• THE FRANCK-HERTZ EXPERIMENT


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    • Abstract: THE FRANCK-HERTZ EXPERIMENTOBJECT: To measure the excitation potential of mercury using the Franck-Hertzmethod.REQUIRED READING: Melissinos & Napolitano Chapter 1.EQUIPMENT LIST: Mercury vapor triode, oven and control unit ( Neva-Klinger

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THE FRANCK-HERTZ EXPERIMENT
OBJECT: To measure the excitation potential of mercury using the Franck-Hertz
method.
REQUIRED READING: Melissinos & Napolitano Chapter 1.
EQUIPMENT LIST: Mercury vapor triode, oven and control unit ( Neva-Klinger
Scientific), rheostat, multimeter, digital storage oscilloscope , Variac.
THEORY: 1. Excitation by electron impact of quantized, bound atomic states. In
1913 Niels Bohr proposed the Bohr model of the atom which assumes that atoms can
exist only in certain bound energy states. This idea was given a powerful boost in 1914
when James Franck and Gustav Hertz performed an experiment that demonstrated the
existence of quantized energy levels in mercury. The experiment involved sending a beam
of electrons though mercury vapor and observing the loss of kinetic energy when an
electron strikes a mercury atom and excites it from its lowest energy state to a higher one.
Figure 1: Energy Levels of Atomic Mercury
Mercury vapor atoms will normally be in their lowest or ground state, with the two
valence electrons occupying a state designated by (6s) 2 (two electrons in n = 6, l = 0
single particle states). The two electrons do not move independently so that three
quantities -- the total spin angular momentum S, the total orbital angular momentum L,
Franck-Hertz 1
and the total angular momentum J -- are constants of the motion designated by quantum
numbers S, L, and J, respectively. Thus the electron states are labeled with the
spectroscopic notation 2S+1LJ. The value of L is denoted by S for L = 0, P for L = 1, etc.
The ground state in mercury is then 1S0.
As shown in Fig. 1, the next levels above the ground state are a "triplet" of levels -- 3P0,
3P1, and 3P2 -- corresponding to single electron states (6s6p) where the electron spins are
parallel. There is a higher singlet 1P1 state with spins antiparallel. In a collision with an
energetic electron the atom could be raised into any of these excited states.
However, in the Franck-Hertz experiment we only observe excitation into the 3P1 state for
the following reason: Once the atom is in an excited state it can return to the ground state
by emission of a photon. This is normally a very rapid process typically taking 10 -8 s.
However, photon de-excitation transitions must satisfy conservation of total angular
momentum. The emitted photon has an angular momentum of 1 which it carries away
from the atom. Thus the total angular momentum of the atom must change by 1, ∆J = ± 1
(remember angular momentum is a vector). Thus de-excitation can occur from the 3P1
and 1P1, states. but not from the 3P0, and 3P2 states, which are metastable -- de-excitation
from a metastable state can only occur by slower processes which typically take on the
order of 10 -3 s. In the Franck-Hertz experiment the electron beam may excite a mercury
atom into the 3P0 or 3P2 state, but then it is stuck there (for a millisecond) and unable to
absorb more energy. On the other hand, if the 3P1 state is excited, it quickly de-excites (in
0.01 microseconds) and the atom is again available to absorb energy from the electron
beam. The 1P1 state is not observed since the 3P1 state is so effective at taking energy
from the electron beam once the electrons reach 4.86 eV that they are not able to acquire
the 6.67 eV needed to excite the 1P1 state.
