Showing posts with label Wave Length. Show all posts
Showing posts with label Wave Length. Show all posts

Debroglie hypothesis

As the Newtonian classical approach to deal with problems involving particles at atomic level failed, the concept of wave mechanics was introduced by Louis Debroglie in 1924 and has been successfully applied to atomic and molecular structures.

Louis Debroglie, a French scientist extended the idea of dual nature of radiation to matter. 

According to Debroglie, a materialistic particle posessess both wave and particle characteristics.  

It is to be known that dual nature of light was explained by combining Planck's expression for energy of photon 

E = h𝜈  --------------------(1)

Einsteins Mass energy relation 

E = mc² ------------------(2)

h𝜈 = mc²------------------(3)

as we know,

𝜈 = C/ƛ  -------------------(4)

From (3) & (4)

ƛ = h/P   --------------------(5)

Just in analogy with equation 5, Debroglie proposed concept of matter waves according to which a material particles of mass M moving with a velocity 'v' should have an associated wave length  ƛ called Debroglie's wavelength given by 

ƛ = h/mv = h/P ----------(6)

'm' - according to theory of relativity is not an invariable quantity as in classical physics.

m = m₀/√ 1-(v²/c²) ------ (7)

If particles like electrons are accelerated to various velocities, we can produce waves of various wavelengths. The higher the velocity of electron, the smaller the debroglie's wavelength.

If electron is accelerated by applying potential V, then work done on electron is eV.

mv²/2 = eV ----------------(8)

mv = P = √ 2meV --------(9)

substituting (9) in (6), we have

ƛ = h/√ 2meV  ------------(10)



X-Ray diffraction techniques have certain limitations. In 1936, "W M Elgasser" suggested that moving Neutrons should have debroglie waves associated with them and therefore could be diffracted. The debroglie wavelength of Neutron moving with most probable speed at 20 is 1.80. this is of order of interplanar spacing in crystals. so neutrons can be diffracted by crystals and can be used to study their structure.
          A beam of thermal neutronsfrom an atomic pile possessing all wavelengths is collimated and allowed to fall upon a single crystal. The diffracted beams are photographed on a photographic plate. A Laue pattern is obtained. The Laue pattern can be used to study the crystal structure. The Laue pattern with Lead clearly shows the greater transparency of matter to Neutrons than X-Rays.

The diffraction patterns are formed in a way similar to that for X-rays. For X-rays of 1Amstrong, one requires energies of order 10^4 eV and for electrons about 10^2 eV.

Neutrons are scattered chiefly by Nuclei of atoms, and since wavelength of Neutrons is much greater than dimensions of scattering nucleus, the atomic scattering factor is nearly independent of  scattering angle. Experimentally it was observed that when a beam of Neutrons from a Radium Beryllium source was diffracted by MgO crystal, a maximum occured where predicted by Bragg's relation. The scattering crossection of Nuclei for thermal Neutrons does not depend on atomic number of element, as it does for X-rays.

The scattering of X-Rays by light elements is relatively weak because X-Ray scattering is done by electrons. The Neutrons can penetrate into the matter very easily enables us to deduce the positions of Hydrogen and Carbon atoms in a number of organic crystals.

A major role of Neutron Diffraction has been investigating the magnetic structure of solids. This is a result of fact that Neutrons possess magnetic moments and that these magnetic moments interact with magnetic moments of scattering atoms of solid. This gives an additional scattering mechanism for Neutrons which often outweighs Nuclear Scattering.

If the atomic moments are randomly oriented as in a paramagnetic solid, the magnetically scattered Neutrons are incoherent in phase leading to a diffuse background. This diffuse background of magnetic scattering is then super imposed on lines produced by scattering.