MIS2502: Data Analytics Advanced Analytics Using R - ppt download

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More Precisely 4.1 - The Energy Levels of the Hydrogen Atom

By observing the emission spectrum of hydrogen and using the connection between photon energy and color first suggested by Einstein (Sec. 4.2), Niels Bohr determined early in the twentieth century what the energy differences between the various energy levels must be. Using that information, he was then able to infer the actual energies of the excited states of hydrogen. A unit of energy often used in atomic physics is the electron volt (eV). Its name derives from the amount of energy gained by an electron when it moves through an electric potential of one volt. For our purposes, however, it is just a convenient quantity of energy, numerically equal to 1.60 x 10-19J (joule)—roughly half the energy carried by a single photon of red light. The minimum amount of energy needed to ionize hydrogen from its ground state is 13.6 eV. Bohr numbered the energy levels of hydrogen starting at the ground state, with level 1 the ground state, level 2 the first excited state, and so on. He found that by assigning zero energy to the ground state, the energy of any state (the nth, say) could be written as follows:   Thus, the ground state has energy E1 = 0 (by our definition), the first excited state has energy    the second excited state has energy    and so on. Notice that there are infinitely many excited states between the ground state and the energy at which the atom is ionized, crowding closer and closer together as n becomes large and En approaches 13.6 eV. EXAMPLE: Using Bohr’s formula for the energy of each electron orbital, we can reverse his reasoning and calculate the energy associated with a transition between any two given states. To boost an electron from the second state to the third, an atom must be supplied with E3 - E2 = 1.89 eV of energy, or 3.03 x 10-19 J. Using the formula E = hf presented in the text, we find that this corresponds to a photon with a frequency of 4.57 x 1014 Hz, having a wavelength of 656 nm and lying in the red portion of the spectrum. (A more precise calculation gives the value 656.3 nm reported in the text.) Similarly, the jump from level 3 to level 4 requires   eV of energy, corresponding to an infrared photon with a wavelength of 1880 nm, and so on. A handy conversion between photon energies E in electron volts and wavelengths λ in nanometers is   The accompanying diagram summarizes the structure of the hydrogen atom. The various energy levels are depicted as a series of circles of increasing radius, representing increasing energy. The electronic transitions between these levels (indicated by arrows) are conventionally grouped into families, named after their discoverers, that define the terminology used to identify specific spectral lines. (Note that the spacings of the energy levels are not drawn to scale here, to provide room for all labels on the diagram. In reality, the circles should become more and more closely spaced as we move outward.) Transitions starting from or ending at the ground state (level 1) form the Lyman series. The first is Lyman alpha (Lyα), corresponding to the transition between the first excited state (level 2) and the ground state. As we have seen, the energy difference is 10.21 eV, and the Lyα photon has a wavelength of 121.6 nm (1216 Å). The Lyβ (beta) transition, between level 3 (the second excited state) and the ground state, corresponds to an energy change of 12.10 eV and a photon of wavelength 102.6 nm (1026 Å). Lyγ (gamma) corresponds to a jump from level 4 to level 1, and so on. The accompanying table shows how we can calculate the energies, frequencies, and wavelengths of the photons in the Lyman series using the formulae given previously. All Lyman-series energies lie in the ultraviolet region of the spectrum. The next series of lines, the Balmer series, involves transitions down to (or up from) level 2, the first excited state. All the Balmer series lines lie in or close to the visible portion of the electromagnetic spectrum. Because they form the most easily observable part of the hydrogen spectrum and were the first to be discovered, these lines are often referred to simply as the "Hydrogen" series and denoted by the letter H. As with the Lyman series, the individual transitions are labeled with Greek letters. An Hα photon (level 3 to level 2) has a wavelength of 656.3 nm and is red, Hß (level 4 to level 2) has a wavelength of 486.1 nm (green), Hγ (level 5 to level 2) has a wavelength of 434.1 nm (blue), and so on. The most energetic Balmer series photons have energies that place them just beyond the blue end of the visible spectrum, in the near ultraviolet. The classification continues with the Paschen series (transitions down to or up from the second excited state), the Brackett series (third excited state), and the Pfund series (fourth excited state). Beyond that point, infinitely many other families exist, moving farther and farther into the infrared and radio regions of the spectrum, but they are not referred to by any special names. A few of the transitions making up the Lyman and Balmer (Hydrogen) series are marked on the figure. Astronomically, these are the most important sequences.

