University of Ljubljana
Faculty of mathematics and physics
Department of physics
Seminar
Marjan Grilj
Thermionic emission
Abstract
________________________________________________________________
Thermionic emission of electrons from metal at high temperature is discussed.
We derive Richardson's law that describes emission of electrons from a metal at
given temperature. Child's law that describes relation between anode current at
given cathode-anode potential is also derived. Other effects at thermionic
emission are mentioned. Simple demonstrative experiment is presented that
shows how to qualitatively check both laws and how to estimate work function
of tungsten.
_________________________________________________________________
Mentor: prof. dr. Gorazd Planinšič
Ljubljana, April 2008
Table of contents
1 Introduction ......................................................................................................................... 2
2 Thermionic emission ........................................................................................................... 3
2.1 Richardson's law........................................................................................................... 3
2.2 Child's law .................................................................................................................... 6
2.3 Related phenomena ...................................................................................................... 8
3 Demonstration experiment .................................................................................................. 9
3.1 Richardson's law......................................................................................................... 10
3.2 Child's law .................................................................................................................. 11
4 Conclusion......................................................................................................................... 13
5 References ......................................................................................................................... 13
1 Introduction
Thermionic emission must not be confused with thermic emission (of black body for
example) where photons are emitted from heated body. In the case of thermionic emission
we observe emitted electrons from a metal due to its high temperature.
Mobile electrons in metals, also called valence electrons, are responsible for electric current
conduction. If we increase the temperature of the metal, electrons start to move faster and
some may have enough energy to escape (evaporate) from the metal. The higher the
temperature, the higher will be the current of escaping electrons. This temperature induced
electron flow is called thermionic emission.
If we use the hot metal from which electrons evaporate as a cathode on a high potential
difference to the anode, then we get electric field between cathode and anode and emission
of electrons can be measured as a anode current.
Besides of showing relations of parameters in thermionic emission, the object of this
seminar is also to show how to estimate charge over mass ratio of an electron and work
function of tungsten with the simple experiment using phenomenon of thermionic emission.
We can take advantage of both mentioned phenomena using ordinary double filament
headlamp bulb with one filament burn-out.
2
2 Thermionic emission
2.1 Richardson's law
Potential barrier at the metal surface tends to prevent free electrons from escaping at low
temperatures. When the metal is heated to sufficiently high temperature, some of free
electrons get enough energy to carry them over potential barrier. With suitable electric field
these electrons can be then drawn away from metal and measured. Electrons emitted by the
filament are drawn to the anode, then the anode current is controlled primarily by the
filament temperature and is practically independent of the potential. Under these conditions
anode current is said to be saturated, and is purely a function of the temperature. [1]
Richardson's law tells us what is the current density Jx [A/m2] of thermically escaped
electrons in the direction perpendicular to heated metal. We can show how the law can be
derived from basic properties of electrons in a metal.
Let us say we have a flow of thermal electrons with charge q in the x direction. We consider
empty space in front of metal plate. Then the current density of this electrons is
J x = ∫ q n ( E ) v x ( E ) dE ,
(1)
where n(E) is the density of electrons in units of [J-1m-3]. vx(E) is the speed of electrons in x
direction. The integral is taken over all electron energies needed for escaping over potential
barrier in x direction.
From statistical physics we know that density of particles can be written as
n( E ) = g ( E ) f ( E ) ,
(2)
where g(E) is density of states and f(E) is the probability for a taken state with energy E. In
case of electrons that are half spin particles both functions have to obey Fermi-Dirac
statistics and are written as
8 2π 3 / 2
g (E) =
m
E,
(3)
h3
f (E) =
1
 E − EF
1 + exp
 kT



,
(4)
where h and k are Planck's and Boltzmann's constants. EF is Fermi energy, that is the
highest energy of electrons in constant confining potential U (figure 2.1). Only electrons
with the highest energies E >> EF can escape from metal, so f(E) approaches Boltzmann's
distribution:
 E − EF 
(5)
f ( E ) = exp −
.
kT 

