FYSP106/1 PHOTOELECTRIC EFFECT
In the exercise you’ll define the ratio of the Planck’s constant to the unit of charge h/e.
The light source is a gas-discharge tube in connection of a slit monochromator used to
select light with different wave lengths. The monochromatic light ejects electrons from
the cathode of a s.c. photocell and they are collected to the anode of the device. The
maximum energy of the photo-electrons will be defined by stopping them with an electric
field (stopping potential) applied between the cathode and the anode. The stopping
potential displayed as a function of the frequency of the light is in principle a straight line
with the linear coefficient corresponding to h/e.
Subject related to this work has been discussed in:
1
•
Alonso - Finn: Fundamental University Physics, Vol. III Quantum and Statistical
Physics pages 11 - 14.
•
Ohanian: Physics, Second edition expanded, pages 1034 – 1037.
PHOTOELECTRIC EFFECT
Light quanta, photons, bombarding a metallic surface can release charge carriers, free
electrons from the metal with following three, observed properties:
•
•
•
The number of ejected electrons (photo current) is proportional to the intensity of
the light beam (fig. 1a).
The maximum kinetic energy of the electrons is directly proportional to the
frequency of the light.
The sensitivity of a photocell (recording device), photo current obtained, depends
on the material of the metallic, cathode exposed to the light, as well, as shown in
fig. 1b. for alkaline metals, qualitatively.
The maximum kinetic energy of the (non relativistic) ejected electrons can be expressed
with
1 2
max
mv = E kin
= hν − eφ ,
2
(1)
where h is the Planck’s constant, ν frequency of the light and eφ, the work function of
the metal (minimum energy needed to release an electron from the surface). Using
max
equation (1) the ratio h/e can be determined measuring Ekin
for several light
frequencies.
Sensitivity
-2-
Figure 1. a) Photoelectric current as a function of potential between anode and
cathode for three different light intensities. b) Spectral sensitivity of light cells
with alkaline -metal cathodes.
The equipment used in this exercise is described in fig. 2, where the monochromatic light
illuminates the cathode of the light cell. The cathode is coated with a metal with a known
work function of equation (1). The current of the photo electrons arriving to the anode is
measured with a sensitive pA-meter. The potential difference V between the cathode and
anode is tuneable and slows down the electrons with an energy eV. When the maximum
kinetic energy of the ejected electrons is eV no current is observed. In fact the potential
field in the equipment is arranged so that a small current of electrons released from anode
to cathode is present and that the saturation of this current (fig. 7) is used to define the
stopping voltage V.
K
U
A
+
V
pA
Figure 2. Schematic diagram of the equipment. The geometry of anode (A) and cathode
(K) depends on the photocell used.
-3If Vs is the voltage preventing any electron just to leave the cathode, then
max
Vs e = Ekin
.
(2)
Eq. (1) reads now
Vs e = hν − eφ .
(3)
Vs displayed as a function of ν is a straight line with the linear coefficient equal to h/e.
Position Vs = 0 gives the minimal frequency ν = ν0 = eφ/h needed to remove a single
electron from the cathode i.e. minimal energy of a single photon needed to
counterbalance the work function (energy of least bound electrons). Introduced by
Einstein eq. (1) is valid strictly, only at the absolute zero temperature (T = 0 K), when the
thermal motion is non existent. In practise our equipment is not sensitive enough to
register the uncertainty (diffuse character of ν0) caused by the thermal motion.
Because of the working principle of the photocell used, the work function of the cathode,
alone, does not define Vs. In fact the ejected electrons “feel” the potential difference
caused by the work functions of cathode and anode materials, both. Schematically this is
shown in fig. 3., where eΦC is the energy needed to release an electron from the cathode
and eΦA the energy needed to release a secondary electron (used for observations) from
the anode.
Figure 3. Definition of the closed loop of electrons between the anode (A) and cathode
(C).
In fig. 3, V’ is the tuneable, external potential between the anode and cathode and V the
stopping potential felt by the secondary electrons. In practice the potential loss caused by
the components (wires, electrodes etc.) is neglected here. Conservation of energy means
that the work done to/by an electron during the closed loop in fig. 3 must be zero:
− eΦ C + eV + eΦ A − eV ' = 0
V − V ' = −(Φ A − Φ C ).
