Quantum Physics

920 Chapter 28 Atomic PhysicsIntensity30KβKα40 50 60 70 80 90λ, pmFigure 28.14 The x-ray spectrumof a metal target consists of a broadcontinuous spectrum (bremstrahlung)plus a number of sharp lines that aredue to characteristic x-rays. The datashown were obtained when 35-keVelectrons bombarded a molybdenumtarget. Note that 1 pm 10 12 m 0.001 nm.1/λFigure 28.15 A Moseley plotof versus Z, where is thewavelength of the K x-ray line ofthe element of atomic number Z.√1/Z28.10 CHARACTERISTIC X-RAYSX-rays are emitted when a metal target is bombarded with high-energy electrons.The x-ray spectrum typically consists of a broad continuous band and a series of intensesharp lines that are dependent on the type of metal used for the target, asshown in Figure 28.14. These discrete lines, called characteristic x-rays, were discoveredin 1908, but their origin remained unexplained until the details of atomicstructure were developed.The first step in the production of characteristic x-rays occurs when a bombardingelectron collides with an electron in an inner shell of a target atom with sufficientenergy to remove the electron from the atom. The vacancy created in theshell is filled when an electron in a higher level drops down into the lower energylevel containing the vacancy. The time it takes for this to happen is very short, lessthan 10 9 s. The transition is accompanied by the emission of a photon with energyequaling the difference in energy between the two levels. Typically, the energyof such transitions is greater than 1 000 eV, and the emitted x-ray photons havewavelengths in the range of 0.01 nm to 1 nm.We assume that the incoming electron has dislodged an atomic electron fromthe innermost shell, the K shell. If the vacancy is filled by an electron droppingfrom the next higher shell, the L shell, the photon emitted in the process is referredto as the K line on the curve of Figure 28.14. If the vacancy is filled by anelectron dropping from the M shell, the line produced is called the K line.Other characteristic x-ray lines are formed when electrons drop from upperlevels to vacancies other than those in the K shell. For example, L lines are producedwhen vacancies in the L shell are filled by electrons dropping from highershells. An L line is produced as an electron drops from the M shell to the L shell,and an L line is produced by a transition from the N shell to the L shell.We can estimate the energy of the emitted x-rays as follows: consider two electronsin the K shell of an atom whose atomic number is Z. Each electron partiallyshields the other from the charge of the nucleus, Ze, so each is subject to an effectivenuclear charge Z eff (Z 1)e. We can now use a modified form of Equation28.18 to estimate the energy of either electron in the K shell (with n 1). We havewhere E 0 is the ground-state energy. Substituting Z eff Z 1 givesE K (Z 1) 2 (13.6 eV) [28.20]As Example 28.5 will show, we can estimate the energy of an electron in an L or anM shell in a similar fashion. Taking the energy difference between these two levels,we can then calculate the energy and wavelength of the emitted photon.In 1914, Henry G. J. Moseley plotted the Z values for a number of elementsagainst , where l is the wavelength of the K line for each element. He foundthat such a plot produced a straight line, as in Figure 28.15. This is consistent withour rough calculations of the energy levels based on Equation 28.20. From hisplot, Moseley was able to determine the Z values of other elements, providing aperiodic chart in excellent agreement with the known chemical properties of theelements.√1/E K m e Z 2 effk e 2 e 42 2 Z 2 eff E 0EXAMPLE 28.5 Characteristic X-RaysGoal Calculate the energy and wavelength of characteristic x-rays.Problem Estimate the energy of the characteristic x-ray emitted from a tungsten target when an electron dropsfrom an M shell (n 3 state) to a vacancy in the K shell (n 1 state).Strategy Develop two estimates, one for the electron in the K shell (n 1) and one for the electron in the M shell(n 3). For the K-shell estimate, we can use Equation 28.20. For the M shell, we need a new equation. There is one