Figure 2
In this experiment electrons emitted by a hot cathode are accelerated through the mercury
Franck-Hertz 2
vapor to a collector electrode. When the accelerating voltage is increased to 4.86 V, the
collector current will drop due to the onset of energy loss of the electrons caused by
collisions with Hg atoms that raise the atoms from their 1P0 ground state to their 3P1
excited state. If the voltage is further increased the current will again increase until 2x4.86
= 9.72 V where it will drop again, as shown in the sketch on the previous page. This is
due to electrons which having lost most of their kinetic energy in the first collision again
being accelerated to 4.86 eV kinetic energy so that in a second they can again excite the
3P1 state. This process will be repeated for 3x4.86 V, 4x4.86 V, etc. As we will discuss
below there are several effects that will shift this pattern of repeated valleys (or peaks,
depending how you look at the graph) to higher voltages. However, the spacing between
the peaks/valleys will remain equal to 4.86 eV. Excitation of states of lower and higher
energy than 4.86 eV, if seen at all, will appear as "shoulders" on the dominant current
drop.
2. Mean free path of electrons. The phenomena involved in this experiment are
influenced strongly by how far, on the average, an electron goes before colliding with a
vapor atom, and producing excitation. The average distance which an electron travels
between collisions, of any type, is called the electron mean free path l . This can be
estimated from :
4 2
l= 2 . (1)
n(πRo )
Ro is the radius of a Hg molecule and n is the number density of the gas, i.e. the number of
molecules per unit volume. (See Halliday and Resnick for a derivation.) Note from Eq.
(1) that l(π Ro )n ≈ 1. This means that l is the average distance an electron will travel
2
before coming within Ro of a mercury atom. In mercury the atomic radius is about 0.15
nm = 1.5 x 10 -8 cm.
In this experiment the Franck-Hertz tube, which contains a drop of mercury, is placed in a
heated oven. The mercury drops evaporates until the vapor is saturated, i.e. the rate of
evaporation equals the rate of condensation. By changing the temperature T of the oven,
you will be able to change the vapor pressure and hence the density. To estimate the mean
free path, we will need to calculate n(T). Mercury is monatomic. From the definition of
gram-atomic weight A, the number of molecules per cubic centimeter is
n = (mass/cm 3)/(mass/atom) = ρ ÷ (A/NA) = ρ NA/A (2)
where ρ is the mass density of the vapor and N A = Avogadro's number.
The Handbook of Chemistry and Physics or International Critical Tables give the pressure
of saturated mercury vapor at various temperatures. If it is assumed that the vapor
behaves approximately as an ideal gas at these low temperatures, then the density can be
calculated by using the general gas law
PV = (m/A)RT, (3)
Franck-Hertz 3
where m is the mass of gas in the volume V and m/A is the number of moles. Then using
ρ = m/V in Eqs. (2) and (3) we find
NA P
n= R T . (4)
In using this equation, be sure that you use a value for R expressed in the same units you
use for P and that T is in Kelvins. Then you can estimate the mean free path at any oven
temperature by using Eq. (4) in Eq. (1) along with the tabulated saturated vapor pressure
P(T). This table is given in the binder of readings available in the lab.
In order to see several peaks and valleys for taking data to determine the excitation
potential, it is necessary, on the one hand, to have the electron mean free path l shorter
than the distance from the cathode to the anode (grid). But on the other hand it must be
larger than the distance between anode and collector. For your experiment vary the oven
temperature to optimize your data and for your report you will prepare a theoretical plot
of l(T ).
3. Contact potential. We saw in the discussion of the photoelectric effect that the work
function W is equal to the photon energy required to extract an electron from a substance
with zero final kinetic energy. The top portion of the diagram above illustrates the concept
of the work function for two metals: In a metal, the valence electrons are not bound to
the metallic ions but can wander freely throughout the metal and can be thought of as a
(Fermi) sea of electrons filling a box to a certain height. The work function is the minimum
energy to remove an electron from the top of this sea to a point far from the metal. If the
two metals shown in the diagram are placed in contact electrons flow from one to the
other until energies of the top of the two Fermi seas (called the Fermi levels) are
equalized. (Question: if the capacitance between the two metals is 10 pF, how many
Franck-Hertz 4
electrons must be transferred between them to equalize the Fermi seas?). A potential
difference ∆V then develops between the two metals which is called the contact potential,
as shown in the diagram. From the diagram we see ∆V = W 1 - W2. For the Franck-Hertz
tube the manufacturer indicates that the contact potential is about 2 V, which you will be
able to verify with your measurements. Note that to have a contact potential, the two
metals do not have to be in intimate physical contact; it is sufficient for there to be a way
for electrons to travel from one metal to the other. In the Franck-Hertz tube the electrons
travel through the vacuum from the cathode to the anode; the two metals do not “touch”.
