 Summer 2015 - Q.5 - Grad Plus

# Summer 2015 – Q.5

5. (a) A slit of width 0.3 mm is illuminated by a light of wavelength 5890 Å. A lens whose focal length is 40 cm forms a Fraunhofer diffraction pattern. Calculate the distance between first dark and the next bright fringe form the axis. [5M]

The distance of first dark band from the centre is λD/a and next bright band is 3λD/2a.
Thus, the distance between first dark and next bright will be,

$latex \begin{array}{l}x=\frac{\lambda D}{2a}\\\\\;\;\;=\frac{5.89\times10^{-5}cm\times40cm}{2\times0.03cm}\\\\\;\;\;=0.0392\;cm\end{array}$

(Note: the focal length f is nearly equal to D, when lens is placed close to the slits.)

(b) An electron is accelerated through 1000 volts and is reflected from a crystal. The first order reflection occurs when glancing angle is 70°. Calculate the interplanar spacing of a crystal. [5M]

We have,

$latex 2d\;\sin\theta=n\lambda$

Where $latex \begin{array}{l}\lambda=\frac{12.26}{\sqrt v}\overset\circ A\\\\\;\;\;=\frac{12.26}{\sqrt{1000}}\\\\\;\;\;=0.3876\;\overset\circ A\end{array}$

Thus, $latex \begin{array}{l}d=\frac{n\lambda}{2\;\sin\theta}\\\\\;\;\;=\frac{1\times0.3876\;\overset\circ A}{2\;\sin\;70}\\\\\;\;\;=0.2062\;\overset\circ A\end{array}$

(c) Explain construction and 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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