The document discusses the principles of X-ray diffraction (XRD) analysis, detailing the generation of X-rays via laboratory sources, specifically sealed-tube and rotating anode systems. It explains the concepts of white radiation and characteristic radiation, including their spectral lines, intensity, and the effects of various anode materials on the resulting wavelengths. Furthermore, it touches on the importance of filters in XRD setups to achieve monochromatic X-ray sources for accurate powder diffraction analysis.

1. X-Rays
The basic understanding of XRD analysis
Chandra Prakash Singh
Electromagnetic Spectrum

2.  Laboratory X-ray sources can be classified into two types: sealed-tube and rotating anode.
 Both may be used to generate monochromatic X-ray radiation and they basically differ only in the intensity of the radiation
produced.
A Typical X-ray Spectrum from a Copper Target
White Radiation
 X-rays are generated when matter is irradiated by a beam
of high-energy charged particles such as electrons.
 In the laboratory, a filament is heated to produce electrons
which are then accelerated in vacuum by a high electric
field in the range 20-60 kV towards a metal target, which
being positive is called the anode.
 The corresponding electric current is in the range 5-
100 mA.
 The process is extremely inefficient with 99% of the
energy of the beam being dissipated as heat in the target.
 The loss of energy of the electrons by collision with the
atoms usually takes place via multiple events. The result is
the production of a continuous spectrum of X-rays known
as white radiation.
Generation of X-Rays

3.  The maximum energy lost, E (max), determines the shortest wavelength, λ(min), that can be obtained according to the
equation
E = e V = h c / λ
Where:
e is the charge on the electron,
V is the accelerating voltage,
h is Planck's constant, and
c is the speed of light.
 A more practical form of this equation is given by
λ = 12.398 / V
Where:
V is in kilovolts and
λ is in Angstroms (1 Å = 0.1 nm).
 Thus, the higher the accelerating voltage of the X-ray generator, the shorter the minimum wavelength that can be obtained.
 The maximum in the intensity of the white radiation occurs at a wavelength that is roughly 1.5× λ(min).
 Longer wavelengths are obtained by multiple-collision processes.
 The total intensity, I(w) of the white radiation is approximately proportional to the filament current, i, the atomic number of
the anode target, Z, and the square of the accelerating voltage, V.

4.  When the energy of the accelerated electrons is higher
than a certain threshold value (which depends on the
metal anode), a second type of spectrum is obtained
superimposed on top of the white radiation. It is called
the characteristic radiation and is composed of discrete
peaks.
 The energy (and wavelength) of the peaks depends
solely on the metal used for the target and is due to the
ejection of an electron from one of the inner electron
shells of the metal atom.
 This results in an electron from a higher atomic level
dropping to the vacant level with the emission of an X-
ray photon characterised by the difference in energy
between the two levels.
Characteristic Radiation

5. The diagram show the electronic energy levels
for a copper atom

6.  The characteristic lines in this type of spectrum are called K, L, M,... and they correspond to transitions to orbitals with
principal quantum numbers 1, 2, 3,...
 When the two orbitals involved in the transition are adjacent (e.g. 2 → 1), the line is called α.
 When the two orbitals are separated by another shell (e.g. 3 → 1), the line is called β.
 Since the transition for β is bigger than for α, i.e. ΔEβ > ΔEα, then λβ < λα.
 This is demonstrated by the values of the Kα and Kα wavelengths in the table below for two common anode materials:
Anode Kα Kβ
Cu 1.54184 Å 1.39222 Å
Mo 0.71073 Å 0.63229 Å
 In the copper X-ray spectrum, only 2 characteristic lines are seen at low-energy resolution.
 However, at higher resolution the Kα1 line is readily seen to be a doublet, which is labelled as Kα1 and Kα2 where
ΔEα1 > ΔEα2.
 The splitting of the 2p orbitals in copper, i.e. the splitting of the energy levels LII and LIII, is very small (0.020 keV) and
so the two wavelengths Kα1 (= 1.54056 Å) and Kα2 (= 1.54439 Å) are very similar.

7. Spectral Line Shape
 The picture is actually a simplified version of reality
since a high-resolution analysis of the spectral lines
of, say, Cu Kα shows that both the α1 and α2 peaks
are distinctly asymmetric.
 An explanation of the origin of this asymmetry is
important in understanding the so-
called fundamental parameter approach to the
profile fitting of powder diffraction data peaks.
 The de-excitation process in which an outer
2p electron fills the inner 1s electron shell is fast
(≈ 10-12 s), but not fast enough to stop double
ionization events.
 In particular, the ejection of the initial 1s electron
can be followed by the loss of one of the 2s or 2p
electrons from the energy levels LI, LII, or LIII.
 The effect of the increased ionization on the atom is
to change slightly the energy gap between the K and
L levels resulting in slightly different wavelengths
for the emitted X-ray photon.
 The resulting peak asymmetry in the spectral
distribution of the Kα lines of copper is shown
in red in the diagram.
The dotted coloured lines represent individual spectral
contributions to the total.

