Bragg’s Law

Lawrence Bragg, an English physicist, developed the Bragg law first. It is a type of special case of the Laue diffraction which is used to calculate the angles of coherent scattering and incoherent scattering from a crystal lattice. When X-rays strike a certain atom, they cause an electronic cloud to move like an electromagnetic wave. Rayleigh scattering occurs when the movement of these charges emits waves of identical frequency but significantly blurred owing to distinct influences. Essentially, the rule explains the interaction between an x-ray light beam and its reflection off a crystal surface.

What is Bragg’s Law?

The relationship between the spacing of atomic planes in crystals and the angles of incidence at which these planes produce the most intense reflections of electromagnetic radiations, such as X-rays and gamma rays, and particle waves, such as those associated with electrons and neutrons, is known as the Bragg law. For maximum intensity, reflected wave trains must remain in phase to induce constructive interference, which occurs when matching points of a wave (e.g., its crests or troughs) arrive at the same spot at the same time. 

The Bragg law is helpful for measuring wavelengths and estimating crystal lattice spacings. To measure a certain wavelength, both the radiation beam and the detector are positioned at an arbitrary angle θ. After that, the angle is adjusted until a strong signal is obtained. The Bragg angle, as it is termed, thus immediately yields the wavelength from the Bragg law. This is the primary method for precisely measuring the energy of X-rays and low-energy gamma rays. The energy of neutrons, which have wave properties according to quantum theory, are typically estimated through Bragg reflection.

Bragg’s Law Mathematical Expression

When x-rays are scattered from a crystal lattice, dispersed intensity peaks are seen that correspond to the circumstances listed below:

• The incidence angle is equal to the scattering angle.
• The difference in pathlength is an integer number of wavelengths.

The maximum intensity requirement provided in Bragg’s equation above allows us to compute information about the crystal structure or, if the crystal structure is known, to predict the wavelength of the x-rays incident onto the crystal.

Mathematically Bragg’s law is expressed in the term of

2d sinθ=nλ

Where,

 λ is the wavelength of the X-ray beam, d is the distance between crystal layers (optical path difference), θ is the incidence angle (angle between incident beam and scattering plane), and n is an integer. X-ray diffraction is used to examine crystal structure using characteristic X-rays. Bragg’s law can be used to calculate lattice size in the Bragg spectrometer.

Bragg’s Law Diagram & Derivation 

Consider homogeneous x-rays of a certain wavelength impinge on a crystal at a glancing bragg angle θ. After deflecting off the lattice planes Y and Z, the incident rays AB and DE travel along BC and EF, respectively. Let’s call the distance between the lattice planes ‘d.’ On DE and EF, BP and BQ are perpendiculars taken from B. As a result, the path difference between waves ABC and DEF is equal to PE plus EQ.

Path difference = PE + EQ

In the △ PBE,

PE/BE = Sine θ

Now,

PE=BE sine θ = d sine θ

In the △ QBE triangle,

EQ/BE = Sine θ

and,

EQ=BE sine θ = d sine θ

Hence,

Path difference = PE + EQ = d sine θ + d sine θ = 2d sineθ

When the path difference 2d sine θ equals the integral multiple of the wavelength of an x-ray which is n, constructive interference occurs between the reflected beams and they reinforce each other. As a result, the intensity of the reflected beam is quite high.

Therefore,

n = 2d sine θ

Where n=1, 2, 3… and so forth

This is referred to as Bragg’s law.

Application of Bragg’s Law

Substantially, there are various applications to consider while discussing the applicability of Bragg’s Law. It has dominated the family of science over time, denoting a variety of intrinsic applications such as,

• Bragg’s Law has been found to be highly beneficial in doing wavelength measurements. In addition, it is used to calculate the lattice spacings of crystals.
• When discussing XRF or WDS, crystals with predetermined d-spacings are used to evaluate the crystal surfaces in a spectrometer.
• In addition to studying crystals, the interplanar spacings in a crystal are used to determine the kind and type of lattices, according to X-ray diffraction.
• It is extremely important in crystallography.

Conclusion 

Bragg’s law is a type of special case of the Laue diffraction which is used to calculate the angles of coherent scattering and incoherent scattering from a crystal lattice. The relationship between the spacing of atomic planes in crystals and the angles of incidence at which these planes produce the most intense reflections of electromagnetic radiations, such as X-rays and gamma rays, and particle waves, such as those associated with electrons and neutrons, is termed as the Bragg law. The maximum intensity requirement provided in Bragg’s equation above allows us to compute information about the crystal structure or, if the crystal structure is known, to predict the wavelength of the x-rays incident onto the crystal. When an X-ray beam strikes a crystal, it is dispersed by individual atoms in the rich atomic planes. As a result, the crystal grating is made up of sets of Braggs planes.