Anisotropic materials

1
Anisotropic materials

• Nesse, 1991 Chapter 5, p. 37-52

2
Anisotropic materials

• Differences isotropic anisotropic mineral
• Velocity of light varies depending on the
  direction of travel through the mineral
• Anisotropic minerals double refraction
• Light that enters anisotropic materials splits
  into 2 rays with different velocities.
• Ray with lower RI fast ray higher RI slow
  ray
• The 2 rays vibrate at right angles to each other
• Each anisotropic material has 1 or 2 directions,
  called optic axes, where the light behaves as if
  it were isotropic

3
Double refraction

• Use clear calcite rhomb (along cleavage plane) on
  paper with dark dots
• The mark will be repeated looking through the
  calcite rhomb
• When the calcite rhomb is viewed through a
  polaroid plate only one row of points is seen

Fig 5.1
4
Double Refraction

• Note the vibration direction of the polarizing
  plate.
• When the plate is rotated 90 degrees the other
  row of points becomes visible
• This shows that the point is transmitted through
  the rhomb by 2 differently polarized rays

Fig 5.1
5
DOUBLE REFRACTION
NOTE Do not confuse propagation direction
vibration direction Propagation Direction the
direction that the light is moving Vibration
Direction the side-to-side oscillation of plane
polarised light
6
DOUBLE REFRACTION
Propagation direction
Vibration direction
7
Interference

• When an anisotropic mineral is placed between
  crossed polars it may show vivid colors.
• These are called INTERFERENCE COLORS, also known
  as BIREFRINGENCE
• How do interference colors form?
• First consider monochromatic light.

8
Interference

• When monochromatic light with velocity V enters
  the anisotropic mineral, it is split into two
  rays that vibrate at right angles and have
  different velocities
• Amplitude of each ray can be determined by vector
  addition

Fig 5.3 (Nesse, 1991)
9
Interference

• Because of the difference in velocity, the slow
  ray lags behind the fast ray
• The distance that the slow ray lags behind when
  both excite the crystal is called Retardation D

10
Interference

• Magnitude of retardation depends on the
  thickness of the crystal d and the difference
  between the velocity of the fast ray Vfast and
  the slow ray V slow
• The time it takes the slow ray to traverse the
  crystal is ts

11
Interference

• tsd/Vs - for slow ray (distance/velocity)
• During time ts the fast ray traverses the crystal
  but also travels the additional distance equal to
  retardation D
• ts d/Vf D/V
• Or d/Vs d/Vf D/V

12
INTERFERENCE

• So d/Vs d/Vf D/V
• Rearranging D d (V/Vs V/Vf)
• V is speed of light in air is essentially the
  speed of light in vacuum.
• Remember Definition of refractive index
  nmV/Vm
• Substituting D d (ns nf)
• Where ns nf is birefringence (d) or
• Difference between the refractive index of the
  slow and the fast ray

13
Interference
When the fast ray the slow ray are resolved
into the vibration direction of the upper
polariser Interference

• a) If the retardation (D) is one wavelength.
• ? i? (i is an integer)
• NOTE When the vector components of the 2 rays
  are resolved in the vibration direction of the
  upper polar, they are in opposite directions and
  cancel out

D1l
Fig 5.4
14
Interference

• b) If the retardation (D) is one-half wavelength
  or
• ? (i ½) ?
• Vector components of both rays resolved in the
  vibration direction of the upper polar, they are
  in the same direction so they constructively
  interfere

D1/2l
Fig 5.4
15
Interference

• Consider the interference pattern formed by
  quartz wedge with monochromatic light.
• Where the retardation is a whole number of
  wavelengths, the slow and fast ray destructively
  interfere at the upper polar and a dark band is
  seen.
• Where the retardation is 1½ wavelength the 2
  rays constructively interfere and light passes
  with maximum intensity