3
Figure 2.1: Free electrons inside metal are treated as in potential well at constant confining poten–
tial U. They fill all available states up to the Fermi energy EF. Electrons with the highest energies
lack exactly W (work function) to leave metal.
We can now write part of the equation (1) as
n( E ) dE =
8 2π 3 / 2
 E − EF 
m
E exp −
 dE .
3
kT 
h

(6)
Let us write the energy as a function of the electron velocity
E=
1 2
mv ⇒
2
E dE =
1
2
m 3 / 2 v 2 dv
(7)
and insert it into (6):
n( E ) dE =
8 2π 3 / 2
 E − EF  1 3 / 2 2
m v dv =
m exp −

3
kT  2
h

=
8π 3
 E − EF
m
exp
−
kT
h3

 2
v dv .

(8)
We can now insert expression (8) into equation (1) to get
8π 3
 E − EF  2
m exp −
v dv =
3
kT 
h

8π
E   E  2
= ∫ q v x ( E ) 3 m 3 exp F exp −
v dv =
h
 kT   kT 
2
E 
 E 
2
= q 3 m 3 exp F  ∫ v x ( E ) exp −
4π v dv .
kT
kT
h




J x = ∫ q vx(E)
(9)
The integral over vx starts from the minimum velocity needed to overcome the confining
potential barrier U (illustrated on figure 2.1), because only electrons with velocity higher
than vx,min will leave metal:
1
2U
2
.
(10)
U = mv x ,min ⇒ v x ,min =
2
m
4
We express energy as a function of the velocity components in all three directions so that
we can integrate over each velocity component. So from equation (9) on we can write
∞
∞
∞
 mv y2 
 mv x2 
 mv z2
2qm 3
 EF 




v
v
v
Jx =
exp
exp
−
d
exp
−
d
exp


∫ x  2kT  x −∫∞  2kT  y −∫∞  − 2kT
h3
 kT v x , min



dv z . (11)

Solving these integrals using known integral
∞
∫ exp(Cx
2
)dx =
−∞
π
(12)
C
we get
Jx =
2qm 3
E 
 U  kT
exp F  exp −

3
h
 kT  m
 kT 
Jx =
2πkT
m
2πkT
,
m
4π qmk 2 2
 E −U 
T exp F

3
h
 kT 
(13)
 W 
J = AR T 2 exp −
,
 kT 
(14)
or shorter
where AR is Richardson's constant with a value AR = 1,2·106 A/m2K2. Work function W = U
– EF is the minimum amount of energy needed for an electron to leave the metal (figure
2.1) and depends on a metal (table 2.1).
Owen W. Richardson1 published the results of his experiments in 1901. The modern form
of this law was demonstrated by Saul Dushman2 in 1923. The equation (14) can be used for
experimental determination of work function of tungsten and some other metals. This is
very simple and the oldest method of measuring work functions, nowadays more precise
method – photoelectric effect is used (more in chapter 2.3).
1
Owen W. Richardson (1879-1959) was physicist from UK who received Nobel prize in physics 1928 for his
work on the thermionic phenomenon and especially for the discovery of the (now called) Richardson's law.
2
Saul Dushman (1883-1954) was Russian-American physical chemist who first derived equation (14)
similarly as it is in this seminar. So the law is sometimes referred as Richardson-Dushman equation.
5
2.2 Child's law
When thermionic electrons are emitted from the metal surface we would like to detect
them. For that we use the evaporating metal as a cathode and anode at a distance x = b so
that they are at a potential difference VA (figure 2.2). This potential attracts electrons from
the cathode to the anode, so the electrons accelerate towards anode where they can be
detected. Electric current IA flows from anode to the cathode and the same holds for the
current density JA [A/m2].
Figure 2.2: Simplified model of cathode-anode apparatus.
Let us suppose we have the setup shown on figure 2.2 in a vacuum. A cloud of thermically
escaped electrons (as shown in chapter 2.1) is formed on the surface of metal, so that the
space around it is unevenly filled with electrons. Space charge density (number density) of
electrons n(x) [m-3] vary with x. Positive potential on x = b pulls these electrons towards the
anode. Electric current density of electrons is
J = −e n( x ) v ( x ) = − J A .
(15)
Kinetic energy of electron equals to
1 2
mv = eV ( x) .
2
(16)
Knowing the V(x) we can calculate velocity of electrons as the function of distance x:
v = v( x) =
2e
V ( x) .
m
(17)
Accelerating electrons constitute a steady current JA, so we can see that n(x) is decreasing
and v(x) is increasing towards anode.
6
We would like to know how space charge density affects electric potential between cathode
and anode. For that we can use Poisson's equation
∂ 2V
ρ ( x)
=−
2
ε0
∂x
(18)
that tells us how the potential V(x) changes in presence of charge density ρ(x). In our case
JA
∂ 2V en( x)
=
=
=
2
ε0
ε 0 v( x)
∂x
Equation have to obey initial condition:
∂V
∂x
V
JA
ε0
2e
V ( x)
m
.
(19)
= 0,
x =0
x =0
=0.
(20)
Solving differential equation we get the potential for every point in region 0 < x < b:
2

