16  Derivation of the Compton Wavelength                  Table of Contents     Previous       Next

The link on the left gives information about Arthur Compton a theorist and experimenter (rare combination) who advanced quantum mechanics via his work in x-ray diffraction.

Definition: The Compton wavelength of a particle is equivalent to the wavelength of a photon whose energy is the same as the rest mass energy of the particle. The Compton wavelength, λ, of a particle is given by λ = h/(mc). Where h is the Planck constant, m is the particle's rest mass, and c is the speed of light. 

The Compton wavelength is understood in the standard model to be the distance over which the mass of a particle exists. In digital wave theory, the Compton wavelength is another type of λ-hop that a Planck Instance (aka neutrino) can make (see Section 30). Usually Planck Instances create light energy, and they are 1. spaced at uniform wavelength intervals, and 2. each appearance moves in the same direction. When we get to wavelengths shorter than that for the most energetic light (10-12 meters) gamma rays, Planck Instances do something “funny”, they reverse direction when they appear. This funniness can be thought of as a spontaneous change of spin (see Section 29). When a Planck Instance hops back and forth at a Compton wavelength it looks as if it is trapped between two mirrors as shown below.

The Derivation of the Compton Wavelength:

DWT likes to make a simple distinction between “light as energy” and “light as mass”.  Light as energy consists of a Planck Instance (the dips in the above diagram) λ-hopping continuously in the same direction.  Light as mass consists of a Planck Instance λ-hopping continuously back and forth at the Compton wavelength, as if it were trapped between mirrors. See Section 30

The mass values associated with the Compton wavelength range from 0 to the Planck mass (see previous section). Low mass values (like an electron) are associated with larger Compton wavelengths. The maximum mass value of the Planck mass is associated with the shortest Compton wavelength.  Compton wavelengths indicate that we are dealing with a type of mass that hops back and forth, this distinguishes it from the longer light wavelengths (gamma rays and longer) which indicates we are dealing with light as energy that travels (hops) in one direction. See section 29.

It is reasonable to assume that the minimum possible Compton wavelength is the Planck length (Section 15). And it has been shown that the Planck mass is the maximum mass of any quantum mechanical particle (see Section 31). We can use these two limiting values to set up a ratio to calculate the Compton Wavelength λ for any quantum mechanical particle. The Compton wavelength is to the Planck length as the particle mass is to the Planck mass. We can put this in the form of an equation:

λ/(Planck Length) = m/(Planck Mass)     where Planck Length = (hG/c3)0.5     Planck Mass = (hc/G)0.5

Solving for λ we get λ = h/(mc), the Compton wavelength for a given rest mass m.

The concept of digital waves has enabled us to derive the Compton wavelength as opposed to having an ad hoc definition.

The Compton wavelength contains the mass energy of the particle.
        a.  e = hf
        b.  e = hc/λ
        c.  If we use the Compton wavelength of an electron λ
c = h/(mec)
        d.  Substituting we get e = m
ec2 , the rest energy of the electron.

The Compton wavelength and its relation to Compton scattering:

Click on the image for more information from its website.

Let’s make a better visualization of this phenomena:

a. Notice the representation of the photon as a continuous thing that wiggles.  DWT thinks it has a better representation of a single photon as a single Planck Instance λ-hopping in one direction.
b. DWT thinks of the target electron as an oscillating Planck Instance with a diameter of the Compton wavelength λc , not as a stationary blob.  In the diagram at left the electron is moving along the line labeled recoil electron.
c. When the photon impinges upon the electron lengthwise (Φ=0) its wavelength λi becomes lengthened to λi + λc .  The photon looses energy via the wavelength change and the electron gains energy via kinetic energy (the recoil).  In this instance all action is along a straight horizontal line.
d. When the photon impinges upon the electron “at an angle Φ” its wavelength λbecomes lengthened to λi + λc(cosΦ).  The projection of the Compton wavelength that the photon sees is the “bump” between the photon and the electron.  In this instance after the bump the photon and the electron move away from each other at an angle. 
e.  In Compton scattering the electron most likely gets hit with a burst of photons (or Planck Instances)

because the electron has too much inertia to move out of the way before the second Planck Instance (if the wavelength is short enough).  And yes both light and matter are not quite pure things or pure waves.  The best model according to DWT is that they are both manifestations of Planck Instances with different wavelengths and different spins (see Section 30).
f.  Compton scattering is usually associated with x-ray crystallography.  It can also be associated with the photo electric effect.  

