5: Thermionic Emission

1 5: Thermionic EmissionPurposeWhile we think of quantum mechanics being best demonstratedin processes that showdiscontinuous change, historically quantum mechanics wasfirst revealed in systems wherea large number of particles washed out the jumps: blackbody radiation and thermionicemission. In this lab you will investigate these two phenomena in addition to classicalspace-charge limited electron Emission : Child s , as demonstrated by their ability to conduct an electric current, contain mobileelectrons. (Most electrons in metals, particularly the core electrons closest to the nucleus,aretightly bound to individual atoms; it is only the outermost valence electrons that aresomewhat free .) These free electrons are generally confined to the bulk of the metal. Asyou learned in E&M, an electron attempting to leave a conductor experiences a strong forceattracting it back towards the conductor due to an image charge:Fx= e24 0(2x)2( )wherexis the distance the electron is from the interface andeis the absolute value of thecharge on an electron.

2 Of course, inside the metal the electric field is zero so an electronthere experiences zero (average) force. You can think of these valence electrons as bouncingaround inside a box whose walls are provided by the image-charge force. (Odd to think:the walls are non-material force fields; the inside of the box is filled with solid metal.)Since temperature is a measure of random kinetic energy, if we increase the temperature ofthe metal, the electrons will be moving faster and some will have enough energy to overcomethe image-charge force (which after all becomes arbitrarily small at large distances from theinterface) and escape. This is electron evaporation . Thehigher the temperature thelarger the current of escaping electrons. This temperatureinduced electron flow is calledthermionic Emission . Starting in 1901, Owen Richardson studied this phenomenon and in1929 he received the Nobel prize in Physics for his hot wire will be surrounded by evaporated electrons. An electric force can pull theseelectrons away from the wire the larger the electric force,the larger the resulting currentof electrons.

3 The precise relationship between the voltageand the resulting current flow101102 Thermionic Emission &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& metalvacuumanodecathodexx = bx = 0acceleratingelectronselectric currentdensity JAVAV = 0 Figure : A planar cathode and a planar anode are separatedby a distanceb. A positivepotential differenceVAattracts electrons from the cathode to the anode, so the speed of theelectronsv(x) increases as they approach the anode. The moving electronsconstitute anelectric current from anode to cathode. The resulting steady current density is called Child s law1(or the Child-Langmuir law, including Langmuir who independentlydiscovered it while working at ). In this experiment youwill measure both Child s Lawand the Richardson s LawConsider a planar interface between a metal (x <0) and vacuum (x >0). Vacuum isin quotes because this region will contain escaped electrons a space charge rather thanbeing totally empty2.

4 The number of electrons per volume ( , the number density) isdenoted this experiment, the metal will be heated ( , its a hotcathode or filament) which willresult in a supply of electrons evaporated from the metal into the vacuum. An additionalconducting sheet (the anode) is located atx=b. A positive potential difference,VA,between the cathode and the anode plane provides a force pulling these electrons from thevicinity of the cathode towards the anode. The result is a stream of moving electrons (acurrent); the number densityn(x) and speedv(x) of these electrons will depend on location,x, between the plates. The negatively charged electrons moving to the right constitute asteady electric current density to the left, , a steady conventional electric current fromthe anode to the cathode:J= en(x)v(x) = JA( )Since the electrons leave the metal with (nearly) zero speedat zero potential, we cancalculate their speed along the path to the anode using conservation of energy:12mv2 eV(x) = 0( )v=r2emV(x)( )1 Clement Dexter Child (1868 1933) Born: Madison, Ohio, , Cornell2In fact a perfect vacuum is not possible, so the word vacuum actually refers simply to a region withrelatively few particles per volumeThermionic Emission103whereV(x) is the potential difference ( voltage ) atxandmis the mass of an the accelerating electrons constitute a steady current ( ,JAdoesn t dependon position),n(x) must decrease as the electrons speed toward the anode.

5 The varyingspace charge density affects the electric potential in the vacuum according to Poisson sequation3: 2V x2= (x) 0=en(x) 0( )Putting these pieces together with have the differential equation:d2 Vdx2=JA 0v(x)=JA 0q2emV(x)( )Since the electric field will be zero at the interface, we havea pair of initial conditions: V x x=0= 0( )V|x=0= 0( )This differential equation looks a bit like Newton s second law:d2xdt2=1mF(x(t))( )as you can see if in Newton s second law you substitute:t xx(t) V(x)1mF(x(t)) JA 0q2emV(x)Recall that force problems are often most simply solved using conservation of energy andthat conservation of energy was proved using an integratingfactor ofdx/dt. If we try theanalogous trick on our voltage problem, we ll multiply Poisson s equation bydV/dx:dVdx d2 Vdx2=JA 0q2emV 12 dVdx( ) 12 dVdx 2! =JA 0q2em V1212! ( )12 dVdx 2=JA 0q2emV1212+ constant( )The initial conditions require the constant to be zero, so12 dVdx 2=JA 0q2emV1212( )3 Poisson s equation is derived in the Appendix to this EmissionordVdx=vuut4JA 0q2emV14( )This differential equation is separable:dVV14=vuut4JA 0q2emdx( )V3434=vuut4JA 0q2emx( )where again the initial conditions require the constant of integration to be zero.