16
Interference

• Using polychromatic light
• Every wavelength will have a different place
  where total cancellation will occur
• The colors that result after the interference by
  the upper polar are called interference colors
• The interference colors depend on
• 1/ the path length or thickness of the
  mineral, t
• 2/ the difference in Refractive Index between
  the fast slow rays (also known as the
  birefringence), nf - ns

17
Interference

• If a quartz wedge is placed between crossed
  polars
• At the thin end thickness and retardation are 0
  and the color is black
• As the thickness increases the color changes from
  gray, to white then to yellow and red and then
  repeating the sequence of blue-green yellow and
  red
• The color produced depends on which colors pass
  through the polars and which are canceled

18
Michel Levy Color Chart
Path difference in nm
19
INTERFERENCE
From Michel-Levy chart Horizontal lines
thickness Vertical lines retardation Birefringen
ce - intersection

• Remember D d (ns nf)
• D (nm) d (nm) x (ns nf)
• 1nm10Å10-6mm
• d (nm) D / (ns nf)
• d 315/0.009 35000nm
• or 0.035mm

20
D d (ns nf)
Interference color of quartz retardation
Birefringence
Thickness 0.03 mm (30 ?m)
21
Extinction

• Anisotropic minerals will go dark (extinct)
  between crossed polars every 90? of stage
  rotation (unless the Optic Axis is vertical).

Fig 5.9
NOTE Extinction occurs when 1 vibration
direction of the mineral is oriented parallel to
upper polariser
22
Extinction

• Measuring extinction angles
• 1/ Align crystal length direction or cleavage
  parallel to cross hair.
• 2/ Note the angle (degrees) on the rim of your
  stage.
• 3/ Cross the nicols and rotate until the grain
  becomes extinct.
• 4/ Note the number of degrees on rim of stage and
  deduct from previous value.
• 5/ Angle of rotation is the extinction angle.

Fig 5.10
23
Extinction

• Categories of extinction
• a) parallel extinction
• b) inclined extinction
• c) symmetrical extinction
• d) no extinction angle

a)
c)
b)
d)
Fig 5.11
24
Fast Slow Ray

• Which vibration direction is the fast or the slow
  ray?

?
ANSWER Use an accessory plate with a fixed path
length.
25

• Sample in 45o position
• Path difference after sample D1
• Accessory plate with fixed path length DA
• Slow ray // slow ray yields additional path
  difference.
• Path difference after accessory plate D2 D1 DA
  or addition
• Interference color will be higher

• Using Polarized light

Fig 5.12
REMEMBER Slow on Slow Addition
26
INTERFERENCE
Birefringence 200 nm
Accessory plates adds 550nm
New birefringence 750nm
27

• Sample in 45o position
• Path difference after sample D1
• Accessory plate with fixed path length DA
• Slow ray // fast ray yields lesser path
  difference.
• Path difference after accessory plate D2 D1- DA
  or subtraction
• Interference color will be lower

• Using Polarized light

REMEMBER
Slow on Fast Subtraction
Fig 5.12
28
INTERFERENCE
Birefringence 200 nm
Accessory plates subtracts 550nm
New birefringence -350nm
The color chart is symmetrical with respect to 0
(so ignore minus)
29
INTERFERENCE
Determining the SLOW FAST ray with Accessory
Plate
Length-fast
Fig 5.13
30
Relief

• Remember the Becke-line Test ?
• In this test, the bright line goes to the higher
  refractive index material
• Now there are 2 Refractive Indices, a fast ray
  a slow ray
• It is now possible to have one refractive index
  higher than balsam or oil (positive relief) and
  one lower than the balsam (negative relief)

31
Relief

• This is called changing relief
• Example Calcite
• Oil n1.550
• Fast ray n1.57, so low relief
• Slow ray n 1.658, so high relief

Fig 5.14
32
Pleochroism

• Change of color on rotation of the stage in plane
  polarized light (so only one vibration direction
  passes)
• This is when the nicols are not crossed and only
  the lower polar in place.
• This effect is known as Pleochroism
• Examples Chlorite (dark green to light green)
  and biotite (dark brown to light brown)

Fig 5.15