Department of Physical Electronics
Faculty of Science, Masaryk University, Brno
Physical laboratory 3
Task C Thermoelectron emission
Tasks
1. Measure the dependence of the anode current on the anode voltage Ia = f(Ua), where Ua
is in the range from −5V to 500V, for two different values of the heat current If and plot
those measuerements in a graph.
• Plot the start-up area of the anode current Ia in ln Ia = f(Ua) coordinates and determine
the electron temperature.
• Use the ln Inas =
√
Ua coordinates to work with the area of the saturated anode current
and determine the current increase due to the Schottky effect. Compare the experimentally
obtained value and the value calculated from (11). You can estimate the electric
field intensity near the surface of the cathode according to (13).
2. Determine the anode voltage Ua, for which the anode current is already saturated, Ia = Inas.
3. By measuring the dependence of the saturated anode current on the neat current Inas = f(If)
determine the work function of tungsten w using the Richardson – Dushman line.
Thermoelectron emission study
Releasing free electrons from the surface of metals is called electron emission. Emission occurs
when electrons gain enough energy for overcoming the attractive forces stopping them from their
emission from the metal’s surface. The electrons can gain this energy by different means. One
of the most common and most effective ways is the heat emission known as thermionic emission,
thermoelectron emission, thermoelectronic emission or thermoemission being the emission of electrons
at high temperatures of the metal and photoemission being the emission of electron due to
short-wavelength irradiation of the metal’s surface. The electrons can also be released due to a
high electric field in the order of 108 −1010 Vm−1 without heating the metal. This case is called an
electron autoemission or cold emission. Emission due to the bombardment of charged particles is
called a secondary emission. This assignment is focused on the study of thermoelectron emission.
The history of heat emission dates back to the 1870s, and nowadays, thermoemission is used for
the generation of electron rays in many electronic devices (electron microscopes, electron tubes,
cathode ray tubes, etc.). It is also used for the analysis of forces binding the electrons in the
matter.
Work function
Metals at a sufficiently high temperature emit electrons. Only electrons with high enough energy
for overcoming the attractive forces between the electron and the metal can be emitted. The
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energy of the attractive forces is called the work function (w). The number of all the electrons
released from the cathode at a given temperature is given by the saturated emission current. Its
magnitude depends on the temperature of the metal constituting the cathode (T) and the work
function. This dependence is shown in the Richardson – Dushman equation
Inas = BT2
exp(−w/kT), (1)
where B is a constant describing the different parameters such as the cathode surface or the
thermionic emission constant and k is the Boltzmann constant. The derivation of this equation
and the formula for the thermionic emission constant can be found for example in[3]. We can
use equation (1) for the measurement of the work function by rewriting it to the form of the
Richardson line. We rearrange the equation and use logarithms
ln(Inas/T2
) = ln B − w/kT. (2)
We denote y = ln(Inas/T2) and x = 1/T. Therefore we get a line in the new coordinates as
y = (−w/k) · x + ln B. (3)
From the slope of this line, we can calculate the work function w of the given metal. Work function
examples for different metals are shown in the following table.
element w [eV] element w [eV]
Cs 1.9 W 4.5
Ba 2.5 Fe 4.7
Th 3.4 Ni 5.1
Ta 4.1 Pt 5.4
Cathode temperature determination
It is necessary to know the temperature of the cathode for plotting the dependence (2). We can
determine this temperature based on the known dependence of the conductor resistance on the
temperature. The tungsten filament resistance (Rt) can be described as
Rt =
ρd
S
(1 + αt), (4)
where ρ = 4.89 × 10−8 Ωm at 0 oC, d is the filament length, S is the filament cross-section,
d/S = 7,76 · 106 m−1, α = 4.83 × 10−3 K−1 is the temperature coefficient of resistivity and t is
the temperature in degrees Celsius. The resistance of the filament can be determined by Ohm’s
law from the value of the measured heat current If and the voltage drop at the cathode Uf.
Distribution of electrons according to their energies
It can be shown that the kinetic energy of electrons emitted from a surface of the metal has a
Maxwellian distribution. The electron energy distribution can be experimentally measured by the
retarding field method. The principle of this method is in introducing an anode voltage Ua with
the variable polarity between a cathode and an anode with a suitable geometric configuration. By
measuring the dependence of the anode current Ia on the anode voltage Ua (for Ua < 0 as well),
one can obtain the electron energy distribution function in its integral shape. If we decrease the
value of the anode voltage from positive to negative values, we decelerate the electrons by the
anode’s electric field. In other words, more and more electrons do not hold true to the condition
1
2
mv2
≥ −eUa (Ua < 0) (5)
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and they, therefore, can not reach the cathode, and they do not take part in the current flowing
between the cathode and the anode. Different cathode temperatures will yield a different number
of electrons of different energy distributions.
