University of Ljubljana Faculty of mathematics and physics Department of physics. Seminar. Marjan Grilj. Thermionic emission

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1 University of Ljubljana Faculty of mathematics and physics Department of physics Seminar Marjan Grilj Thermionic emission bstract 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 eperiment is presented that shows how to qualitatively check both laws and how to estimate work function of tungsten. Mentor: prof. dr. Gorazd Planinšič Ljubljana, pril 008

2 Table of contents 1 Introduction... Thermionic emission....1 Richardson's law.... Child's law Related phenomena... 8 Demonstration eperiment Richardson's law Child's law Conclusion References Introduction Thermionic emission must not be confused with thermic emission (of black body for eample) 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 eperiment using phenomenon of thermionic emission. We can take advantage of both mentioned phenomena using ordinary double filament headlamp bulb with one filament burn-out.

3 Thermionic emission.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 [/m ] 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 direction. We consider empty space in front of metal plate. Then the current density of this electrons is q n( E) v ( E) de, (1) where n(e) is the density of electrons in units of [ -1 m - ]. v (E) is the speed of electrons in direction. The integral is taken over all electron energies needed for escaping over potential barrier in direction. From statistical physics we know that density of particles can be written as n ( E) g( E) f ( E), () 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 π g( E) m / E, () h f 1 ( E), (4) E EF 1+ ep kt where h and k are Planck's and Boltzmann's constants. E F is Fermi energy, that is the highest energy of electrons in constant confining potential U (figure.1). Only electrons with the highest energies E >> E F can escape from metal, so f(e) approaches Boltzmann's distribution: E EF f ( E) ep. (5) kt

4 Figure.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 E F. Electrons with the highest energies lack eactly W (work function) to leave metal. We can now write part of the equation (1) as 8 π E EF n( E)dE m / E ep de. (6) h kt Let us write the energy as a function of the electron velocity E 1 1 mv EdE m / v dv (7) and insert it into (6): 8 π E EF 1 n( E)dE m / ep m / v dv h kt We can now insert epression (8) into equation (1) to get 8 E EF π m ep v dv. (8) h kt 8π E EF q v( E) m ep v dv h kt 8π EF E q v( E) m ep ep v dv h kt kt E E q F m ep v( E)ep 4π v dv. (9) h kt kt The integral over v starts from the minimum velocity needed to overcome the confining potential barrier U (illustrated on figure.1), because only electrons with velocity higher than v,min will leave metal: 1 U U mv, min v,min. (10) m 4

5 We epress 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 qm E F mv y mv z ep v ep dv ep dv y ep dv z h kt kt kt kt v,min. (11) Solving these integrals using known integral ep( C )d π C (1) we get qm EF U kt πkt πkt ep ep, h kt kt m m m π qmk h 4 T EF U ep kt (1) or shorter W RT ep, (14) kt where R is Richardson's constant with a value R 1, 10 6 /m K. Work function W U E F is the minimum amount of energy needed for an electron to leave the metal (figure.1) and depends on a metal (table.1). Owen W. Richardson 1 published the results of his eperiments in The modern form of this law was demonstrated by Saul Dushman in 19. The equation (14) can be used for eperimental 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.). 1 Owen W. Richardson ( ) was physicist from UK who received Nobel prize in physics 198 for his work on the thermionic phenomenon and especially for the discovery of the (now called) Richardson's law. Saul Dushman ( ) was Russian-merican 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

6 . 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 b so that they are at a potential difference V (figure.). This potential attracts electrons from the cathode to the anode, so the electrons accelerate towards anode where they can be detected. Electric current I flows from anode to the cathode and the same holds for the current density [/m ]. Figure.: Simplified model of cathode-anode apparatus. Let us suppose we have the setup shown on figure. in a vacuum. cloud of thermically escaped electrons (as shown in chapter.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() [m - ] vary with. Positive potential on b pulls these electrons towards the anode. Electric current density of electrons is en( ) v( ). (15) Kinetic energy of electron equals to 1 mv ev ( ). (16) Knowing the V() we can calculate velocity of electrons as the function of distance : e v v( ) V ( ). (17) m ccelerating electrons constitute a steady current, so we can see that n() is decreasing and v() is increasing towards anode. 6

