths under such ultraviolet excitation. A sample of ruby, for example, will fluoresce very efficiently at \(\lambda\approx694\) nm in the deep red, a sample of crystal or glass doped with, say, the rare-earth ion terbium, Tb³⁺, will fluoresce at \(\lambda\approx540\) nm (bright green), and a liquid sample of Rhodamine 6G dye will fluoresce bright orange.

Since ruby was the very first laser material, and is still a useful and instructive laser system, let us examine its fluorescence in more detail.

Figure 1.8 shows a more sophisticated version of such an experiment, in which a scanning monochromator plus an optical detector are used to examine the ruby fluorescent emission in more detail.

The lower trace shows the two very sharp (for a solid) and very closely spaced deep-red emission lines that will be observed from a good-quality ruby sample cooled to liquid-helium temperature. (At higher temperatures these lines will broaden and merge into what appears to be a single emission line.)

Figure 1.9 shows the crystal structure of ruby. Ruby consists essentially of lightly doped sapphire, AI₂O₃, with the darker spheres in the figure indicating the Al³⁺ ions. (The lattice planes shown in the figure are ~ 2.16 Å apart.)

Sapphire is a very hard, colorless (when pure), transparent crystal which can be grown in large and optically very good samples by flame-fusion techniques.

The transparency of pure sapphire in the visible and infrared means that its Al³⁺ and O^2- atoms, when they are bound into the sapphire crystal lattice, have no absorption lines from their ground energy levels to levels anywhere in the infrared or visible regions.

Indeed, no optical absorption appears in pure sapphire below the insulating band gap of the crystal in the ultraviolet. 

We can, however, replace a significant fraction (several percent) of the Al³⁺ ions in the lattice by chromium or Cr³⁺ ions. The sapphire lattice as a result acquires a pink tint at low chromium concentrations, or a deeper red color at higher concentrations, and becomes what is called "pink ruby."

The individual chromium ions, when they are bound into the sapphire lattice, have a set of quantum energy levels that are associated with partially filled inner electron shells in the Cr³⁺ ion. These energy levels are located as shown in Figure 1.10.

The chromium ions can then absorb incident light in broad wavelength bands extending across much of the visible and near ultraviolet, by making transitions upward from the ground or \(^4A_2\) Cr³⁺ energy level to the series of broad bands or groups of levels labeled \(^4F\) and \(^2F\) in Figure 1.10.

The chromium ions that are excited up into these levels then drop down by rapid nonradiative processes to the two sharp \(^2E\) levels shown in the figure. From there, these ions relax across the remaining energy gap down to the ground state by almost totally radiative relaxation, emitting the deep-red fluorescent emission characteristic of ruby.

(The two sharp \(^2E\) levels are often called the \(R_1\) and \(R_2\) levels, with most of the fluorescent emission coming from the lower or \(R_1\) level. The two very sharp emission lines shown in Figure 1.10 then represent the separate transitions from the \(R_1\) level down to the two closely spaced sublevels of the \(^4A_2\) ground level.)

Energy Levels in Solids: Rare Earth Ions

Figures 1.11 and 1.12 show how a typical rare-earth ion such as Nd³⁺ or Tb³⁺ can be bonded into an irregular glassy lattice structure, together with the quantum energy levels associated with a trivalent terbium Tb³⁺ ion when such an ion is dispersed at low concentration, either in a glass or in a crystal structure (for example, CaF₂).

Note that the energy levels of rare-earth ions such as Tb³⁺ or Nd³⁺ are associated with the electrons in the partially filled \(4f\) inner shell of the rare-earth atom.

In nearly all solid materials, these inner electrons are well shielded, by surrounding outer filled electron shells, from the crystalline Stark effects caused by the bonds to surrounding atoms in the crystal or glass material.

Hence the quantum energy levels of such rare-earth ions are almost unchanged in many different crystalline or glass host materials.

Almost any material containing small amounts of Tb³⁺, for example, will fluoresce with the same brilliant green color around 540 nm, and materials containing Nd³⁺ all fluoresce strongly around 1.06 μm in the near infrared.

There are also several other such rare-earth ions, including Dy²⁺, Tm²⁺, Ho³⁺, Eu³⁺, and Er³⁺, that make good to excellent laser materials.

Optical Pumping of Atoms

All of these minerals illustrate another basic method for pumping or exciting atoms into upper energy levels, that is, through the absorption of light at an appropriate pumping wavelength.

The high-pressure mercury lamp used as the excitation source in a "mineral light" emits a broad continuum of visible and ultraviolet wavelengths.

As shown in Figures 1.8 and 1.12, some of these wavelengths will coincide with the transition frequencies from the lowest or ground levels of the chromium or terbium ions (nearly all the ions are located at ground level when in thermal equilibrium) up to some of the higher energy levels of these ions.

These ions can thus absorb radiation ("absorb photons") from the UV light source at these particular frequencies, and as a result be lifted up to various of the upper levels. This excitation is enhanced by the fact that in solids the higher energy levels are often rather broad bands of levels.

The absorption linewidths of the ruby and terbium absorption lines are thus relatively broad, permitting reasonably efficient absorption of the continuum radiation from the mercury lamp.

Once they are lifted upward by this so-called "optical pumping," the ions in each case then relax or fluoresce down to lower energy levels, as shown in Figure 1.12, emitting a relatively sharp fluorescence at two or three visible wavelengths as they drop from upper to lower levels.

Spontaneous Energy Decay or Relaxation 

Let us discuss a little more the spontaneous decay or relaxation process we have introduced here.

Suppose that a certain number \(N_2\) of such atoms have been pumped into some upper energy level \(E_2\) of an atom or molecule, whether by electron collision in a gas like helium, or by optical pumping in a solid like ruby, or by some other mechanism.

