4.2 The Formation of Spectral Lines

The Formation of Spectral Lines

By the start of the twentieth century, physicists had accumulated substantial evidence that light sometimes behaves in a manner that cannot be explained by the wave theory. As we have just seen, the production of absorption and emission lines involves only certain very specific frequencies or wavelengths of light. This would not be expected if light behaved like a continuous wave and matter always obeyed the laws of Newtonian mechanics. Other experiments conducted around the same time strengthened the conclusion that the notion of radiation as a wave was incomplete. It became clear that when light interacts with matter on very small scales, it does so not in a continuous way but in a discontinuous, "stepwise" manner. The challenge was to find an explanation for this unexpected behavior. The eventual solution revolutionized our view of nature and now forms the foundation for all of physics and astronomy—indeed for virtually all modern science.

ATOMIC STRUCTURE

To explain the formation of emission and absorption lines, we must understand not just the nature of light but also the structure of atoms—the microscopic building blocks from which all matter is constructed. Let's start with the simplest atom of all—hydrogen. A hydrogen atom consists of an electron with a negative electrical charge, orbiting a proton carrying a positive charge. The proton forms the central nucleus (plural: nuclei) of the atom. The hydrogen atom as a whole is electrically neutral. The equal and opposite charges of the proton and the orbiting electron produce an electrical attraction that binds them together within the atom.

How does this picture of the hydrogen atom relate to the characteristic emission and absorption lines associated with hydrogen gas? If an atom emits some energy in the form of radiation, that energy has to come from somewhere within the atom. Similarly, if energy is absorbed, it must cause some internal change. It is reasonable (and correct) to suppose that the energy emitted or absorbed by the atom is associated with changes in the motion of the orbiting electron.

	Figure 4.8 Bohr Atom An early-twentieth-century conception of the hydrogen atom pictured its electron orbiting the central proton in a well-defined orbit, rather like a planet orbiting the Sun. Two electron orbits of different energies are shown. (a) The ground state. (b) An excited state.

The first theory of the atom to provide an explanation of hydrogen's observed spectral lines was set forth by the Danish physicist Niels Bohr in 1912. Now known simply as the Bohr model of the atom, its essential features are as follows. First, there is a state of lowest energy—the ground state—which represents the "normal" condition of the electron as it orbits the nucleus. Second, there is a maximum energy that the electron can have and still be part of the atom. Once the electron acquires more than that maximum energy, it is no longer bound to the nucleus, and the atom is said to be ionized; an atom missing one or more of its electrons is called an ion. Third, and most important (and also least intuitive), between those two energy levels the electron can exist only in certain sharply defined energy states, often referred to as orbitals.

This description of the atom contrasts sharply with the predictions of Newtonian mechanics, which would permit orbits with any energy, not just at certain specific values. In the atomic realm such discontinuous behavior is the norm. In the jargon of the field, the orbital energies are said to be quantized. The rules of quantum mechanics, the branch of physics governing the behavior of atoms and subatomic particles, are far removed from everyday experience.

	Figure 4.9 Modern Atom The modern view of the hydrogen atom sees the electron as a "cloud" surrounding the nucleus. The same two energy states are shown as in Figure 4.8.

In Bohr's model, each electron orbital was pictured as having a specific radius, much like a planetary orbit in the solar system, as shown in Figure 4.8. However, the modern view is not so simple. Although each orbital doeshave a precise energy, the electron is now envisioned as being smeared out in an "electron cloud" surrounding the nucleus, as illustrated in Figure 4.9. We cannot tell "where" the electron is—we can only speak of the probability of finding it in a certain location within the cloud. It is common to speak of the average distance from the cloud to the nucleus as the "radius" of the electron's orbit. When a hydrogen atom is in its ground state, the radius of the orbit is about 0.05 nm (0.5 Å). As the orbital energy increases, the radius increases, too. For the sake of clarity in the diagrams that follow, we will represent electron orbitals as solid lines, but always bear in mind that Figure 4.9 is a more accurate depiction of reality.

Atoms do not always remain in their ground state. An atom is said to be in an excited state when an electron occupies an orbital at a greater than normal distance from its parent nucleus. An atom in such an excited state has a greater than normal amount of energy. The excited state with the lowest energy (that is, the one closest in energy to the ground state) is called the first excited state, that with the second-lowest energy is the second excited state, and so on. An atom can become excited in one of two ways: by absorbing some energy from a source of electromagnetic radiation or by colliding with some other particle—another atom, for example. However, the electron cannot stay in a higher orbital forever; the ground state is the only level where it can remain indefinitely. After about 10^-8 s, an excited atom returns to its ground state.

