30.7

X-Rays

X-rays were discovered by Wilhelm K. Roentgen (1845–1923), a Dutch physicist who performed much of his work in Germany. X-rays can be produced when electrons, accelerated through a large potential difference, collide with a metal target made from molybdenum or platinum, for example. The target is contained within an evacuated glass tube, as Figure 30-19 shows. Example 9 discusses the relationship between the wavelength of the emitted X-rays and the speed of the impinging electrons.

Figure 30-19 In an X-ray tube, electrons are emitted by a heated filament, accelerate through a large potential difference V, and strike a metal target. The X-rays originate when the electrons interact with the target.

The physics of X-rays.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 9 | | | X-Rays and Electrons

The highest-energy X-rays produced by an X-ray tube have a wavelength of . What is the speed of the electrons in Figure 30-19 just before they strike the metal target? Assume that the speed is much less than the speed of light in a vacuum. Reasoning We can find the electron’s speed from a knowledge of its kinetic energy, since the two are related. Moreover, it is the electron’s kinetic energy that determines the energy of any photon. The energy of any photon, on the other hand, is directly proportional to its frequency, as discussed in Section 29.3. But we know from our study of waves (see Section 16.2) that the frequency of a wave is inversely proportional to its wavelength. Thus, we will be able to find the speed of an impinging electron from the given X-ray wavelength.
Knowns and Unknowns The following table summarizes what we know and what we seek:

Description | Symbol | Value
Wavelength of X-ray photon |  |
Unknown Variable |  |
Speed of electron just before it strikes target | v | ?

Modeling the Problem

| Step 1 Kinetic Energy
| The kinetic energy of an electron is the energy it has because of its motion. If the speed of the electron is much less than the speed of light in a vacuum, the kinetic energy KE is given by Equation 6.2 as , where m and v are its mass and speed. Solving this expression for the speed gives Equation 1 at the right. The mass of the electron is known but its kinetic energy is not, so this will be evaluated in Steps 2 and 3.

| (1)

| Step 2 Energy of a Photon
| An X-ray photon is a discrete packet of electromagnetic-wave energy. The photon’s energy E is given by E = hf, where h is Planck’s constant and f is the photon’s frequency (see Equation 29.2). The energy needed to produce an X-ray photon comes from the kinetic energy of an electron striking the target. We know that the X-ray photons have the highest possible energy. This means that all of an electron’s kinetic energy KE goes into producing an X-ray photon, so KE = E. Substituting KE = E into E = hf gives

| (2)

This result for the energy of the photon can be substituted into Equation 1, as indicated at the right. The frequency is not known, but Step 3 discusses how it can be obtained from the given wavelength.

| Step 3 Relation between Frequency and Wavelength
| Electromagnetic waves such as X-rays travel at the speed c of light in a vacuum. According to Equation 16.1, this speed is related to the frequency f and the wavelength by . Solving for the frequency gives

All the variables on the right side of this equation are known, so we substitute it into Equation 2 for the kinetic energy, as shown in the right column.

Solution Algebraically combining the results of the three steps, we have

The speed of an electron just before it strikes the metal target is

Related Homework: Problem 38

A plot of X-ray intensity per unit wavelength versus the wavelength looks similar to Figure 30-20 and consists of sharp peaks or lines superimposed on a broad continuous spectrum. The sharp peaks are called characteristic lines or characteristic X-rays because they are characteristic of the target material. The broad continuous spectrum is referred to as Bremsstrahlung (German for “braking radiation”) and is emitted when the electrons decelerate or “brake” upon hitting the target.

Figure 30-20 When a molybdenum target is bombarded with electrons that have been accelerated from rest through a potential difference of 45 000 V, this X-ray spectrum is produced. The vertical axis is not to scale.

In Figure 30-20 the characteristic lines are marked and because they involve the n = 1 or K shell of a metal atom. If an electron with enough energy strikes the target, one of the K-shell electrons can be knocked out. An electron in one of the outer shells can then fall into the K shell, and an X-ray photon is emitted in the process. Example 10 shows that a large potential difference is needed to operate an X-ray tube so that the electrons impinging on the metal target will have sufficient energy to generate the characteristic X-rays.

| Example 10 | | | Operating an X-Ray Tube

Strictly speaking, the Bohr model does not apply to multiple-electron atoms, but it can be used to make estimates. Use the Bohr model to estimate the minimum energy that an incoming electron must have to knock a K-shell electron entirely out of an atom in a platinum (Z = 78) target in an X-ray tube.
Reasoning According to the Bohr model, the energy of a K-shell electron is given by Equation 30.13, , with n = 1. When striking a platinum target, an incoming electron must have at least enough energy to raise the K-shell electron from this low energy level up to the 0-eV level that corresponds to a very large distance from the nucleus. Only then will the incoming electron knock the K-shell electron entirely out of a target atom.
Solution The energy of the Bohr n = 1 level is

In this calculation we have used 78 − 1 rather than 78 for the value of Z. In so doing, we account approximately for the fact that each of the two K-shell electrons applies a repulsive force to the other. This repulsive force tends to balance the attractive force of one nuclear proton. In effect, one electron shields the other from the force of that proton. Therefore, to raise the K-shell electron up to the 0-eV level, the minimum energy for an incoming electron is . One electron volt is the kinetic energy acquired when an electron accelerates from rest through a potential difference of one volt. Thus, a potential difference of 81 000 V must be applied to the X-ray tube.

Problem solving insight Equation 30.13 for the Bohr energy levels can be used in rough calculations of the energy levels involved in the production of X-rays. In this equation, however, the atomic number Z must be reduced by one, to account approximately for the shielding of one K-shell electron by the other K-shell electron.

