Scientific Background

A Brief History of the Quantum Theory of Light

            Before the quantum theory of light, light was widely believed to be an electromagnetic wave. According to the theory of James Clerk Maxwell, an alternating current in a wire would set up fluctuating electric and magnetic fields, and the resulting electromagnetic waves would behave in every way like light – they would be reflected, refracted, and would travel at the same speed in a vacuum as light would. Maxwell’s theory also predicted that the frequency (the rate at which a pattern repeats) of the electromagnetic wave would be equal to the frequency of the oscillations in the electric current. Thus it was thought that visible light was also a type of electromagnetic disturbance created by submicroscopic oscillators (or resonators, as some called them) within the matter, which emitted light at a frequency equal to the frequency of their motion. At the time, the exact nature of these oscillators was unknown.
            Although it was impossible to generate current oscillations anywhere near the frequency of visible light, Heinrich Hertz, a great German scientist, was able to experimentally verify that electromagnetic waves could be reflected, refracted, focused, polarized, and so on; that is to say they behaved in every way like light.
            The confirmation was a great victory for Maxwell, and the electromagnetic model of light emission was rapidly applied to explain many of the scientific problems of the day. This was the classical theory of light.
            In many of the cases, Maxwell’s theories and equations could be successfully applied to explain various phenomena involving electromagnetic radiation, but there were some instances in which the classical theory of light did not agree with experiment.
             One of the scientific puzzles in the late 19th century was the “blackbody” problem. The problem was to predict the radiation intensity at a given wavelength emitted by a glowing solid at a specific temperature. Although formulas existed – most notably, Wien’s law - they did not agree with experiment at the lower frequencies (higher wavelengths). It was not until Max Planck, a brilliant German physicist, came onto the scene, that an accurate formula for the radiation intensity was discovered. Maxwell was able to interpolate between Wien’s law (which he knew worked well at higher frequencies), and recent experimental results, to find the famous “blackbody” formula for spectral density.
             However, as often happens in science, the answer posed more questions than it solved. According to the classical theory of light, the submicroscopic oscillators at a frequency f which emitted the radiation, could have any energy and could change their amplitude continuously as energy is radiated away. However, Planck’s formula forced the assumption that an oscillator with frequency f could only have a value for energy that was an integral multiple of hf, where h is Planck’s constant (). That is,

      where       n = 1, 2, 3, 4 . . .

meaning that an oscillator can’t change its energy by just any fraction, but by a multiple of a specific value that depends on its frequency, which was very difficult to explain classically.
            The next big development came when the so-called ‘photoelectric’ effect began to be seriously studied. The photoelectric effect occurs when light comes into contact with a material (a metallic emitter), and knocks out electrons, thus creating an electric current, also called a photoelectric current. In 1902, a physicist named Philip Lenard made a series of discoveries that were completely unexpected classically. He found that the maximum kinetic energy of the emitted electrons in no way depended on the intensity of light, that maximum kinetic energy depends linearly on frequency (independent of intensity), and that there was no measurable time lag between the incidence of the light on the metal and the start of the electric current (the photocurrent). Classically, of course, it was expected that a brighter light would give an emitted electron a higher kinetic energy, because brightness implied bigger light wave amplitude, and thus more energy. It was also expected that time would be required for the light wave to transfer energy to the electron, thus creating the mentioned time lag.
            The year 1905 was a very important year for physics; it was the year that Albert Einstein, hitherto unknown to the scientific community, published three of his famous papers. One of them, titled “A Heuristic Point of View About the Generation and Transformation of Light,” brilliantly explained the implications of Planck’s formula, as well as the photoelectric effect. Albert Einstein reached the conclusion that light itself is composed of packets of energy, or discontinuous quanta (particles of light also referred to as photons), as opposed to a continuous stream of energy in the form of a wave. This very convincingly explained the photoelectric effect, as a photon could transfer its energy almost instantly to an electron, and a more intense light would simply mean that there are no photons. It also explained the linear dependency of the maximum kinetic energy of the emitted electron on the frequency of the incident light. Einstein acknowledged that Maxwell’s theory of light worked very well across large distances, but that a different model might be necessary to explain phenomena at a very small, or quantum, level. Einstein’s view of light at a quantum level soon became generally accepted by the scientific community, and he later received a Nobel Prize in physics for this theory.
            To summarize, the quantum theory of light regards light as composed of particles called photons, each photon carrying an energy proportional to its frequency. This is quite ironic, as frequency is generally considered to be a property of a wave. In fact, in many cases light is still treated as a wave today. Paradoxically, it is often necessary to invoke the wave nature of light to explain its particle behavior, and the other way around. This particle-wave duality of light is generally accepted in modern science, and will be so until a unifying theory can be found. 