3
 9J
 4
A

 x3 .
V ( x) =

2e 
 4ε 0 m 


(21)
We know that at point x = b, the potential is V(b) = VA. So from equation (21) current
density can be expressed as
 4ε
J A =  02
 9b
2e  3
V A 2 .
m
(22)
Equation (22) describes the current density flow JA at any given voltage VA. The result is
known as Child's law or Child-Langmuir3 law.
This relation can be used for experimental determination of specific charge e/m of an
electron. (See third chapter for details).
Small correction should be made here. Electrons are not emitted by the filament with zero
3
initial velocity. They have thermal energy kT per electron. That potential difference
2
should be added to the plate voltage, and is about 0,3 V at T = 2000 K.
3
After Clement D. Child (1868-1933) and Irving Langmuir (1881-1957).
7
2.3 Related phenomena
There are many other phenomena happening on small scale accompanying thermionic
emission when real experiment is done. Let's only mention some of them.
•
The work function in fact slightly depends on temperature due to thermal expansion
of the atom lattice. Richardson's law doesn't take this into consideration.
•
Vacuum in the tube where experiment is placed is of course not a real vacuum.
Bulbs that are used in experiment (chapter 3) are usually filled with argon gas under
negative pressure. Gas atoms may impede electrons at lower voltages on the path to
the anode, but when the field is stronger atoms get excited and may themselves
cause some anode current.
•
Surface of the filament is contaminated with other atoms besides tungsten and the
filament itself has some impurity. This is affecting emission probability and work
function.
•
At higher voltages, strong electric field reduces the barrier of electron emission so
the work function decreases. Due to so called Schottky4 emission electrons can
tunnel through the barrier even if they lack the energy required classically to
overcome potential barrier. Microscopic peaks on the surface of the filament
enhances the electric field near the peak slightly beyond that calculated for a smooth
surface.
Many techniques have been developed based on two physical effects to measure the
electronic work function of metals. Besides methods based on thermionic emission,
methods based on photoemission are used due to its better precision and practical
implementation.
Photoelectron emission spectroscopy (PES) is the general term for spectroscopic techniques
based on the outer photoelectric effect. The surface of a solid metal is radiated with
ultraviolet light and the kinetic energy of the emitted electrons is analyzed. As UV light
have an energy E = hν lower than 100 eV it is able to extract mainly valence electrons. The
resulting spectrum reflects the electronic structure of the sample providing information on
the density of states, the occupation of states and the work function.
4
Walter H. Schottky (1886-1976) was a German physicist.
8
Table 2.1: Work functions of some common elements expressed in units of eV [6]:
Ag 4,26 Al 4,28 As 3,75 Au 5,10 B 4,45 Ba
Be 4,98 Bi 4,22 C 5,00 Ca 2,87 Cd 4,22 Ce
Co 5,00 Cr 4,50 Cs 2,14 Cu 4,65 Eu 2,50 Fe
Ga 4,20 Gd 3,10 Hf 3,90 Hg 4,49 In 4,12 Ir
K 2,30 La 3,50 Li 2,90 Lu 3,30 Mg 3,66 Mn
Mo 4,60 Na 2,75 Nb 4,30 Nd 3,20 Ni 5,15 Os
Pb 4,25 Pt 5,65 Rb 2,16 Re 4,96 Rh 4,98 Ru
Sb 4,55 Sc 3,50 Se 5,90 Si 4,85 Sm 2,70 Sn
Sr 2,59 Ta 4,25 Tb 3,00 Te 4,95 Th 3,40 Ti
Tl 3,84 U 3,63 V 4,30 W 4,55 Y 3,10 Zn
2,70
2,90
4,50
5,27
4,10
4,83
4,71
4,42
4,33
4,33
3 Demonstration experiment
With demonstrative experiment we can qualitatively verify the form of both derived
equations. We can also show how work function of tungsten and e/m ratio of an electron
can be measured.
In the simplest form of experiment we can use double filament made of tungsten car
headlamp bulb with one filament burn-out (figure 3.1). That can be achieved simply with
too high current through a filament. The broken filament support can be used as an extra
electrode providing an anode for the directly-heated cathode. [2,4,5]
Figure 3.1: Left: Headlamp bulb with one filament burn-out on which anode current is measured.
Right: Scheme circuit of an experiment.
In the experiment we used light bulb with given characteristics: Narva 12 V, 40 W. The
surface S and length of filament l are specified by manufacturer (S = 1,3·10-5 m2, l = 31,7
mm). Distance from anode to cathode is measured to approximately b = 3 mm.
From the circuit of an experiment (figure 3.1) it is seen that U0 is used for heating tungsten
