# Photons exert power on a mirror

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### Chapter 4

The momentum of the photon

So far we have got to know the first particle properties of light by looking at the quantization of the energy of light. We are now investigating situations in which one can prove that light also carries an impulse or, in other words, that photons also have an impulse.

In Section 4.1 we will first carry out a (classic) calculation that shows that an electromagnetic wave when it is reflected on a mirror exerts a pressure, the so-called radiation pressure, on it. This will give us a first clue about the momentum of photons. However, the historical route to the expression for the momentum of the photon did not lead via this calculation, but via an experiment carried out by Arthur Holly Compton around 1922. The effect he observed, the so-called Compton effect, will then be discussed in Section 4.2. The observation of the Compton effect provided the first experimental evidence of the momentum of a photon.

### 4.1 The radiation pressure

We calculate the radiation pressure that an electromagnetic wave exerts on a flat plate when it is absorbed (see Fig. 4.1).

An electromagnetic plane wave with an electric field strength and magnetic flux density

applies to on a plate. First we calculate the momentum transfer from the electromagnetic wave to an electron in the absorbing material.

An electrical force acts on the electron in the plate

In addition to this force, the electron experiences a damping in the material, which is caused by so-called mobility or mobility is characterized. In balance between and damping is the so-called drift speed with which the electrons move through the material described by the following equation

We now assume that the time in which this equilibrium is established is much shorter than the period in which the electromagnetic field strength oscillates. Under this assumption, we get an average drift velocity of for the electrons

with which the electron moves along the negative y-direction. We keep this averaging or neglect of the field strength oscillations for the following calculations. As a result, all of the following variables are to be considered under this averaging. For the sake of simplicity, we will omit the averaging line.

In addition to the electrical force, a magnetic force, which comes from the magnetic flux density, also acts on the electron

So the electron feels an additional force along the positive z-direction (). This term is responsible for the radiation pressure. With (4.5) and the relationship we get from it

With the relationship

between a force and the impulse the following equation results

For work per unit of time , which is produced by the electric field at the electron, we get with (4.9)

Finally, integration provides for the momentum transfer of an electromagnetic wave on an electron of the plane plate

i.e. when an electromagnetic wave is absorbed in a material, an impulse is transmitted to it. The size of the momentum transfer is determined by the work caused by the electromagnetic wave on the material and by the speed of light .

If one considers the case of the reflection of the radiation instead of absorption in the plate, then there is an additional momentum transfer of the same amount to the interacting surface, i.e. the reflection on a mirror results in a momentum transfer of:

In the last chapters we saw that we can view light as a stream of particles (photons). If one wants to interpret the momentum transfer during the reflection of an electromagnetic wave at a mirror with this idea of photons, one must assume that these have a momentum and like particles are elastically reflected by the mirror. In this case, the component of the photon pulse, which is perpendicular to the mirror, changes its sign and transmits twice as much momentum as in the case of absorption.

The momentum of a photon we get from the energy an electromagnetic wave that is an integral multiple of the energy corresponds to a photon

Inserting in (4.12) yields

So we get the following result

A photon has the momentum

Between impulse and energy of the photon, the following relationship applies

To conclude these considerations, we come back to the concept of radiation pressure and give a formula for it. We consider the case of absorption. The radiation pressure that is on the mirror surface works, we can express it as follows

in which corresponds to the radiation intensity that is absorbed by the mirror.

### 4.2 The Compton Effect

As mentioned at the beginning, the historical way of expressing the momentum of a photon did not lead via the calculation carried out in Section 4.1, but via an experiment that Compton carried out around 1922^{1}.

#### 4.2.1 The Compton experiment

Compton applied the radiation from an X-ray tube with a molybdenum anode (see Section 3.2.1) directly to a piece of graphite^{2} fall (see Fig. 4.2). He then analyzed the spectrum at an angle of scattered radiation with a Bragg spectrometer (see Section 3.5) and compared it with the spectrum of the incident radiation. He used calcite as a spectrometer crystal () and an ionization chamber as a detector^{3}.

Fig. 4.3 shows the measurement result of the historical experiment by Compton, which was published in the Phyiscal Review in 1923. The dashed line shows the measured spectrum of molybdenum. The solid line shows the spectrum of molybdenum after it is at an angle of was scattered on graphite.

You can see the continuous braking spectrum and the two dominant spectral lines and . The shape of the two spectra is essentially the same. However, the following essential result results: The spectrum of the scattered radiation is shifted towards longer wavelengths compared to the spectrum of the direct (unscattered) radiation. This wavelength shift is called the Compton shift.

In contrast to the scattering on which the Bragg reflection at the spectrometer crystal is based, what Compton observed could not be coherent scattering. Obviously there are other scattering processes as well.

