Birefringence and Retardation IN THIS LECTURE Isotropic versus

Birefringence and Retardation IN THIS LECTURE – – – Isotropic versus Anisotropic Minerals Double Refraction Interference Phenomena Retardation Birefringence Interference at the Upper Polar

Isotropic versus Anisotropic • Anisotropic minerals differ from isotropic minerals because: – the velocity of light varies depending on the direction through the mineral; – they show double refraction. • When light enters an anisotropic mineral it is split into two rays of different velocity which vibrate at right angles to each other. • In anisotropic minerals there are one or two directions, through the mineral, along which light behaves as though the mineral were isotropic, ie the light is not split into two rays. This direction or these directions are referred to as the optic axis. • Hexagonal and tetragonal minerals have one optic axis and are optically UNIAXIAL. • Orthorhombic, monoclinic and triclinic minerals have two optic axes and are optically BIAXIAL.

Double Refraction • Double refraction develops in anisotropic minerals because light travelling through the mineral is split into two rays which vibrate at right angles to each other. • Two images corresponding to the two rays are produced.

Double Refraction in Calcite

Determining Vibration Direction • The vibration direction of the two rays which exit the calcite rhomb can be determined using a piece of polarising film. • The polarising film has a single vibration direction and as such only allows light, which has the same vibration direction as the filter, to pass through the filter to be detected by your eye.

Polarisation and Calcite

Polarisation and Calcite

Fast and Slow Rays • It is possible to measure the index of refraction for the two rays using the immersion oils, and one index will be higher than the other. • The ray with the lower index is called the fast ray – recall that n = Vvac/Vmedium If • n. Fast Ray = 1. 486, then v. Fast Ray = 2. 02 x 1010 m/sec The ray with the higher index is the slow ray – If n. Slow Ray = 1. 658, then v. Slow Ray = 1. 8 1 x 1010 m/sec

Development of Retardation • We have already seen that due to differences in velocity, the slow ray lags behind the fast ray, and the distance represented by this lagging after both rays have exited the crystal is called the retardation – D • The magnitude of the retardation is dependant on the thickness (d) of the mineral and the differences in the velocity of the slow (Vs) and fast (Vf) rays. • The time it takes the slow ray to pass through the mineral is given by: ts = d Vs Eqn 1

Development of Retardation • during this same interval of time the fast ray has already passed through the mineral and has travelled an additional distance = retardation. ts = • d + Vs D Eqn 2 V Substituting equation (1) into equation (2) yields d Vs = d Vf + D V Eqn 3

Development of Retardation • Rearranging equation (3) gives D=d( V Vs - V Vf ) D = d ( ns – nf) and since V Vs Eqn 4 = ns

Birefringence • The relationship (ns – nf) is called birefringence and is given the Greek lower case symbol d (delta). • d represents the difference in the indices of refraction for the slow and the fast rays. • In anisotropic minerals the path along the optic axis, exhibits zero birefringence, others show maximum birefringence, but most show an intermediate value. • The maximum birefringence is characteristic for each mineral. • Birefringence may also vary depending on the wavelength of the incident light.

Interference at the Upper Polar • The colours for an anisotropic mineral observed in thin section under crossed polars are called interference colours and are produced as a consequence of splitting the light into two rays on passing through the mineral. • Now look at the interference of the fast and slow rays after they have exited the anisotropic mineral with the fast ray is ahead of the slow ray by some amount = D • Interference phenomena are produced when the two rays are resolved into the vibration direction of the upper polar.

Recall…

Upper Polar Interference (1) • Light passing through lower polar, plane polarized, encounters sample and is split into fast and slow rays. • If the retardation of the slow ray = 1 whole wavelength, the two waves are IN PHASE. • When the light reaches the upper polar, a component of each ray is resolved into the vibration direction of the upper polar. • Because the two rays are in phase, and at right angles to each other, the resolved components are in opposite directions and destructively interfere and cancel each other. • Result is no light passes the upper polar and the grain appears black.

Upper Polar Interference (1) Upper Polar The two rays exiting the mineral are in phase with the slow ray lagging behind the fast ray by one whole wavelength Mineral Sample Lower Polar The vector R, the sum of the two rays, is at right angles to the vibration direction of the upper polariser => no light passes the upper polar

Upper Polar Interference (2) • If the retardation of the slow ray behind the fast ray = ½ a wavelength, the two rays are OUT OF PHASE, and can be resolved into the vibration direction of the upper polar. • Both components are in the same direction, so the light constructively interferes and passes the upper polar.

Upper Polar Interference (2) Upper Polar The two rays exiting the mineral are out of phase with the slow ray lagging behind the fast ray by half a wavelength Mineral Sample The vector R, the sum of the two rays, is parallel to the vibration direction of the upper polariser Lower Polar => light passes the upper polar