Continuous and characteristic X-ray spectra
Explain how bremsstrahlung and characteristic lines form an X-ray spectrum and how the accelerating voltage sets the cut-off wavelength, as optional enrichment.
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The X-rays from a tube form a smooth continuous spectrum with a few sharp lines on top of it. This page explains where each part comes from and how each one responds to the tube voltage, the tube current and the target metal. X-ray spectra go slightly beyond the H2 syllabus.
Two processes in the same target
Bremsstrahlung means braking radiation. When a fast electron is deflected and slowed in the electric fields of the target, radiation can carry away a variable amount of its kinetic energy. Different interactions give different photon energies, producing a continuous background.
Characteristic X-rays arise when an inner-shell vacancy is filled by an electron from a higher-energy shell. A radiative transition emits a photon with energy equal to the difference between the two atomic levels. Those gaps depend on the target element, so the spectrum can contain narrow, target-specific lines superimposed on the background. Vacancy filling can also transfer energy to another electron instead of emitting a photon.
Both processes occur during electron bombardment. The appearance of characteristic lines does not replace or remove bremsstrahlung.
Guided practice 1
Continuous or characteristic?
Try the calculation
Try this before viewing the solution
Hints
Hint 1
choose the energy relation
Identify the smooth background and the narrow lines, then connect each to its production mechanism.Show solution step by step
Energy of A
Method
A carries 12.4 keV.Reason
Use energy equal to hc/λ.Working
Answer
EA = 1240/0.100 = 12400 eV.Energy of B
Method
B carries 15.5 keV.Reason
Its shorter wavelength gives greater energy.Working
Answer
EB = 1240/0.080 = 15500 eV.Classify A
Method
A belongs to the continuous background in the stated spectrum.Reason
Its wavelength matches neither characteristic line.Working
Answer
0.100 nm is between the listed lines.Interpret B
Method
B’s wavelength alone does not establish its mechanism.Reason
Bremsstrahlung can contribute at the characteristic-line wavelength too.Working
Answer
The line adds photons above the background; it does not remove that background.
Quick check
Check your understanding 1 Lines on a background
A tube’s characteristic lines disappear when its voltage is reduced below the energy needed to create the relevant inner-shell vacancies. Can a continuous X-ray background remain? Explain the two processes separately.
Show answer
Yes. The incoming electrons can still slow down in the target and emit bremsstrahlung with a range of photon energies. The missing characteristic lines require vacancies that are no longer created at the reduced electron energy. Losing those lines does not remove the separate bremsstrahlung process, though its energy limit and output change with voltage.
Read the voltage limit separately from the lines
Copper spectra below and above the K-shell threshold
The 8.5 kV background begins at about 0.146 nm and has no K-series spikes because its electron energy is below the approximately 9 keV vacancy threshold. The 20 kV background begins at 0.0620 nm and has representative copper K alpha one and K beta one lines near 0.154 and 0.139 nm.
Scroll across the figure to read all labels.
View figure data
| Series | Photon wavelength (nm) | Energy output per wavelength interval (relative units) |
|---|---|---|
| 8.5 kV: continuous background | 0.000000 | 0.0000000 |
| 8.5 kV: continuous background | 0.002000 | 0.0000000 |
| 8.5 kV: continuous background | 0.004000 | 0.0000000 |
| 8.5 kV: continuous background | 0.006000 | 0.0000000 |
| 8.5 kV: continuous background | 0.008000 | 0.0000000 |
| 8.5 kV: continuous background | 0.010000 | 0.0000000 |
| 8.5 kV: continuous background | 0.012000 | 0.0000000 |
| 8.5 kV: continuous background | 0.014000 | 0.0000000 |
| 8.5 kV: continuous background | 0.016000 | 0.0000000 |
| 8.5 kV: continuous background | 0.018000 | 0.0000000 |
| 8.5 kV: continuous background | 0.020000 | 0.0000000 |
| 8.5 kV: continuous background | 0.022000 | 0.0000000 |
| 8.5 kV: continuous background | 0.024000 | 0.0000000 |
| 8.5 kV: continuous background | 0.026000 | 0.0000000 |
| 8.5 kV: continuous background | 0.028000 | 0.0000000 |
| 8.5 kV: continuous background | 0.030000 | 0.0000000 |
| 8.5 kV: continuous background | 0.032000 | 0.0000000 |
| 8.5 kV: continuous background | 0.034000 | 0.0000000 |
| 8.5 kV: continuous background | 0.036000 | 0.0000000 |
| 8.5 kV: continuous background | 0.038000 | 0.0000000 |
| 8.5 kV: continuous background | 0.040000 | 0.0000000 |
| 8.5 kV: continuous background | 0.042000 | 0.0000000 |
| 8.5 kV: continuous background | 0.044000 | 0.0000000 |
| 8.5 kV: continuous background | 0.046000 | 0.0000000 |
| 8.5 kV: continuous background | 0.048000 | 0.0000000 |
| 8.5 kV: continuous background | 0.050000 | 0.0000000 |
| 8.5 kV: continuous background | 0.052000 | 0.0000000 |
| 8.5 kV: continuous background | 0.054000 | 0.0000000 |
| 8.5 kV: continuous background | 0.056000 | 0.0000000 |
| 8.5 kV: continuous background | 0.058000 | 0.0000000 |
| 8.5 kV: continuous background | 0.060000 | 0.0000000 |
| 8.5 kV: continuous background | 0.062000 | 0.0000000 |
| 8.5 kV: continuous background | 0.064000 | 0.0000000 |
| 8.5 kV: continuous background | 0.066000 | 0.0000000 |
| 8.5 kV: continuous background | 0.068000 | 0.0000000 |
| 8.5 kV: continuous background | 0.070000 | 0.0000000 |
| 8.5 kV: continuous background | 0.072000 | 0.0000000 |
| 8.5 kV: continuous background | 0.074000 | 0.0000000 |
| 8.5 kV: continuous background | 0.076000 | 0.0000000 |
| 8.5 kV: continuous background | 0.078000 | 0.0000000 |
| 8.5 kV: continuous background | 0.080000 | 0.0000000 |
| 8.5 kV: continuous background | 0.082000 | 0.0000000 |
| 8.5 kV: continuous background | 0.084000 | 0.0000000 |
| 8.5 kV: continuous background | 0.086000 | 0.0000000 |
| 8.5 kV: continuous background | 0.088000 | 0.0000000 |
| 8.5 kV: continuous background | 0.090000 | 0.0000000 |
| 8.5 kV: continuous background | 0.092000 | 0.0000000 |
| 8.5 kV: continuous background | 0.093000 | 0.0000000 |
| 8.5 kV: continuous background | 0.094000 | 0.0000000 |
| 8.5 kV: continuous background | 0.096000 | 0.0000000 |
| 8.5 kV: continuous background | 0.098000 | 0.0000000 |
| 8.5 kV: continuous background | 0.100000 | 0.0000000 |
| 8.5 kV: continuous background | 0.102000 | 0.0000000 |
| 8.5 kV: continuous background | 0.104000 | 0.0000000 |
| 8.5 kV: continuous background | 0.106000 | 0.0000000 |
| 8.5 kV: continuous background | 0.108000 | 0.0000000 |
| 8.5 kV: continuous background | 0.110000 | 0.0000000 |
| 8.5 kV: continuous background | 0.112000 | 0.0000000 |
| 8.5 kV: continuous background | 0.114000 | 0.0000000 |
| 8.5 kV: continuous background | 0.116000 | 0.0000000 |
| 8.5 kV: continuous background | 0.118000 | 0.0000000 |
| 8.5 kV: continuous background | 0.120000 | 0.0000000 |
| 8.5 kV: continuous background | 0.122000 | 0.0000000 |
| 8.5 kV: continuous background | 0.124000 | 0.0000000 |
| 8.5 kV: continuous background | 0.126000 | 0.0000000 |
| 8.5 kV: continuous background | 0.128000 | 0.0000000 |
| 8.5 kV: continuous background | 0.130000 | 0.0000000 |
| 8.5 kV: continuous background | 0.132000 | 0.0000000 |
| 8.5 kV: continuous background | 0.134000 | 0.0000000 |
| 8.5 kV: continuous background | 0.136000 | 0.0000000 |
| 8.5 kV: continuous background | 0.138000 | 0.0000000 |
| 8.5 kV: continuous background | 0.139241 | 0.0000000 |
| 8.5 kV: continuous background | 0.140000 | 0.0000000 |
| 8.5 kV: continuous background | 0.142000 | 0.0000000 |
| 8.5 kV: continuous background | 0.144000 | 0.0000000 |
| 8.5 kV: continuous background | 0.145882 | 0.0000000 |
| 8.5 kV: continuous background | 0.146000 | 0.0004169 |
| 8.5 kV: continuous background | 0.148000 | 0.0072035 |
| 8.5 kV: continuous background | 0.150000 | 0.0134540 |
| 8.5 kV: continuous background | 0.152000 | 0.0192101 |
| 8.5 kV: continuous background | 0.154000 | 0.0245100 |
| 8.5 kV: continuous background | 0.154079 | 0.0247104 |
| 8.5 kV: continuous background | 0.156000 | 0.0293888 |
| 8.5 kV: continuous background | 0.158000 | 0.0338784 |
| 8.5 kV: continuous background | 0.160000 | 0.0380083 |
| 8.5 kV: continuous background | 0.162000 | 0.0418054 |
| 8.5 kV: continuous background | 0.164000 | 0.0452946 |
| 8.5 kV: continuous background | 0.166000 | 0.0484986 |
| 8.5 kV: continuous background | 0.168000 | 0.0514384 |
| 8.5 kV: continuous background | 0.170000 | 0.0541333 |
| 8.5 kV: continuous background | 0.172000 | 0.0566011 |
| 8.5 kV: continuous background | 0.174000 | 0.0588582 |
| 8.5 kV: continuous background | 0.176000 | 0.0609199 |
| 8.5 kV: continuous background | 0.178000 | 0.0628000 |
| 8.5 kV: continuous background | 0.180000 | 0.0645116 |
| 8.5 kV: continuous background | 0.182000 | 0.0660665 |
| 8.5 kV: continuous background | 0.184000 | 0.0674759 |
| 8.5 kV: continuous background | 0.186000 | 0.0687500 |
| 8.5 kV: continuous background | 0.188000 | 0.0698983 |
| 8.5 kV: continuous background | 0.190000 | 0.0709296 |
| 8.5 kV: continuous background | 0.192000 | 0.0718521 |
| 8.5 kV: continuous background | 0.194000 | 0.0726734 |
| 8.5 kV: continuous background | 0.196000 | 0.0734004 |
| 8.5 kV: continuous background | 0.198000 | 0.0740398 |
| 8.5 kV: continuous background | 0.200000 | 0.0745976 |
| 8.5 kV: continuous background | 0.202000 | 0.0750795 |
| 8.5 kV: continuous background | 0.204000 | 0.0754907 |
| 8.5 kV: continuous background | 0.206000 | 0.0758362 |
| 8.5 kV: continuous background | 0.208000 | 0.0761204 |
| 8.5 kV: continuous background | 0.210000 | 0.0763477 |
| 8.5 kV: continuous background | 0.212000 | 0.0765219 |
| 8.5 kV: continuous background | 0.214000 | 0.0766469 |
| 8.5 kV: continuous background | 0.216000 | 0.0767259 |
| 8.5 kV: continuous background | 0.218000 | 0.0767623 |
| 8.5 kV: continuous background | 0.218824 | 0.0767656 |
| 8.5 kV: continuous background | 0.220000 | 0.0767591 |
| 8.5 kV: continuous background | 0.222000 | 0.0767189 |
| 8.5 kV: continuous background | 0.224000 | 0.0766445 |
| 8.5 kV: continuous background | 0.226000 | 0.0765383 |
| 8.5 kV: continuous background | 0.228000 | 0.0764026 |
| 8.5 kV: continuous background | 0.230000 | 0.0762394 |
| 8.5 kV: continuous background | 0.232000 | 0.0760509 |
| 8.5 kV: continuous background | 0.234000 | 0.0758388 |
| 8.5 kV: continuous background | 0.236000 | 0.0756049 |
| 8.5 kV: continuous background | 0.238000 | 0.0753508 |
| 8.5 kV: continuous background | 0.240000 | 0.0750781 |
| 8.5 kV: continuous background | 0.242000 | 0.0747882 |
| 8.5 kV: continuous background | 0.244000 | 0.0744824 |
| 8.5 kV: continuous background | 0.246000 | 0.0741620 |
| 8.5 kV: continuous background | 0.248000 | 0.0738281 |
| 8.5 kV: continuous background | 0.250000 | 0.0734819 |
| 8.5 kV: continuous background | 0.252000 | 0.0731244 |
| 8.5 kV: continuous background | 0.254000 | 0.0727565 |
| 8.5 kV: continuous background | 0.256000 | 0.0723791 |
| 8.5 kV: continuous background | 0.258000 | 0.0719931 |
| 8.5 kV: continuous background | 0.260000 | 0.0715993 |
| 8.5 kV: continuous background | 0.262000 | 0.0711984 |
| 8.5 kV: continuous background | 0.264000 | 0.0707912 |
| 8.5 kV: continuous background | 0.266000 | 0.0703782 |
| 8.5 kV: continuous background | 0.268000 | 0.0699601 |
| 8.5 kV: continuous background | 0.270000 | 0.0695374 |
| 8.5 kV: continuous background | 0.272000 | 0.0691107 |
| 8.5 kV: continuous background | 0.274000 | 0.0686805 |
| 8.5 kV: continuous background | 0.276000 | 0.0682472 |
| 8.5 kV: continuous background | 0.278000 | 0.0678114 |
| 8.5 kV: continuous background | 0.280000 | 0.0673733 |
| 8.5 kV: continuous background | 0.282000 | 0.0669334 |
| 8.5 kV: continuous background | 0.284000 | 0.0664921 |
| 8.5 kV: continuous background | 0.286000 | 0.0660497 |
| 8.5 kV: continuous background | 0.288000 | 0.0656065 |
| 8.5 kV: continuous background | 0.290000 | 0.0651627 |
| 8.5 kV: continuous background | 0.292000 | 0.0647188 |
| 8.5 kV: continuous background | 0.294000 | 0.0642748 |
| 8.5 kV: continuous background | 0.296000 | 0.0638312 |
| 8.5 kV: continuous background | 0.298000 | 0.0633880 |
| 8.5 kV: continuous background | 0.300000 | 0.0629455 |
| 8.5 kV: continuous background | 0.302000 | 0.0625039 |
| 8.5 kV: continuous background | 0.304000 | 0.0620634 |
| 8.5 kV: continuous background | 0.306000 | 0.0616242 |
| 8.5 kV: continuous background | 0.308000 | 0.0611863 |
| 8.5 kV: continuous background | 0.310000 | 0.0607500 |
| 8.5 kV: continuous background | 0.312000 | 0.0603154 |
| 8.5 kV: continuous background | 0.314000 | 0.0598826 |
| 8.5 kV: continuous background | 0.316000 | 0.0594517 |
| 8.5 kV: continuous background | 0.318000 | 0.0590228 |
| 8.5 kV: continuous background | 0.320000 | 0.0585961 |
| 8.5 kV: continuous background | 0.322000 | 0.0581716 |
| 8.5 kV: continuous background | 0.324000 | 0.0577495 |
| 8.5 kV: continuous background | 0.326000 | 0.0573297 |
| 8.5 kV: continuous background | 0.328000 | 0.0569124 |
| 8.5 kV: continuous background | 0.330000 | 0.0564976 |
| 8.5 kV: continuous background | 0.332000 | 0.0560853 |
| 8.5 kV: continuous background | 0.334000 | 0.0556758 |
| 8.5 kV: continuous background | 0.336000 | 0.0552689 |
| 8.5 kV: continuous background | 0.338000 | 0.0548647 |
| 8.5 kV: continuous background | 0.340000 | 0.0544633 |
| 8.5 kV: continuous background | 0.342000 | 0.0540648 |
| 8.5 kV: continuous background | 0.344000 | 0.0536691 |
| 8.5 kV: continuous background | 0.346000 | 0.0532762 |
| 8.5 kV: continuous background | 0.348000 | 0.0528862 |
| 8.5 kV: continuous background | 0.350000 | 0.0524992 |
| 20 kV: continuous background | 0.000000 | 0.0000000 |
| 20 kV: continuous background | 0.002000 | 0.0000000 |
| 20 kV: continuous background | 0.004000 | 0.0000000 |
| 20 kV: continuous background | 0.006000 | 0.0000000 |
| 20 kV: continuous background | 0.008000 | 0.0000000 |
| 20 kV: continuous background | 0.010000 | 0.0000000 |
| 20 kV: continuous background | 0.012000 | 0.0000000 |
| 20 kV: continuous background | 0.014000 | 0.0000000 |
| 20 kV: continuous background | 0.016000 | 0.0000000 |
| 20 kV: continuous background | 0.018000 | 0.0000000 |
| 20 kV: continuous background | 0.020000 | 0.0000000 |
| 20 kV: continuous background | 0.022000 | 0.0000000 |
| 20 kV: continuous background | 0.024000 | 0.0000000 |
| 20 kV: continuous background | 0.026000 | 0.0000000 |
| 20 kV: continuous background | 0.028000 | 0.0000000 |
| 20 kV: continuous background | 0.030000 | 0.0000000 |
| 20 kV: continuous background | 0.032000 | 0.0000000 |
| 20 kV: continuous background | 0.034000 | 0.0000000 |
| 20 kV: continuous background | 0.036000 | 0.0000000 |
| 20 kV: continuous background | 0.038000 | 0.0000000 |
| 20 kV: continuous background | 0.040000 | 0.0000000 |
| 20 kV: continuous background | 0.042000 | 0.0000000 |
| 20 kV: continuous background | 0.044000 | 0.0000000 |
| 20 kV: continuous background | 0.046000 | 0.0000000 |
| 20 kV: continuous background | 0.048000 | 0.0000000 |
| 20 kV: continuous background | 0.050000 | 0.0000000 |
| 20 kV: continuous background | 0.052000 | 0.0000000 |
| 20 kV: continuous background | 0.054000 | 0.0000000 |
| 20 kV: continuous background | 0.056000 | 0.0000000 |
| 20 kV: continuous background | 0.058000 | 0.0000000 |
| 20 kV: continuous background | 0.060000 | 0.0000000 |
| 20 kV: continuous background | 0.062000 | 0.0000000 |
| 20 kV: continuous background | 0.064000 | 0.1979599 |
| 20 kV: continuous background | 0.066000 | 0.3610068 |
| 20 kV: continuous background | 0.068000 | 0.4951214 |
| 20 kV: continuous background | 0.070000 | 0.6051778 |
| 20 kV: continuous background | 0.072000 | 0.6951678 |
| 20 kV: continuous background | 0.074000 | 0.7683750 |
| 20 kV: continuous background | 0.076000 | 0.8275122 |
| 20 kV: continuous background | 0.078000 | 0.8748293 |
| 20 kV: continuous background | 0.080000 | 0.9121992 |
| 20 kV: continuous background | 0.082000 | 0.9411863 |
| 20 kV: continuous background | 0.084000 | 0.9631013 |
| 20 kV: continuous background | 0.086000 | 0.9790459 |
| 20 kV: continuous background | 0.088000 | 0.9899482 |
| 20 kV: continuous background | 0.090000 | 0.9965926 |
| 20 kV: continuous background | 0.092000 | 0.9996430 |
| 20 kV: continuous background | 0.093000 | 1.0000000 |
| 20 kV: continuous background | 0.094000 | 0.9996629 |
| 20 kV: continuous background | 0.096000 | 0.9971313 |
| 20 kV: continuous background | 0.098000 | 0.9924564 |
| 20 kV: continuous background | 0.100000 | 0.9859860 |
| 20 kV: continuous background | 0.102000 | 0.9780175 |
| 20 kV: continuous background | 0.104000 | 0.9688051 |
| 20 kV: continuous background | 0.106000 | 0.9585665 |
| 20 kV: continuous background | 0.108000 | 0.9474880 |
| 20 kV: continuous background | 0.110000 | 0.9357295 |
| 20 kV: continuous background | 0.112000 | 0.9234281 |
| 20 kV: continuous background | 0.114000 | 0.9107013 |
| 20 kV: continuous background | 0.116000 | 0.8976498 |
| 20 kV: continuous background | 0.118000 | 0.8843601 |
| 20 kV: continuous background | 0.120000 | 0.8709063 |
| 20 kV: continuous background | 0.122000 | 0.8573515 |
| 20 kV: continuous background | 0.124000 | 0.8437500 |
| 20 kV: continuous background | 0.126000 | 0.8301479 |
| 20 kV: continuous background | 0.128000 | 0.8165846 |
| 20 kV: continuous background | 0.130000 | 0.8030933 |
| 20 kV: continuous background | 0.132000 | 0.7897023 |
| 20 kV: continuous background | 0.134000 | 0.7764353 |
| 20 kV: continuous background | 0.136000 | 0.7633121 |
| 20 kV: continuous background | 0.138000 | 0.7503493 |
| 20 kV: continuous background | 0.139241 | 0.7423916 |
| 20 kV: continuous background | 0.140000 | 0.7375605 |
| 20 kV: continuous background | 0.142000 | 0.7249568 |
| 20 kV: continuous background | 0.144000 | 0.7125470 |
| 20 kV: continuous background | 0.145882 | 0.7010508 |
| 20 kV: continuous background | 0.146000 | 0.7003383 |
| 20 kV: continuous background | 0.148000 | 0.6883360 |
| 20 kV: continuous background | 0.150000 | 0.6765440 |
| 20 kV: continuous background | 0.152000 | 0.6649652 |
| 20 kV: continuous background | 0.154000 | 0.6536012 |
| 20 kV: continuous background | 0.154079 | 0.6531570 |
| 20 kV: continuous background | 0.156000 | 0.6424528 |
| 20 kV: continuous background | 0.158000 | 0.6315200 |
| 20 kV: continuous background | 0.160000 | 0.6208022 |
| 20 kV: continuous background | 0.162000 | 0.6102982 |
| 20 kV: continuous background | 0.164000 | 0.6000063 |
| 20 kV: continuous background | 0.166000 | 0.5899242 |
| 20 kV: continuous background | 0.168000 | 0.5800497 |
| 20 kV: continuous background | 0.170000 | 0.5703798 |
| 20 kV: continuous background | 0.172000 | 0.5609117 |
| 20 kV: continuous background | 0.174000 | 0.5516421 |
| 20 kV: continuous background | 0.176000 | 0.5425678 |
| 20 kV: continuous background | 0.178000 | 0.5336852 |
| 20 kV: continuous background | 0.180000 | 0.5249907 |
| 20 kV: continuous background | 0.182000 | 0.5164809 |
| 20 kV: continuous background | 0.184000 | 0.5081519 |
| 20 kV: continuous background | 0.186000 | 0.5000000 |
| 20 kV: continuous background | 0.188000 | 0.4920216 |
| 20 kV: continuous background | 0.190000 | 0.4842129 |
| 20 kV: continuous background | 0.192000 | 0.4765701 |
| 20 kV: continuous background | 0.194000 | 0.4690897 |
| 20 kV: continuous background | 0.196000 | 0.4617679 |
| 20 kV: continuous background | 0.198000 | 0.4546011 |
| 20 kV: continuous background | 0.200000 | 0.4475858 |
| 20 kV: continuous background | 0.202000 | 0.4407183 |
| 20 kV: continuous background | 0.204000 | 0.4339953 |
| 20 kV: continuous background | 0.206000 | 0.4274133 |
| 20 kV: continuous background | 0.208000 | 0.4209689 |
| 20 kV: continuous background | 0.210000 | 0.4146589 |
| 20 kV: continuous background | 0.212000 | 0.4084800 |
| 20 kV: continuous background | 0.214000 | 0.4024291 |
| 20 kV: continuous background | 0.216000 | 0.3965031 |
| 20 kV: continuous background | 0.218000 | 0.3906990 |
| 20 kV: continuous background | 0.218824 | 0.3883438 |
| 20 kV: continuous background | 0.220000 | 0.3850137 |
| 20 kV: continuous background | 0.222000 | 0.3794445 |
| 20 kV: continuous background | 0.224000 | 0.3739884 |
| 20 kV: continuous background | 0.226000 | 0.3686427 |
| 20 kV: continuous background | 0.228000 | 0.3634048 |
| 20 kV: continuous background | 0.230000 | 0.3582720 |
| 20 kV: continuous background | 0.232000 | 0.3532418 |
| 20 kV: continuous background | 0.234000 | 0.3483117 |
| 20 kV: continuous background | 0.236000 | 0.3434792 |
| 20 kV: continuous background | 0.238000 | 0.3387419 |
| 20 kV: continuous background | 0.240000 | 0.3340977 |
| 20 kV: continuous background | 0.242000 | 0.3295441 |
| 20 kV: continuous background | 0.244000 | 0.3250791 |
| 20 kV: continuous background | 0.246000 | 0.3207005 |
| 20 kV: continuous background | 0.248000 | 0.3164063 |
| 20 kV: continuous background | 0.250000 | 0.3121943 |
| 20 kV: continuous background | 0.252000 | 0.3080627 |
| 20 kV: continuous background | 0.254000 | 0.3040096 |
| 20 kV: continuous background | 0.256000 | 0.3000330 |
| 20 kV: continuous background | 0.258000 | 0.2961312 |
| 20 kV: continuous background | 0.260000 | 0.2923023 |
| 20 kV: continuous background | 0.262000 | 0.2885448 |
| 20 kV: continuous background | 0.264000 | 0.2848569 |
| 20 kV: continuous background | 0.266000 | 0.2812370 |
| 20 kV: continuous background | 0.268000 | 0.2776834 |
| 20 kV: continuous background | 0.270000 | 0.2741948 |
| 20 kV: continuous background | 0.272000 | 0.2707695 |
| 20 kV: continuous background | 0.274000 | 0.2674061 |
| 20 kV: continuous background | 0.276000 | 0.2641032 |
| 20 kV: continuous background | 0.278000 | 0.2608594 |
| 20 kV: continuous background | 0.280000 | 0.2576734 |
| 20 kV: continuous background | 0.282000 | 0.2545438 |
| 20 kV: continuous background | 0.284000 | 0.2514694 |
| 20 kV: continuous background | 0.286000 | 0.2484489 |
| 20 kV: continuous background | 0.288000 | 0.2454811 |
| 20 kV: continuous background | 0.290000 | 0.2425649 |
| 20 kV: continuous background | 0.292000 | 0.2396991 |
| 20 kV: continuous background | 0.294000 | 0.2368826 |
| 20 kV: continuous background | 0.296000 | 0.2341143 |
| 20 kV: continuous background | 0.298000 | 0.2313931 |
| 20 kV: continuous background | 0.300000 | 0.2287180 |
| 20 kV: continuous background | 0.302000 | 0.2260880 |
| 20 kV: continuous background | 0.304000 | 0.2235022 |
| 20 kV: continuous background | 0.306000 | 0.2209595 |
| 20 kV: continuous background | 0.308000 | 0.2184591 |
| 20 kV: continuous background | 0.310000 | 0.2160000 |
| 20 kV: continuous background | 0.312000 | 0.2135814 |
| 20 kV: continuous background | 0.314000 | 0.2112024 |
| 20 kV: continuous background | 0.316000 | 0.2088621 |
| 20 kV: continuous background | 0.318000 | 0.2065598 |
| 20 kV: continuous background | 0.320000 | 0.2042946 |
| 20 kV: continuous background | 0.322000 | 0.2020658 |
| 20 kV: continuous background | 0.324000 | 0.1998727 |
| 20 kV: continuous background | 0.326000 | 0.1977144 |
| 20 kV: continuous background | 0.328000 | 0.1955903 |
| 20 kV: continuous background | 0.330000 | 0.1934996 |
| 20 kV: continuous background | 0.332000 | 0.1914418 |
| 20 kV: continuous background | 0.334000 | 0.1894160 |
| 20 kV: continuous background | 0.336000 | 0.1874217 |
| 20 kV: continuous background | 0.338000 | 0.1854582 |
| 20 kV: continuous background | 0.340000 | 0.1835250 |
| 20 kV: continuous background | 0.342000 | 0.1816213 |
| 20 kV: continuous background | 0.344000 | 0.1797467 |
| 20 kV: continuous background | 0.346000 | 0.1779005 |
| 20 kV: continuous background | 0.348000 | 0.1760822 |
| 20 kV: continuous background | 0.350000 | 0.1742912 |
| 20 kV: copper Kα₁ | 0.154079 | 0.6531570 |
| 20 kV: copper Kα₁ | 0.154079 | 1.5500000 |
| 20 kV: copper Kβ₁ | 0.139241 | 0.7423916 |
| 20 kV: copper Kβ₁ | 0.139241 | 1.2000000 |
This comparison uses the same copper target and tube current at 8.5 kV and 20 kV. The continuous cut-off is calculated from λ min = hc/(eV): approximately 0.146 nm and 0.0620 nm respectively. The copper Kα and Kβ line positions are approximately 0.154 nm and 0.139 nm. The background is an ideal unfiltered model, and peak heights are schematic; this is not measured output from a particular tube.
The vertical axis represents energy output per wavelength interval, not photon energy or a count of electrons. Its height depends on how the spectrum is represented. Real windows, filtration and detector response alter the recorded shape; do not use the illustrated peak heights to infer line strengths.
Check your understanding 2 An endpoint is different from an average
Most incoming electron energy becomes target heat. Does that mean the ideal relation eV = hc/λ min should use the average emitted photon energy? State what the relation describes and its assumptions.
Show answer
No. The relation describes the upper photon-energy limit, assuming negligible initial electron kinetic energy, steady voltage and negligible recoil correction. Typical interactions share energy with other processes and produce lower-energy photons. Their average energy is not the endpoint. Voltage variation, attenuation and detector response matter when interpreting a real measured spectrum.
A vacancy needs more energy than the emitted line
To produce copper K-series radiation by electron impact, the incoming electron must have enough kinetic energy to eject a K-shell electron. Copper’s K-shell binding energy is about 9.0 keV. This exceeds the Kα photon energy, about 8.05 keV, because the filling electron was itself bound in a higher shell.
At 8.5 kV, the available energy exceeds the Kα photon energy but is still below the K-vacancy threshold. The continuum can extend to that photon energy, yet the characteristic K lines are absent. At 20 kV, K vacancies can be produced and K lines can appear. Not every impact produces a vacancy or an X-ray.
These are rounded values for copper. Only the main Kα and Kβ lines are shown; real spectra have further structure.
Guided practice 2
Doubling the accelerating voltage
Problem
Try this before viewing the solution
Hints
Hint 1
separate the mechanisms
Voltage fixes the maximum electron/photon energy; target energy-level gaps fix characteristic lines.Show solution step by step
Cut-off
Method
λ min halves.Reason
λ min = hc/(eV) is inversely proportional to voltage.Working
Answer
V × 2 ⇒ λ min/2Characteristic lines
Method
The characteristic peak wavelengths remain the same.Reason
The unchanged target retains the same inner-shell energy gaps.Working
Answer
Δ E target unchanged
Quick check
Spot the mistake 1
Cut-off or characteristic line?
Learner claim
Try this before viewing the solution
Show solution step by step
Classify the feature
Method
λ min is the continuous bremsstrahlung cutoff.Reason
It represents the maximum photon energy available from eV.Working
Answer
E max = hc/λ minCalculate energy
Method
Use the eV·nm shortcut.Reason
Units remain paired.Working
Answer
E max = 1240/0.015 = 8.27 × 10⁴ eV = 82.7 keVMap to voltage
Method
The required accelerating voltage is about 82.7 kV, or 83 kV.Reason
One electron gains V eV through V volts.Working
Answer
V ≈ 83 kV
Common mistake
Try it yourself 1
Changing the target material
Problem
Try this before viewing the solution
Hints
Hint 1
assign each feature to its control
Orient by matching the cutoff to eV and the lines to target atomic energy gaps.Show solution step by step
Cut-off
Method
λ min remains approximately unchanged.Reason
The unchanged accelerating voltage fixes the maximum electron and photon energy.Working
Answer
λ min = hc/(eV)Characteristic peaks
Method
Characteristic wavelengths change, and some line series may not be excited at the chosen voltage.Reason
Different target atoms have different atomic energy gaps and vacancy thresholds.Working
Answer
Δ E atomic changes ⇒ λ line changes
Quick check
Predict changes one control at a time
| Change | Continuous spectrum | Characteristic lines |
|---|---|---|
| Raise voltage at fixed tube current and target | Shorter-wavelength limit; output usually rises | Positions stay fixed when the same lines are present; new lines can appear above vacancy thresholds |
| Raise tube current at fixed voltage and target | Output increases; ideal cut-off stays fixed | Intensities can increase; positions stay fixed |
| Change target at fixed voltage | Ideal cut-off stays fixed; background shape and yield can change | Available line energies and vacancy thresholds change |
The cutoff is an energy limit, not a characteristic peak. A characteristic line identifies an atomic energy gap, not the accelerating voltage on its own.
Optional: the background model used in the graph
An ideal, unattenuated thick-target Kramers model has energy output per photon-energy interval proportional to E max -E for 0 < E < E max. Converting to wavelength requires the factor |dE/dλ| = hc/λ². The displayed background is therefore proportional to (E max -hc/λ)/λ² for λ ≥ λ min, and zero below the cut-off. Both voltages use the same scale factor. The two characteristic spikes are added schematically at the target’s line wavelengths.
This model explains the plotted shape and quantity; solving it is not an H2 requirement.
Shell labels
K-series lines end in a K-shell vacancy: Kα commonly denotes an L-to-K transition, and Kβ an M-to-K transition. Different subshells give further closely spaced lines. These labels describe photon transitions, while the K-shell binding energy sets the electron-impact threshold for creating the vacancy.
Common mistakes
- Calling the cut-off wavelength a characteristic line. The cut-off is set by the voltage; the lines are set by the target element.
- Saying the characteristic lines move when the voltage changes. Their wavelengths depend only on the target.
- Thinking any voltage above the Kα photon energy produces K lines. The electrons must have enough energy to remove a K-shell electron.
- Reading the height of the graph as photon energy. It shows the energy output per wavelength interval.
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Syllabus and review details
No official syllabus alignment is listed for this lesson.