More about → More Precisely 4.1 - The Energy Levels of the Hydrogen Atom

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.

More about → 4.2 The Formation of Spectral Lines

4.1 Spectral Lines

Spectral Lines

In Chapter 3 we saw something of how astronomers can analyze electromagnetic radiation received from space to obtain information about distant objects. A vital step in this process is the formation of a spectrum—splitting the incoming radiation into its component wavelengths. But in reality, no cosmic object emits a perfect blackbody spectrum like those discussed earlier. (Sec. 3.4) All spectra deviate from this idealized form—some by only a little, others by a lot. Far from invalidating our earlier studies, however, these deviations contain a wealth of detailed information about physical conditions in the source of the radiation. Because spectra are so important, let's examine in more detail how astronomers obtain and interpret them.

Radiation can be analyzed with an instrument known as a spectroscope. In its most basic form, this device consists of an opaque barrier with a slit in it (to define a beam of light), a prism (to split the beam into its component colors), and an eyepiece or screen (to allow the user to view the resulting spectrum). Figure 4.1 shows such an arrangement. The research instruments calledspectrographs, or spectrometers, used by professional astronomers are rather more complex, consisting of a telescope (to capture the radiation), a dispersing device (to spread it out into a spectrum), and a detector (to record the result). Despite their greater sophistication, however, their basic operation is conceptually similar to the simple spectroscope shown in the figure.

Figure 4.1 Spectroscope Diagram of a simple spectroscope. A small slit in the mask on the left allows a narrow beam of light to pass. The light passes through a prism and is split up into its component colors. The resulting spectrum can be viewed through an eyepiece or simply projected onto a screen.

In many large instruments the prism is replaced by a device called a diffraction grating, consisting of a sheet of transparent material with many closely spaced parallel lines ruled on it. The spacing between the lines is typically a few microns (10^-6 m), comparable to the wavelength of visible light. The spaces act as many tiny openings, and light is diffracted as it passes through the grating (or is reflected from it, depending on the design of the device). (Discovery 3-1) Because different wavelengths of electromagnetic radiation are diffracted by different amounts on encountering the grating, the effect is to split a beam of light into its component colors. You are probably more familiar with diffraction gratings than you think—the "rainbow" of colors seen in light reflected from a compact disk is the result of precisely this process.

EMISSION LINES

	Figure 4.2 Continuous and Emission Spectra When passed through a slit and split up by a prism, light from a source of continuous radiation (a) gives rise to the familiar rainbow of colors. By contrast, the light from excited hydrogen gas (b) consists of a series of distinct bright spectral lines called emission lines. (The focusing lenses have been omitted for clarity.)

The spectra we encountered in Chapter 3 are examples of continuous spectra. A lightbulb, for example, emits radiation of all wavelengths (mostly in the visible range), with an intensity distribution that is well described by the blackbody curve corresponding to the bulb's temperature. (Sec. 3.4) Viewed through a spectroscope, the spectrum of the light from the bulb would show the familiar rainbow of colors, from red to violet, without interruption, as presented in Figure 4.2(a).

Not all spectra are continuous, however. For instance, if we took a glass jar containing pure hydrogen gas and passed an electrical discharge through it (a little like a lightning bolt arcing through Earth's atmosphere), the gas would begin to glow—that is, it would emit radiation. If we were to examine that radiation with our spectroscope, we would find that its spectrum consists of only a few bright lines on an otherwise dark background, quite unlike the continuous spectrum described for the incandescent light bulb. Figure 4.2(b) shows this schematically. A more detailed rendering of the spectrum of hydrogen appears in the top panel of Figure 4.3. The light produced by the hydrogen in this experiment does not consist of all possible colors but instead includes only a few narrow, well-defined emission lines—narrow "slices" of the continuous spectrum. The black background represents all the wavelengths notemitted by hydrogen.

