p class="Normal" style=" text-align: right;">Advanced Lab I

Wave-Particle Duality- The Nature of Light

Abstract

In this
lab two experiments and a simulated x-ray diffraction measurement will
be done. The first experiment is similar to the one Davisson and Germer
performed and demonstrated that electrons interfere with themselves
as if they were waves. The second experiment demonstrates the particle
nature of an electron by measuring their trajectory in a magnetic field
to give the e/m ratio. For the x-ray diffraction simulation, diffraction
lines are identified from a NaCl crystal and where used to predict the
diffraction lines from a graphite sample. The determination of the short
wavelength limit of the x-ray spectrum from an x-ray tube will be discussed
and its interpretation in terms of the particle nature of x-rays will
be noted.

Introduction

Picture
1: Louis deBroglie

Experiments
that measure properties of the electron show that it has wave-like and
particle-like properties.  Depending on the experiment, one will
find evidence for the different interpretations of the electron. 
As a fundamental assumption of quantum mechanics, that has been established
experimentally, by Davisson and Germer, show that electrons behave like
waves.  It was also shown by J.J. Thompson that electrons possess
particle properties. 

It will be demonstrated by diffraction
from an ordered crystal that electrons do indeed diffract, just like
waves.  DeBroglie's relations for the energy dependence of the
electron's wavelength will be verified and measurements of the spacing
between carbon atoms in graphite will be performed.

As an example of particle behavior,
measurement of the e/m ratio of an electron will show that it indeed
has a particle nature. An electron in a uniform magnetic field will
orbit in a plane perpendicular to the field with a radius determined
from Newton's Laws. A Helmholtz coil is used to produce a nearly uniform
magnetic field. Electrons are produced from a hot cathode and focused
into a narrow beam that is injected into the field with a non-zero angular
momentum. The electron trajectory is visible because the tube is has
a low partial pressure of argon. Some electrons colliding with the argon
will cause electronic excitations that decay and produce visible light. 
This looks pretty cool. 

Safety reasons will hinder the actual
experiment using x-ray diffraction.  But instead what was allowed
was a simulated experiment that demonstrates how crystallography is
actually done.

Theoretical Background

Electron Diffraction Experiment- wave properties of the electron:

Due to the contribution of deBroglie,
we know that electrons behave sometimes as waves, with a simple relation
between the electrons momentum and wavelength:  = h/p.  This is a interesting equation because h
is just Plancks constant, telling us that the wavelength of an electron
is very simply, related inversely to the electrons momentum. 
When electrons are given a certain energy, say, sufficient energy to
light up a phosphorous screen, their momentum can be expressed in terms
of this energy, namely, eV = p<sup>2/2m.  And you can see, using our knowledge
that an electrons wavelength is inversely proportional to its wavelength,
and our energy relation, we can express the electrons wavelength by: 

  
 

Eq. (1)

Hopefully its painfully obvious
that m and e are the mass and charge of the electron.  A diffraction
grating, such as a plane of graphite crystals, could perhaps diffract
electron waves into certain angles, as given by Braggs Law:

  
 

Eq. (2)

Now it can be mentioned that indeed,
in this experiment, electrons are shot from an electron gun, emitted
from a cathode, focused by a focus element, accelerated to a graphite
target, and they then light up a phosphorus screen!  This graphite
target, is of course, a plane of graphite atoms, crystals in fact, that
act a diffraction grating. In Braggs Law, for our case, d is the separation of the atom planes, not the individual
atoms, but the planes of carbon layers that make up the graphite.

Graphite is cool because it not
only is good for writing down physics notes, but because it has a hexagonal
structure.  Kind of like a stop sign, only with six sides. 
Now, to understand what kind of diffraction we are going to see when
these electron waves scatter from the graphite, a couple of nice figures
are necessary.  The six sided structure of graphite is shown below,
along with the geometry of Braggs scattering law.

 

Figure 1: Bragg scattering geometry

 

Figure 2: Hexagonal (a) and Cubic (b) lattices. a<sub>0 is the bond distance between atoms.

Now, we want to know what kind of
diffraction we will see.  It looks like, from Figure 1 and Figure
2, that if we have 6 atoms that make up the graphite, we will see 6
spots on the screen.  Six spots that is, for each d spacing.  But actually, graphite isnt so perfect. 
It is a little disordered.  The planes of hexagonal rings are weakly
bonded to each other.  What this means, is because of the weak
inter-plane bonding, there is a random orientation to the planes. 
Material that has this property is given an interesting name, its called
powdered crystals.  So, because of randomly orientated planes,
we dont see a beautiful 6 spots for each d spacing, but instead a large set of 6 diffraction beams for
each d spacing, or two rings, which is still good enough. 

Electron
Charge to Mass Ratio Experiment- particle properties of the electron:

About 100 years ago, the atom was
still thought to be indivisible, but fortunately at this time, J.J.
Thomson was studying cathode rays. To Thomson, cathode rays were a
strange form of energy emitted from metals heated to very high temperatures.
Thomson figured the form of energy was some kind of uniform set of particles,
maybe an ionized atom or molecule.  He figured that he could bend
a beam of cathode rays with a magnetic field and determine what the
mysterious particles charge to mass ratio was. He found that regardless
of how strong the field was, q/m was always the same, meaning he had
discovered a new particle! This new particle was indeed the electron. 
From his experiment he found that it must have a small amount of negative
charge and an incredibly small mass. To follow him up, Robert Millikan
determined the charge of the electron and later received the Noble prize
for his work.

This discovery led to a whole new
branch of physics. Soon after this, the protons and neutrons were discovered.
These sort of techniques were applied to these new particles and as
these experiments were performed using higher energies, new particles
were found.

So lets begin the basic physics. 
The force acting on a charge, e, when moving perpendicular to a magnetic
field, B, with a velocity, v, is given by the equation

                                                        F = Bev      Eq. (3)

The direction of the force is perpendicular
to the direction of travel of the electron, causing the electron to
move in a circular path with a radius, r, such that

                                      
     Eq. (4)

The kinetic energy of an electron
when accelerated through a potential difference V is given by the equation
                                           

eV
= 1/2mv<sup>2      Eq. (5) 

Combining equations (4) and (5)
gives                                             

     Eq.
(6)

To produce the magnetic field a
pair of coils, known as Helmholtz coils are used to create a nearly
uniform field at the midpoint of the two rings.  This convenient
way of producing relatively uniform B fields, can be shown to result
in a field (by adding two coils with current going through them in the
same direction at a distance apart equal to the radius of one loop),
of