| 3. Towards the Standard Model |
The early physicists thought that the atom was indivisible. In fact the word atom means "can't be cut". Then came two important discoveries:
Thompson's early model of the atom was the plum-pudding, with electrons studded in a matrix of positive charge. Neutrons had not been discovered at that time.
Neils Bohr developed the nuclear atom model, with the positive charge in the middle and the electrons orbiting a certain distance out.
It was Ernest Rutherford who did much to develop the idea of the nuclear atom with his famous alpha-scattering experiment.
Rutherford was using alpha particles as nuclear bullets to smash up the atoms; he wanted to see atoms bursting like watermelons.
But…
Instead of bits of atom, Rutherford found that a small proportion of the alpha particles were deflected, while an even smaller proportion bounced back.
From analysis of these observations he concluded:
Most of the atom was empty space;
The positive charge was concentrated in a very small space;
The radius of the nucleus was in the order of 3 × 10-14 m;
The alpha particles that were deflected back had to be travelling in a line with the nucleus.
Rutherford’s estimates were not far out. Later research has shown the nuclear radius to be in the order of 1.5 × 10-14 m. However the boundary is not sharp, but rather fuzzy, as the nucleus is a very dynamic entity.
It was the Eighteenth Century physicist Charles Coulomb who worked to discover the laws of electrostatic attraction and repulsion, which bear his name. He used little pith balls [be careful how you say it]; nowadays we use expanded polystyrene balls.
In the picture below we have two identically charged light balls. They are repelling, because they have the same charge.
We can see that there are three forces in equilibrium:
The weight acting vertically downwards;
A force acting horizontally;
A resultant force, producing tension in the light cord.
The weight W = mg is the easiest of the forces to measure. Then we can say that:
mg = T cos q;
F = T sin q;
We can rearrange these to give a simple expression:
F = T sin q = tan q
mg T cos q
So this gives us:
F = mg tan q
Despite the not so difficult theory, the experiment is a bit of a swine to set up. But Coulomb persisted and found that:
the force was proportional to each of the charges;
the force was inversely proportional to the distance between the charges.
In physics code we can write:
F µ Q1Q2
r2
Adding in the constant of proportionality we write:
F = k Q1Q2
r2
With F in newtons, Q in coulombs, and r in metres, the constant k is written:
1
4pe0
The term e 0 (pronounced "epsilon-nought") is called the permittivity of free space. It has the value of 8.85 × 10-12 F m-1. Epsilon is a Greek letter 'e'.
Our constant has a value:
1 = 9.0 × 109 N m2 C-2
4pe0
So we can write:
F = 1 × Q1Q2
4pe0 r2
If we plot a graph of force against distance, the area under the graph gives us the energy.
If we use logarithmic scales on both axes, we get a straight line of gradient -2, since the force follows an inverse square law.
This graph tells us that as the alpha particle approaches the nucleus, its kinetic energy falls, and is converted to electrostatic potential energy. Eventually it stops (kinetic energy = 0). All its energy is potential. This is transferred back to kinetic as the alpha particle shoots back.
By calculus it can be shown that the increase in electrostatic potential energy is given by:
DE = k Q1Q2
r
An electric field is a region in which a charged object experiences a force. The electric field of a point charge is radial.
The field of a spherical charge is also radial, as long as the distance is much bigger than the radius. For simplicity, we treat all charges as point charges. If we double the distance from the centre, we see that the concentration of field lines drops to a quarter. This indicates that the electric field strength varies as an inverse square law.
The electric field strength is defined as force per unit charge:
E = F
Q
This definition gives the units of force as being newtons per coulomb (N C-1). We have seen that in a uniform field, the electric field strength was given as:
E = V
d
So the units were volts per metre (V m-1). Therefore we can say that:
1 N C-1 = 1 V m-1
The full equation is:
E = k Q
r2
Where there are opposite charges, the electric field lines are like this:
Not dissimilar to a magnetic field. Hardly surprising as magnetic forces are really a version of the electromagnetic force. Like magnetic field lines, the closer the electric field lines are the stronger the electric field.
When the charges are repelling the picture is like this:
We can plot equipotentials which are voltage contours of the electric field.
We have seen momentum before, and know that:
Momentum is the product between mass and velocity. p = mv
Direction is important.
Momentum is always conserved in collisions.
We also saw that for a perfectly elastic collision, kinetic energy is conserved.
We will now look at a special case when a particle hits another particle of identical mass that is stationary. The simplest case is when the moving particle hits the stationary particle head-on. The first particle stops and the second moves off at the same velocity as the first.
If the first particle strikes the second at an angle, the two particles move off in paths that are 90 o to each other.
We can prove this as follows.
We can express the kinetic energy in terms of momentum. We know that:
p = mv
Ek = 1/2mv2
To get the v2 bit, we need to square the first equation:
p2 = m2v2
Then divide both sides by 2m:
p2 = m2v2 = mv2
2 2m 2
So this gives us:
Ek = p2
2m
Since momentum is conserved, we can write:
p = p1 + p2
The bold script tells us that this is the vector sum of the two momenta.
Since kinetic energy also is conserved, we can write:
So we can get rid of the 2m terms downstairs to give us:
Since this is Pythagoras, the angle between the two paths must be at right angles. Tedious derivation, but you might be asked about it in the exam.
One of the key discoveries in the early Twentieth Century was that light, long considered to be a wave, had particle properties. Einstein produced the concept as the light as "a wave in a box", called a photon. It seemed to be very reasonable to suggest that particles had wave like properties, and this was confirmed by a member of the Belgian royal family, Prince Louis de Broglie (1892 – 1987). (The name is pronounced "de Broy".)
Any particle of mass m would have an associated wavelength, the de Broglie Wavelength, that could be worked out by:
Electrons are diffracted at certain angles by a very thin layer of graphite to produce rings. The ring spacing fits very well the model predicted by the Bragg Equation:
l = 2d sin q
This equation was worked out by the father and son team of William and Lawrence Bragg. It applied to diffraction of X-rays, which are, of course, electromagnetic waves. Therefore we can say that electrons are showing wave-like properties.
We can find the momentum of the electrons easily enough:
Kinetic energy = electrical energy supplied = charge × voltage (between the cathode and the anode)
1/2 mv2 = eV
mv2 = 2eV
(mv)2 = 2meV
p = mv = Ö(2meV)
We can combine this with the de Broglie relationship:
What is the significance of this? We can resolve objects with a microscope to about 1 wavelength of light, roughly 0.2 mm. So to resolve anything smaller than this, we need shorter wavelengths. We can get those by accelerating particles like electrons in order to get short de Broglie wavelengths. The higher voltages that we apply, the more resolution we can get. However, if we start to get to very high voltages, and then working out the speed of the electrons, we find that they are travelling faster than the speed of light in a vacuum, which can't happen.
Instead we see relativistic effects, whereby the mass of particles increases as the speed gets close to the speed of light.
The structure of materials has been interpreted from diffraction patterns. The picture below shows the pattern from a single slit.
We can work out the angle of diffraction using a simple equation:
sin q = l
a
where q is the angle, l is the wavelength and a is the width of the aperture.
We can use diffraction patterns in a similar way to to interpret the spacing between atoms and their arrangement in crystal lattices.
Methods similar to this were used to resolve the nucleus to the proton level, and about 50 years ago, the size of the proton was determined to be about 5.6 × 10-15 m. However it was viewed as a fuzzy billiard ball, and there was no idea that it had any ultrastructure.
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