One can, therefore, verify the validity of the Maxwellian electron energy distribution by measuring
the VA characteristics in the region of negative anode voltage. One can also determine the
Figure 1: a - integral, b - differential shape of the electron energy distribution. Region c is the
region of the start-up current, region d is the region of the saturated current.
emitted electron temperature from the VA characteristic. The start-up region can be described
as
Ia = I0 exp
eUa
kT
. (6)
The electron temperature can be determined after performing the logarithm of the previous formula
from the slope a = e
kT in the linear region of the dependence ln Ia = f(Ua) in the start-up
current region. This temperature can be compared to the one determined in the task focused on
the work function calculation.
Schottky effect
The presence of a strong electric field near the metal surface causes a decrease in the cathode work
function. Figure 2 shows the potential near the surface of the metal. Without the presence of the
magnetic field, the potential has an approximately rectangular step shape. For the electron to be
released from the metal, it has to have energy greater than the work function w. The value of the
work function is given by the depth of the Fermi level, that is by the difference of the potential of
the Fermi level of the vacuum potential.
If an electric field of the intensity E is introduced, its potential is superimposed with the step
potential. The step potential is transformed into a potential barrier of finite thickness. Also, the
height of the barrier is lower. Therefore, the electron emission is facilitated by two processes. On
the one hand, the work function decreases to w , and on the other hand, the finite thickness of
the barrier gives rise to a non-zero probability of the electron to tunnel through the barrier. The
shape of the superimposed potential also shows that the electrons are accelerated away from the
cathode from a certain distance.
It can be shown that the work function of the electron in a metal w is decreased by wp at the
presence of an electric field as per
wp =
e3E
4π 0
(7)
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Metal Vacuum
Figure 2: Shottky effect.
and therefore the new value of the work function w is
w = w − wp = w −
e3E
4π 0
, (8)
where 0 is the permittivity of vacuum. Richardson – Dushman equation of the saturated emission
current becomes
Inas = BT2
exp(−w /kT) = BT2
exp(−w/kT) exp(wp/kT) = Inas exp(wp/kT) , (9)
where Inas is the saturated emission current without the presence of the electric field. From this
we obtain
ln Inas = ln Inas + wp/kT (10)
and we can see that Inas > Inas. Therefore, decrease in the work function by wp leads to an
increase a of the saturated emission current. After substitution of wp, we can write
ln Inas = ln Inas +
e3
4π 0k2T2
×
√
E. (11)
Formula 11 contains the electric field intensity near the surface of the cathode. Numerical methods
are necessary to determine the exact value of the electric field in the present configuration.
However, the electric field can be approximately estimated by assuming that in the closest vicinity
of the cathode, it does not significantly differ from the electric field in a cylindrical capacitor. In
such a case, the electric field intensity E near the surface of a cylindrical electrode with radius R
can be described by
E = Ua
1
r ln(R/r)
, (12)
where Ua is the anode voltage. In our simplified case, the anode has a virtual radius D corresponding
to the distance of the hot cathode and the anode. It is obvious that the real geometry
significantly differs from a cylindrical capacitor. Numerical calculations have shown that formula
Physical laboratory 3 manuals (version May 10, 2023) 5
Figure 3: The VA characteristic in the ln I and U
1/2
a coordinates.
(12) needs to be multiplied by a factor of (L − D)/D, where D is the distance of the hot cathode,
and the anode and L is the distance between the anode and the planar cold part of the cathode
placed behind the hot cathode that is used to homogenise the electric field between the electrodes.
The electric field intensity near the present hot cathode can be, therefore, calculated according to
E = Ua
L − D
D
1
r ln(R/r)
. (13)
As the electric field intensity is proportional to the anode voltage Ua, the logarithm of the anode
current ln Inas must be directly proportional to the anode voltage square root
√
Ua, see figure 3.
Measurement setup description
The measurement is carried out with the apparatus connected as per figure 5. The measurement
is semi-automatic due to a computer-controlled power source Agilent E3631A that also serves as
a measurement device. Agilent E3631A has several outputs. The 0–6 V, 0–5 A output is used
as the current source in the cathode circuit serving for the heating of the tungsten filament.
A hot enough filament emits electrons due to the thermionic emission. These electrons can be
collected at an anode depending on the potential difference between the cathode and the anode
(anode voltage). The anode voltage can be controlled by the 0 – 20V output of the power source.
The voltage range is not high enough to carry out the whole experiment. Therefore, a voltage
converter multiplying the output voltage by the factor of 25 is connected to this output. A switch
can disconnect this converter for more precise measurement in the region of low anode voltages.