RADIATION AS PARTICLES

Because electrons may exist only in orbitals having specific energies, atoms can absorb only specific amounts of energy as their electrons are boosted into excited states. Likewise, they can emit only specific amounts of energy as their electrons fall back to lower energy states. Thus, the amount of light energy absorbed or emitted in these processes must correspond precisely to the energy difference between two orbitals. The atom's quantized energy levels require that light must be absorbed and emitted in the form of distinct "packets" of electromagnetic radiation, each carrying a specific amount of energy. We call these packets photons. A photon is, in effect, a "particle" of electromagnetic radiation.

The idea that light sometimes behaves not as a continuous wave but as a stream of particles was proposed by Albert Einstein in 1905 to explain a number of experimental results then puzzling physicists. Further, Einstein was able to quantify the relationship between the two aspects of light's double nature. He found that the energy carried by a photon had to be proportional to thefrequency of the radiation:

photon energy    radiation frequency.

Thus, for example, a "deep red" photon having a frequency of 4 x 10¹⁴ Hz (or a wavelength of approximately 750 nm) has half the energy of a violet photon of frequency of 8 x 10¹⁴ Hz (wavelength = 375 nm), and 500 times the energy of an 8 x 10¹¹ Hz (wavelength = 375 µm) microwave photon.

The constant of proportionality in the above relation is now known as Planck's constant, in honor of the German physicist Max Planck, who determined its numerical value. It is always denoted by the symbol h, and the equation relating photon energy E to radiation frequency f is usually written

E = h f.

Like the gravitational constant G and the speed of light c, Planck's constant is one of the fundamental physical constants of the universe.

In SI units, the value of Planck's constant is a very small number: h = 6.63 x 10^-34 joule seconds (J. s). Consequently, the energy of a single photon is tiny. Even a very-high-frequency gamma ray (the most energetic type of electromagnetic radiation) with a frequency of 10²² Hz has an energy of just (6.63 x 10^-34) x 10²²  7 x 10^-12 J—about the same energy carried by a flying gnat. Nevertheless, this energy is more than enough to damage a living cell. The basic reason that gamma rays are so much more dangerous to life than visible light is that each gamma-ray photon typically carries millions, if not billions, of times more energy than a photon of visible radiation.

The equivalence between photon energy and photon frequency, or wavelength, completes the connection between atomic structure and atomic spectra. Atoms absorb and emit radiation at characteristic wavelengths determined by their own particular internal structure. Because this structure is unique to each element, the colors of the absorbed and emitted photons—that is, the spectral lines we observe—are characteristic of that element and only that element. The spectrum we see is thus a unique identifier of the atom involved.

Many people are confused by the idea that light can behave in two such different ways. To be truthful, modern physicists don't yet fully understand why nature displays this wave–particle duality. Nevertheless, there is irrefutable experimental evidence for both of these aspects of radiation. Environmental conditions ultimately determine which description—wave or stream of particles—better fits the behavior of electromagnetic radiation. As a general rule of thumb, in the macroscopic realm of everyday experience, radiation is more usefully described as a wave, whereas in the microscopic domain of atoms it is best characterized as a series of particles.

THE SPECTRUM OF HYDROGEN

Figure 4.10 illustrates schematically the absorption and emission of photons by a hydrogen atom.Figure 4.10(a) shows the atom absorbing a photon and making a transition from the ground state to the first excited state, then emitting a photon of precisely the same energy and dropping back to the ground state. The energy difference between the two states corresponds to an ultraviolet photon, of wavelength 121.6 nm (1216 Å).

