two models for the description of light the corpuscular theory of light stating that light can be...
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Two models for the description of light
The corpuscular theory of light stating that light can be regarded as a stream of particles of
discrete energy called photons. Their energy E is defined by:
E = h
The wave theory of light stating that light can be treated as a wave with an electrical and
magnetic field, each described by a vector. The magnitude of the electric field vector Y at
position x and time t and the amplitude ao (constant) is given by:
Y = aosin[(2)(x+vt)]
The velocity (v) is related to the frequency () by the equation:
v =
The description of most atomic processes such as absorption, fluorescence, and the photo-
electric effect require the photon approach.
The electro-magnetic wave intensity (I) is proportional to the square of the amplitude:
I = a
K is a constant of proportionality and depends upon the properties of the medium containing
the wave.
Polarisation of Light
circularly polarized
linearly polarized
Four principal interactions of light with matter
Ignoring fluorescence, the interactions of light with matter can be expressed
thus:Io = Ireflected + Iscattered + Iabsorbed + Itransmitted
transparentmaterial
translucentmaterial
Refractive Index and Polarizability of Matter
Refraction and the refractive index
When light enters a transparent medium of different refractive index, n, it is refracted (Snell’s Law):
n = sin sinangles of incident & refraction,
respectively)
sin1 / sin2 = n2 / n1
The velocity of a light wave changes when light enters a transparent medium of different refractive index but not the frequency:
velocity = n = c / v = vacsubs
Graphical change of wavelength with change of n.
At the critical angle c, the emerging
ray travels exactly along the surface.
Exceeding this angle results in total
reflection (no light is lost). The critical
angle is given by:
sin c = n(low) / n(high)
Total Internal Reflection
Dispersion and Colour
The refractive index of a transparent solid varies with wavelength.This is called dispersion.
Polarisation by Reflection
The reflectivity or reflectance of a surface is
given by:
Rs = [sin(1 – 3) / sin (1 + 3)]2
Rp = [tan(1 – 3) / tan (1 + 3)]2
The Brewster’s angle
Birefringence of Optically Anisotropic Matter
Birefringence
A nematic phase, for example, is essentially a one-dimensionally ordered
elastic fluid in which the molecules are orientationally ordered along the
director.
The nematic phase is birefringent due to the anisotropic nature of its physical
properties. Thus, a light beam entering into a bulk nematic phase will be split
into two rays, an ordinary ray and an extraordinary ray (along the director).
These two rays will be deflected at different angles and travel at different
velocities through the mesophase, depending on the principal refractive
indices. If the extraordinary ray travels at a slower velocity than the ordinary
ray, the phase has a positive birefringence.
We can write for most optically uniaxial calamitic mesophases:
ne > no with n = ne-no
Double Refraction and Birefringence of an Anisotropic Transparent Medium
The relationship between the magnitude
of n’e and the angle that the ray makes
with the optic axis is:
1 / (n’e)2 = cos2 / no2 + sin2 / ne
2
Snell’s Law: sin1 / sin2 = n2 / n1
Birefringence and the Indicatrix
Molecular Theory of Refractive Indices
Lorentz local field for an isotropic medium: Eloc= [(+ 2) / 3] E
is the mean permittivity
Using = n2 derived from the Maxwell’s
Equations, the Lorenz-Lorentz expression relates
the refractive index to the mean molecular
polarizability: n2 – 1 / n2 + 2 = N / 30
where N is the number density (d NA / M ), the
mean polarizability, and 0 = 8.86 10-12 As/Vm
ne2 – 1 / n2 + 2 = N / 30 no
2 – 1 / n2 + 2 = N / 30
with n2 = 1/3 (ne2 + 2no
2)
Anisotropic Molecular Polarizability
Schlieren Texture of a Nematic Phase
Defect Textures in Thermotropic Liquid Crystals
Textures of a SmA Phase
Textures of a SmC Phase
broken focal-conic schlieren
Textures of a Colh Phase
Mosaic Texture of a SmB Phase
Since the nematic phase can be treated as an elastic continuum fluid, three
possible elastic deformations of its structure are possible:
The splay deformation, the twist deformation, and the bend deformation.
The elastic constants associated with them are k11, k22, and k33, respectively.
Deformations in Thermotropic Liquid Crystals
Mesophase Homogeneous (planar) alignment
Homeotropic (orthogonal) alignment
Mechanical shearing other
Nematic N schlieren extinct black shears easily Brownian flashes
SmA focal-conic, polygonal defects
extinct black shears to homeotropic
Cubic D-phase
extinct black extinct black viscous grows in squares or rectangles
SmC focal-conic broken schlieren (4 brushes) shears to schlieren Brownian motion
SmB focal-conic extinct black shears to homeotropic
Mosaic possible
SmI focal-conic broken schlieren shears viscous schlieren diffuse
Crystal B mosaic extinct black shears viscous grain boundaries
SmF mosaic schlieren, mosaic shears viscous grain boundaries
Crystal J mosaic mosaic very viscous grain boundaries
Crystal G mosaic mosaic very viscous grain boundaries
Crystal E mosaic shadowy mosaic very viscous grain boundaries
Crystal H mosaic mosaic very viscous grain boundaries
Crystal K mosaic mosaic very viscous grain boundaries
Natural textures exhibited by calamitic LCs(as seen between crossed polarizers)
Mesophase Paramorphotic textures
SmC broken focal-conic from SmA focal-conic; schlieren from SmA homeotropic; sanded schlieren from cubic D-phases
SmI focal-conic broken, chunky defects from SmA or C focal-conic; schlieren from schlieren SmC or SmA homeotropic
SmB focal-conic from SmA focal-conic; clear focal-conic defects from broken SmC focal-conic; extinct homeotropic from SmA homeotropic or SmC schlieren
Crystal B clear focal-conic from focal-conic SmA, B or C; homeotropic from homeotropic SmA and B or SmC schlieren
SmF broken focal-conic from focal-conic SmA, B, C, I or crystal B; schlieren mosaic from homeotropic SmA and B or SmC and I schlieren
Crystal J Broken pseudo focal-conic fans, chunky from focal conic domains SmA, B, C, I, and F or crystal B; mosaic from homeotropic SmA, B, and crystal B or SmC and I schlieren or SmF schlieren mosaic
Crystal G
Crystal E
Crystal H
Crystal K
Paramorphotic textures associated with calamitic LCs(as seen between crossed polarizers)