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Section 4.6 Fluorescence and Phosphorescence

Many materials can absorb one wavelength of EM radiation and emit EM radiation with a different wavelength. Uranine dye (sodium fluorescein), for example, readily absorbs near-UV radiation, but emits green light. The principle — which is called “fluorescence” — is quite simple, and is very common in materials with more than two possible electronic energy levels.

Figure 4.19. Fluorescence: Energy-level diagram for a three-level atom, showing absorption of a photon with large energy and emission of two photons with smaller energy.

Figure 4.19 shows the basic idea for fluorescence. A large energy photon (e.g., corresponding to ultraviolet light) is absorbed by a low energy electron, which jumps up more than one energy level. The electron can later spontaneously drop back down to a lower energy level, but in addition to dropping back down directly to the original state, it can also drop back down to an intermediate state and then later drop to the original state. For a three-level system like in Figure 4.19, there are three different energies possible for the emitted light: \(E_\text{ ph } = \Delta E_{31}\text{,}\) \(\Delta E_{32}\text{,}\) or \(\Delta E_{21}\text{.}\)

So, why does black light illumination cause certain pigments to glow at different colors? Assume that the pigment has an energy level structure similar to that in Figure 4.19. An incoming photon from the black light — which is really near-UV radiation with a “color” just beyond the visible (although there is usually a little violet light that comes out of a black light as well) — has enough energy to cause an electron in the ground (\(n = 0\)) state to become excited to the \(n = 2\) state. After an undetermined amount of time, the electron spontaneously drops back down to a lower energy level. If it drops directly back to the \(n = 0\) level, it emits another near-UV photon, which you don't really see. But if it drops first to the \(n = 1\) level and then later to the \(n = 0\) level, it emits two lower-energy photons, one of which (and maybe both) has energy in the visible part of the EM spectrum. (For many fluorescent pigments, one photon will be in the visible range, e.g., orange or green, and the other will be an infrared photon which you can't see.) So, that's why some materials fluoresce at a different color than they are illuminated with.

Fluorescent pigments can be found in many applications. Of course, there are many toys with fluorescent pigments, including Crayola\(^{TM}\) crayons (with fluorescent colors such as screamin' green, atomic tangerine, and unmellow yellow). Fluorescent dyes are also used in laundry detergents to make clothes appear brighter (that's why white clothes glow when illuminated by black light).

But fluorescence is also a very important tool used in scientific studies. Fluorescent “tags” are often used in biological studies. The basic idea is that if you can attach a fluorescent marker to a particular molecule in a biological system, then if you illuminate your sample with near-UV radiation, only the tagged regions will fluoresce in the visible, enabling you to see (usually under a microscope) the particular molecules in question. In recent years, biologists have developed a fluorescent protein called “green-fluorescent protein” (or GFP for short) that can be attached to various molecules in a living cell. Better yet, techniques have been developed to create targeted mutations in microorganisms that result in the GFP tag being an inherited property of the organism (i.e., they are born fluorescing.)

Fluorescence is also being used to revolutionize surgery with a new (as of 2017) technique referred to as “fluorescence image guided surgery.” The idea is quite simple: a fluorescent dye that preferentially bonds to cancerous cells is injected into a patient. During surgery, the region is illuminated with high-frequency light (possibly near-UV), and the fluorescent dye attached to the cancerous tumor glows, enabling the surgeon to cut away the minimum amount of cancerous tissue without removing healthy tissue. This is a technique that figures to significantly improve survival rates for cancer patients during the next decade.

Another very new technology is that of the “quantum dot,” which is a micro-engineered particle-in-a-box. We talk a lot about the particle-in-a-box problem because it is easy to solve, but these things are actually being made!! And the nice thing about a quantum dot is that the energy level structure can be altered simply by changing the size of the quantum dot (i.e., the width of the square-well potential).

