(a) Show that the values of a at which intensity maxima for single-slit diffraction occur can be found exactly by differentiating Eq. 36-5 with respect to a and equating the result to zero, obtaining the condition $\mathit{t}\mathit{a}\mathit{n}\mathit{\alpha }\mathbf{=}\mathbf{\alpha }$. To find values of a satisfying this relation, plot the curve $\mathbf{y}\mathbf{=}\mathbf{tan\alpha }$ and the straight line $\mathbf{y}\mathbf{=}\mathbf{\alpha }$ and then find their intersections, or use $\mathbf{\alpha }$calculator to find an appropriate value of a by trial and error. Next, from $\mathbf{\alpha }\mathbf{=}\mathbf{(}\mathbf{m}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{)}\mathbf{\pi }$, determine the values of $\mathit{m}$ associated with the maxima in the singleslit pattern. (These m values are not integers because secondary maxima do not lie exactly halfway between minima.) What are the (b) smallest and (c) associated , (d) the second smallest $\mathbf{\alpha }$(e) and associated , (f) and the third smallest (g) and associated ?

As the intensity increases, the diffraction maximum becomes narrower as well as more intense. When you have 600 slits, the maxima are very sharp and bright and permit high-resolution separation of the maxima for different wavelengths. Such a multiple-slit is called a diffraction grating.

In single slit diffraction,

Intensity is given as $I=I_m\frac{{\text{sin}}^2\alpha }{{\alpha }^2}$

Differentiating the above with respect to a ,

$\frac{dI}{d\alpha }=2I_m\frac{\sin \alpha }{{\alpha }^3}\left(\alpha \cos \alpha -\text{sin}\alpha \right)$

For maxima and minima$\frac{dI}{d\alpha }=0$

So either

$\alpha =0$or $\text{sin}\alpha =0$or $\alpha \cos \alpha -\text{sin}\alpha =0$

But $\alpha =0$so $\text{sin}\alpha =0$

So $\alpha =m\pi$

Again, if $\alpha \cos \alpha -\text{sin}\alpha =0$

Or $\tan \alpha =\alpha$

Since ${\left(\frac{d^{2}I}{d{\alpha }^2}\right)}_{\alpha =\tan \alpha }=negative$

There is a maxima at $\tan \alpha =\alpha$.

Let $y=\tan \alpha$ where $y=\alpha$

From the graph,

The smallest value of $\alpha =0$.

Hence, the value is$\alpha =0$.

As,$\tan \alpha =\alpha$,

localid="1664272360306" $\alpha =\left(m+\frac{1}{2}\right)\pi$

For central maximum, $\alpha =0$

Hence, the value is $\left(m=-\frac{1}{2}\right)$

The second smallest $\alpha =4.493rad$

Hence, the value is$\alpha =4.493rad$.

Associated $\left(m\right)=\frac{a}{\pi }-\frac{1}{2}$

$\left(m\right)=0.93$

Hence, the value is $\left(m\right)=0.93$

The third smallest $\left(a\right)=7.725rad$

Hence, the value is $\left(a\right)=7.725rad$

Associated $\left(m\right)=\frac{a}{\pi }-\frac{1}{2}$

$\left(m\right)=1.96$

Hence, the value is $\left(m\right)=1.96$