The line in Figure 30-20 arises when an electron in the n = 2 level falls into the vacancy that the impinging electron has created in the n = 1 level. Similarly, the line arises when an electron in the n = 3 level falls to the n = 1 level. Example 11 determines an estimate for the wavelength of platinum.

Need more practice?

Interactive LearningWare30.2
High-energy electrons impinge on a metal target in an X-ray tube. The target is a mixture of two metals, one of which is cobalt. Both metals emit X-ray photons, and the wavelength that originates from the cobalt is 1.247 times as great as that produced by the unknown metal. What is the unknown metal?
Related Homework: Problem 34

| Example 11 | | |

The Characteristic X-Ray for Platinum Use the Bohr model to estimate the wavelength of the line in the X-ray spectrum of platinum (Z = 78). Reasoning This example is very similar to Example 4, which deals with the emission line spectrum of the hydrogen atom. As in that example, we use Equation 30.14, this time with the initial value of n being ni = 2 and the final value being nf = 1. As in Example 10, a value of 77 rather than 78 is used for Z to account approximately for the shielding effect of the single K-shell electron in canceling out the attraction of one nuclear proton.
Solution Using Equation 30.14, we find that

This answer is close to an experimental value of .

Most people think of X-rays as being used in the medical and dental professions. Here, radiologist Steven Sirr demonstrates for violinist Desiree Ruhstrat the X-ray procedure called computed tomography (CT) on her violin. CT scans can reveal defects, information about how instruments are made, and distinctive “fingerprints” that can aid in identifying lost or stolen instruments. (©AP/Wide World Photos)

Another interesting feature of the X-ray spectrum in Figure 30-20 is the sharp cutoff that occurs at a wavelength of on the short-wavelength side of the Bremsstrahlung. This cutoff wavelength is independent of the target material but depends on the energy of the impinging electrons. An impinging electron cannot give up any more than all of its kinetic energy when decelerated by the metal target in an X-ray tube. Thus, at most, an emitted X-ray photon can have an energy equal to the kinetic energy KE of the electron and a frequency given by Equation 29.2 as , where h is Planck’s constant. But the kinetic energy acquired by an electron in accelerating from rest through a potential difference V is eV, according to earlier discussions in Section 19.2; V is the potential difference applied across the X-ray tube. Thus, the maximum photon frequency is . Since , a maximum frequency corresponds to a minimum wavelength, which is the cutoff wavelength :

| (30.17)

Figure 30-20, for instance, assumes a potential difference of 45 000 V, which corresponds to a cutoff wavelength of

The medical profession began using X-rays for diagnostic purposes almost immediately after their discovery. When a conventional X-ray is obtained, the patient is typically positioned in front of a piece of photographic film, and a single burst of radiation is directed through the patient and onto the film. Since the dense structure of bone absorbs X-rays much more than soft tissue does, a shadow-like picture is recorded on the film. As useful as such pictures are, they have an inherent limitation. The image on the film is a superposition of all the “shadows” that result as the radiation passes through one layer of body material after another. Interpreting which part of a conventional X-ray corresponds to which layer of body material is very difficult.

The physics of CAT scanning.

The technique known as CAT scanning, or CT scanning, has greatly extended the ability of X-rays to provide images from specific locations within the body. The acronym CAT stands for computerized axial tomography or computer-assisted tomography, and the shorter version CT stands for computerized tomography. In this technique a series of X-ray images are obtained as indicated in Figure 30-21. A number of X-ray beams form a ”fanned out” array of radiation and pass simultaneously through the patient. Each of the beams is detected on the other side by a detector, which records the beam intensity. The various intensities are different, depending on the nature of the body material through which the beams have passed. The feature of CAT scanning that leads to dramatic improvements over the conventional technique is that the X-ray source can be rotated to different orientations, so that the fanned-out array of beams can be sent through the patient from various directions. Figure 30-21a singles out two directions for illustration. In reality many different orientations are used, and the intensity of each beam in the array is recorded as a function of orientation. The way in which the intensity of a beam changes from one orientation to another is used as input to a computer. The computer then constructs a highly resolved image of the cross-sectional slice of body material through which the fan of radiation has passed. In effect, the CAT scanning technique makes it possible to take an X-ray picture of a cross-sectional “slice” that is perpendicular to the body’s long axis. In fact, the word “axial” in the phrase “computerized axial tomography” refers to the body’s long axis. Figure 30-22a shows three-dimensional CAT scans of parts of the human anatomy.

Figure 30-21 (a) In CAT scanning, a “fanned-out” array of X-ray beams is sent through the patient from different orientations. (b) A patient in a CAT scanner. (Laurent/American Hospital of Paris/Photo Researchers, Inc.)

Figure 30-22 These three-dimensional CAT scans show an image of (a) a skull and brain and (b) the pelvis and part of the spine, including the intervertebral discs. (a, ©GJLP/Phototake; b, ©Collection CNRI/Phototake)

Check Your Understanding 4

Which one of the following statements is true?(a)The wavelength can be smaller than the cutoff wavelength . (b)The wavelength is produced when an electron undergoes a transition from the n = 1 energy level to the n = 2 energy level. (c) The wavelength is always smaller than the wavelength for a given metal target. (The answer is given at the end of the book.)
Background: X-rays are produced when high-energy electrons collide with a metal target contained within an evacuated glass tube. As a result of the collisions, electrons in the atoms of the target undergo transitions between atomic energy levels.
For similar questions (including calculational counterparts), consult Self-Assessment Test 30.2, which is described at the end of Section 30.9.

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