Download 1 The Nature of Light: Wave versus Particle Light travels in a

The Nature of Light: Wave versus Particle
Light travels in a vacuum with a velocity c=3x 108 m/s. When light travels through matter,
its velocity is less than this and is given by
v = c/n
where n is the index of refraction of the substance. The value of the index of refraction
depends on both the composition of the substance and the color of the light.
In the early nineteenth century, Thomas Young described interference experiments that
could be explained only by assuming that light was a wave. By the end of the nineteenth century
nearly all the known properties of light were explained by assuming that light consists of an
electromagnetic wave.
By electromagnetic wave we mean that
a. Light can be produced by accelerating an electric charge.
b. Light has an electric and magnetic field associated with it.
c. The velocity of light traveling in a vacuum is given by electromagnetic theory in terms of
parameters measured in the laboratory for "ordinary" electric and magnetic fields.
In the early twentieth century, light was discovered to have both particle properties and
electromagnetic wave properties at the same time. This rather disconcerting discovery was
followed in a few years by the discovery that matter also has wave properties.
A traveling wave of light can be described by a function of the form f(x - ct) which
represents a disturbance traveling along the x axis in the positive direction. If the wave is sinusoidal
then the period, frequency and wavelength can be defined:
ν = 1/T
c = λν
Each particle of light called photon has energy E. The energy of each photon is related to
the frequency ν by the relation
E=hν= hc/λ
where h is called Planck's constant having the numerical value
h = 6.63 x 10-34 J s
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h = 4.14x10-15 eVs
It is sometimes useful to use the number ђ :
ђ=1.05x10-34 Js
ђ=0.66x10-15 eVs
In terms of the angular frequency ω = 2пν,
E = ђω
The full electromagnetic spectrum includes radio waves, microwaves, infrared, visible and
ultraviolet light, X-rays, and y-rays. The photons of visible light have energy of a few eV, X-ray
photons are approx. 104 times more energetic, while γ-ray photons, originating from atomic nuclei,
are usually even more energetic.
The property of light that determines its color is the frequency or the energy of each photon.
Visible light covers a narrow range of frequencies between 4.0 – 7.5 x 1014 Hz, corresponding to a
wavelength range of 400 – 750 nm.
The Origin of Atomic Spectra
The simplest system that can emit or absorb light is an isolated atom. An atom can change
from one energy level to another by emitting or absorbing a photon with energy equal to the energy
difference between the levels. One of the experimental proofs of this phenomenon is the so called
Frank Hertz experiment. Let the energy levels be labeled by i = 1, 2, 3, . . . , with the energy of the
ith state being Ei. There is a lowest possible internal energy for the atom; when the atom is in this
state (ground state), no further energy loss can take place. If Ei is greater than the ground state
energy, then the atom can loose energy by emitting a photon of energy (Ei - Ef) and exist in a lower
energy state Ef , perhaps different from the ground state of the atom.
It is possible to calculate the energies belonging to each level by techniques of quantum
mechanics. All these techniques are based on Schrödinger’s equation:
Hop Ψ = E Ψ
In the hydrogen atom the energy of the nth level is given as
2
En = −
13.6
n2
(eV )
An energy level diagram in the slide show displays these energies and some transitions
between them. It must be noted that the energy levels depend not only on the integer n = 1, 2, 3,4,..,
which is called the principal quantum number, but on other quantum numbers, as well.
Transitions between various energy levels in hydrogen result in the spectrum for hydrogen
displaying different light intensities vs. wavelength.
In general, the energy of an atom depends on the values of five quantum numbers for each
electron in the atom. These quantum numbers are:
n = 1, 2, 3....
principal quantum number
l = 0, 1, 2, . . . , (n - 1)
orbital angular momentum quantum number
s=½
spin quantum number
ml = -l, -(l - 1), . . . , l
“z component” of orbital angular momentum
ms = -1/2, +1/2
“z component” of spin
The internal energy of the atom is the sum of the kinetic and potential energy of each
electron. The energy of each electron depends on the values of its quantum numbers. It is
influenced by the electric field of the nucleus and all the other electrons. There are also magnetic
interactions between electrons and between each electron and the nucleus, because the moving
charges generate magnetic fields. No two electrons in the atom can have the same values for all
their quantum numbers (Pauli’s exclusion principle).