filament, varied in interval [0 V, 8 V], which makes filament glow in visible light. We also
measure current I0 so we can determine resistance R of filament using Ohm's law. UA is
9
anode (accelerating) voltage varied in interval [0 V, 300 V]. Measured anode current IA is
normally in mA region.
To check Richardson's equation (14) it is also necessary to determine the absolute
temperature of the emitter (tungsten filament). This is not a trivial problem due to various
reasons. The temperature of the filament is not uniform throughout its entire length and it's
attached to the supporting rods. Filaments dimensions are very small and unreachable in
glass light bulb.
One way of temperature determination (that was used in this experiment) is from
temperature dependence of resistance unique to tungsten. According to [3] the best fit for
tungsten is
T = 112 + 202 x − 1.81x 2 ,
(23)
where x is the ratio of the hot resistance to that at 293 K. We have to note that (23) does not
have special physical meaning, it is only fitted function commonly used in practice.
Temperature can be alternatively determined with Stefan-Boltzmann's law, which says that
the power radiated from the surface of a hot body is:
P = σT 4 Sε T ,
(24)
where σ is Stefan's constant, S is the surface area of the body and ε T is emissivity of a
body (about 0,3 for tungsten). Problem with emissivity is that the value is slightly
temperature dependant and would need to be set for each temperature. The harder problem
is exact measuring of power P, so we used first method of determining temperature.
3.1 Richardson's law
 W 
Richardson's equation (14) can be written as I A = AT 2 exp −
 , where A is Richardson
 kT 
constant multiplied with surface of filament S (we know that JA = IA/S).
We measured anode current IA and temperature of filament T via its resistance using
equation (23). The measurements were made in region U0[1 V, 8 V] at three different anode
voltages UA as shown on plot (figure 3.2). The plot reveals the shape of Richardson
function at the temperatures that are little under operational for given light bulb (12 V, 40
W). Besides qualitative validation of Richardon's relationship, it also allows us to
approximately determine the work function of the filament sample by fitting measured data
with Richardson function.
10
Figure 3.2: Logarithmic plot of anode current IA versus filament temperature measurements at three
constant anode voltages UA. Black line is Richardson function for work function W = 4,7 eV.
Measurements at higher anode voltage (UA = 300 V) agrees with tabular work function for
tungsten (table 2.1) where we find W = 4,55 eV, so the error is luckily small. As said before
greater error represents determination of temperature. Results calculated from StefanBoltzmann's law gives up to 200 K higher results for the temperature of filament.
We also see that at temperatures up to 2000 K approximately the same amount of electrons
arrive to the anode, independent of their kinetic energy (all measurements lie on Richardson
function). But at higher temperatures anode current IA starts to drop for electrons that are on
low voltage acceleration (measurements begin to deviate from Richardson function).
Apparently UA < 300 V wasn't enough for all electrons to accelerate to the anode. The
higher potential difference, more electrons will be accelerated towards anode.
3.2 Child's law
With the same demonstration we can test Child's law (22): J A
2
 4ε
=  02
 9b
2
2e 
3
 UA .
m
We can show anode current versus anode voltage IA(UA) dependence and approximately
determine factor in brackets, where b is the distance from anode to the cathode. JA is current
density defined as IA/S where we measure anode current IA at given surface area of the
filament S.
3
Calculating the logarithm of Child's law gives us ln I A = ln k '+ ln U A , where k' is factor in
2
brackets multiplied with the surface of filament S. Anode current was measured at different
11
node potentials ranging from 10 V to about 140 V. Experiment was repeated at four
different filament voltages U0. We see that measurements in plot (figure 3.3) are confirming
2
I 
2
3
Child's relation J A =  A  ∝ U A .
 S 
Figure 3.3: Logarithmic plot of anode current IA versus anode voltage UA measurements at four
different filament voltages U0. Black line is function f(x) = k' x3/2, where k' = 3,4·10-6 A/V3/2.
 4ε
2e 
According to Child's law k' must be equal to k ' =  02
 S which equals (using natural
m
9
b


