12 X-ray diffraction- I

Introduction

X-rays were discovered by German physicist Rontgen in 1895. The nature of these rays was completely unknown at that time, hence, they were named as X-rays. X-rays are invisible, travels in straight lines and affect the photographic film in the same way as light. They have much more penetration power than light and can thus, easily pass through the human body, wood, metals and other opaque objects. Radiography was started long back to study the internal structure of opaque objects even before the understanding of nature of X-rays was developed. In 1912, the phenomenon of X-ray diffraction by crystals was discovered which proved the wave nature of X-rays and provided a new method for investigating the fine structure of matter. Radiography is considered to be one of the most important tools having a wide range of applications but it can resolve the sizes of the order of 10-7 mm. On the other hand, diffraction is an indirect method which is capable of revealing the details of internal structure of the order of 10-7 mm. Thus, in this module, X-ray diffraction is discussed.

The continuous spectrum of X-rays

X-rays are produced when an electrically charged particle such as electrons having a sufficient kinetic energy rapidly decelerates. The radiation is produced in an X-ray tube which contains a source of electrons and two metal electrodes. The high voltage maintained across these electrodes (of the order of 10 kV), rapidly draws the electrons to the anode or target, where they strike with a very high velocity. This results in the production of X-rays at the point of impact which radiate in all directions. If e is the charge on the electron and V is the voltage across the electrodes, then the kinetic energy of the electrons on impact on target is given by the equation,

Where, ‘m’ is the mass of an electron, and ‘v’ is its velocity before impact. Most of the kinetic energy of electrons is converted into heat, and less than 1% is transformed into X-rays. Upon analysis of the rays coming from the target, it is found that the rays consist of a mixture of different wavelengths and the variation of intensity with wavelength depends on the tube voltage. Figure1 shows the X-ray spectrum of molybdenum at different tube voltages.

It can be observed that for a given tube voltage, the intensity of X-rays is zero upto a certain wavelength, called the short-wavelength limit (λSWL), increases rapidly upto a maximum and then, decreases with no sharp limit on the long wavelength side. When the tube voltage in increased, the intensity of all wavelengths increases and both the short-wavelength limit and position of maximum shift to shorter wavelengths. This is because with increase in tube voltage, the number of photons produced per second as well as the average energy per photon, both increases. The radiation represented in the curves is called polychromatic, continuous, or white radiation, since, similar to white light, it is made up of several wavelengths. White radiation is also called as Bremsstrahlung or braking radiation as it is caused by the deceleration of electrons. The continuous spectrum of X-rays results due to the rapid deceleration of electrons hitting the target since any decelerated charge emits energy. Electrons are decelerated in different ways, some of them lose their complete energy in one impact and some lose fraction of their kinetic energy in each impact. Photons of maximum energy or minimum wavelength are produced when the electrons are stopped in one impact so that they transfer all their energy. In equation form,

The area under the curve resulting from X-ray spectrum gives the total X-ray energy emitted per second. The total X-ray energy per second also depends on the atomic number Z of the target and on the tube current i (measure of electrons striking the target per second) and is given by

?_{????.????????} = ????^?,where A is a constant of proportionality and ‘m’ is a constant with a value of about 2. Heavy metals like tungsten are preferred as target material when large amounts of white radiation is desired. It is important to mention that the material of the target affects the intensity but not the wavelength distribution of the continuous spectrum.

The characteristic spectrum

When the voltage on an X-ray tube is raised above a certain critical value which is a characteristic of the target material, sharp intensity maxima appear at certain wavelengths superimposed on the continuous spectrum (Figure 1). Since the peaks are very narrow and the wavelengths are characteristic of the target material used, they are called the characteristic lines. These lines fall into several sets, referred to as K, L, M, N etc. in order of their increasing wavelength. All the lines together form the characteristic spectrum of the metal used as the target.