Figure 4.10 Atomic Excitation (a) Diagram of a photon being absorbed by a hydrogen atom (left), causing the momentary excitation of that atom (center) into its first excited state. After about 10-8 s, the atom returns to its ground state, accompanied by the emission of a photon of the same energy as the original photon (right). (b) Absorption of a higher-energy photon may also boost the atom into a higher excited state, from which there may be several possible paths back to the ground state. (Remember that the sharp lines used for the orbitals here and in similar figures that follow are intended merely as a schematic representation of the electron energy levels and are not meant to be taken literally. In actuality, electron orbitals are "clouds," as shown in Figure 4.9.) As ultraviolet photons from a hot star pass through surrounding hydrogen gas, many are absorbed by the gas, boosting its atoms into excited states. Electrons in the second excited state can fall to the first excited state on their way back to the ground state (the lower path in part b). This transition produces radiation in the visible region of the spectrum—the 656.3-nm red glow that is characteristic of excited hydrogen gas. The object shown in the inset, designated N81, is an emission nebula: an interstellar cloud consisting largely of hydrogen gas excited by extremely hot star (as seen in white at the center). (Inset:NASA)

Absorption may also boost an electron into an excited state higher than the first excited state. Figure 4.10(b) depicts the absorption of a more energetic (higher-frequency, shorter-wavelength) ultraviolet photon, this one having a wavelength of 102.6 nm (1026 Å). Absorption of this photon causes the atom to jump to the second excited state. As before, the atom returns rapidly to the ground state, but this time it can do so in one of two possible ways:

1. It can proceed directly back to the ground state, in the process emitting an ultraviolet photon identical to the one that excited the atom in the first place.

2. Alternatively, the electron can cascade down one orbital at a time. If this occurs, the atom will emit two photons: one with an energy equal to the difference between the second and first excited states, and the other with an energy equal to the difference between the first excited state and the ground state.

Either possibility can occur, with roughly equal probability. The second step of this cascade process (2), produces a 121.6-nm ultraviolet photon, just as in Figure 4.10(a). However, the first transition—the one from the second to the first excited state—produces a photon with a wavelength of 656.3 nm (6563 Å), which is in the visible part of the spectrum. This photon is seen as red light. An individual atom—if one could be isolated—would emit a momentary red flash. This is the origin of the red line in the hydrogen spectrum shown in Figure 4.3.

The inset in Figure 4.10 shows an astronomical object whose red coloration is the result of precisely this process. As ultraviolet photons from a young hot star pass through the surrounding cool hydrogen gas out of which the star recently formed, some photons are absorbed by the gas, boosting its atoms into excited states or ionizing them completely. The 656.3-nm red glow characteristic of excited hydrogen gas results as the atoms cascade back to their ground states. This process is called fluorescence.

Absorption of additional energy can boost the electron to even higher orbitals within the atom. As the excited electron cascades back down to the ground state, the atom may emit many photons, each with a different energy and hence a different wavelength, and the resulting spectrum shows many spectral lines. In a sample of heated hydrogen gas, at any instant atomic collisions ensure that atoms are found in many different excited states. The complete emission spectrum therefore consists of wavelengths corresponding to all possible transitions between those states and states of lower energy.

In the case of hydrogen, all transitions ending at the ground state produce ultraviolet photons. However, downward transitions ending at the first excited state give rise to spectral lines in or near the visible portion of the electromagnetic spectrum (Figure 4.3). The energy levels and spectrum of hydrogen are discussed in more detail in More Precisely 4-1.

KIRCHOFF'S LAWS EXPLAINED

Let's reconsider our earlier discussion of emission and absorption lines in terms of the model just presented. In Figure 4.7 a beam of continuous radiation shines through a cloud of hydrogen gas. The beam contains photons of all energies, but most of them cannot interact with the gas—the gas can absorb only those photons having just the right energy to cause a change in an electron's orbit from one state to another. All other photons in the beam—with energies that cannot produce a transition—do not interact with the gas at all but pass through it unhindered. Photons having the right energies are absorbed, excite the gas, and are removed from the beam. This is the cause of the dark absorption lines in the spectrum of Figure 4.7(b). These lines are direct indicators of the energy differences between orbitals in the atoms making up the gas.

The excited gas atoms return rapidly to their original states, each emitting one or more photons in the process. We might think, then, that although some photons from the beam are absorbed by the gas, they are quickly replaced by reemitted photons, with the result that we could never observe the effects of absorption. In fact, this is not the case, for two reasons. First, while the photons not absorbed by the gas continue on directly to the detector, the reemitted photons can leave in anydirection. Most of the reemitted photons leave at angles that do not take them through the slit and on to the detector, and so they are effectively lost from the original beam. Second, as we have just seen, electrons may cascade back to the ground state, emitting several lower-energy photons instead of a single photon equal in energy to the one originally absorbed.