8. Spectral Intensity
 The intensity of the Kα1 peak is almost exactly double the intensity of the Kα2 peak.
 The intensity of a K line is given approximately by the formula
IK = c i (V - VK)n
Where
i is the electron beam current,
c is a constant, and
VK is the excitation potential of the K line (as given earlier by VK = 12.398 [kV/Å] / λ ).
 The exponent n is approximately 1.5, but drops towards 1.0 when V > 2VK.
 The ratio IK : Iwhite is a maximum when the accelerating voltage V is approximately 4× the excitation potential VK.
 For a Cu Kα anode, where VK is 8.0 kV, run with a typical operating voltage of 40 kV, the Kα line is approximately
90× more intense than the white radiation of a similar wavelength.
 Thus the white radiation from a copper anode is too weak to be of any practical use for powder diffraction in the
laboratory.
 What about the intensity of the Kβ radiation?
Again considering a copper anode, the intensity of the Kα lines is approximately 5 times that of Kβ. Hence, all
instrumental setups are optimized around the Kα radiation, and preferably around Kα1 when high resolution
monochromators are used as part of the X-ray optics.

9. Choice of X-ray Target
 The wavelength, λ, of the characteristic line giving rise to a particular transition is given by Moseley's Law:
1 / λ = c (Z - σ)2
Where
c and σ are constants, and
Z is the atomic number of the metal used for the anode.
 From this equation it can seen that as the atomic number of the target increases, then the wavelength of the characteristic
radiation decreases.
 Since the target has to be metallic (so that it conducts electrons) and has to have a reasonably high melting point (40 kV
at 30 mA generates 1.2kW of heat), this limits the choice of anode material to chromium (Cr), iron (Fe), cobalt (Co),
copper (Cu), molybdenum (Mo), and a few other less commonly used materials for X-ray powder diffraction.
 The table below shows the Kα radiation for each element:
Anode Cr Fe Co Cu Mo Ag
Kα (Å) 2.29 1.94 1.79 1.54 0.71 0.56
 Copper anodes are by far the most common since copper gives the shortest wavelength above 1 Å.
 The wavelengths provided by molybdenum and silver are normally too short for most powder diffraction work in the
laboratory.
 Short wavelengths both scatter weakly and contract the diffraction pattern towards low Bragg angles with consequent loss
of d spacing accuracy and resolution.

10.  Metal foil filters are one way of achieving this.
 The photograph shows the typical metals used to filter X-
rays produced by a sealed X-ray tube, i.e. Ni, Fe, Mn, V,
or Zr.
 Filters preferentially reduce the intensity of the Kβ line in
the X-ray spectrum compared to Kα as explained below.
 Note that absorption filters cannot be used to remove the
unwanted Kα2 component from Kα radiation.
 Filters exploit the X-ray absorption edge of the particular
element.
 At wavelengths longer than the absorption edge (i.e. just
above the edge), the absorption of the X-rays is
considerably less than for wavelengths shorter than the
absorption edge (i.e. just below the edge) as shown below
for nickel metal:
X-Ray Filters
 The spectrum from a sealed X-ray tube is composed of several X-ray lines.
 Laboratory powder diffraction requires an X-ray source that is essentially monochromatic and so the Kβ line in the X-ray
spectrum needs to be removed.

11.  Note that the filter also removes much of the high energy
background radiation.
 The choice of filter material depends upon the choice of anode
material in the X-ray tube as shown in the following table:
Anode Cu Co Fe Cr Mo
Filter Ni Fe Mn V Zr
 From the table it can be seen that the ideal choice
of material for an X-ray filter is a metal whose
atomic number, Z, is one less than that of the
anode target metal for first row transition metals
(or two less for second row transition metals).
 The absorption edge of nickel metal at 1.488 Å lies between the Kα (λ = 1.542 Å) and Kβ (λ = 1.392 Å) X-ray spectral lines
of copper. Hence nickel foil of an appropriate thickness can be used to reduce the intensity of the Cu Kβ X-rays as shown:

12.  The optimum thickness, x of the filter can be determined from the mass-absorption law:
I(λ) / Io(λ) = exp{− (μ / ρ)λ ρx}
Where:
(μ / ρ) is the mass absorption coefficient at the wavelength λ,
ρ is the density of the material, which for nickel metal is 8.92 g/cm3,
I(λ) and Io(λ) are the transmitted and incident X-ray intensities, respectively.
 The mass absorption coefficients of nickel for Cu Kα and Cu Kβ are 49.2 and 286 cm2/g, respectively.
 The table below shows the percentage transmission for various thicknesses of nickel foil:
Thickness (cm) I / Io (%) for Cu Kα I / Io (%) for Cu Kβ Reduction Ratio
0.0010 64.5 7.8 8
0.0015 51.8 2.2 24
0.0020 41.6 0.6 68
0.0025 33.4 0.2 197
 It can be seen from the table that the optimum thickness has to be a compromise between reducing the intensity of the
unwanted Cu Kβ and reducing the intensity of the desired Cu Kα.
 Most commercial systems employing a nickel filter with a copper anode target will choose the thickness of the foil so as
to give a reduction ratio in the range 25:1 to 50:1, i.e. foils between 15 and 20 µm thick.
 From the table, it can be seen that this range of foil thickness will diminish the desired radiation by approximately a
factor of 2

13. Thanks
Chandra Prakash Singh