Minimum deviation
In a prism, the angle of deviation () decreases with increase in the angle of incidence () up to a particular angle. This angle of incidence where the angle of deviation in a prism is minimum is called the minimum deviation position of the prism and that very deviation angle is known as the minimum angle of deviation (denoted by , , or ). The angle of minimum deviation is related with the refractive index as: formula_1 This is useful to calculate the refractive index of a material. Rainbow and halo occur at minimum deviation. Also, a thin prism is always set at  minimum deviation. Formula. In minimum deviation, the refracted ray in the prism is parallel to its base. In other words, the light ray is symmetrical about the axis of symmetry of the prism. Also, the angles of refractions are equal i.e. . And, the angle of incidence and angle of emergence equal each other (). This is clearly visible in the graph below. The formula for minimum deviation can be derived by exploiting the geometry in the prism. The approach involves replacing the variables in the Snell's law in terms of the Deviation and Prism Angles by making the use of  the above properties. From the angle sum of formula_2, formula_3 formula_4 formula_5 Using the exterior angle theorem in formula_6, formula_7 formula_8 formula_9 formula_10 formula_11 This can also be derived by putting in the prism formula: From Snell's law, formula_12 formula_13 formula_14 This is a convenient way used to measure the refractive index of a material(liquid or gas) by directing a light ray through a prism of negligible thickness at minimum deviation filled with the material or in a glass prism dipped in it. Worked out examples: Answer: 37°, 49° Solution: Here, , Plugging them in the above formula , formula_15 formula_16 formula_17 formula_18 formula_19 formula_20 formula_21 Also, formula_22 This is also apparent in the graph below. Answer: 60° Solution: Here, formula_23 formula_24 Using the above formula, formula_25 formula_26 formula_27 formula_28 Also, the variation of the angle of deviation with an arbitrary angle of incidence can be encapsulated into a single equation by expressing in terms of in the prism formula using Snell's law: formula_29 Finding the minima of this equation will also give the same relation for minimum deviation as above. For thin prism. In a thin or small angle prism, as the angles become very small , the sine of the angle nearly equals the angle itself and this yields many useful results. Because and are very small, formula_30 formula_31 Interestingly, using a similar approach with the Snell's law and the prism formula for an in general thin-prism ends up in the very same result for the deviation angle. Because , and are small, formula_32 From the prism formula, formula_33 Thus, it can be said that a thin prism is always in minimum deviation. Experimental determination. Minimum deviation can be found manually or with spectrometer. Either the prism is kept fixed and the incidence angle is  adjusted or the prism is rotated keeping the light source fixed. Minimum angle of dispersion. The minimum angle of dispersion for white light is the difference in minimum deviation angle between red and violet rays of a light ray through a prism. For a thin prism, the deviation of violet light, formula_34 is formula_35 and that of red light, formula_36 is formula_37. The difference in the deviation between red and violet light, formula_38 is called the Angular Dispersion produced by the prism. Applications. One of the factors that causes a rainbow is the bunching of light rays at the minimum  deviation angle that is close to the rainbow angle (42°). It is also responsible for phenomena like halos and sundogs, produced by the deviation of sunlight in mini prisms of hexagonal ice crystals in the air bending light with a minimum deviation of 22°.