An interference pattern is observed in a vacuum

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Chapter 2
The photo effect

In Chapter 1 we looked at how wave properties of light come to light in interference and diffraction experiments. In this chapter we consider the photo effect, which cannot be explained in the classic wave pattern of light. The photo effect shows that light also has the character of a particle, namely that the energy of electromagnetic radiation is quantized.

2.1 Experimental observation of the photo effect

The photo effect was first observed in 1839 by Alexandre Edmond Becquerel. In 1886 Heinrich Hertz and his assistant Wilhelm Hallwachs then carried out the first systematic investigations. In 1900, Philipp Lenard was the first to quantitatively investigate the photoelectric effect in a high vacuum apparatus.

We first investigate the photoelectric effect using an experiment that Hallwax carried out around 1888 (see Fig. 2.1). A zinc plate that has previously been positively or negatively charged with a voltage source is irradiated with the white light of an arc lamp. The light may pass through a filter (e.g. window glass or a quartz disk) before it hits the zinc plate. The charge on the zinc plate is measured with an electrometer during the experiment1 measured.



Fig. 2.1: Photoeffect experiment by Hallwax around 1888: A charged zinc plate is irradiated with light, which passes through a filter beforehand. An electrometer measures the charge on the plate.

The following observations are made in this experiment:

  • When the positively charged zinc plate is irradiated with light, the charge remains constant regardless of the frequency of the light or the filter used.
  • When the negatively charged zinc plate is irradiated with light, the plate is discharged. The rate at which the electrometer discharges (the photocurrent) increases with the intensity of the light. The discharge can be stopped by using a glass plate as a filter, since the part of the spectrum emitted by the light source that is relevant for the photo effect is apparently absorbed. A quartz disk as a filter, on the other hand, cannot prevent the discharge.

From the experimental observations we can draw the following conclusions:

  • Exposure to light removes negative charge carriers (electrons) from the zinc plate.
  • From the observation of the experiment using various filters it follows that the escape of electrons depends on the wavelength of the light. In the case of a zinc plate, the ultraviolet spectrum is responsible for the effect. A glass plate absorbs this while quartz lets through it.
  • If the plate is sufficiently positively charged, further electrons are prevented from escaping due to the strong Coulomb interaction with the positive charges on the plate.

The release of electrons when the surface of a metal or other solid body is irradiated with light is called a photo effect.

2.1.1 Measurement of the photo effect in a vacuum

The photo effect can be quantitatively investigated in a vacuum apparatus in which the released electrons can be detected as an electric current (see Fig.2.2):



Fig. 2.2: Quantitative investigation of the photo effect based on the irradiation of an alkali metal with light. The text deals with the various components of the experimental set-up.

  • The experiment is carried out in a high vacuum so that the electrons that have escaped can move as freely as possible due to collisions with the surrounding gas.
  • Instead of zinc, an alkali metal is often used for this experiment, in which the photo effect can be demonstrated in visible light (i.e. lower frequencies than in the ultraviolet).
  • In the vacuum tube there is a collector electrode made of a more noble metal opposite the alkali metal, which takes on the function of the photocathode.
  • The experiment is done with monochromatic2Light carried out. This can be generated, for example, by a mercury vapor lamp, the light of which is spectrally split with a prism so that the photocathode is only illuminated by the light of a selected spectral line at a fixed frequency.
  • The so-called opposing field method is used to determine the energy of the released electrons. By irradiating the alkali metal with monochromatic light, electrons get from the photocathode to the collector electrode (anode). A photocurrent, which is transported by the released electrons, is measured. A counter voltage (acceleration voltage) can now be applied between the photocathode and anode so that the electrons are slowed down by the electrical field of the counter voltage and no longer reach the collector electrode if the voltage is high enough.

In performing this experiment, the following observations are made:

  • The photocurrent depends on the applied acceleration voltage and the intensity3 of the incident light (see Fig. 2.3). It can be observed that the current saturates at sufficiently large positive acceleration voltages. In doing so, it reaches a limit value , which is proportional to the number of electrons released. In the case of small (negative) counter voltages, a current is first measured, which is at disappears, i.e.. It follows that the electrons that leave the alkali metal due to the irradiation apparently have a maximum kinetic energy of have.

  • The saturation current increases proportionally to the intensity of the incident light, since at higher intensities more electrons are released per time and area, see schematic representation in Fig. 2.4.

    Fig. 2.4: The saturation current shows a linear dependence on the intensity of the incident light.

  • The most important observation is that the maximum kinetic energy of the exited electrons linearly with the frequency depends on the incident light (see Fig. 2.5). As already discussed in a), is a measure of the maximum kinetic energy of the leaked electrons. It is observed that regardless of the intensity of the incident light is and only of the frequency depends on the incident light. The constant of proportionality is independent of the material and can be Js to be determined. It is called Planck's quantum of action. There is a minimum frequency below which no electrons are released. depends on the irradiated material. This observation can be interpreted as follows: In order to release the electrons from the metal, work has to be done. This becomes work function called and depends on the irradiated material. This also makes the connection .