(4)
-4Term ΦA - ΦC is called the contact-potential difference and usually the metals are
selected so that ΦC < ΦA. Equation (4) is valid for each frequency, so it must be V = 0,
when the photoelectrons have energy enough just to release secondary electrons from the
cathode. This condition means that the character of V changes from decelerating to
accelerating potential. This corresponds to the saturation of the pA-current measured.
Normalising the (pA)-curves measured for various frequencies to a single, common
saturation point one finds for the contact-potential difference V' = ΦA - ΦC.
Now equations (1) and (3) give stopping potential corrected with the contact-potential
difference
Vs = Vs' + ( Φ A − Φ C ) ,
which is equivalent to
1
e Vs' = mv 2 = hν − eΦ C ,
2
Hence
h
Vs' = ν − Φ A .
e
(5)
So, if you plot values of the uncorrected stopping potential Vs as a function of frequency
ν, you’ll obtain a straight line with the linear coefficient equal to h/e. The ordinate at ν= 0
corresponds now to the work function of the anode instead of the one of the cathode
(explain why). The contact-potential difference can not be defined (why?), but a rough
estimate for the work function of the anode is possible to achieve (explain why and give
possible sources of error).
2. EQUIPMENT
Figure 4. Experimental arrangement.
-5The key element of the monochromator (MP-1018B) is a blaze-type reflection grade (fig.
5). A reflection grade enhances certain multiples of reflected wavelengths and kills other
ones based on the principle of interference. Together with diffraction the grade is a mirror
for the enhanced rays so the reflected angle equals the entrance one.
diffracted
light
polychromatic
light
α
β
d
Figure 5. "Blaze"-grade
The direction of the grade can be controlled manually or with a stepping motor. The
reflected light can be read with an accuracy of one tenth of a nanometer. The widths of
the openings for incoming and outgoing light can be tuned with an accuracy of
micrometers.
The light source is an Hg/10 – gas-discharge tube, which has strongest intensities at
wavelengths of 313, 365.0, 404.7, 435.8, 546.1, 577.0 and 579.1 nm. One can also use
the Cd/Hg –tube, which gives additional lines. Due to the strong ultraviolet background
radiation of the tubes, try to avoid looking directly to the tubes.
The power source of the photocell is a battery equipped with a potentiometer to tune the
voltage desired. The current of the secondary electrons is measured with sensitive pAmeter. The stopping voltage can be measured with a general-purpose digital meter.
3. MEASUREMENTS
In order to check the calibration of the monochromator locate first the spectral lines of
the Hg/10 – light source using the given reference spectrum. Using several different
wavelengths measure the current of the secondary (photo)electrons as a function of the
stopping potential. Work according to following steps:
1. Switch the light source on and let it stabilise for few minutes.
2. Set the width of the opening in the monochromator to about 300 µm.
-63. Use the Keithley 485 pA-meter (brown, digital instrument) with Auto – scale to
measure the current.
4. Optimise the incoming light intensity using one strong line by searching for a
maximum current reading. Here the sensitivity should be on 10-9 A scale. With the same
method, find out accurate values for the wavelengths of the spectral lines.
intensity
5. Using at least seven different lines (from Hg or Cd) measure current vs. stopping
voltage curves. Start from zero voltage and continue with 0.1 V increments until reaching
and monitoring the linear part of the current. Close to this point use increments of e.g.
0.02 V.
Figure 6. Spectrum of a Hg - light source.
4. ANALYSIS OF RESULTS
Plot the current as a function of the potential for each wavelength used. An example how
to estimate the stopping potential for each frequency is given in fig. 7. Display the
stopping potential as a function of the frequency. Calculate a least squares fit to the data
points and include the result (with error) into the figure. Give your experimental value of
h/e with error and compare it to values in literature. Write a report explaining theory,
equipment, measurements and results of the work. In the report, also answer the
following questions:
-7-
1) With what materials does the photoelectric effect occur?
2) Can the stopping potential be negative?
3) What determines the critical frequency, below which the phenomenon doesn’t
happen?
4) What can you say about the determination of the work function with the
device given, if the anode is made of platinum and the cathode of potassium?
Figure 7. Estimation of the stopping potential.