The ionization energy is the amount of energy required to remove an electron from the atom
when the atom is in its ground state. For hydrogen the ionization energy is 13.6 eV.
Molecular Energy Levels
In addition to internal energy, an atom can have kinetic energy of translation with three
degrees of freedom. The translational kinetic energy is also quantized; but as long as the atom is not
confined to a very small volume, the levels are so closely spaced that they can be ignored.
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Two atoms together have six degrees of freedom, because each can move in threedimensional space. If the atoms are bound together, however, their motions are not independent.
One can speak of the three degrees of freedom for translation of the molecule as a whole (center of
mass motion) and also the vector displacement of one atom from the other (See figure in slide
show). Vector r locates the center of mass of the two atoms.
The two atoms can have two types of relative motion:
a. They can rotate about some axis with angular velocity ω.
b. They can vibrate: move back and forth along the line connecting their centers.
Both motions give rise to quantized energy levels. Energy difference between rotational
energy levels are much smaller than those of the vibrational ones.
The atoms in the molecule can vibrate back and forth along their line of centers. In classical
physics, if two masses are held at a fixed distance apart, by a spring for example, work must be
done to push the masses closer together or to pull them apart. In both cases potential energy is
increased. At the equilibrium separation of the atoms in the molecule the potential energy is a
minimum. In the slide show the potential energy of a sodium ion and a chloride ion is displayed as
a function of their nuclear separation.
Changes in electronic quantum numbers within an atom and of vibrational and rotational
quantum numbers within the molecule lead to and result in a complicated spectrum of the molecule.
When the electronic quantum numbers change, the shape of the curve of energy as a function of
nuclear separation also changes.
If we consider transitions in which the quantum numbers of atomic electrons change; these
may involve frequencies in the visible, the ultraviolet, or the infrared, depending on the spacing of
the levels.
Instrumental arrangements for measuring emission, absorption and fluorescence (or
scattering)
1. Emission spectrophotometry
The emission flame spectrophotometer is used for quantitative measurement of the
concentration of elements such as sodium, potassium, or chlorine in serum or urine. The sample is
diluted and atomized in a flame. The temperature of the flame is high enough so that some of the
atoms are excited to energy levels above the ground state. They then decay, emitting spectral lines
characteristic of that element. The instrument can be calibrated so that spectral line intensity
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measures concentration, although dilution factor, atomization efficiency, and flame temperature
must all be carefully controlled. A schematic arrangement of such an apparatus is shown in the
slide show. Emission of hot solids and liquids results in a continuous spectrum. Energy levels split
due to the interaction of the atoms, band structure is formed. Emission of hot gases result in a line
spectrum. Absorption of light by cold gases also result in a line spectrum.
2. Absorption spectrophotometry
Absorption spectrometry uses the absorption of light by atoms in their ground state to
measure the concentration of the atoms. The absorption is proportional to the number of
photons present, and also to the number of absorbing atoms. The light intensity falls as
I = I 0 e − µx
where x is the thickness of the absorber. The absorption coefficient µ depends on both wavelength
of the light and the concentration of the absorbing molecules or atoms.
3. Fluorescence. (Will be discussed in detail in forthcoming lectures.)
4. Raman scattering
Raman scattering is the scattering of visible light (for which the biological preparation is
more likely to be transparent), in which the emerging photon does not keep its original energy, but
has lost or gained energy corresponding to a rotational or a vibrational transition. The effect gained
practical importance only recently, because lasers can provide a source of visible light with
sufficient stability so that the Raman lines can be resolved. In Raman scattering, a photon gains or
loses energy due to a change in the vibrational or rotational state of the scattering molecule. The
spectrum contains a very intense line with no energy change and two weak lines on either side of
the intense line due to Raman scattering.
Raman scattering has been used to study e.g. the conformation of the antibiotic
valinomycin, which drastically affects the permeability of cell membranes, in an effort to
understand the mechanism for the permeability changes.
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