  • In addition, one observes that the distribution of the electron energy is independent of the intensity of the incident light and that electrons after a few nanoseconds ( s), i.e. almost instantaneously, triggered after switching on the light, even at very low intensities ().

2.2 Explanation of the photo effect

The photo effect cannot be explained conclusively with the means of classical physics, as we will see in the following by means of a few simple estimates.

2.2.1 Classic expectation

We consider a metal surface () that with light of performance W is irradiated (see Fig. 2.6). The aim is to determine the mean energy consumption per electron per unit of time on the metal surface and from this the time it takes until all electrons are released from the surface of the metal on average. From the experiment (see Section 2.1.1) we know that this triggering takes place almost instantaneously and that this time is in the order of magnitude of s lies.



Fig. 2.6: Arrangement to attempt a classic explanation of the photo effect: A metal surface of the surface will be with light of achievement W irradiated.

The number of electrons on a metal surface can be estimated as follows. The diameter of an atom Å. Assuming that one electron per atom, which is the area Å occupies, is relevant for the charge density on the surface, results in the number of electrons on the surface of the metal

This gives the mean energy consumption per electron and time unit on the metal surface

The work function of a typical metal is on the order of 1 eV. If the energy of the incident light were evenly distributed to all electrons on the surface, it would It takes s until all electrons have absorbed enough energy on average to overcome the work function. Under this assumption, an electron would only have an energy of on average in 1 ns eV, far too little to overcome the workforce.

This result is in clear contradiction to the observations from the experiment that this triggering occurs almost instantaneously. Therefore, a new approach to explaining the photoelectric effect is needed here, which was provided by Albert Einstein, for which he received the Nobel Prize in Physics in 1921.

2.2.2 Declaration according to Einstein

Light of frequency can the energy on an electron transferred, where is Planck's quantum of action. In order to free an electron from the metal, the work function must be are expended. The kinetic energy of the electron detached from the light is a maximum

From this Einstein concluded that light is of frequency feel like a particle of energy behaves. These particles are called light quanta or photons. A single photon can transfer all of its energy to a single electron. When leaving the metal, the electron becomes the work function overcome and the remaining energy as kinetic energy take up

The process of releasing a single electron from a metal surface through interaction with a single photon can be easily represented in an energy level diagram (see Fig. 2.7). We will come across such energy level diagrams frequently in the further course of the book, as they are very helpful in understanding numerous problems in quantum mechanics.



Fig. 2.7: Energy level diagram: a photon of energy meets an electron of a metal and transfers all of its energy to it. This enables the electron to do the work function to overcome and it leaves the metal with a kinetic energy .

2.3 Single photon detectors

The photo effect is used today in many ways in technical applications when small light intensities, down to individual photons, are to be detected sensitively, for example on the charged-coupled device (CCD) microchip of a digital camera.

With the first technically realized single-photon detectors it became possible to detect a single electron released by a metal and thus also a single photon as an electrical impulse. Such single-photon detectors are built roughly according to the scheme outlined in Fig. 2.8.



Fig. 2.8: Photon counter: Detection of individual photons. The individual points are dealt with in the text.

The single-photon detector works as follows: (1) A photon generates an electron by means of a photoelectric effect. (2) The electron is then focused on the so-called dynodes with the help of an electromagnetic lens. A dynode is an electrode from a series of individual electrodes. An accelerating voltage is applied across the entire series of dynodes. A partial voltage thus drops between a pair of dynodes according to the number of individual dynodes and electrons are accelerated from dynode to dynode. If an electron hits the surface of a dynode, more electrons are emitted and the current is increased. A dynode fulfills the properties of both a cathode and an anode, since it emits and absorbs electrons. (3) The first electron is due to the positive voltage to a kinetic energy of accelerated, where W corresponds to the work function of a dynode. This creates secondary electrons4 triggered. By stringing together several dynodes under the condition the number of electrons is multiplied to a measurable level. (4) For the measurement, the secondary electrons are captured by a collector anode and a current pulse triggered by an electron or photon can be detected. Each current pulse thus corresponds to a single incident photon.

In modern semiconductor electronics, individual photons can generate freely moving charge carriers in the semiconductor if their energy is greater than the energy gap of the semiconductor. The charges generated in this way can then be detected electrically. It is no longer necessary to remove the charges from the material in a vacuum. In 2009, Willard Boyle and George Elwood Smith were awarded the Nobel Prize in Physics for the further development of this technology into a CCD detector, as it is used in many digital cameras.

2.4 Particle and wave character of electromagnetic radiation

On the one hand, when we observed the photoelectric effect, we recognized that electromagnetic radiation has particle properties. In appropriately selected experiments it is shown that the energy of radiation is in the form of individual quanta of energy , which we call photons, is quantized. On the other hand, diffraction and interference experiments (see Chapter 1) show that light also behaves like a wave in suitable experiments. At first glance, this might seem contradictory. However, this is a fundamental property of light that was first recognized in the context of the development of quantum mechanics, i.e. light has both the character of waves and particles. What exactly we mean by this dual nature is explained in more detail in this section using an experiment.