7 We would like to know how space charge density affects electric potential between cathode and anode. For that we can use Poisson's equation V ρ( ) ε 0 (18) that tells us how the potential V() changes in presence of charge density ρ(). In our case V en( ) ε 0 ε v( ) 0 ε 0. (19) e V ( ) m Equation have to obey initial condition: V 0 0, V 0. (0) 0 Solving differential equation we get the potential for every point in region 0 < < b: 9 V ( ) e 4ε 0 m 4. (1) We know that at point b, the potential is V(b) V. So from equation (1) current density can be epressed as 4ε 0 e V 9b m. () Equation () describes the current density flow at any given voltage V. The result is known as Child's law or Child-Langmuir law. This relation can be used for eperimental 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 initial velocity. They have thermal energy kt per electron. That potential difference should be added to the plate voltage, and is about 0, V at T 000 K. fter Clement D. Child ( ) and Irving Langmuir ( ). 7

8 . Related phenomena There are many other phenomena happening on small scale accompanying thermionic emission when real eperiment is done. Let's only mention some of them. The work function in fact slightly depends on temperature due to thermal epansion of the atom lattice. Richardson's law doesn't take this into consideration. Vacuum in the tube where eperiment is placed is of course not a real vacuum. Bulbs that are used in eperiment (chapter ) 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 ecited 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. t higher voltages, strong electric field reduces the barrier of electron emission so the work function decreases. Due to so called Schottky 4 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. s UV light have an energy E hν lower than 100 ev it is able to etract 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 ( ) was a German physicist. 8

9 Table.1: Work functions of some common elements epressed in units of ev [6]: g 4,6 l 4,8 s,75 u 5,10 B 4,45 Ba,70 Be 4,98 Bi 4, C 5,00 Ca,87 Cd 4, Ce,90 Co 5,00 Cr 4,50 Cs,14 Cu 4,65 Eu,50 Fe 4,50 Ga 4,0 Gd,10 Hf,90 Hg 4,49 In 4,1 Ir 5,7 K,0 La,50 Li,90 Lu,0 Mg,66 Mn 4,10 Mo 4,60 Na,75 Nb 4,0 Nd,0 Ni 5,15 Os 4,8 Pb 4,5 Pt 5,65 Rb,16 Re 4,96 Rh 4,98 Ru 4,71 Sb 4,55 Sc,50 Se 5,90 Si 4,85 Sm,70 Sn 4,4 Sr,59 Ta 4,5 Tb,00 Te 4,95 Th,40 Ti 4, Tl,84 U,6 V 4,0 W 4,55 Y,10 Zn 4, Demonstration eperiment With demonstrative eperiment 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 eperiment we can use double filament made of tungsten car headlamp bulb with one filament burn-out (figure.1). That can be achieved simply with too high current through a filament. The broken filament support can be used as an etra electrode providing an anode for the directly-heated cathode. [,4,5] Figure.1: Left: Headlamp bulb with one filament burn-out on which anode current is measured. Right: Scheme circuit of an eperiment. In the eperiment we used light bulb with given characteristics: Narva 1 V, 40 W. The surface S and length of filament l are specified by manufacturer (S 1, 10-5 m, l 1,7 mm). Distance from anode to cathode is measured to approimately b mm. From the circuit of an eperiment (figure.1) it is seen that U 0 is used for heating tungsten filament, varied in interval [0 V, 8 V], which makes filament glow in visible light. We also measure current I 0 so we can determine resistance R of filament using Ohm's law. U is 9