These atoms will then spontaneously drop down or relax to lower energy levels, giving up their excess internal energy in the process (Figure 1.13).

The rate at which atoms spontaneously decay or relax downward from any upper level \(N_2\) is given by a spontaneous energy-decay rate, often called \(\gamma_2\), times the instantaneous number of atoms in the level, or

\[\tag{4}\left.\frac{\text{d}N_2}{\text{d}t}\right|_\text{spon}=-\gamma_2N_2\equiv-N_2/\tau_2\]

If an initial number of atoms \(N_{20}\) are pumped into the level at \(t=0\), for example, by a short intense pumping pulse, and the pumping process is then turned off, the number of atoms in the upper level will decay exponentially in the form

\[\tag{5}N_2(t)=N_{20}e^{-\gamma_2t}=N_{20}e^{-t/\tau_2}\]

where \(\tau_2\equiv1/\gamma_2\) is the lifetime of the upper level \(E_2\) for energy decay to all lower levels.

The lifetime of the \(R\) levels in the ruby crystal happens to be long enough (about 4 msec), and the visible fluorescence strong enough, that we can rather easily demonstrate this kind of exponential decay by using the simple apparatus shown in Figure 1.15.

The pulsed stroboscopic light source emits a broadband flash of visible and ultraviolet light about 60 μsec long. This flash of light optically pumps the Cr³⁺ ions in the ruby sample up to upper levels, from which they very rapidly decay to the metastable \(R\) levels.

These levels then decay to the ground level by emitting visible red fluorescence with a decay time \(\tau\approx4.3\) msec.

(Similar fluorescence lifetime measurements can also be made for any of the other materials we have mentioned, but some of the lifetimes are much shorter, and the fluorescent intensities much smaller, making the experiment more difficult.)

Radiative and Nonradiative Relaxation

There are actually two quite separate kinds of downward relaxation that occur in these solid-state materials, as well as in most other atomic systems.

One mechanism is radiative relaxation, which is to say the spontaneous emission of electromagnetic or fluorescent radiation, as we have already discussed. We can usually measure this emitted radiation directly, with some suitable kind of photodetector.

The other mechanism is what is commonly called nonradiative relaxation. In terbium, for example, when the terbium ions relax from higher energy levels shown in Figure 1.12 down into the \(^5D_4\) level, they get rid of the transition energy not by radiating electromagnetic radiation somewhere in the infrared, but by setting up mechanical vibrations of the surrounding crystal lattice.

To put this in another way, the excess energy is emitted as lattice phonons, or as heating of the surrounding crystal lattice, rather than as electromagnetic radiation or photons—hence the term nonradiative relaxation.

This kind of nonradiative emission is usually difficult to measure directly, since it mostly goes into a very small warming up of the surrounding medium. This same kind of nonradiative relaxation process also allows excited ruby atoms to relax down into the \(^2E\) levels.

The total relaxation rate \(\gamma\) on any given transition will thus be, in general, the sum of a radiative or fluorescent or electromagnetic part, described by a purely radiative decay rate that we often write as \(\gamma_\text{rad}\); plus a nonradiative part, with a nonradiative decay rate that we often write as \(\gamma_\text{nr}\).

The total or measured decay rate for atoms out of the upper level will then be the sum of these, or \(\gamma_\text{tot}\equiv\gamma_\text{rad}+\gamma_\text{nr}\).

The actual numerical values for these rates, and the balance between radiative and nonradiative parts, will in general be different for every different atomic transition, and may depend greatly on the immediate surroundings of the atoms, as we will discuss in much more detail later.

The one certain thing is that atoms placed in an upper level will decay downward, by some combination of radiative and/or nonradiative decay processes.

Nonradiative relaxation can be a particularly rapid process for relaxation across some of the smaller energy gaps for rare-earth ions and other absorbing ions in solids, as we will see in more detail later.

For example, in terbium as in many other rare-earth ions, there may be many rather closely spaced levels or bands at higher energies; but then the energy gap down from the lowest of these upper levels (the \(^5D_4\) level in terbium) to the next lower group of levels may be larger than the frequency \(\hbar\omega\) of the highest phonon mode that the crystal lattice can support.

As a result, the terbium ion cannot relax across this gap very readily by nonradiative processes, i.e., by emitting lattice phonons, since the lattice cannot accept or propagate phonons of this frequency.

Instead the atoms relax across this gap almost entirely by radiative emission, i.e., by spontaneous emission of visible fluorescence. Across other, smaller gaps, however, the nonradiative relaxation rate is so fast that any radiative decay on these transitions is completely overshadowed by the nonradiative rate.

This behavior is typical for many other rare-earth ions in crystals and glasses. Following optical excitation to high-lying levels, the atoms relax by rapid nonradiative relaxation into some lower metastable level, from which further nonradiative relaxation is blocked by the size of the gap to the next lower level.

Efficient fluorescent emission from here to the lower levels then occurs, followed by further fast nonradiative relaxation across any remaining energy gaps to the ground level.

The nonradiative decay time of the atoms via phonon emission across the smaller energy gaps may be in the subnanosecond to picosecond range—too fast to be easily measured—and the average lifetime of the same rare-earth ions in their metastable levels, before they radiate away their energy and drop down, is typically between a few hundred μsec and a few msec.

We will see later that in many rare-earth samples it is possible, by pumping hard enough, to actually build up enough of a population inversion between the metastable level and lower levels to permit laser action on these transitions.

Several different rare-earth atoms can thus be used as good optically pumped solid-state lasers (though terbium itself is not among the best of these).

The next tutorial introduces single element lens.