5: Thermionic Emission - Tools for Science

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.)

2 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. 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.)

3 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.

4 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 ).

5 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. 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.

6 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.

7 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.

8 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. 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.

9 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.

10 The cathode is heated so electronevaporation is possible,and a potential differenceVAattracts electrons from the cathode to the anode. 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.