A different viewpoint on the photoelectric effect:

The Standard Physics Story: The photoelectric effect is by now the "classic" experiment, which demonstrates the quantized nature of light: when applying monochromatic light to a metal in vacuum one finds that electrons are released from the metal. This experiment confirms the notion that electrons are confined to the metal, but can escape when provided sufficient energy, for instance in the form of light. However, the surprising fact is that when illuminating with long wavelengths (typically larger than 400 nm) no electrons are emitted from the metal even if the light intensity is increased. On the other hand, one easily observes electron emission at ultra-violet wavelengths for which the number of electrons emitted does vary with the light intensity.

The DWT interpretation of the photoelectric effect:  The DWT explanation is very similar to the standard physics story above, but with a better visualization of how the photons bump the electron.

1.  Free electrons in metals are prevented from escaping because they are held by the force provided
     by the protons in the nucleus of the metal atoms.  To overcome this force (work function), energy
     needs to be given to the electron.  If enough energy is provided to the electron, it can escape the
     metal surface.

     Why is wavelength important?  Before Einstein explained the photoelectric effect, it was thought

     that energy effects were only a function of a photons intensity (amplitude). But Einstein explained
     how the energy effects needed to take into account the photon’s frequency. This turned out to be
     the starting point of quantum theory.
2.  DWT thinks it can provide some clarity to the photoelectric effect.  It is usually thought that a
     photon’s shorter wavelengths hit the electron harder. This is not quite right, each wavelength gives
     the same impact to the electron. And it is the frequency of the impacts that cause the electron to
     become free of the metal. The wavelength of the light is important because it is the spacing of the
     Planck Instances.  A single Planck Instance can impart some energy to the electron, but  in most
     instances it is not enough to free it from the surface of the metal.  Only if the electron is
     continuously hit by Planck Instances (spaced closely enough) will it get free of the surface. This is
     why the light wavelength has a threshold value.

3.  Why is 400nm the magic wavelength?  Because it provides the minimum repetition rate to pump
     the electron out of the surface of the metal.  All the metals require approximately 400nm light to
     initiate the photoelectric effect.  Wavelengths shorter than 400nm will pump the electron more
     frequently and give it greater and greater escape velocities.  At wavelengths longer than 400nm
     the electron will fall back too much before the next pump (Planck Instant hit) and it will never
     escape the surface of the metal.  By the way 400nm light is way longer than the approximate
     average 0.28nm diameter of the metal atoms in the periodic table.

4.  A sustained photoelectric effect?  We could try to create a type of perpetual motion machine if
     we could could get a photons shorter than 400nm to free an electron from a metal and then direct
     this photon back onto the metal surface to release another electron.  This type of perpetual motion
     is not possible because the photon after each electron encounter gains wavelength (as in Compton
     scattering detailed above) and will eventually not be short enough to hit the electrons frequently
     enough to release them from the metal surface.  It could be some fun to test this theory by using
     an integrating sphere with a metallic interior with a fast pulsed deuterium lamp (150 nm) and
     measure the duration (decay) of the photoelectric effect after a short wavelength light pulse.

Using the Compton Wavelength to define the Graviton (the QM essence of Gravity):
     Arthur Compton essentially labeled an ordinary photon’s wavelength to be a Compton wavelength when it showed evidence of having mass like properties, like being able to move electrons. Ordinarily photons when propagating in space do not have mass. Following this lead of Compton, DWT postulates that the graviton is in essence a Compton wavelength (the mediator particle of gravity). This graviton Compton wavelength is essentially a photon with some special attributes:

    a. It connects two masses via a single wavelength hop back and forth.
    b. It’s wavelength can be thousands of light years long.
    b. It is quantized like photons are quantized.

For more on the Compton wavelength as a graviton see: Section 17, Section 18Section 19, and Section 20.

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