Compton interpreted the measurement result as follows:

An incident photon with the energy and the impulse collides elastically with an electron from the scattering body. In doing so, it loses part of its energy and momentum. The impulses and energies of the observed particles can be determined by considering the energy and momentum conservation of the entire system.

Before we deal with the calculation of this wavelength shift (Compton shift), let's first consider another more modern experiment on the Compton effect, in which significantly larger shifts in the wavelengths can be observed.

#### 4.2.2 Compton effect with gamma radiation

A source is used in this experiment (see Fig. 4.4) Cs source. The emitted high-energy photons (gamma rays), which arise during radioactive decay, are directed onto a thin copper rod, where some of the photons are elastically scattered by electrons. The one at an angle Scattered photons are measured with a scintillation counter consisting of a NaI crystal with a subsequent photomultiplier^{4} proven energy-resolved.

The result of the measurement is shown in Fig. 4.5 in a so-called polar diagram. Every scattering angle is the ratio between the photon energy at the corresponding scattering angle and the photon energy of direct radiation assigned by the respective distance between the measurement curve and the zero point.

The result obtained confirms the result of the historical measurement (see Section 4.2.1): The frequency (wavelength) of the scattered radiation is shifted compared to the direct radiation. Another finding is that the maximum of the under measured spectrum occurs at the same frequency (wavelength) as the maximum in the spectrum of direct radiation.

Refined experiments show that in the spread spectrum not only at the shifted wavelength a maximum can be observed, but also at the wavelength of direct radiation. We will go into the explanation of this observation and the calculation of the Compton shift in more detail in the next section.

#### 4.2.3 Calculation of the Compton shift

To calculate the Compton shift, we start from Compton's interpretation (see Section 4.2.1). An incident photon with the energy and the impulse collides elastically with an electron from the scattering body (see Fig. 4.6). The conservation of energy and momentum applies to the entire photon and electron system. We assume that the electron hit by the photon is unbound and was originally at rest. This is fulfilled to a good approximation in a metal, since the conduction electrons are only weakly bound compared to the energy of the incident photons and carry a small pulse.

From the conservation of energy results

From the conservation of momentum we get

With

From (4.18) we get by squaring

Division by results

From (4.19) we get

in which the angle between and designated.

Equating (4.22) and (4.23) gives

Shortening and inserting (4.20) yields after some transformations for the Compton shift

##### Remarks:

- m is called the Compton wavelength of the electron. This Compton wavelength of the electron corresponds to the Compton shift at . It only depends on the mass of the photon's scattering partner and on natural constants.
- The Compton shift is independent of the wavelength of the incident radiation. The Compton shift becomes more pronounced the smaller the wavelength of the incident radiation. Therefore, the Compton effect is hardly measurable for visible light, but the effect becomes large for X-rays or gamma radiation.

#### 4.2.4 Compton scattering and coherent scattering

As mentioned in Section 4.2.2, in refined experiments in the scatter spectrum a maximum is observed not only at the shifted wavelength , but also in terms of wavelength of direct radiation. This means that in addition to Compton scattering, there is also coherent scattering. This is due to the fact that the electrons are bound differently in the scattering body, or more precisely: The scattering body contains different types of electrons:

- Quasi-free electrons (conduction electrons),
- weakly bound electrons (in the outer shells of the atoms),
- strongly bound electrons (in the inner shells of atoms).

If the electron hit by the photon is so strongly bound to an atom that the energy that the photon transfers to it is not enough to tear it out of the atom, there is no Compton scattering, but essentially coherent scattering. Or in other words: the larger the number of strongly bound electrons compared to the number of weakly bound electrons, the more dominant the coherent scattering compared to Compton scattering and vice versa.

More precise investigations show that the following rule can be found for the ratio of the intensity of the shifted radiation (Compton scattering) to the intensity of the unshifted radiation (coherent scattering): The ratio increases with it

- decreasing atomic weight of the scattering body,
- decreasing wavelength of the incident radiation,
- increasing scattering angle.

Qualitatively, these three observations can be explained as follows:

- The smaller the atomic weight, the greater the fraction of the number of electrons that are weakly bound and that can be regarded as free in the event of a collision.
- The higher the wavelength and thus also the energy of the incident photon, the sooner the collided electron can be considered free.
- The greater the scattering angle, the greater the energy transferred to the electron and the sooner the bond can be neglected.

#### 4.2.5 Difference between photo effect and Compton effect

In the Compton effect, a photon performs a fully elastic collision with a quasi-free electron. Thereby it gives up part of its energy to the electron. After the collision, the scattered photon has a lower energy and thus also a lower frequency than before the collision. In contrast to this, with the photo effect (see Chapter 2) a photon gives up all of its energy to an electron and disappears in the process. The impulse of the entire system of photon and electron cannot be retained under these circumstances and therefore a third collision partner (usually an atomic nucleus) must be present to which an impulse can be transferred. The photo effect can only occur with bound electrons.

### 4.3 Summary

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