Summer 2016 - Q.5 - Grad Plus

Summer 2016 – Q.5

5(a)A diffraction grating used at normal incidence gives a yellow line (λ = 6000A°) in a certain spectral order superimposed on a blue line (λ = 4800A°) of next higher order if the angle of diffraction is sin-1(3/4), calculate the grating element? [5M]

We have,

$latex \begin{array}{l}\left(a+b\right)\;\sin\theta=n\lambda_{yellow}\\\\\left(a+b\right)\;\sin\theta=\left(n+1\right)\lambda_{blue}\end{array}$

Dividing

$latex \begin{array}{l}1=\frac{n\lambda_{yellow}}{\left(n+1\right)\lambda_{blue}}\\\\n6000=\left(n+1\right)4800\\\\n6=4.8n+4.8\\\\or\\\\n=4\end{array}$

From first equation,

$latex \begin{array}{l}=(a+b)\;x\;3/4\;\\\\=\;4\;x\;6\;x\;10-5\;cm\;\\\\=\;32\;x\;10-5\;cm\end{array}$

(b)Derive one dimensional time dependent Schrodinger’s equation for matter waves? [5M]

Schrodinger’s wave equation determines the motion of atomic particle. Consider a microparticle in motion which is associated with wave function Ψ. The Ψ represents the field of the wave, say. The classical wave equation is of the form,

$latex \frac{\partial^2y}{\partial x^2}=\frac1{\nu^2}\frac{\displaystyle\partial^2y}{\displaystyle\partial t^2}$

A solution of this equation is,

$latex y\left(x,\;t\right)=Ae^{i\left(kx-wt\right)}$

For a microparticle the w and k can be replaced with E and p using Einstein and de-Broglie relations as,

$latex E=\hslash\omega\;and\;p=\hslash k,$

Also replacing y(x, t) by $latex \psi$(x, t) we may write

$latex \psi\left(x,\;t\right)=Ae^\frac{-\left(Et-px\right)}\hslash$

Differentiating with respect to t, we get,

$latex \frac{\partial\psi}{\partial t}=-\frac{i}{\hslash}E\psi$

Differentiating twice with respect to x, we get,

$latex \frac{\partial^2\psi}{\partial x^2}=-\frac{p^{2}}{\hslash}\psi$

For a free particle we have

$latex E=\frac{p^2}{2m}$

and in case of a particle moving in force field characterized by potential energy V, we have

$latex \frac{p^2}{2m}=E-V.$

Multiplying above equation by $latex \psi,$

$latex \frac{p^2}{2m}\psi=E\psi-V\psi$

Substituting for $latex E\psi\;and\;p^2\psi,$ and  rearranging we get

$latex -\frac{\hslash^2\partial^2\psi}{2m\partial x^2}+V\psi=i\hslash\frac{\partial \psi }{\partial t}$

Which is time dependent Schrodinger’s wave equation, where $latex i\hslash\frac\partial{\partial t}=E,$ the energy operator.

Therefore,

$latex -\frac{\hslash^2\partial^2\psi}{2m\partial x^2}+V\psi=E\psi$

This is time independent Schrodinger’s wave equation.

(c)With neat diagram, explain construction & working of Atomic Force Microscope [5M]

Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very high-resolution
type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a
nanometer, more than 1000 times better than the optical diffraction limit. The AFM is one of the foremost tools for imaging, measuring, and manipulating matter at the nanoscale.

The information is gathered by “feeling” the surface with a mechanical probe. Piezoelectric
elements that facilitate tiny but accurate and precise movements on (electronic) command enable the very precise scanning. In some variations, electric potentials can also be scanned using conducting cantilevers. In newer more advanced versions, currents can even be passed through the tip to probe the electrical conductivity or transport of the underlying surface, but this is much more challenging with very few research groups reporting reliable data.

The AFM consists of a cantilever with a sharp tip (probe) at its end that is used to scan the specimen
surface. The cantilever is typically silicon or silicon nitride with a tip radius of curvature on the order of nanometers. When the tip is brought into proximity of a sample surface, forces between the tip and the sample lead to a deflection of the cantilever according to Hooke’s law.

Depending on the situation, forces that are measured in AFM include mechanical contact force,
van der Waals forces, capillary forces, chemical bonding, electrostatic forces, magnetic forces (see
magnetic force microscope, MFM), Casimir forces, solvation forces, etc. Along with force, additional
quantities may simultaneously be measured through the use of specialized types of probe.

If the tip was scanned at a constant height, a risk would exist that the tip collides with the surface,
causing damage. Hence, in most cases a feedback mechanism is employed to adjust the tip-to-sample distance to maintain a constant force between the tip and the sample. Traditionally, the sample is mounted on a piezoelectric tube that can move the sample in the z direction for maintaining a constant force, and the x and y directions for scanning the sample.

Alternatively a ‘tripod’ configuration of three piezo crystals may be employed, with each
responsible for scanning in the x,y and z directions. This eliminates some of the distortion effects seen with a tube scanner. In newer designs, the tip is mounted on a vertical piezo scanner while the sample is being scanned in X and Y using another piezo block. The resulting map of the area z = f(x, y) represents the topography of the sample.

The AFM can be operated in a number of modes, depending on the application. In general,
possible imaging modes are divided into static (also called contact) modes and a variety of dynamic (or non-contact) modes where the cantilever is vibrated.

Fig :- Schematic diagram of AFM

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