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Diffraction grating experiment

What is diffraction?

“The bending of light waves around the corners of an opening or obstacle and spreading of light waves into geometrical shadow is called diffraction.”Diffraction effect depends upon the size of obstacle.Diffraction of light takes place if the size of obstacle is comparable to the wavelength of light.Light waves are very small in wavelength,i.e,from 4×10-7 m to 7 × 10 -7 m.If the size of opening or obstacle is near to this limit,only then we can observe the phenomenon of diffraction.

Types of diffraction:

Diffraction of light can be divided into two classes:

  • Fraunhoffer Diffraction
  • Fresnel Diffraction

Fraunhoffer Diffraction:

In Fraunhoffer diffraction:

  • Source and the screen are far away from each other.
  • Incident wave fronts on the diffracting obstacle are plane.
  • Diffraction obstacle give rise to wave fronts which are also plane.
  • Plane diffracting wave fronts are converged by means of a convex lens to produce diffraction pattern.

Fresnel Diffraction:

In Fresnel diffraction:

  • Source and screen are not far away from each other.
  • Incident wave fronts are spherical.
  • Wave fronts leaving the obstacles are also spherical.
  • Convex lens is not needed to converge the spherical wave fronts.

Diffraction of light:

In Young’s double slit experiment for the interference of light,the central region of the fringe system is bright.If light travels in straight path,the central region should appear dark i.e.,the shadow of the screen between the two slits.Another simple experiment can be performed for exhibiting the same effect.diffraction

Consider that a small and smooth ball of about 3 mm in diameter is illuminated by a point source of light.The shadow of the object is received on a screen as shown in figure.The shadow of the spherical object is not completely dark but has a bright spot at its centre. According to Huygens’s principle ,each point on the rim of the sphere behaves as a source of secondary wavelets which illuminate the central region of the shadow.

These two experiments clearly show that when light travels past an obstacle,it does not proceed exactly along a straight path,but bends around the obstacle.The phenomenon is found to be prominent when the wavelength of light is compared with the size of the obstacle or aperture of the slit.The diffraction of light occurs,in effect,due to the interference between rays coming from different parts of the same wave front.

Diffraction due to Narrow slit:diffraction due to narrow slit

Figure shows the experimental arrangement for studying diffraction of light due to narrow slit.The slit AB of width d is illuminated by a parallel beam of monochromatic light of wavelength λ.The screen S is placed parallel to the slit for observing the effects of the diffraction of light.A small portion of the incident wavefront passes through the narrow slit.Each point of this section of the wavefront sends out secondary wavelets to the screen.These wavelets then interfere to produce the diffraction pattern.It becomes simple to deal with rays instead of wavefronts  as shown in figure.In this figure,only nine rays have been drawn whereas actually there are a large number of them.Let us consider rays 1 and 5 which are in phase on in the wavefront AB.When these reach the wavefront AC,ray 5 would have a path difference ab say equal to λ/2.Thus,when these two rays reach point p on the screen;they will interfere destructively.Similarly,each pair 2 and 6,3 and 7,4 and 8 differ in path by λ/2 and will do the same.But the path difference ab=d/2 sinθ.

The equation for the first minimum is,then

d/2 sinθ =λ/2

or d sinθ=λ

In general,the conditions for different orders of minima on either side of centre are given by:

d sinθ =mλ

where m=± (1,2,3,….)

The region between any two consecutive minima both above and below O will be bright.A narrow slit,therefore,produces a series of bright and dark regions with the first bright region at the centre of the pattern.

Diffraction grating:

“The diffraction grating is a useful devise for analyzing light sources.It consists of a large number of equally spaced parallel slits.”Its working principle is based on the phenomenon of diffraction.The space between lines act as slits and these slits diffract the light waves there by producing a large number of beams which interfere in such a way to produce spectra.

A transmission grating can be made by cutting parallel lines on a glass plate with a precision ruling machine.The space between the lines are transparent to the light and hence act as separate slits.A reflection grating can be made by cutting parallel lines on the surface of a refractive material.Gratings that have many lines very close to each other can have very small slit spacing’s .For example,a grating ruled with 5000 lines/cm has a slit spacing d=1/5000 cm=2.00×10-4 cm.diffraction grating

A section of a diffraction grating is illustrated in figure.A plane wave is incident from the left,normal to the plane of the grating.A converging lens brings the rays together at point P.The pattern observed on the screen is the result of the combined  effects of interference and diffraction.Each slit is produces diffraction,and the diffracted beams interfere with one another to produce the final pattern.