The net result of these processes is that some of the original energy is channeled into photons of many different colors, moving in many different directions. A second detector looking at the cloud from the side would record the reemitted energy as an emission spectrum, as in Figure 4.7(c). (A spectrum of the object shown in the inset of Figure 4.10, called an emission nebula, would show the same thing.) Like the absorption spectrum, the emission spectrum is characteristic of the gas, not of the original beam.

Absorption and emission spectra are created by the same atomic processes. They correspond to the same atomic transitions. They contain the same information about the composition of the gas cloud. In the laboratory we can move our detector and measure both. In astronomy we cannot change our vantage point, so the type of spectrum we see depends on our chance location with respect to both the source and the intervening gas cloud.

MORE COMPLEX SPECTRA

All hydrogen atoms have basically the same structure—a single electron orbiting a single proton—but, of course, there are many other kinds of atoms, each kind having a unique internal structure.The number of protons in the nucleus of an atom determines the element that it represents. Just as all hydrogen atoms have a single proton, all oxygen atoms have eight protons, all iron atoms have 26 protons, and so on.

The next simplest element after hydrogen is helium. The central nucleus of the most common form of helium is made up of two protons and two neutrons (another kind of elementary particle having a mass slightly larger than that of a proton but having no electrical charge). Two electrons orbit this nucleus. As with hydrogen and all other atoms, the "normal" condition for helium is to be electrically neutral, with the negative charge of the orbiting electrons exactly canceling the positive charge of the nucleus (Figure 4.11a).

Figure 4.11 Helium and Carbon (a) A helium atom in its normal ground state. Two electrons occupy the lowest-energy orbital around a nucleus containing two protons and two neutrons. (b) A carbon atom in its normal ground state. Six electrons orbit a six-proton, six-neutron nucleus, two in an inner orbital, with the other four at a greater distance from the center.

More complex atoms contain more protons (and neutrons) in the nucleus and have correspondingly more orbiting electrons. For example, an atom of carbon, shown in Figure 4.11(b), consists of six electrons orbiting a nucleus containing six protons and six neutrons. As we progress to heavier and heavier elements, the number of orbiting electrons increases, and the number of possible electron transitions rises rapidly. The result is that very complicated spectra can be produced. The complexity of atomic spectra generally reflects the complexity of the atoms themselves. A good example is the element iron, which contributes nearly 800 of the Fraunhofer absorption lines seen in the solar spectrum (Figure 4.4).

Atoms of a single element such as iron can yield many lines for two main reasons. First, the 26 electrons of a normal iron atom can make an enormous number of different transitions among available energy levels. Second, many iron atoms are ionized, with some of their 26 electrons stripped away. The removal of electrons alters an atom's electromagnetic structure, and the energy levels of ionized iron are quite different from those of neutral iron. Each new level of ionization introduces a whole new set of spectral lines. Besides iron, many other elements, also in different stages of excitation and ionization, absorb photons at visible wavelengths. When we observe the entire Sun, all these atoms and ions absorb simultaneously, yielding the rich spectrum we see.

	Figure 4.12 Emission Nebula The visible spectrum of the hot gases in a nearby star-forming region known as the Omega nebula (M17). Shining by the light of several very hot stars, the nebula produces a complex spectrum of bright and dark lines (bottom), also shown here as an intensity trace from red to blue (center). (ESO)

The power of spectroscopy is most apparent when many different gases are mixed together because it enables us to study one kind of atom or ion to the exclusion of all others simply by focusing on specific wavelengths of radiation. By identifying the superimposed absorption and emission spectra of many different atoms, we can determine the cloud's composition (and much more—see Section 4.4). Figure 4.12 shows an actual spectrum observed coming from a cosmic object. As in Figure 4.10, the characteristic red glow of this emission nebula comes from the H transition in hydrogen, the nebula's main constituent. 

Spectral lines occur throughout the entire electromagnetic spectrum. Usually, electron transitions among the lowest orbitals of the lightest elements, such as hydrogen and helium, produce visible and ultraviolet spectral lines. Transitions among very highly excited states of hydrogen and other elements can produce spectral lines in the infrared and radio parts of the electromagnetic spectrum. Conditions on Earth make it all but impossible to detect these radio and infrared features in the laboratory, but they are routinely observed coming from space. Electron transitions among lower energy levels in heavier, more complex elements produce X-ray spectral lines. These lines have been observed in the laboratory; some have also been observed in stars and other cosmic objects.

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