X-ray Diffraction Analysis Principle Instrument and Applications I Definition, Methods, XRD analysis, and 5 Advantages.

What is X-ray diffraction ?

The Definition of x-ray diffraction is: Diffraction of light means the bending of light around the corner of an obstacle. It is a fact that for diffraction to occur.

The size of the obstacle should nearly be equal to the wavelength of light used. X-ray, like other electromagnetic rays, can also be diffracted, but for the diffraction of X-ray. The size of the obstacle should be a few angstroms (approx 1 Å) Which is approximately the wavelength of X rays. Since the atomic spacing in the Crystal is nearly a few  Å.

Crystal provides the best alternative for the study of the diffraction of X-ray. The diffraction of X-rays, in consequence, provides valuable information about the structure and properties of crystalline and noncrystalline materials.

X ray Diffraction Principle

X-ray diffraction is based on constructive interference of monochromatic X-rays and a crystalline sample. These X-rays are generated by a cathode ray tube, filtered to produce monochromatic radiation, collimated to concentrate, and directed toward the sample.

When a monochromatic x-ray incident occurs on a crystal. The atomic electrons in the Crystal are sent into vibration. With the same frequency as that of the frequency of the incident ray and are accelerated. These Accelerated electrons then emit the radiation of the same frequency as that of incident x-rays in all directions.

If the wavelength of incident radiation is large compared to the dimensions of the Crystal. Then the radiated X-ray are in phase with each. But since the atomic dimension are nearly equal to the wavelength of X-Ray.

The radiation emitted by the electrons is out of phase with each other. These radiations may interfere constructively or destructively producing a diffraction pattern(i.e. maxima and minima) in certain directions. 

Bragg’s Law of Diffraction

In order to explain the diffraction of x rays , W.L. Bragg considered the X-ray diffraction from a crystal as a problem of reflection of X-rays from the atomic planes of the crystal in accordance with the laws of reflection.

Consider a set of parallel atomic planes of the Crystal with Miller Indices [hkl], such that the distance between the two successive planes is d. Let a parallel beam of monochromatic X-rays of wavelength λ be incident on the Plane at a glancing angle θ such that the incident rays lie in the plane of the paper.

Let AP and BQ be two parallel incident rays that are reflected from points P and Q on the Crystal planes and travel along with PA’ and QB’ respectively. If the path difference between APA’ and BQB’ is an integral multiple of λ, there will be constructive interference and a maximum will be observed.

Laue Patterns: Laue Method of X ray Diffraction

In 1912, Max Von Laue, a German theoretical physicist feet suggested that. If a crystal consists of a regular and orderly arrangement of atoms. The planes of the Crystal should have a spacing of the order of 1 Armstrong. And a crystal should behave as a neutral three-dimensional diffraction grating for the diffraction of X-rays. Which also have a wavelength of about 1 Armstrong.

It was, therefore, predicted that if a beam of inhomogeneous X-Ray is made to fall on a crystal. A series of spots arranged in a geometric fashion. About the center of incident beam should be obtained in the diffraction pattern. Obtained on a photographic plate list on other side of the Crystal. 

This was actually verified experimentally and a definite arrangement of atoms in the Crystal was established. The spots obtained on the photographic plate when the experiment was performed with ZnS crystal. These sports were named as Laue spot. And the pattern of spot obtained on the photographic plate was named as Laue pattern. The Laue pattern of ZnS crystal is as shown in fig.

X-ray Diffraction Methods

The phenomenon of x-ray diffraction is useful for the determination of structure of solid. And as well as for the study of the X-ray spectroscopy. Bragg’s law is widely used for both these applications. For applying bragg’s law for crystal structure determination. It is required that λ and θ must be matched properly.

To do so experimentally, another continuous range of wavelength λ or θ is provided. So that the value of λ Is arbitrarily chosen for a given value of the orientation ‘θ’. Three methods are generally adopted for the study of crystal structure. These are Laue Method, Rotating Crystal method, and powder method.

Laue Method 

It is one of the important methods. Used for the study of crystal structure and is mostly used for determination of Crystal symmetry.

In this method, a beam of polychromatic X-rays of wavelengths ranging from 0.2Å to 2Å is allowed to fall on a small crystal of dimension 1 mm * 1mm * 1mm, placed on a goniometer.

The goniometer can be rotated to change the orientation of the crystal with respect to the beam of X-rays. Generally, the beam is allowed to fall perpendicular to the plane of the crystal under study. While passing through the crystal.

The X-ray falls on different Bragg’s planes having a spacing d. And making different angles ‘θ’ with the incident direction of X-rays.

For some value of d, λ and θ, which satisfy the Bragg’s condition, 2d sinθ = nλ. constructive interference takes place and increase in intensity takes place at certain directions producing a diffraction pattern. This diffraction pattern may be observed by placing a photographic plate on the other side of the crystal.