 

A Brief Introduction to the Special Theory of  Relativity

            In 1887, Albert Michelson and Edward Morley showed in the famous Michelson-Morley experiment that the speed of light is always constant, that regardless from what perspective or reference frame the speed of light is observed, it will always have the same value for all observers. This value is usually denoted by c, and it equals to approximately  meters per second. However, this discovery had very deep and profound implications for the way in which we describe motion.
            In Newtonian mechanics, also called classical mechanics, it is generally assumed that time is absolute, that is to say that a time interval between two events measured from one frame of reference is the same as from another frame of reference. However, in his famous special theory of relativity, Einstein worked out some of the implications of the Michelson-Morley experiment, which included time dilation, mass dilation, length contraction, and so on, which also influenced the way in which energy, momentum, and other such properties would be calculated. This ultimately resulted in a set of relativistic equations describing the properties of objects that are moving close to the speed of light relative to some reference frame. To give an example, according to classical mechanics, if an object accelerates uniformly it should eventually surpass the speed of light; according to special relativity however, as the object approaches the speed of light from some frame of reference, the mass of the object from that same reference frame will approach infinity, thus it will take an infinite amount of energy to reach the speed of light.
            Generally, classical mechanics are used to describe motion in our everyday lives such as the acceleration of a car, or the speed of an airplane, but in fields like particle physics where particles can move at a speed of about 0.9999 c, it becomes necessary to use relativistic equations in order to describe their motion and their other properties.

 

Mass Units

            One of the most famous physics equations that’s recognized almost universally is Einstein’s; m is the relativistic rest mass (the object’s mass from its own reference frame), E is the rest energy, and c is the speed of light, which is universal for all objects in all reference frames. According to this equation, energy and mass go hand in hand, and mass may be represented in terms of energy.
One of the common units of energy in particle physics and other physics fields is the electron-Volt, or eV for short. One eV may be defined as the kinetic energy that a single unbound electron gains when it passes through a potential difference of one Volt in a vacuum. Some of the common versions of the unit include KeV (1,000 eV), or the Kilo-electron-Volt; and MeV (1,000,000 eV), the Mega-electron-Volt. In this project, I mostly worked with these units of energy.
            Thus, from , or , we see that by dimensional analysis, the units , or  are in fact units of mass, as a unit of energy is divided by a unit of velocity squared. This is often abbreviated to simply MeV or KeV, the “divided by c squared” part is automatically assumed when discussing mass.
By definition, 1 eV/c2 is equivalent to 1.783×10−36 kg, but overall, in particle physics and in many other fields, it proves to be very convenient to simply leave the mass in terms of energy, as the two are often interchanged.

 

Measuring the mass of an electron

The Compton effect

            The Compton scattering effect was first observed by American Arthur Holly Compton in 1922. This effect occurs when a photon strikes an electron, knocking the electron from its orbit. This is very similar to the photoelectric effect, except that the electron absorbs only a portion of the photon’s energy as opposed to all of it, and the rest is released in the form of another photon with a longer wavelength. Thus the re-emitted photon and the electron are said to scatter at angles  and  respectively to the original incident photon.
            Let’s turn to the derivation of the equation that describes the relationship between the frequency of the original photon, the frequency of the scattered photon, and the angle at which the scattered photon is re-emitted. This final equation will also give a hint as to how an electron’s mass can be determined by using the Compton scattering effect.

Diagram 1. A vector representation of the Compton scattering effect. Here, p and E are the momentum and energy of the incoming photon, p’ and E’ are the momentum and energy of the re-emitted photon, pe and Ee are the momentum and energy of the outgoing electron.

            The outgoing electron’s relativistic energy may be expressed by the equation

                                                                                                                          (1)

            The expression for the conservation of energy in this case is:

                                                                                                                         (2)

            This equation simply states that the total energy of the photon and electron before they collide equals to their total energy after. In other words, energy cannot be created or destroyed.

            From Diagram 2 below, it is evident that the equation for the conservation of momentum in this case is:

                                                                                                                    (3)

Diagram 2. Conservation of momentum. A vector diagram representing the incoming photon, and the outgoing photon and electron.