6 Finally:V(x) = 9JA4 0q2em 23x43( )Of course,V(b) is the anode voltageVA, so we can rearrange this equation to show Child slaw:JA="4 09b2r2em#V32A( )Much of Child s law is just the result of dimensional analysis, , seeking any possibledimensionally correct formula forJA. Our differential equation just involves the followingconstants with dimensions (units) as shown:b:L( )VA:EQ=ML2/T2Q( ) 0r2em k:Q2 ELQ12M12=Q52M32L3/T2( )JA:Q/TL2( )where the dimensions are:L=length,T=time,M=mass,E=energy, andQ=charge. Tomake a dimensionally correct formula forJA, we just eliminate theMdimension which wecan only do with the combination:VAk23:Q23T23( )We can then get the right units forJAwith: VAk23 32b2=kb2V32A:Q/TL2( )Thus the only possible dimensionally correct formula isJA kb2V32A( ) Thermionic Emission105lbanodecathode:hot filament, radius aanodecathode142 Figure : Coaxial cylinders: an inner wire (radiusa) and outer cylindrical anode (radiusb), form a vacuum tube diode. The cathode is heated so electronevaporation is possible,and a potential differenceVAattracts electrons from the cathode to the anode.

7 The speedof the electronsv(r) increases as they approach the anode. The moving electronsconstitutea steady electric current from anode to cathode. Since the same current is spread out overlarger areas, the current density,J, between the cylinders must be proportional to 1 exact proportionality constant, found from the differential equation, is (as usual) is nothugely different from have derived Child s law for the case of infinite parallel plates, but you will be testingit in (finite length) coaxial cylinders. The inner wire (radiusa) is the cathode; the outercylinder (radiusb) is the anode. Your cylinder with have some length , but we will belowconsider infinite length coaxial cylinders. Note that dimensional considerations require thatthe anode current per length should be given by a formula like:I/ j kbV32A( )although we could have an arbitrary function of the radius ratio:b/aon the Poisson s equation4we have: 2V=J 0v(r)=I2 r 0v(r)=j2 r 0q2emV 12( )Using the Laplacian in cylindrical coordinates we find: 2V r2+1r V r=j2 r 0q2emV 12( )There is no known formula for the solution to this differential equation, but we can makeconsiderable progress by writing the differential equationin terms of dimensionless quanti-4 Poisson s equation is derived in the Appendix to this Emissionties:r/a= ( )V= ja2 0q2em 23f( )( )yielding: 2f 2+1 f =f ( ) +1 f ( ) =1 f 12( )with initial conditions:f(1) = 0( )f (1) = 0( )We can numerically solve this differential equation usingMathematica:NDSolve[{f [p]+f [p]/p==1/(p Sqrt[f[p]])}, f[1]==0, f [1]==0, {f},{p,1,200}]It s actually not quite that simple.

8 The cathode, at = 1, is actually a singular point ofthe differential equation ( ,f (1) = ). However the situation very near the cathode iswell approximated by the planar case, where we ve shown:V(x) = 9JA4 0q2em 23x43= 9I2 a 4 0q2em 23(r a)43= 9ja2 4 0q2em 23 r aa 43= 94 23 ja2 0q2em 23 r aa 43( )So, near the cathode ( , slightly larger than 1):f( ) 94 23( 1)43( )We can use this approximation to start our numerical differential equation solution at anon-singular point (like = ).Real devices are designed withb/a= anode 1. The behavior offfor large can bedetermined by findingAand for whichf=A is a solution to the differential finds:f= 94 23( )A useful approximation for the range: 100< b/a <1000 is:f= 94 23+ 2( )(For example, the device used in lab hasb/a= For this value, the differentialequation givesf= ; the above approximation gives:f= ) Thermionic Emission10750100150200102030405060 f2468101214246810 f( ) exactf( ) for f( ) for 1 Figure : The plot on the left displays the dimensionless voltagefobtained by numericalsolution to the differential equation.

9 The plot on the right compares various approximationsforfto this numerical recover Child s law by rearranging ( ):2 0q2ema VAf(b/a) 32=j=I/ ( )Note:Langmuir s original work (Phys. , 347 (1923)) on this subject is expressed interms of where: 2( ) 49f32 = 1( 1)2 1( )So:8 0 q2em9b 2V32A=I( ) 2= for the device used in s LawMost any thermal process is governed by the Boltzmann factor:exp EkT =e E/kT( )wherekis the Boltzmann constant. Approximately speaking the Boltzmann factor ex-presses the relative probability for an event requiring energy Ein a system at (absolute)temperatureT. Clearly if E kT, the probability of the event happening is low. If anelectron requires an energyW(called the work function) to escape from the metal, TheBoltzmann factor suggests that this would happen with relative probabilitye W/kT. Thusyou should expect that the current emitted by a heated metal would follow:I e W/kT( )108 Thermionic EmissionClearly you should expect different elements to have different work functions, just as differ-ent atoms have different ionization potentials.

10 What is surprising is that the proportionalityfactor in the above equation includes a universal constant that is, a constant that just de-pends on the properties of electrons (and, most importantly, Planck s constant,h) and doesnotdepend on the type of material. (This situation is similar tothat of blackbody radia-tion, in which photons rather than electrons are leaving a heated body, which was Planck stopic in discovering his constant. We will take up this topicon page 113.) Thermionicemission probes the quantum state of the electrons statistically, whereas the photoelectriceffect probes much the same physics electron by electron. (The next level problem is toexplain why this universal constant (the Richardson constant,A) in fact does depend a biton the material.) To show:J=AT2e W/kT( )whereA=4 emk2h3= 106A/m2K2( )Quantum Theory: Free Electron GasInstead of thinking about electron particles bouncing around inside a box, de Broglie invitesus to consider standing waves of electron probability amplitude: =Nexp(ikxx) exp(ikyy) exp(ikzz) =Neik r( )Recall5that vector~kis the momentum,p=mv, of the electron and~=h/2.