| 4. Particle Beams and Accelerators |
The idea of accelerators in particle physics is to give particles more momentum, hence reducing the de Broglie wavelength. The easiest way to accelerate a particle is to attract an electron that is boiled off by thermionic emission and attract it to an anode. We have seen that done in the cathode ray tube in a TV set.
We can use protons which have 1800 times the mass of the electron. For a given speed, they have 1800 times the momentum, and a de Broglie wavelength of 1800 times less. They should resolve objects 1800 times smaller.
We can easily measure the speed of the particles by considering their kinetic energy given by:
Ek = ½mv2
The energy acquired is:
DE = eV
For a typical accelerating voltage of 50 000 V, the energy gained is:
DE = 1.6 × 10-19 C × 50000 V = 8.0 × 10-15 J
The speed can be worked out from:
v2 = 2Ek
m
v2 = 2 × 8.0 × 10-15 J
9.11 × 10-31 kg
v2 = 1.71 × 1016 m2 s-2
v = 1.31 × 108 m s-1
If we were to increase the voltage by 100 times to 5 × 106 V (5 MV), which is quite possible, we would increase the speed by 10 times to 1.31 × 109 m s-1. However Einstein's Special Relativity Theory tells us that nothing can travel faster than the speed of light. This tells us two things:
That laws and models that we use in normal physics do not apply to all situations.
There are strange things that happen when things travel towards the speed of light.
We find that as a particle approaches the speed of light, some of its kinetic energy is converted to mass. The particle gets heavier. This is called a relativistic effect. This should not be a surprise since mass and energy are interchangeable according to Einstein's famous equation:
E = mc2
Joules and kilograms in this context are rather clumsy to work with. So we use electron-volts (eV). They are a unit of energy, not voltage. This should not be too difficult since we know that:
energy (J) = charge (C) × voltage (V)
The electron-volt is the energy gained by unit charge accelerated through a potential difference of 1 volt.
1 eV = 1.6 × 10-19 J
Our electron above accelerated through 50 000 V would have an energy of 50 000 eV or 50 keV. A lot easier than 8.0 × 10-15 J.
Many accelerators can accelerate particles to energies of giga electron-volts (GeV) (109 eV) or even tera-electron-volt (TeV) (1012 eV).
The units found in particle physics are often expressed in very odd looking units that are based on the rest mass of the particle. If we were able to convert all an electron's mass into energy according to E = mc2, we would get:
E0 = 9.11 × 10-31 kg × (3.0 × 108 m s-1)2
= 8.20 × 10-14 J = 5.12 × 105 eV
This is 0.512 MeV. Since E = mc2, it does not take a genius to rearrange this to give the mass:
m = E/c2
So we can say that the rest mass of the particle is 0.512 MeV/c2. Heavier particles has rest masses of GeV/c2. These strange looking units allow particle physicists to see if they have enough energy for a collision without having to do any calculations.
A CRT is an accelerator, because electrons are attracted from a cathode to an anode, and their speed increases. The electron microscope works on much the same principles, and it can resolve down to the atomic level, about 10-10 m.
To resolve further, we need to get down to de Broglie wavelengths of 10-15 m. Higher powered machines are needed. The picture shows a linear accelerator (LINAC).
Ions have a heavier mass than electrons, so have a greater momentum (hence de Broglie wavelength). They are attracted to the first of the drift tubes by an opposite charge. The drift tube is connected to an alternating supply. As the ion passes through the polarity changes, so the ions is repelled by the first and attracted to the second. Then the polarity is changed again. The effect of this is that the little brutes are first pulled in then kicked up the backside out. By time they get to the end, they are shifting like greased lightning.
The longest machine is 3 km long. For a machine that requires millimetric precision in setting up, building it across an active fault line did not show a great deal of foresight.
The cyclotron is much more compact, as it send the particles in a spiral path.
When we put charged particles into a magnetic field, we find that they are subject to a force at right angles to the direction of the travel. The force always remains at right angles, so the resulting path is circular. The radius depends on the square of the speed. In a cyclotron, the speed increases as the electron passes between the poles, so the path is a spiral with increasing radius. Then the charged particles come out tangentially in a straight line.
We will look at the physics in more detail...
Consider a mass m travelling at a velocity v. A force F is applied at 90o.
If the force continues to be applied at 90o, the resulting path is circular.
The linear speed remains constant, but the direction changes. There is a change in velocity, hence acceleration.
The angular velocity, physics code w is how big an angle is turned in one second in radian. The symbol w is omega, a Greek letter long 'ō'.
One radian is the angle that subtends an arc whose length is the same as the radius.
q rad = s/r.
1 revolution is 2p radians.
For small angles in radians, q » sin q » tan q.
w rad s-1 is the angular velocity. w = 2pf Þ linear speed v = wr = 2pfr m/s.
The direction of the velocity is tangential.
A common bear-trap
is to fail to convert revolutions per minute to radians per second. Divide the
rpm by 60, then multiply the answer by 2p.
Acceleration is always towards the centre of the circle and is given by:
a = w2r.
or
a = v2/r
Centripetal force is described by the formula:
F = mv2
r
or
F = mw2r
The force acts towards the centre of the circle.
Examples
Satellite orbiting the Earth (gravity provides the centripetal force)
Vehicle going a bend (friction)
Electron orbiting the nucleus (electrostatic attraction).
There is no such thing as centrifugal force!
We have seen how the force on a charged particles is given by:
F = Bqv sin q
The physics codes are:
B - magnetic field strength (T);
F - force (N);
q - charge (C);
v - velocity (m/s);
q - angle with the magnetic field (usually 90o, so sin q = 1).
The direction of the force is given by Fleming's Left Hand Rule. Since the force is always at 90o to the direction of the velocity, the path is circular. Whether the circular path is clockwise or anticlockwise depends on:
The direction of the field;
The charge of the particles.
We can produce an expression to give us the radius of the track:
Bqv = mv2/r
One of the v terms cancels out to give:
Bq = mv/r
So we can write:
r = mv
Bq
Since the equation for momentum is:
p = mv
We can write:
r = p
Bq
The greater the momentum, the greater the radius of the path. Alternatively a bigger magnetic field is needed to keep it on a track of the same radius.
Particles are tracked in huge and complex detectors with many millions of euros. But the paths they take follow these simple principles of physics.
|
|
|
|
