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Different energy levels are shown in the form of horizontal lines. The line at the bottom shows the energy level for n is equal to one for the K shell. At a distance above this line, another horizontal line shows the energy level for n is equal to two for the shell L. Similarly, other lines are shown for the shells M and N. As we move from bottom to the top, the distance between the lines decreases, and near the end a few lines are shown very close to each other. Each level is labeled according to the characteristic x ray of the shell.
A characteristic x ray is emitted when an electron fills an inner-shell vacancy, as shown for several transitions in this approximate energy level diagram for a multiple-electron atom. Characteristic x rays are labeled according to the shell that had the vacancy and the shell from which the electron came. A K α size 12{K rSub { size 8{α} } } {} x ray, for example, is produced when an electron coming from the n = 2 size 12{n=2} {} shell fills the n = 1 size 12{n=1} {} shell vacancy.

Characteristic x-ray energy

Calculate the approximate energy of a K α size 12{K rSub { size 8{α} } } {} x ray from a tungsten anode in an x-ray tube.

Strategy

How do we calculate energies in a multiple-electron atom? In the case of characteristic x rays, the following approximate calculation is reasonable. Characteristic x rays are produced when an inner-shell vacancy is filled. Inner-shell electrons are nearer the nucleus than others in an atom and thus feel little net effect from the others. This is similar to what happens inside a charged conductor, where its excess charge is distributed over the surface so that it produces no electric field inside. It is reasonable to assume the inner-shell electrons have hydrogen-like energies, as given by E n = Z 2 n 2 E 0 n = 1, 2, 3, ... size 12{ left (n=1, 2, 3 "." "." "." right )} {} . As noted, a K α size 12{K rSub { size 8{α} } } {} x ray is produced by an n = 2 size 12{n=2} {} to n = 1 size 12{n=1} {} transition. Since there are two electrons in a filled K size 12{K} {} shell, a vacancy would leave one electron, so that the effective charge would be Z 1 size 12{Z - 1} {} rather than Z size 12{Z} {} . For tungsten, Z = 74 size 12{Z="74"} {} , so that the effective charge is 73.

Solution

E n = Z 2 n 2 E 0 size 12{E rSub { size 8{n} } = - { {Z rSup { size 8{2} } } over {n rSup { size 8{2} } } } E rSub { size 8{0} } } {} n = 1, 2, 3, ... gives the orbital energies for hydrogen-like atoms to be E n = ( Z 2 / n 2 ) E 0 size 12{E rSub { size 8{n} } = - \( Z rSup { size 8{2} } /n rSup { size 8{2} } \) E rSub { size 8{0} } } {} , where E 0 = 13.6 eV size 12{E rSub { size 8{0} } ="13" "." 6"eV"} {} . As noted, the effective Z size 12{Z} {} is 73. Now the K α size 12{K rSub { size 8{α} } } {} x-ray energy is given by

E K α = Δ E = E i E f = E 2 E 1 , size 12{E rSub { size 8{k rSub { size 6{α} } } } =ΔE=E rSub {i} size 12{ - E rSub {f} } size 12{ {}=E rSub {2} } size 12{ - E rSub {1} }} {}

where

E 1 = Z 2 1 2 E 0 = 73 2 1 ( 13.6 eV ) = 72.5 keV size 12{E rSub { size 8{1} } = - { {Z rSup { size 8{2} } } over {1 rSup { size 8{2} } } } E rSub { size 8{0} } = - { {"73" rSup { size 8{2} } } over {1} } "13" "." 6" eV"= - "72" "." 5" keV"} {}

and

E 2 = Z 2 2 2 E 0 = 73 2 4 ( 13.6 eV ) = 18.1 keV. size 12{E rSub { size 8{2} } = - { {Z rSup { size 8{2} } } over {2 rSup { size 8{2} } } } E rSub { size 8{0} } = - { {"73" rSup { size 8{2} } } over {4} } "13" "." 6" eV"= - "18" "." 1" keV"} {}

Thus,

E K α = 18 .1 keV ( 72.5 keV ) = 54.4 keV.

Discussion

This large photon energy is typical of characteristic x rays from heavy elements. It is large compared with other atomic emissions because it is produced when an inner-shell vacancy is filled, and inner-shell electrons are tightly bound. Characteristic x ray energies become progressively larger for heavier elements because their energy increases approximately as Z 2 size 12{Z rSup { size 8{2} } } {} . Significant accelerating voltage is needed to create these inner-shell vacancies. In the case of tungsten, at least 72.5 kV is needed, because other shells are filled and you cannot simply bump one electron to a higher filled shell. Tungsten is a common anode material in x-ray tubes; so much of the energy of the impinging electrons is absorbed, raising its temperature, that a high-melting-point material like tungsten is required.

Medical and other diagnostic uses of x-rays

All of us can identify diagnostic uses of x-ray photons. Among these are the universal dental and medical x rays that have become an essential part of medical diagnostics. (See [link] and [link] .) X rays are also used to inspect our luggage at airports, as shown in [link] , and for early detection of cracks in crucial aircraft components. An x ray is not only a noun meaning high-energy photon, it also is an image produced by x rays, and it has been made into a familiar verb—to be x-rayed.

Questions & Answers

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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