After some experimentation we would also find that although we could alter the intensity of the lines—for example, by changing the amount of hydrogen in the jar or the strength of the electrical discharge—we could not alter their color (in other words, their frequency or wavelength). The pattern of spectral emission lines is a property of the element hydrogen. Whenever we perform this experiment, the same characteristic colors result.

Figure 4.3 Elemental Emission The emission spectra of some well-known elements. In accordance with the convention adopted throughout this text, frequency increases to the right. (Wabash Instrument Corp.)

By the early nineteenth century scientists had carried out similar experiments on many different gases. By vaporizing solids and liquids in a flame, they extended their inquiries to include materials that are not normally found in the gaseous state. Sometimes the pattern of lines was fairly simple, sometimes it was very complex, but it was always unique to that element. Even though the origin of the lines was not understood, researchers quickly realized that the lines provided a one-of-a-kind "fingerprint" of the substance under investigation. They could detect the presence of a particular atom or molecule (a group of atoms held together by chemical bonds—see Sec. 4.4) solely through the study of the light it emitted. Scientists have accumulated extensive catalogs of the specific wavelengths at which many different hot gases emit radiation. The particular pattern of the light emitted by a gas of a given chemical composition is known as its emission spectrum. Examples of the emission spectra of some common substances are shown in Figure 4.3.

	Figure 4.4 Solar Spectrum This visible spectrum of the Sun shows hundreds of dark absorption lines superimposed on a bright continuous spectrum. Here, the scale extends from long wavelengths (red) at the upper left to short wavelengths (blue) at the lower right. (AURA)

ABSORPTION LINES

When sunlight is split by a prism, at first glance it appears to produce a continuous spectrum. However, closer scrutiny with a spectroscope shows that the solar spectrum is interrupted vertically by a large number of narrow dark lines, as shown in Figure 4.4. We now know that many of these lines represent wavelengths of light that have been removed (absorbed) by gases present either in the outer layers of the Sun or in Earth's atmosphere. These gaps in the spectrum are called absorption lines.

The English astronomer William Wollaston first noticed the solar absorption lines in 1802. They were studied in greater detail about 10 years later by the German physicist Joseph von Fraunhofer, who measured and cataloged over 600 of them. They are now referred to collectively asFraunhofer lines. Although the Sun is by far the easiest star to study, and so has the most extensive set of observed absorption lines, similar lines are known to exist in the spectra of all stars.

Figure 4.5 Absorption Spectrum When cool gas is placed between a source of continuous radiation, such as a hot lightbulb, and a detector, the resulting spectrum consists of a continuous spectrum crossed by a series of dark absorption lines. These lines are formed when the intervening gas absorbs certain wavelengths (colors) from the original beam. The absorption lines appear at precisely the same wavelengths as the emission lines that would be produced if the gas were heated to high temperatures.

At around the same time as the solar absorption lines were discovered, scientists found that absorption lines could also be produced in the laboratory by passing a beam of light from a source that produces a continuous spectrum through a cool gas, as shown in Figure 4.5. They quickly observed an intriguing connection between emission and absorption lines: The absorption lines associated with a given gas occur at precisely the same wavelengths as the emission lines produced when the gas is heated.

As an example, consider the element sodium, whose emission spectrum appears in Figure 4.3. When heated to high temperatures, a sample of sodium vapor emits visible light strongly at just two wavelengths—589.9 nm and 589.6 nm—lying in the yellow part of the spectrum. When a continuous spectrum is passed through some relatively cool sodium vapor, two sharp, dark absorption lines appear at precisely the same wavelengths. The emission and absorption spectra of sodium are compared in Figure 4.6, clearly showing the relation between emission and absorption features.

