An Experiment on the Velocity Distribution of Thermionic Electrons

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An experiment on the velocity distribution of thermionic electrons Onofrio Rosario Battaglia, Claudio Fazio, ͒ Ivan Guastella, and Rosa Maria Sperandeo-Mineoa Dipartimento di Fisica e Tecnologie Relative, University of Palermo Physics Education Research Group (UOP_PERG), 90128 Palermo, Italy ͑Received 8 March 2010; accepted 8 July 2010͒ This paper describes an undergraduate experiment that yields the velocity distribution of thermionic electrons by analyzing the I-V characteristics of diodes and triodes. The experiment allows students to focus on the distribution function more than on difﬁculties arising from the complexity of thermionic emission. By using a simple model, the velocity distribution of thermionic electrons emitted by the vacuum tube cathode can be described by Maxwell’s distribution. © 2010 American Association of Physics Teachers. ͓DOI: 10.1119/1.3471937͔

I. INTRODUCTION lindrical symmetry is given in the Appendix.͒ In our approximation the retarding potential Vr =V influences only one The Maxwell distribution is an important topic in thermal ͑ ͒ component of the electron velocity say vz , and an electron physics courses. Maxwell derived this distribution law in 2 / Ն 1 2 will reach the anode surface if vz −2eV m 0, where e and 1860, but Miller and Kusch performed the first rigorous m are the electron charge and mass, respectively. experimental demonstration almost 100 years later. Other ex- 3 It follows that the electron current I in the z direction periments have been performed that use mechanical velocity perpendicular to the vacuum tube surface S is analyzers to select particles in a certain velocity range from a ϱ large range of velocities. It was pointed out by Germer4 that ͑ ͒ ͑ ͒ the experiments of Richardson5 and Brown6 on the initial I = eS͵ n vz vzdvz, 2 ͱ / velocity distribution of thermionic electrons “…establish the 2eV m ͑ ͒ conclusion that thermionic electrons are emitted into a high where n vz indicates the number of electrons having a ve- vacuum with velocity components according to Maxwell’s locity in the interval vz and vz +dvz. distribution.” We write In vacuum tubes, such as diodes and triodes, the selection ␧ 2/ ͑ ͒ of the emitted electrons in a given velocity range can be = mvz 2, 3 performed by using retarding potentials and measuring the ␧ and d =mvzdvz. By this change of variables, we can express variations in the electron current. In this way, an experiment, n͑v ͒ as n͑⑀͒ and obtain aimed at showing the velocity distribution of a system of z ϱ particles and verifying its Maxwellian form, can be per- e I = S͵ n͑␧͒d␧, ͑4͒ formed by analyzing the current variations in a vacuum tube m at a temperature appropriate for thermionic emission. eV In this paper, we report on such an experiment whose in- where n͑⑀͒ is normalized so that terpretation and analysis allow students to focus on the dis- ϱ tribution function more than on difficulties arising from the m ͵ n͑␧͒d␧ = n = I , ͑5͒ complexity of thermionic emission. The necessary equip- 0 0 0 Se ment is available in most undergraduate laboratories. and n0 is the number of thermionic electrons reaching the II. MODEL anode per unit time and unit surface area when V=0 and the electron current is I0. Free electrons in metals at temperature T yield a current If the retarding potential V is further increased to V+⌬V,a 5 density J0 described by the Richardson equation, further drop in the anode current will take place given by ϱ ϱ J = AT2e−e␾/kT, ͑1͒ e 0 ⌬I = Sͫ͵ n͑␧͒d␧ − ͵ n͑␧͒d␧ͬ. ͑6͒ where ␾ is the height of the barrier above the Fermi level m eV e͑V+⌬V͒ and A=120.4 A cm−2 K−2. 7 8 The second integral on the right-hand side of Eq. ͑6͒ can be Many theoretical and experimental papers have studied ⑀ the thermionic current in vacuum tubes when a retarding expanded around eV= . We keep only the first two terms, ͑ and in the limit of ⌬V→0, we have potential Vr obtained, for example, by fixing the anode potential at zero and making the cathode potential positive͒ is dI n͑␧͒ ϰ − . ͑7͒ applied. An electric field prevents the electrons with energy ␧ ⑀Ͻ d eVr from reaching the anode. To model the electron dynamics, we consider a diode with According to Eq. ͑7͒, the velocity distribution can be ob- a simple geometry and a cathode-anode spacing small tained by differentiating the anode current with respect to the enough to minimize the effects of the space charge ͑see Fig. anode retarding voltage at each value of the voltage ͑energy͒. 1͒. For these conditions, plane parallel geometry may by Thus, measurements of the anode current as a function of the assumed as a good approximation. ͑The calculation for cy- retarding voltage, coupled with a suitable method of numeri-