The waves from all slits are in phase as they leave the slits.However,for some arbitrary direction θ measured from the horizontal, the waves must travel different path lengths before reaching point p. From figure, we note that the path difference’ δ ‘ between rays from any two adjacent slits is equal to d sin θ. If this path difference is equal to one wavelengths or some integral multiple of a wavelength, then waves from all slits are in phase at point P and a bright fringe is observed. Therefore, the condition for maxima in the interference pattern at the angle θ is.

d sin θ =m‍λ

Where m=0,1,2,3,4…

We can use this expression to calculate the wavelength if we know the grating spacing and the angle 0. If the incident radiation contains several wavelength, the m th-order maximum for each wavelength occurs at a specific angle. All  wavelengths are seen at θ =0, corresponding to m=0, the zeroth-order maximum (m=1) is observed at the angle that satisfies the relationship sin θ =λ/d: the second-order maximum (m=2) is observed at a larger angle θ, and so on.

intensity distribution in diffraction grating

The intensity distribution for a diffraction grating obtained with the use of a monochromatic source. Note the sharpness of the principle maxima and the broadness of the dark areas. This is in contrast to the broad bright fringes characteristic of the double-slit interference pattern. Because the principle maxima are so sharp, they are very much brighter than double-slit interference maxima.

Grating element:

Distance between two consecutive slits (lines) of grating is called grating element.Grating element ‘d’ is calculated as:

Grating element =Length of grating/Number of lines

Dispersion and resolving power:

Dispersion:

The ability of a grating to produce spectra that permit precise measurement of wavelengths is determined by two intrinsic properties of grating.

  • The separation Δθ between the spectral lines that differ in wavelength by small amount Δλ.
  • The width or sharpness of the lines.

The dispersion D of the grating is defined as:

“The angular separation Δθ per unit wavelength Δλ is called the dispersion D of the grating.”

D = Δθ/Δλ

For lines of nearly equal wavelengths to appear as widely as possible,we would like our grating to have the largest possible dispersion.

Since the grating equation is:

d Sinθ =mλ

Differentiating the above equation we have:

d cosθdθ = mλ

Now in terms of small differences the above equation we have:

d cosθ dθ =m dλ

Now in terms of small differences the above relation can be written as:

d cosθ Δθ =mΔλ

Δθ/Δλ =m/d cosθ

D = m/d cosθ

From the above relation we see that the dispersion D increases as spacing between the slits ‘d’ decreases.We can also increase the dispersion by working at higher order ( large m).Note that the dispersion does not depend on the number of rulings N.

Resolving power:

“The resolving power of an instrument is its ability to reveal minor details of the object under examination.”

Resolving Power of  Diffraction Grating:

“The resolving power of grating is a measure of how effectively it can separate or resolve two wavelengths in a given order of their spectrum”.

The diffraction grating is most useful for measuring accurately. Like the prism, the diffraction grating can be used to disperse a spectrum into its wavelength components. The grating is the more precise device if we want to distinguish two closely spaced wavelengths.

According to Rayleigh’s Criterion ” For two nearly equal wavelengths λ1 and λ2 between which a diffraction grating can just barely distinguish, the resolving power R of the grating is defined as:

R = λ/Δλ

Thus,a grating that has a high resolving power can distinguish small differences in wavelength.

Where  λ = λ1 + λ2 /2 and  Δλ = λ1 – λ

Thus,resolving power increases with  increasing order number and with increasing number of illuminated slits.If the lines are to be narrow, the angular separation δθ is small, then corresponding wavelengths interval Δλ must be small, and by equation(1) the resolving power must be large. To find the physical property of the grating that determines resolving power R,we write the spacing between nearby lines as:

derivation of diffraction grating

⇒R = Nm

Thus resolving power increases with the order number m and number of lines N.Resolving power is independent of the separation d of the slits.

Diffraction of X-rays by Crystals:

The wavelength of any electromagnetic wave can be determined if a grating of the proper spacing i.e. of the order of wavelength λ of the wave, is available. X-rays are electromagnetic waves of very short wavelengths ( of the order of 0.1 nm). It would be impossible to construct a grating having such a small spacing by the cutting process. However: the atomic spacing in a solid is known to be about 0.1 nm. In 1913, Max von Laue suggested that the regular array of atoms in a crystal solid could act as a three-dimensional diffraction patterns are complex because of the three-dimensional diffraction grating for X-rays. Subsequent experiment confirmed this prediction. The diffraction patterns are complex because of the three dimensional nature of the crystal. Nevertheless, x-ray diffraction has proved to be an invaluable technique for studying crystalline structures and for understanding the structure of matter.Diffraction of x rays by crystals

A collimated beam of x-rays is incident on a crystal. The diffracted beams are very intense in certain directions, corresponding to constructive interference from waves reflected from layers of atoms in the crystal. The diffracted beams can be detected by a photographic film, and they form array of spots known as a Laue pattern. One can deduce the crystalline structure by analyzing the positions and intensities of the various spots in the pattern.

diffraction x rays pattern

Bragg’s Equation:

Suppose that an x-ray beam is incident at an angle θ on one of the planes. The beam can be reflected from both the upper plane and the lower plane travels farther than the beam reflected from the upper plane. It is clear that beam 2 travels a greater distance than beam 1 after reflection from atoms of the plane. Thus the distance BC+CD is the effective path difference between the two reflected beams 1 and 2.

Watch also:

Watch also:

  • Reflection of light
  • Light
  • Refraction of light
  • Interference

 

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