A quantum dot is a a semiconductor that is manufactured to be so small (about 10 — 50 atoms in diameter) that its charge carriers experience effects of quantum confinement, including quantized energy levels. The result is that the filled valence band and empty conduction band for the semiconductor are replaced by a smaller number of discrete energy levels, similar to those of the particle-in-a-box. But since the electrons are confined (due to the small size of the dot), the minimum conduction band energy is shifted upward (away from the band) gap by a particle-in-box energy. Similarly, the maximum valence band energy is shifted downward. The result is the lowest energy \(E_\text{ ph }\) of an emitted photon for a quantum dot is

\begin{equation} E_\text{ ph } =E_g+\eta\, \frac{h^2}{8m_\text{e} R^2}\text{,}\label{eq_QDot}\tag{4.23} \end{equation}

where \(m_\text{e}\) is the electron mass and \(\eta\) is a factor that includes all of the material-specific semiconductor effects. 1  Since there are higher, open energy levels an electron in the quantum dot can absorb smaller wavelength (higher energy) photons and then fluoresce by emitting larger wavelength (smaller energy) photons as it drops back down in energy.

A quantum dot made from a CdTe semiconductor (with band gap energy \(1.38\Xunits{eV}\) and \(\eta = 20.5\)) is fabricated with a radius \(4.22\Xunits{nm}\text{.}\) Calculate the longest wavelength emitted by this quantum dot.

Solution.

The longest wavelength corresponds to the smallest energy ((4.23)). Since, \(E_\text{ ph } = hc/\lambda\text{,}\) and using the trick employed in Example 4.4 of multiplying the numerator and denominator of the second term by \(c^2\text{,}\) it follows that:

\begin{align*} \frac{hc}{\lambda} =\mathstrut \amp E_g+\eta\, \frac{h^2}{8m_\text{e} R^2}\\ =\mathstrut \amp E_g + \eta\, \frac{(hc)^2}{8(m_\text{e} c^2) R^2}\\ =\mathstrut \amp 1.38\Xunits{eV} + 20.5\, \frac{(1240\Xunits{eV \cdot nm})^2} {8 \times (511\times 10^3\Xunits{eV})\times (4.22\Xunits{nm})^2}\\ =\mathstrut \amp 1.81\Xunits{eV}\text{.} \end{align*}

Rearranging this gives

\begin{align*} \lambda =\mathstrut \amp \frac{hc}{1.81\Xunits{eV}}\\ =\mathstrut \amp \frac{1240\Xunits{eV \cdot nm}}{1.81\Xunits{eV}}\\ =\mathstrut \amp 684\Xunits{nm}\text{.} \end{align*}

One final question in this chap: when will an excited electron drop back down in energy, emitting a photon? The answer: you can never know. There is nothing in quantum theory that enables you to predict the precise moment when the emission process will happen. However, it is possible to calculate a probability that an emission process will occur within a certain time range, similar to how we can calculate a probability for a particle to be found in a certain region, and it is also possible to define a half-life for emission processes; i.e., the duration over which (on the average) half of the atoms in an excited state will have decayed back to the ground state, emitting photons.

The half-life for spontaneous emission can vary dramatically from one material to another. Usually, the half-life is a very short time; e.g., nanoseconds and even picoseconds. But there are some materials that have unnaturally long half-lives. In these cases, the material will continue to emit light. Some composites made from SrAl\(_2\)O\(_4\) have been made with half-lives of several minutes, which means that an excited sample of this material will emit light for quite a long time after it is excited.

This is the principle (called phosphorescence) behind “glow-in-the-dark” pigments and materials. You “charge up” the material by exposing it to visible or near-UV light, promoting electrons in the pigment to excited energy levels. These electrons then drop back to the lower energy levels, but over the course of several minutes. During that time, the pigment appears to “glow,” even if there are no lights around.

The constant \(\eta\) includes deviations from the standard particle-in-box energies due to the fact that this box isn't empty; rather, the electron is moving within a semiconductor.