The electrons that travel through the Franck-Hertz tube are released by thermionic
emission from the very hot cathode as follows: The surface of the Fermi sea that we have
shown in the diagram is appropriate for a temperature of absolute zero. At higher
temperatures thermal agitation raises some of the electrons above the Fermi level. At
room temperature the average thermal energy is kT ~ 0.025 eV, which is not enough for
electrons to overcome the work function and escape. But when the cathode is raised to a
high temperature (~1500 K) a few electrons have enough energy to escape, i.e. to be
thermionically emitted. Ideally for the Franck-Hertz experiment we would like these
electrons to have zero kinetic energy so that this initial kinetic energy will not interfere
with our measurement of the change in kinetic energy due to inelastic collisions.
However, experimental measurements show that for cathode temperatures in the 1500 K
to 2500 K range thermionic electrons have a distribution of kinetic energies peaking at 0.2
to 0.3 eV. This will affect the location of the first peak in the current, but not the spacing
of subsequent peaks.
EQUIPMENT: The essential components are a mercury thyratron tube, a variable power
supply for the cathode heater, anode and collector voltage supplies with necessary meters,
a sensitive dc amplifier for observing the small collector currents, and an oven with
thermometer and controls to heat the thyratron to an appropriate temperature to produce
the desired vapor pressure.
1. The oven. The oven consists of a small steel cabinet with a heating element to
uniformly heat the tube (and all connections leading to it). A 300 watt heating element is
mounted on the bottom of the housing. An adjustable thermostat in the oven regulates the
temperature of the oven. Since the thermostat functions imperfectly (1-3 degrees drift), it
is preferable to bypass it by setting it to maximum temperature and instead limiting the
power delivered to the oven with a Variac (a variable auto transformer that steps down
the 110 V AC line voltage). This approach gives adequate stability. A hole in the top of
the cabinet is provided for a thermometer.
2. Thyratron tube. A thyratron is a vacuum tube in which a drop of mercury was added
before sealing. Such tubes, therefore, contain saturated mercury vapor (vapor in
equilibrium with the liquid) at a pressure corresponding to the temperature of the bulb.
The Franck-Hertz tube, a triode, has three plane-parallel electrodes which provide a
uniform electric field to accelerate the electrons. The electrodes are the cathode (K),
anode (A) and collector (M). The cathode is indirectly heated with a heater electrode (H
and K) using a nominal voltage of 6.3 volts AC. (Note that the rheostat used to vary the
cathode heater current should be connected to H, not to K which is grounded.) The anode
Franck-Hertz 5
is a perforated screen 8 mm beyond the cathode that is held at a positive potential V a
relative to the cathode in order to accelerate the electrons thermionically emitted by the
heated cathode. The screen allows most of the electrons to pass through the anode. The
collector lies a small distance (~2 mm) beyond the anode and is negatively charged relative
to the anode with a voltage V ret, which acts to retard (slow) the electrons that pass
through the anode. These electrons (charge e) can make it to the collector, if as they pass
through the anode their kinetic energy is greater than eVret. Otherwise, they slow, stop,
and reverse direction to return to the anode.
3. Operation of the Franck-Hertz tube to observe inelastic scattering through
changes in the collector current. There are two lengths of interest -- the cathode anode
separation (8 mm) and the mean free path l(T ) -- and two voltages -- the accelerating
voltage V a between the anode and cathode and the retarding voltage V ret between the
anode and collector. It is important to understand how they affect the experiment.
The value of l and hence the temperature of the oven is not critical as long as l


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