Origin of the characteristic spectrum

As discussed, the continuous spectrum results from the rapid deceleration of electrons after striking the target, whereas, the characteristic spectrum originates from the atoms of target material. In order to understand this, consider an atom having a nucleus surrounded by electrons in various shells. The shells are designated as K, L, M,…. corresponding to the value of principal quantum number n=1,2,3….. If any of the electrons bombarding the target has sufficient kinetic energy, it can knock an electron out of the K shell, leaving the atom in an excited, high-energy state. One of the outer electrons immediately falls into the K-shell vacancy, release energy and again comes back in normal state. The emitted energy is in the form of a radiation of particular wavelength and is in fact, characteristic K radiation. The K-shell vacancy may be filled by an electron lying in any one of the outer shells giving rise to a series of K lines. line results from filing of electron from L shell and line results from filling of electron from M shell. Since, it is more probable for a L-shell electron to fill the K-shell vacancy in comparison to a M-shell electron, therefore, line is stronger than a line. Similarly, L characteristic lines originate when a L-vacancy is filled by its outer shell electrons. The electron transition in an atom are shown in figure 2. If WK is the work required in order to knock out a K-electron, then the kinetic energy of the bombarding electrons should be necessarily equal to

Therefore, the characteristic spectrum originates only when the tube voltage is above some critical voltage. Since, L-shell electron is relatively farther from nucleus in comparison to K-shell, therefore, L-excitation energy is less than K and K characteristic spectrum is always accompanied with L, M etc. radiation.

Figure 2: Electronic transition in an atom

The longer wavelength lines are easily absorbed and thus, only the K lines are useful in X-ray diffraction. There are several lines in the K set, of which the strongest ones are ??1,??2 and ??1, ??1: 0.709 Å, ??2 = 0.71 Å and ??1 = 0.632 Å. The ?1 and ?2 components have wavelengths so close together that they are not resolved as separate lines; if resolved, they are called the ?? doublet and if not resolved, simply the ?? line. Similarly, ??1 is simply referred to as ?? line. ??1 is twice as strong as ??2. In general, the intensity ratio of ??1 to ??1 depends on the atomic number of target. The intensity of any characteristic line, measured above the continuous spectrum depends both on the tube current I and the amount by which the applied voltage V exceeds the critical; excitation voltage for that line. For a K line, the intensity is approximately given by

?_????? = ??(? − ??)^? where, B is a proportionality constant, V_K is the excitation voltage and n is a constant with a value of about 1.5. The existence of the strong sharp ?? line is highly useful for X-ray diffraction as it is used as the monochromatic radiation for most of the diffraction experiments.

X-ray Diffraction

The phenomenon of X-ray diffraction basically is an interaction between X-rays and the geometry of crystals. For many years, crystallographers had accumulated knowledge about crystals, chiefly by measurement of interfacial angles, chemical analysis, and determination of physical properties and there was a little knowledge of interior structure. Von Laue proposed that if crystals are made up of regularly spaced atoms which might act as scattering centers for x-rays, and if x-rays are electromagnetic waves of wavelength about equal to the interatomic distance in crystals, then it should be possible to diffract x-rays by means of crystals. Followed by this, various experiments were conducted to test this hypothesis. In 1912, the two English physicists W.H. Bragg and W.L. Bragg analyzed the Laue experiment and successfully expressed the necessary conditions for diffraction in a simple form known as Bragg’s law after the name of the physicists who proposed it.

Bragg’s law

Consider a section of the crystal where the atoms are arranged on a set of parallel planes A, B, C, D, normal to the plane of the paper and spaced a distance ‘d’ apart. Assume that a beam of perfectly parallel, perfectly monochromatic x-rays of wavelength λ is incident on this crystal at an angle θ, called the Bragg angle (θ is the angle measured between the incident beam and the particular crystal planes under consideration). Our aim is to find out whether this incident beam of x-rays will be diffracted by the crystal and, if so, under what conditions. Diffraction is essentially a scattering phenomenon and a diffracted beam may thus be defined as a beam composed of a large number of scattered rays mutually reinforcing one another. X-rays are scattered by atoms in all directions and in certain directions, the scattered beams will be completely in phase i.e. they interfere constructively giving rise to diffraction effect. Consider rays 1 and 1a in the incident beam that strikes atoms K and P in the first plane of the atoms and are scattered in all directions as shown in figure 3. Scattered beams are completely in phase only for in the directions 1′ and 1a’ because the path difference between the rays 1K1′ and 1aP1a’ is equal to

?? − ?? = ?? cos ? − ?? cos ? = 0

Similarly, the rays scattered by all atoms in the first plane in a direction parallel to 1′ are in phase and add their contributions to the diffracted beam. This will be true for all planes separately.