10 anode (accelerating) voltage varied in interval [0 V, 00 V]. Measured anode current I is normally in m 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 eperiment) is from temperature dependence of resistance unique to tungsten. ccording to [] the best fit for tungsten is T , () where is the ratio of the hot resistance to that at 9 K. We have to note that () 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, (4) T where σ is Stefan's constant, S is the surface area of the body and ε T is emissivity of a body (about 0, 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 eact measuring of power P, so we used first method of determining temperature..1 Richardson's law Richardson's equation (14) can be written as W I T ep, where is Richardson kt constant multiplied with surface of filament S (we know that I /S). We measured anode current I and temperature of filament T via its resistance using equation (). The measurements were made in region U 0 [1 V, 8 V] at three different anode voltages U as shown on plot (figure.). The plot reveals the shape of Richardson function at the temperatures that are little under operational for given light bulb (1 V, 40 W). Besides qualitative validation of Richardon's relationship, it also allows us to approimately determine the work function of the filament sample by fitting measured data with Richardson function. 10

11 Figure.: Logarithmic plot of anode current I versus filament temperature measurements at three constant anode voltages U. Black line is Richardson function for work function W 4,7 ev. Measurements at higher anode voltage (U 00 V) agrees with tabular work function for tungsten (table.1) where we find W 4,55 ev, so the error is luckily small. s said before greater error represents determination of temperature. Results calculated from Stefan- Boltzmann's law gives up to 00 K higher results for the temperature of filament. We also see that at temperatures up to 000 K approimately 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 I starts to drop for electrons that are on low voltage acceleration (measurements begin to deviate from Richardson function). pparently U < 00 V wasn't enough for all electrons to accelerate to the anode. The higher potential difference, more electrons will be accelerated towards anode.. Child's law With the same demonstration we can test Child's law (): 4ε 0 e U 9b m. We can show anode current versus anode voltage I (U ) dependence and approimately determine factor in brackets, where b is the distance from anode to the cathode. is current density defined as I /S where we measure anode current I at given surface area of the filament S. Calculating the logarithm of Child's law gives us ln I ln k' + lnu, where k' is factor in brackets multiplied with the surface of filament S. node current was measured at different 11

12 node potentials ranging from 10 V to about 140 V. Eperiment was repeated at four different filament voltages U 0. We see that measurements in plot (figure.) are confirming Child's relation I U. S Figure.: Logarithmic plot of anode current I versus anode voltage U measurements at four different filament voltages U 0. Black line is function f() k' /, where k', /V /. 4ε e ccording to Child's law k' must be equal to k' 0 S which equals (using natural 9b m constants and given characteristics of light bulb) to, /V /. Calculating e/m from k' we get the value e/m 1, s/kg while real value of e/m is 1, s/kg. pparently error is quite large here, which can be eplained with limited accuracy of given characteristics of tungsten filament. Critical factor S /b 4 gives the main error to our result. If one have eact parameters given by manufacturer, it is possible to measure e/m with errors smaller than 5 % or 6 %. [] From figures. we can also see that there is no direct relationship between heating voltage U 0 (i.e. temperature of a filament) and anode current I observed at this point. pparently the effect can be only seen at higher accelerating voltages and higher temperatures of filament (see chapter.1). Simple calculation shows the number of electrons per second from cathode to anode: I de dt N dt I e ,6 10 s 15 /s. 1

13 4 Conclusion Thermionic emission is widely used, for eample 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 eperiment 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 eact characteristics from manufacturer. Eperiment can be epanded 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. (196). [] S. Brody, S. Singer, Eperiment on thermionic emission of electrons, mer.. Phys. 8 (1970). [] H.. ones, I. Langmuir, The characteristics of tungsten filaments as functions of temperature, GE Rev., vol. 0, pp (197). [4]. Dodd, n eperiment on electron emission, mer.. Phys. 9 (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 (008). [7]. zooz, n eperiment on thermionic emission, Eur.. Phys. 8 (007). 1

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