Another switch is used to change the voltage polarity on the electrodes. The anode voltage is
monitored by a Protek 506 voltmeter connected to the computer through a serial port. The anode
current is monitored by an Agilent 34410A digital ammeter connected to the computer by USB.
During the measurement, the measured data are plotted as well as saved to a text file for later
analysis.
Experimental procedure
• You will measure with voltages up to 500 V, mind your own safety first!
• Do not exceed the maximum allowed values of the heating current and anode voltage.
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1. Connect the circuits as per figure 4 and wait for a check by the teacher before turning
anything on.
2. Run „Termoemise“ (MTE) from the desktop.
3. Connect all the measurement devices (Agilent E3631A, Agilent 34410A a Protek 506) in the
menu Zařízení (Devices) and sub-menu Připojit (Connect). The taskbar at the bottom of
the window (see fig. 6) will indicate a successful connection of the devices.
4. Click Zapnout výstupy (Turn on outputs). This will activate the power source outputs and
the pull bars and switches for the anode current and anode voltage in the software.
5. Choose the correct experiment in menu Měření (Measurement). Výstupní práce is for the
measurement of the work function by measuring the dependence of the saturated ion current
on the heating current or Schottkyho efekt for the measurement of the Schottky effect for
the measurement of the dependence of the anode current on the anode voltage.
6. While measuring the Schottky effect measure the VA characteristic of the diode (Ia = f(Ua))
for two different values of the heating current If. The heating current must not exceed 2 A!
Consider the measurement range of the anode voltage Ua before the experiment.
7. Set the desired values of the heating current and wait for several minutes (at least 10) for
its stabilisation.
8. Use the manual switch to change the polarity of the applied voltage. You need to use
the voltage converter to use a voltage higher than 20 V. Make sure that Ua = 0 V before
engaging the converter. Then use the switch „zesilovač“ in the software.
9. Save the measured data by clicking Uložit měření (Save measurement) in the Soubor (File)
menu after each task.
10. Determine the anode voltage Ua corresponding to the saturated anode current region from
the VA characteristic.
+
V
A
V
+
x25
PS1
PS2
Uf
If
Ua
Ia
=
OUT
IN
GND
1x
25x
+
-
A
Figure 4: Electrical diagram of the diode connection used for the study of thermoemission.
Physical laboratory 3 manuals (version May 10, 2023) 7
1
2
3
4
5
6
a
Figure 5: Diagram of the diode connection used for the study of thermoemission. 1 – Agilent
E3631A power source, 2 – voltage converter, 3 – voltage polarity and voltage converter switch, 4
– Protek 506 multimeter, 5 – Agilent 34410A multimeter, 6 – transparent diode cover: a) vacuum
tube containing the hot cathode and planar anode.
11. While measuring the work function in the Výstupní práce regime, measure the Inas = f(If)
dependence at a chosen anode voltage Ua. Always wait several minutes after every change
of the heating current If for its stabilisation.
12. After you finish the measurement, click Vypnout výstup (turn off the output) and within
the Zařízení (Devices) menu and the Odpojit (Disconnect), disconnect all the measurement
devices.
Notes
• Maximum heating current: If = 2 A
• Maximum anode voltage: Ua = 500 V (while using the voltage converter)
• Cathode: tungsten (W), tabulated value of the work function: w = 4.5 eV
• Cathode diameter: 2r ≈ 0.09 mm, cathode length: d = 50 mm, cathode geometry factor
d/S = 7,76 · 106 m−1
• Anode radius: R = 17 mm
• Distance between the anode and the hot cathode: D = 15 mm
• Distance between the anode and the cold cathode: L = 25 mm
References
[1] T. Chudoba a kol.: Fyzikální praktikum III., skripta Přír. fak. UJEP v Brně, SPN Praha 1986.
[2] L.N. Dobrecov: Elektronová a iontová emise, Nakladatelství ČSAV, Praha 1956.
[3] O. Vybíhal: Automatizace měření ve fyzikálním praktiku, bakalářská práce, PřF MU 2011.
https://is.muni.cz/auth/th/175317/prif_b/
Physical laboratory 3 manuals (version May 10, 2023) 8
1
2
3
4
5
6
7
Figure 6: The user interface of the MTE software. 1 – main menu, activation of the connected
devices and measurement start, 2 – heating current control, 3 – anode voltage control and (zesilovač
= turn on the 25 times voltage converter), 4 – data registered by the individual measurement
devices, 5 – taskbar indicating possible connection errors, 6 – graph of the dependence of the
anode current on the heating current used for the measurement of the work function, 7 – graph
of the dependence of the anode current on the anode voltage used for the measurement of the
Schottky effect.