constants and given characteristics of light bulb) to 3,4·10-6 A/V3/2.
Calculating e/m from k' we get the value e/m = 1,2·1011 As/kg while real value of e/m is
1,8·1011 As/kg. Apparently error is quite large here, which can be explained with limited
accuracy of given characteristics of tungsten filament. Critical factor S2/b4 gives the main
error to our result. If one have exact parameters given by manufacturer, it is possible to
measure e/m with errors smaller than 5 % or 6 %. [2]
From figures 3.3 we can also see that there is no direct relationship between heating voltage
U0 (i.e. temperature of a filament) and anode current IA observed at this point. Apparently
the effect can be only seen at higher accelerating voltages and higher temperatures of
filament (see chapter 3.1).
Simple calculation shows the number of electrons per second from cathode to anode:
I=
de
⇒
dt
10 −3 A
N IA
=
≈
≈ 6 ⋅1015 /s.
−19
dt e0 1,6 ⋅ 10 As
12
4 Conclusion
Thermionic emission is widely used, for example in cathode tubes, electron microscopes,
also in triodes for precise regulation of electron flow. Basically everywhere we need narrow
stream of electrons.
The basics of thermionic emission has been described in this seminar. Richardson's and
Child's law have been derived from basic knowledge of electrons in metal.
We showed pedagogical demonstration of verification of both laws and showed how is it
possible to obtain work function of tungsten filament and e/m ratio. Demonstrative
experiment can be done easily using two filament car light bulb, voltage sources and
microampere meters. For higher precision of results it is good to use bulb diode with exact
characteristics from manufacturer.
Experiment can be expanded analyzing how magnetic disturbance affects anode current. It
can be shown that even Earth's magnetic field noticeably affects anode current. Second
open problem is more precise temperature determination of the filament.
5 References
[1] C. Wall, R. Levine, Physics laboratory manual, Prentice-hall, Inc. (1962).
[2] S. Brody, S. Singer, Experiment on thermionic emission of electrons, Amer. J. Phys. 38
(1970).
[3] H. A. Jones, I. Langmuir, The characteristics of tungsten filaments as functions of
temperature, GE Rev., vol. 30, pp. 354-361 (1927).
[4] J. Dodd, An experiment on electron emission, Amer. J. Phys. 39 (1971).
[5] R. Walker, Thermionic emission using car headlamp bulbs, Physics Education (1976).
[6] D. R. Lide, CRC Handbook of Chemistry and Physics, CRC Press/Taylor
and Francis, Boca Raton, FL (2008).
[7] A. Azooz, An experiment on thermionic emission, Eur. J. Phys. 28 (2007).
13