            As well, it can be seen from Diagram 2, that  can be eliminated from the above equation by using the Cosine Law to represent p, p’, and  in terms of . This gives

                                                                                                             (4)

            Ironically, at this point it is necessary to invoke a wave property of light in order to explain its particle behavior. The energy of a photon may be represented by hf where h is Planck’s constant, and f is the frequency of the photon. Also, units of energy divided by units of velocity will give units of momentum. Therefore,  and may be substituted into the above equation to give:

                                                                                    (5)

            hf may also be substituted into equation 2 to represent the energy of the outgoing electron in terms of its mass and the frequency of the two photons:

                                                                                                                         (6)
           
            Now, it is possible to substitute equations 5 and 6 into equation 1, to derive the final equation, which shows the relationship between  and the incoming and re-emitted photons:

              

            Simplifying of the above equation gives:

           This final equation beautifully relates the angle, frequency, and the shifted frequency of light.

            However, the Compton scattering equation may also be slightly modified to give a valuable insight into how an electron’s mass may be measured using the Compton effect. Since energy is also equal to hf, the above equation can also be written in terms of energy by dividing both sides by Planck’s constant, h. This gives:

    or    

            It is evident that the latter equation also takes the form of the equation of a line, namely , where y and x are coordinates, b is the y-intercept, and m is the slope of the line. In this case, the slope of the line is . Therefore, if the energies for the Compton scattering effect can be measured at various angles for a known source, and the values for the inverse of the scattered photon’s energy plotted against , then the inverse of the slope of the resulting line will yield the relativistic rest mass of the electron. Of course, it is also possible to take just one measurement of  for any angle for a known source and simply substitute that value into the equation, but this will likely produce a large error in the final result, as an error of just 1 KeV in  would drastically alter the calculated mass of the electron (especially at the smaller angles).
 
            So how can a photon’s energy be measured?

The Scintillation Detector

            When an excited nucleus decays to its ground state, it can emit gamma rays of well-defined energies. Gamma rays are photons usually emitted by radioactive decay of nuclei, and have the smallest wavelength (and as such, the most energy) on the electromagnetic spectrum.

Figure 1. As the excited atom decays to a lower energy state, it releases energy in the form of gamma radiation.
(http://www.euronuclear.org/info/encyclopedia/e/excitedstate.htm)

            The easiest method to measure the energy of such a photon is to use a scintillation detector (Diagram 3). Typically, a scintillation detector consists of a crystal that reacts to gamma rays, a photocathode, dynodes & anode, amplifier, and the related power and computer components.

Diagram 3. A schematic diagram of a NaI (Tl) scintillation detector. (http://www.physics.mun.ca/~cdeacon/labs/3900/gamma.pdf)

            Once a gamma ray comes into contact with the crystal, an electron is ejected which produces light as it loses energy inside the crystal. The new light then strikes the photocathode, producing electrons, which are accelerated towards the first dynode. Upon striking the dynode, secondary electrons are emitted, and are accelerated towards the next dynode, then a third, and so on. The electric pulse grows in size as the process continues, until it reaches the anode. The size of the final pulse depends on the energy of the original incident gamma ray.
            This process repeats for every gamma ray that strikes the scintillation detector, and the resulting electric pulses are in turn fed into a Multichannel Analyzer, MCA for short, which sorts them according to size. The pulses are stored according to channel numbers (sometimes also referred to as bin numbers), which correspond to the size of the pulse. For example, if a photon with an energy of 352 KeV comes in contact with the detector, then the number of counts in the channel number labeled 352 KeV will increase by one.
            As the radioactive source continues to decay, the number of counts (how many pulses were ‘stored’ in the channel) for the channels will rise, and will produce a spectrum on the computer of channel number vs. counts. The peaks and other properties of the resulting spectrum will tell a great deal about the gamma rays and their source.
            Of course, there are many different types of scintillation detectors, and the correct type usually depends on application. However, there are two important factors to consider when deciding what type of detector to use: resolution and efficiency. There are often peaks in a given spectrum, and the resolution is the detector’s ability to distinguish between two or more peaks. For instance, what may appear to be just one peak at low resolution may actually turn out to be two peaks that are close together at higher resolution. On the other hand, a detector’s efficiency is characterized by how many of the incoming gamma rays will be detected.