X-rays can be diffracted by crystals just in the same way as the visible light is diffracted by a diffraction grating; in other words, we can say that crystals can be used as diffraction gratings for the diffraction of X-rays. This important concept was first conceived by Von Laue in 1912, and subsequently tested by Freidrich and Knipping who demonstrated that an X-ray beam passing through a single crystal was indeed broken up into a collection of diffracted beams.
Our immediate goal with the diffraction of these rays by crystals is only in connection with the direct exploration of the interior of the crystals; that is, in connection with the fixations of the position of the atoms on the crystal lattice, the measurement of distances between atoms and the associated internal symmetry.
Such a study is possible because of the fact that the intensities of diffracted beams and their directions are related to the atomic arrangements in crystals. Thus, measurements of their intensities and directions would provide the desired information about crystals.
The Diffraction Phenomenon of X-Ray:
Diffraction occurs when a wave encounters a series of regularly spaced obstacles that (a) are capable of scattering the wave, and (b) have spacing that are comparable in magnitude to the wavelength. Furthermore, diffraction is a consequence of specific phase relationships that are established between two or more waves that have been scattered by the obstacles.
Consider waves 1 and 2 in Fig. 2.62 (a) which have the same wavelength (λ) and are in phase at point O-O’. Now, suppose that both waves are scattered in such a way that they traverse different paths. The phase relationship between waves, which will depend upon the difference in path length, is important. One possibility results when this path length difference is an integral number of wavelengths.
As given in Fig. 2.62 (a), these scattered waves (now labeled 1′ and 2′) are still in phase. They are said to mutually reinforce (or constructively interfere with) one another; and, when amplitudes are added, the wave shown on the right side of the figure results. This is manifestation of diffraction, and we refer to a diffracted beam as one composed of a large number of scattered waves that mutually reinforce one another.
Other phase relationships are possible between scattered waves that will not lead to this mutual reinforcement. The other extreme is that demonstrated in Fig. 2.62 (b) wherein the path length difference after scattering is some integral number of half wavelengths.
The scattered waves are out of phase—that is, corresponding amplitude cancel or destructively interfere (i.e., the resultant wave has zero amplitude), as indicated on the extreme right side of the Fig. 2.62. Of course, phase relationships intermediate between these two extremes exist, resulting in only partial reinforcement.
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To start with consider the influence of the X-rays on an atom. When the atom is exposed to a monochromatic beam of X-rays, the electric field vector of the radiation forces its electrons to carry out harmonic vibrations of a frequency equal to that of the incident beam and thus to undergo acceleration. These accelerated charges in turn re-emit the radiation at the frequency of their vibration, that is, at the incident wave frequency.
The emitted wave has a spherical wave front centered about the atom, so the energy goes off in all directions. Now, since in a crystal we are concerned with a group of atoms arranged in a regular pattern, let us consider a row of identical atoms upon which there falls a plane X-ray wave normally, according to Fig. 2.63.
Each atom of the row emits radiation. Assuming the incident wave crests to be parallel to the row of atoms, the envelope of the wavelets emitted by individual atoms forms new wave crests and one can see that besides a beam propagated in the same direction as the incident beam, there are beams in few other (specific) directions also.
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Thus, although the individual atoms scatter (re-emit) radiation in all directions, there are only a few directions in which these wavelets reinforce each other to produce plane waves. These waves are said to be produced by diffraction and are designated as zero order, first order, and second order etc., diffracted beams.
Condition for X-Ray Diffraction (The Bragg Law ):
W.L. Bragg investigated the conditions for X-ray diffraction by means of a model which gives the correct mathematical results in a very simple way. He found that the directions of diffracted beams can be accounted for by making the assumption that the X-rays are secularly (mirror-like) reflected from parallel atomic planes in the crystal and the multiple reflections interfere constructively in those directions. It is common practice, therefore, to interchange the words diffraction and reflection of X-rays.
Suppose the horizontal lines in Fig. 2.64 represent parallel planes of atoms which partly reflect incident X-radiation, the distance between successive planes being d. Assuming the truth of Snell’s law (i.e., incident beam, reflected beam, and normal are in one plane for mirror reflection; and angle of incidence equals angle of reflection) the path difference for rays 1 and 2 reflected from adjacent planes is the length PQR. Now, the rays reflected from adjacent planes will be in phase and their amplitudes will reinforce to produce a strong reflection only if this path difference is equal to an integral number n of wavelengths. Thus, the Bragg condition for diffraction is derived as-
PQ + QR = nλ
Also from Fig. 2.64 PQ = QR = d sin θ
∴ 2d sin θ = nλ … (xiii)
This is known as Bragg’s law, n being the order of diffraction. It is now obvious that there are only certain directions θ, in which the reflections of a given wavelength λ from all parallel planes add up in phase to give a strong reflected (diffracted) beam.
We also conclude accordingly that a beam of monochromatic X-rays incident on a crystal with an arbitrary angle θ is in general not reflected. Also, because sin θ ≤ 1 wavelengths λ ≤ 2d are essential if the Bragg reflection is to occur. Since d ≈ 10-8 cm., this condition is equivalent to λ ≤ 10-8 cm. It is for this reason that X-rays are most useful for crystal analysis.
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The Bragg equation can be used for determining the lattice parameters of cubic crystals. Let us first consider the value that n should be assigned.
A second order reflection from (100) planes should satisfy the following Bragg condition:
2λ = 2d100 sin θ
λ = d100 sin θ … (xiv)
Similarly, a first order reflection from (200) planes should satisfy the following condition:
λ = 2d200 sin θ … (xv)
For cubic unit cell the interplanar spacing of (100) planes is twice that for (200) planes. So, Equations (xiv) and (xv) are identical. For any incident beam of X-rays, the Bragg angle θ would be the same, as the two sets of planes in question are parallel. Therefore, the two reflections will superimpose on each other and cannot be distinguished. By a similar argument, it can be shown that the third order reflection from (100) planes will superimpose on the first order reflection from (300) planes.
In view of such superimposition, there is no need to consider the variations in n separately; instead, we take to n to be unity for all reflections from parallel sets of planes such as (100), (200), (300), (400), etc. In a crystal, it may turn out, for example, that there is no (200) plane with atoms on it. Then, what is designated as a (200) reflection actually refers to the second order reflection from (100) planes.
Now from Bragg’s law for first order reflection i.e. n = 1, we have,
2d sin θ = λ
Or, d = λ/2 sin θ
Or, d = α (1/sin θ)
Therefore, d100 : d110 : d111 = 1/sin θ100 : 1/sin θ110 : 1/sin θ111
Hence by measuring glancing angles at which reflections occur for different planes, we can determine the interplanar spacings knowing the wavelengths of X-rays used. Thereafter comparing the ratios of these interplanar spacings (d100: d100: d111) with Table 2.12, we can determine the structure of the given cubic crystal.
Advantages of Using X-Rays:
X-rays are used for producing diffraction effects in crystals due to the following reasons:
1. The light of longer wavelength gives rise to the familiar effect of optical refraction and reflection and so cannot be used to explore the structure of crystals on an atomic scale.
2. Radiation of wavelength shorter than X-rays, on the other hand, is diffracted through inconveniently small angles.
3. The wavelength of X-rays is comparable to the interatomic distance in actual crystals.
4. X-rays are scattered elastically without change of wavelength by the charged particles of the atoms.
5. X-rays can be produced either by the deceleration of electrons in metal targets or by exciting the electrons in the atoms of the target elastically. The former method produces a broad continuous spectrum, whereas sharp lines are obtained in the later method.