The diffraction pattern obtained on the photographic plate. Consists of a symmetrical arrangement of spot depending upon the symmetry property of the crystal lattice. The Laue method is generally used to determine the symmetry.

For example, for a crystal with four-fold symmetry and axis parallel to the beam of X-ray, four fold rotation of the crystal would produce identical Laue patterns.

Conclusion

It may be noted that the Laue method cannot be used for the determination of crystal structure. It is because that out of the continuous range of wavelengths. A number of wavelengths may be reflected in different order. From a single plane producing and overlapping of certain reflections at a single spot. Thus a number of reflections May be missing in the Laue pattern.

Rotating crystal method 

In this method, a single crystal of dimension 1 mm, is mounted on a rotating spindle. Such that the axis of rotation of the spindle coincides with either of the axis of the crystal. A beam of monochromatic X-rays is incident on the Crystal perpendicular to the axis of rotation of the spindle.

The spindle is covered by a Hollow cylindrical holder having its axis collinear with the axis of spindle. Such that the crystal lies at the centre of this cylindrical holder. For obtaining the diffraction pattern.

A photographic plate is attached inside the cylindrical holder along with its surface. It may be noted that generally the vertical Axis is taken as the rotation axis.

During the course of rotation of the Crystal, x-ray will be diffracted. Whenever the orientation of the Crystal plane with respect to X rays. In such a way that the Bragg’s Condition is satisfied. All those of the Crystal.

Which are parallel to the axis of the rotating spindle. Diffract the incident ray into a horizontal plane perpendicular to the axis. Whereas, there will be no diffraction from the plane which contains the incident beam of X-rays.

However, all those planes which are inclined at a certain angle to the axis of rotation will produce reflection above and below the horizontal plane, thereby producing Line or layers above and below the plane depending upon the inclination of the plane. These lines are called the layer line and are produced due to the Diffraction spot aligned along a line.

Conclusion 

if we consider the c-axis of the Crystal system as the axis of rotation. Then the reflection from all planes {hk0}. i.e. The Miller planes with l=0. Will produce the zero layer whereas the reflection from {hk1} and {hk1 }.

Planes will produce the layer Above and below the zero layers and equally spaced from the zero layers. Since the distance between the Layer is found to depend only on the separation of the lattice point on the c-axis. Therefore the distance c can be directly be calculated using the relation c sin θ = λ. Where θ is the angle of inclination of the plane with the rotation axis.

By mounting The Crystal along a, b and c axis turn-by-turn and getting the rotational photograph. The Crystal translation vector and hence the size of the unit cell of the Crystal can be Found.

Powder Crystal Method

Bragg’s method and the rotating Crystal Method required the precise mounting of a single crystal on a certain crystal Axis. Which is a tedious task to do.To overcome this difficulty, powder crystal method is used.

This method was developed independently by Deby, Scherrer, and Hull. In this method, the crystalline material is ground to powder formShow that the Crystallites assume random orientation.

A small sample of this powder is placed in a small capillary tube. Made of non-Diffracting material or the sample is just stuck on a hair with the help of non-diffracting binding material. And is placed in the path of a fine monochromatic beam of X-rays.

The principle involved in the crystal method is that since there are a large number of crystallites with random orientation. All possible diffraction plain shall be available for bragg’s diffraction to occur. i.e. For the diffraction to occur in accordance with the condition 2d sinθ = nλ. λ Being a constant in this case.

Therefore, the reflection will occur from the family of parallel planes which are inclined to the x-ray beam at different angles.

Also, the higher order of reflection apart from the first order, second-order reflection. Will also be produced. Since for a given value of angle θ. A large number of orientations of a given family of planes is possible.

The X-ray diffracted corresponding to the perpendicular value of d and θ will Lie on the surface of a cone. Whose Axis lie along the direction of the incident beam. And apex is at the sample with semi-vertical angle 2θ as shown in fig. 

X-rays are made monochromatic by passing through a filter. These monochromatic rays are collimated into a fine beam by passing them through two lead slits. This collimated beam is now made to fall on the Powder specimen.

Taken in a capillary tube made of non-diffracting material. And suspended along the axis of the cylindrical camera having a photo film attached around its inner surface.

This camera is called Debye – Sherrer camera. And surrounds the Crystal powder completely in all directions up to an angle of 180 degrees. With the direction of incident X-rays.

Since the width of the photo film is small. Only a small part of the circular ring is formed on it. When the film is developed and laid flat on a table, the shape of the Rings is obtained.

Applications of x-ray Crystallography

• Used to identify the fine-grained minerals such as clays and mixed layer clays that are difficult to determine optically.
• To determination of unit cell dimensions.
• To determine crystal structures.
• It is used to make textural measurements, such as the orientation of grains, in a polycrystalline sample.

Advantages and Disadvantages of X-ray Powder Diffraction