Figure 4.6 Sodium Spectrum (a) The characteristic emission lines of sodium. The two bright lines in the center appear in the yellow part of the spectrum. (b) The absorption spectrum of sodium. The two dark lines appear at exactly the same wavelengths as the bright lines in the sodium emission spectrum.

KIRCHHOFF'S LAWS

 The analysis of the ways in which matter emits and absorbs radiation is called spectroscopy.One early spectroscopist, the German physicist Gustav Kirchhoff, summarized the observed relationships among the three types of spectra—continuous, emission line, and absorption line—in 1859. He formulated three spectroscopic rules, now known as Kirchhoff's laws, governing the formation of spectra:

1. A luminous solid or liquid, or a sufficiently dense gas, emits light of all wavelengths and so produces a continuous spectrum of radiation.

2. A low-density hot gas emits light whose spectrum consists of a series of bright emission lines.These lines are characteristic of the chemical composition of the gas.

3. A cool thin gas absorbs certain wavelengths from a continuous spectrum, leaving darkabsorption lines in their place superimposed on the continuous spectrum. Once again, these lines are characteristic of the composition of the intervening gas—they occur at precisely the same wavelengths as the emission lines produced by that gas at higher temperatures.

Figure 4.7 Kirchhoff's Laws A source of continuous radiation, here represented by a lightbulb, is used to illustrate Kirchhoff's laws of spectroscopy. (a) The unimpeded beam shows the familiar continuous spectrum of colors. (b) When the source is viewed through a cloud of hydrogen gas, a series of dark hydrogen absorption lines appears in the continuous spectrum. These lines are formed when the gas absorbs some of the bulb's radiation and reemits it in random directions. Because most of the reemitted radiation does not go through the slit, the effect is to remove the absorbed radiation from the light that reaches the screen at left. (c) When the gas is viewed from the side, a fainter hydrogen emission spectrum is seen, consisting of reemitted radiation. The absorption lines in (b) and the emission lines in (c) have the same wavelengths.

Figure 4.7 illustrates Kirchhoff's laws and the relationship between absorption and emission lines. When viewed directly, the light source, a hot solid (the filament of the bulb), has a continuous (blackbody) spectrum. When the light source is viewed through a cloud of cool hydrogen gas, a series of dark absorption lines appear, superimposed on the spectrum at wavelengths characteristic of hydrogen. The lines appear because the light at those wavelengths is absorbed by the hydrogen. As we will see later in this chapter, the absorbed energy is subsequently reradiated into space, but in all directions, not just the original direction of the beam. Consequently, when the cloud is viewed from the side against an otherwise dark background, a series of faint emission lines is seen. These lines contain the energy lost by the forward beam. If the gas was heated to incandescence, it would produce stronger emission lines at precisely the same wavelengths.

ASTRONOMICAL APPLICATIONS

By the late nineteenth century, spectroscopists had developed a formidable arsenal of techniques for interpreting the radiation received from space. Once astronomers knew that spectral lines were indicators of chemical composition, they set about identifying the observed lines in the solar spectrum. Almost all the lines in light from extraterrestrial sources could be attributed to known elements. For example, many of the Fraunhofer lines in sunlight are associated with the element iron, a fact first recognized by Kirchhoff and coworker Robert Bunsen (of Bunsen burner fame) in 1859. However, some unfamiliar lines also appeared in the solar spectrum. In 1868, astronomers realized that those lines must correspond to a previously unknown element. It was given the name helium, after the Greek word helios, meaning "Sun." Not until 1895, almost three decades after its detection in sunlight, was helium discovered on Earth. (A laboratory spectrum of helium is included in Figure 4.3.)

Yet for all the information that nineteenth-century astronomers could extract from observations of stellar spectra, they still lacked a theory explaining how the spectra themselves arose. Despite their sophisticated spectroscopic equipment, they knew scarcely any more about the physics of stars than did Galileo or Newton. To understand how spectroscopy can be used to extract detailed information about astronomical objects from the light they emit, we must delve more deeply into the processes that produce line spectra.

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