1302 Am. J. Phys. 78 ͑12͒, December 2010 http://aapt.org/ajp © 2010 American Association of Physics Teachers 1302 ء / Fig. 3. Plot of the ratio I I against the anode retarding potential Va for are in the range of ءdifferent ﬁlament temperatures. The values of I ␮ 20–800 A for Tf varying in the range of 1170–1420 K.

we can neglect this potential barrier, but for high current Fig. 1. Sketch of the structure of a diode. densities, we have to take it into account because it affects the shape of the I-V characteristic curve. If we assume that the emitted electrons leave the cathode cal derivation, can lead to the evaluation of the electron ve- at temperature T with velocities distributed according to the ͑ ͒ locity distribution. Maxwell’s distribution, the function n vz will be the positive Many factors have to be taken into account to explain the velocity part of the Gaussian distribution for each component 9,10 diode I-V characteristics of thermionic emission. In par- of the electron velocity, ticular, Dodd9 pointed out that the actual cathode-anode po- ͑ / ͒ 2/ ␧/ ͑ ͒ − 1 2 mvz kT − kT ͑ ͒ tential difference equals the sum of the external applied re- n vz = Ce = Ce , 10 tarding potential V and the cathode-anode contact potential, a where C is a constant, and we have made the change of ⌬␾=͑␾ −␾ ͒. Thus, electrons reaching the anode must have c a variable in Eq. ͑3͒. Hence, for the simple case of planar energy geometry, Eq. ͑4͒ can be written as ␧Ն ͑ ⌬␾͒ ͑ ͒ eVr = e Va + . 8 ϱ e ␧/ / I = SC͵ e− kTd␧ = I e−eVr kT Because the term ⌬␾ can be considered to be constant at a 0 m eV given temperature T, it will not influence the differentiation r / ء /␾͒⌬ ͑ −e Va+ kT −eVa kT ͑ ͒ of Eq. ͑7͒, and therefore we obtain = I0e = I e , 11 ء ͒ ͑ ␧ d = edVa. 9 where I0 is the value of the current at Vr =0 and I is a constant given by The other factor discussed in Ref. 9 is related to the effects ͒ ͑ ␾/kT⌬− ء of the space charge between the cathode and the anode, I = I0e . 12 which generates a potential barrier ␦ that affects the actual If we take into account the cylindrical symmetry and indicate cathode-anode potential difference. For low current densities, by rc and ra the cathode and anode radius, respectively, the current that reaches the surrounding anode at the retarding potential Vr is given by ͓ −␩ ␩/͑ 2 ͒ 1/2 ␩/͑ / 2͒ 1/2͔ I = I0 Re erf͑ R −1 ͒ +1−erf͑ 1−1R ͒ , ͑13͒ / ␩ where R=ra rc, is the reduced retarding potential defined as ␩ / ͑ ͒ = eVr kT, 14 and erf͑x͒ is the error function, x 2 2 erf͑x͒ = ͵ e−x dx. ͑15͒ ␲1/2 0 The derivation of Eq. ͑13͒ was given by Schottky and Fig. 2. The diode circuit diagram. V1 is a dc power supply giving 1 V, Rp is 7 a potentiometer giving ͑0–470͒ ⍀. V2 is a variable power supply giving Lindsay. A simplified derivation of Eq. ͑13͒ is given in the ͑ ͒ ⍀ 0–30 V, and Renv =3.2 at Tenv =300 K. Appendix.

1303 Am. J. Phys., Vol. 78, No. 12, December 2010 Battaglia et al. 1303 Fig. 4. Normalized distribution functions obtained from the numerical derivatives of data in Fig. 3 for four filament temperatures. The derivative values are ͑ 2 / ͒ ͑ ͒ ͑ ