X-ray tubes and tube spectra

X-rays were discovered by W. Rontgen in Würzburg, Germany, in 1895. The modern x-ray tube operates in the same manner as the one that Rontgen used. Electrons are "boiled" off of a heated thoriated tungsten filament and accelerated toward a target or anode where the x-rays are produced. The Coolidge-type x-ray tube is sealed and operates under a relatively high vacuum. The focusing cup, held at a negative potential, helps to direct the electrons toward the anode. To adjust the x-ray beam intensity, the tube current and/or the tube voltage can be changed, but the IV power is limited because of the extensive heat development in the target material. About 99% of the energy input is converted to heat. As a result, the anode must be water-cooled. The energy of the electrons as they hit the target is given by eV = hc/ because of their dual wave-particle nature. = hc/eV = 12.4/V (V in kV and in Å).

White or continuous radiation:

White or continuous radiation is produced by the deceleration of electrons as they hit the target. (Decelerating charged particles give up their energy in the form of radiation.) This radiation is also called Bremstrahlung or "braking" radiation. A spectrum of continuous radiation is obtained since the electrons can exchange any or all of their energy upon interaction with the atoms in the target. The shortest wavelength (_SWL) in the spectrum arises from an exchange of all the energy of an electron, which is a function of the applied voltage. The total intensity of the continuous radiation is I_conts = AiZV^~2 , where A is a constant, i is the tube current, and Z is the atomic number of the target. There is only one common technique that specifically employs this type of radiation - the Laue technique for single crystals; an x-ray tube with a target of high atomic number is generally used for Laue photographs.

Characteristic radiation:

Characteristic radiation is produced through the ejection, upon collision, of an electron from one of the energy levels of the atoms of the target material; the x-rays are produced when the empty level is filled again by a transition of an electron from a higher level to the empty level. (Decelerating charged particles give up their energy in the form of radiation.) This type of radiation is characteristic of the target material. Different target materials give off characteristic x-rays of different wavelengths according to Moseley's law

where C and are constants. Transitions that terminate on the K level (and in a few cases the L levels) give rise to x-rays with wavelengths in the useful range of about 0.5-5 Å. There are two L K transitions (which give K₁ and K₂ x-rays) and one MK transition (which gives K x-rays) allowed which are important.

The wavelengths of the K₁ and K₂ x-rays are so close together that they are frequently not very well resolved in the peaks in x-ray diffraction patterns, and therefore, to interpret such measurements, a weighted average of the wavelengths is used:

A minimum voltage (V_crit = 12.4/ _K, where _K is the wavelength of the "K edge") on the x-ray tube is required in order to obtain emission of the characteristic radiation since the incoming electrons must have an energy great enough to eject the electrons from the atoms of the target material. For example, to calculate the minimum voltage at which a iron target x-ray tube must be operated in order to produce the characteristic x-rays for iron, we look up the K edge for iron and calculate:

12.4 Å/kV

1.743 Å = 7.11 kV.

The intensity of the characteristic radiation is given by I_char = Ai(V-V_crit)^~1.5. This equation shows that the intensity of the characteristic radiation at V < V_crit is zero, and that it increases as V increases above V_crit.

The characteristic and continuous spectra superimpose upon one another, and the resulting wavelength distribution is what the sample usually sees.

SS#_________________________________

Grade_____________/10

MatSE 430 - Problem 1 - Due 8/31/00
X-ray Spectra

The problem below will help you fix on the above concepts, and help you understand the setup and the operating conditions for your x-ray generators.

a. Sketch the curve of intensity versus wavelength for a Cu target X-ray tube operated at 5 milliamperes and 5, 10, and 25 kilovolts.

To sketch the continuous radiation component, note the following.

For the characteristic lines of the spectra, refer to the appropriate appendix in Jenkins & Snyder or Cullity. Also:

b. How would the curve for 5 kV change if the X-ray tube current is raised to 10 mA?

Problem 2

Braggs' law calculations

a. Find 2_hkl for d_hkl = 2.00, 1.50, and 0.707 Å for CuK₁ radiation.

b. For CuK₁ radiation, suppose that the smallest 2 at which a measurement can be made is 10°. What, then, is the largest d that can be measured? If 2_max = 160°, what is the smallest d that can be measured?

c. For a cubic crystal, the interplanar spacings can be calculated from the lattice parameter (the length of the cubic cell edge), a, and the Miller indices (hkl) of the planes according to the equation:

Combine this equation with Braggs' law:

and determine the 2 angles at which the first three reflections for nickel metal occur for CuKa₁ radiation. For this F cubic metal, the first three reflections are (111), (200), and (220), and a = 3.5239 Å.

Problem 3

More Braggs' law calculations

Calculate and list in order of increasing angle the (hkl) and 2 values for the first three reflections for the following, using
CuK radiation.

a. primitive cubic, a = 3.00 Å

The next two calculations involve a tetragonal lattice. The unit cell is not a cube; you can picture a tetragonal unit cell as a cube that has been either squashed or extended along one lattice translation direction. As a result, one of the lattice translation distances (c) is no longer equal to the other two (|a| = |b|). The equation which relates h, k, l, a, and c to d_hkl is:

b. primitive tetragonal, a = 2.00, c = 3.00 Å

c. primitive tetragonal, a = 3.00, c = 2.00 Å