2.4.1 Interference of single photons at the double slit

We consider an experiment that reveals both the wave and particle properties of light in a single experiment (see Fig. 2.9).



Fig. 2.9: Experimental setup for the experiment Observation of the wave and particle character of light during diffraction at the double slit. The text deals with the individual parts of the test arrangement.

A laser is used as the light source, the light of the wavelength nm at a power of mW generated. This corresponds approximately to a number of Photons that are emitted by the laser every second. The emitted light beam is split up with mirrors and beam splitters to different measuring devices in the three arms of the structure, which we will now go into in more detail.

The particle character of light is demonstrated with the first apparatus. For this purpose, the light intensity is reduced so much by means of an absorber that a photon only emerges from the absorber every few seconds. The detection of individual photons then takes place with a photon counter, which we got to know in Section 2.3.

In the second apparatus, without reducing the intensity of the light emitted by the laser, the known diffraction of light at the double slit takes place, analogous to Section 1.2, in which the known interference pattern is observed, which makes the wave properties of light visible.

In the third apparatus, the laser beam is weakened again to such an extent that only a few photons per unit of time are present in the beam. These photons then pass through a double slit. A single photon detector detects the individual photons that have passed through the double slit in a spatially resolved manner in the image plane. This is possible, for example, by using an image intensifier5 combined with a CCD camera6that registers the individual photons in the image plane.

In this experiment we observe how individual photons in the image plane seem to be detected one after the other at random locations. If we collect more and more photons with the detector, we observe that the familiar interference pattern of the double slit is slowly emerging. The number of photons that hit a certain point on the screen seems to be given by the square of the amplitude of the interfering electromagnetic wave that passes through the double slit.

The wave and particle properties of electromagnetic radiation can thus be reconciled as follows: The quantity given as intensity in the classical theory of electromagnetic waves (proportional to the square of the amplitude of the field) is proportional to the probability that a photon will be detected at the relevant location in space can. This observation will prove to be an important element of quantum theory as the book progresses.

Next, we examine how this interpretation is to be concretely understood using the example of diffraction at the slit (see Section 1.1).

2.4.2 Statistical interpretation

The diffraction intensity behind a slit is to be understood as a statistic according to the interpretation given above. I.e.the diffraction intensity is a measure of the frequency with which photons under the relevant diffraction angle can be observed (see Fig. 2.10).



Fig. 2.10: The dual nature of light using the example of diffraction at a slit: photons hit a slit and are registered behind it by a detector. The frequency distribution behind the gap is given by the intensity distribution calculated from the wave pattern.

We notice that the detector detects individual photons at every angle setting. Fractions of photons are never detected. A photon is therefore not split up and is also not distributed over the diffraction area when it is detected.

Another important point is that the intensity distribution behind the gap does not result from interference between different photons. Even if there is only a single photon in the apparatus at any point in time during the experiment, the same interference phenomena are observed.

In other words: With electromagnetic radiation one can make diffraction experiments in which the intensity is so small that only a single photon is in the apparatus at a time. In this case, too, the frequency distribution of the detected photons is given by the one calculated on the basis of the wave image. With these considerations we have arrived at a statistical interpretation of the intensity.

Diffraction of microwaves at the slit

We consider the diffraction of microwaves, i.e. the diffraction of light of wavelength cm (frequency 1 / s), at a gap across the width cm. The radiant power incident on the gap be of the order of magnitude W. For the energy of a microwave photon results

The following number of photons therefore fall per second on the gap

i.e. a relatively large number. Thus, when microwaves are diffracted at the slit, a large number of detected photons is obtained even for a short measurement, and the quantum nature of the radiation is not easy to prove, since the detected photons can no longer be temporally resolved.

Diffraction of X-rays at the slit

As a second example we consider the diffraction of X-rays, i.e. the diffraction of light of wavelength cm (frequency 1 / s), at a gap across the width cm. The radiant power incident on the gap is of the order of magnitude W. For the energy of an X-ray photon results

The following number of photons therefore fall per second on the gap

At first glance you can see that you have to measure significantly longer than with the microwave experiment in order to obtain good statistics when measuring the angular distribution behind the gap.

Finally, we look at the question of how many photons are in the apparatus at the same time. We assume that the distance that the photons have to pass through is 10 cm. From this it follows for the time that a photon is in the apparatus

Since per second only When photons arrive, it is therefore very unlikely that there will be more than one photon in the apparatus at the same time.

2.5 Summary

When a metal plate is irradiated with light, electrons are released. This effect is called the photo effect and was a first indication of the quantization of light, i.e. that light shows particle character. The classic wave model fails to explain and needs a new model. Here are the most important results for Einstein's interpretation of the photo effect:

Another finding is that light has both wave and particle properties. Which property appears depends on the experiment carried out.

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