Now, we need to find out the condition for constructive interference of rays scattered by atoms in different planes. Consider rays 1 and 2, which are scattered by atoms K and L of the lattice. The path difference between rays 1K1′ and 2L2′ is

?? + ?? = ? sin ? + ? sin ? = 2? sin ? (4)

Scattered rays 1′ and 2′ will be completely in phase (i.e. interfere constructively) if this path difference is equal to integral multiple of wavelength, λ i.e.

This relation was formulated by W.L. Bragg and is known as Bragg’s law. The relation gives the necessary condition for the diffraction to take place. ‘n’ refers to the order of diffraction For fixed values of λ and d, the diffraction may take place at several angles of incidence 1, 2,  3 …corresponding to the values of n=1,2,3…..In all other directions of space, the scattered beams are out of phase leading to destructive interference. For same order and interplanar spacing, angle θ decreases as wavelength decreases. Thus, glancing angles are used to study diffraction using γ-rays.

Since, angle of incidence is equal to the angle of reflection, diffraction of x-rays by crystals may be considered as similar to the reflection of visible light by mirrors. Therefore, planes of atoms act as mirrors which reflect the X-rays. However, diffraction and reflection differ fundamentally in three aspects:

1. The diffracted beam from a crystal is built up of rays scvattered by all atoms of the crystal which lie in the path of incident beam, whereas, the reflection of visible light takes place in a thin layer from the surface only.
2. The diffraction of x-rays takes place only at certain angles of incidence which satisfy Bragg’s law, whereas, the reflection of visible light can take place at any angle of incidence.
3. The intensity of reflected light by a good mirror is almost 100% whereas, the intensity of diffracted x-ray beam is generally very small compared to that of incident beam.

Two important geometrical facts relating to Bragg’s law are:

1. The incident beam, normal to the diffraction plane and the diffracted beam are coplanar.
2. The angle between the diffracted beam and the transmitted beam is always 2θ. This angle is known as diffraction angle and it is the angle, which is measured experimentally.

Since sin θ cannot be greater than unity, therefore from Bragg’s law

This implies that diffraction occurs only when the wavelength of wave motion is of the same order of magnitude as the repeat distance between scattering centres. For diffraction, the smallest value of n=1 as n=0 corresponds to the beam diffracted in the same direction as incident beam which cannot be observed. Therefore, the condition for diffraction at an angle 2θ is < 2 . Since d is of the order of 3 Å or less, therefore λ cannot exceed about 6 Å. So, visible light cannot be used for crystal diffraction (for visible light λ≈ 10-7 m). If λ is very small, specialized equipment is required to measure the diffraction angles.

Since the coefficient of λ is unity, therefore, a reflection of any order can be considered as a first order reflection from planes which may be real or fictitious, spaced at a distance (1/n) of the previous spacing (i.e. d’ = d/n).

So λ = 2d’ sin θ          (9)

References:

1. Elements of X-ray diffraction by B.D. Cullity and S.R. Stock, 3rd edition, Addison-Wesley Publishing Company, Inc.
2. X-ray diffraction crystallography by Yoshio Waseda, Eiichiro Matsubara, Kozo Shinoda, Springer Science & Business Media (2011).
3. Structure Determination by X-ray Crystallography by Mark Ladd, Rex Palmer, Springer Science & Business Media (2014).
4. Introduction to Solid State Physics by Charles Kittel, 8th edition, Wiley India (2016).