In the last studio session you observed diffraction and interference. You studied the diffraction and interference patters produced by single and multiple slits and verified that one can predict the positions of the maxima and minima in these patterns by assuming that light is an EM wave. Where crests meet crests and troughs meet troughs we predict and observe maxima or bright regions, and where crests meet troughs we predict and observe minima or dark regions.

**So light is a wave, these experiments settle
this? Not so fast!**

In this session you will simulate an experiment that suggests that light is a particle. You will investigate the photoelectric effect. To eject an electron from a metal surface a certain amount of energy Φ must be supplied to this electron. Φ is called the work function of the metal. (If no energy were required to free the electrons, they would just leave ordinary pieces of metal.) In the wave picture the energy of the light beam does not depend on the frequency, but only on the intensity, which is proportional to the square of the amplitude. Einstein explained the photoelectric effect by postulating that an electron can only receive the large amount of energy necessary to escape the metal from the EM wave by absorbing a single photon. If this photon has enough energy, the electron is freed. Excess energy appears as kinetic energy of the electron. The maximum kinetic energy of the electron is given by E = hf - Φ. If the photon does not have enough energy, then the electron cannot escape the metal.

In this session you will direct light with different wavelength onto a metal surface and measure the kinetic energy of the photoelectrons ejected from the metal as a function of the frequency of the light used to eject the electrons. You will measure the work function of the metal and also determine the value of Planck's constant from your data. This experiment reveals the "particle nature" of light. A second experiment in the studio session will reveal the "wave nature" of electrons. You will observe and analyze electron diffraction.

Equipment needed:

- Demonstration equipment

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions. This log will become your lab report. Address the points highlighted in blue. Answer all questions.

**Experiment 1**

You will use an on-line simulation
from the University of Colorado PhET group.

Link to the simulation:
http://phet.colorado.edu/en/simulation/photoelectric

Explore the interface. There are some non-obvious controls.

- You can select Show photons in the Options menu to show the light beam as composed of individual photons.
- You can select Control photon number instead of intensity in the Options menu to change the Intensity slider to a Number of photons slider.
- You can use the camera icon to take a snapshot of the graphs so that you can compare graphs for different settings.
- You can Pause the simulation and then use Step to incrementally analyze.

In the simulation photons strike a metal cathode and eject
electrons. Electrons are ejected with a range of energies, up to a maximum
energy. The electrons are collected on the anode and then flow back from
the anode to the cathode through a wire. The current in the wire is
measured. Electrons
are negatively charged. If the anode is at a positive voltage compared to
the cathode, electrons are attracted and gain energy. If the anode is at a
negative voltage compared to the cathode, electrons are repelled and loose
energy. Let V_{ca} = V_{cathode} - V_{anode}.
If this potential difference V_{ca} (energy per charge) becomes large
enough, electrons will no longer reach the anode and current will no longer flow
in the wire.
It is convenient to measure electron energy in units of electron volt (eV). In SI units 1 eV = 1.6*10^{-19 }J.

If the potential difference between
the cathode and anode is x volts (V_{ca} = x V), then electrons ejected from the
cathode need an energy of at least x eV to overcome this potential difference
and to reach the anode. By determining voltage V_{ca} needed to reduce the
current in the wire to zero, we can determine the maximum energy
of the ejected electrons.

The maximum energy E of electrons that reach the anode in eV has the same
numerical value as the voltage in V_{ca}.

(a) Exploration:

For a Sodium target discuss:

- For a
**fixed number of photons and zero battery voltage**, how does the number of photoelectrons ejected depend on the wavelength? Does every photon eject an electron? Does the probability of ejection change with wavelength? Discuss! - For a
**fixed wavelength and zero battery voltage**, how does the current depend on the light intensity? Discuss! - For a
**fixed wavelength and light intensity**, how does the current depend on the battery voltage? - For a
**fixed wavelength and light intensity**, do all ejected electrons have the same energy? How can you measure the maximum energy of the ejected electrons.

(b) Measurement:

Use a Sodium target. Set the
intensity to 100%. For the wavelengths listed in the table below, find the
maximum energy of the ejected electrons by finding the battery voltage V_{ca}
that just prevents the most energetic electrons from reaching the anode.

Note: The text box under the battery displays -V_{ca}.

Wavelength (nm) | Frequency (s) | Maximum Electron Energy (eV) |
---|---|---|

150 | 2.00e15 | |

200 | 1.50e15 | |

300 | 1.00e15 | |

400 | 0.75e15 | |

500 | 0.60e15 |

Prepare a graph of the maximum electron energy versus
the frequency of the light. Choose a X-Y Scatter plot. Plot maximum electron energy on the vertical axis and frequency on the
horizontal axis.

Prediction: E = hf - Φ

This equation is of the form y = ax + b, with a being the slope and b being the
y-intercept. The slope of the plot of E versus f will yield Planck's constant
and the y-intercept will yield the work function of the metal cathode.

Add a trendline and insert the equation for the trendline into your plot. Use
the slope of your trendline to find Planck's constant h and the intercept value
to find the work function Φ of Sodium. If
there are data points that seem to deviate to far from the trendline, repeat
those measurements until you are satisfied that you have done your best.

(Since the electron energy is measured in eV and the
frequency in seconds, Planck's constant h will have units of eV s and the work
function will have units of eV. Convert Planck's constant to SI units (J s) by
multiplying the value you obtained from the slope by 1.6*10^{-19}J/eV.

Insert your table into your Word document.

- What value did you obtain for h in units of eV s and J s?
- How does this value compare with the accepted value
h = 6.626*10
^{−34}J s = 4.136*10^{−15}eV s? - Describe how the maximum energy of the photoelectrons depends on the wavelength of the incident light.
- Defend whether this experiment supports a wave or a quantum model of light based on your lab results.

**Experiment 2**

How does matter behave on a scale of a few nanometers or smaller? Is
its behavior governed by Newton's laws or by a wave equation? The de
Broglie relations associate a wavelength λ = h/p = h/√(2mE) with each particle
of momentum p. For an electron which has been accelerated through a
potential difference of 5 kV and therefore has a kinetic energy of 5000 eV =
8*10^{-16} J, this wavelength is λ = 1.74*10^{-11} m. Can
we measure λ?

If a beam of accelerated electrons passes through a thin crystal, the crystal planes can act like a diffraction grating. (For a grating d sinθ = mλ.) Different crystal planes can produce different diffraction patterns. Planes that are spaced farther apart produce a narrower patterns. You will observe the diffraction pattern produced when 5 keV electrons pass through graphite.

Graphite has the crystal structure shown below.

In this orientation, the d100 planes produce a horizontal pattern.

d100 = 2.10*10^{-10} m

In this orientation, the d110 planes produce a horizontal pattern.

d110 = 1.21*10^{-10
} m.

The graphite in our apparatus is not a single crystal, but a polycrystalline powder. The orientation of the various crystals is random. Powder diffraction produces a pattern of concentric rings. The powder diffraction patters is just the superposition of the patterns produced by the individual crystals with random orientations.

Assume for a single crystal with a fixed orientation we observe the pattern below. The diffraction pattern is rotated by the same angle as the crystal.

Then for 4 crystals with 4 different orientations, we also observe 4 different orientations of the diffraction patterns. The individual diffraction patterns plotted in the same color as the corresponding crystal start to add up to rings.

For 40 randomly oriented crystals, powder rings become clearly visible.

In our experiment accelerated electron with 5 keV
kinetic energy pass through a graphite target in an evacuated tube and
hit a fluorescent screen. We observe the ring pattern on this
screen.

Two rings are clearly visible. These correspond to the
first-order maxima produced by the d100 and the d110 planes. All
other rings are too dim or at angles too large to observe with our
apparatus.
You will measure the diameter of
the two visible rings and calculate the angles θ, given the distance L =
13.5 cm from the target to the screen.

Using the known plane spacings d100 = 2.10*10^{-10} m
and d110 = 1.21*10^{-10}
m you will then use
d sinθ = λ
to experimentally determine the de Broglie wavelength λ of the 5 keV electrons.

Schematic sketch off the apparatus:

L = 13.5 cm (distance between graphite foil and screen)

D = diameter of a diffraction ring observed on the screen

θ = diffraction angle

sinθ = (D/2)/(L^{2} + (D/2)^{2})^{½}

Make the measurements and fill in the table below. Measure the center-to-center distance for each bright ring. The scale on the picture is a mm scale.

1st ring (d100) | 2nd ring (d110) | |
---|---|---|

d (m) | 2.10E-10 | 1.21E-10 |

L (m) | 0.135 | 0.135 |

D (m) | ||

sinθ | ||

λ |

Discuss your results.

- Do the two values for λ agree within experimental uncertainty?
- What do you think contributes most to the experimental uncertainty?
- Is your experimental value of λ close to the expected de Broglie wavelength of the electrons?
- Does this experiment convince you that electrons do not behave like classical particles, or can you think of a classical explanation for your results?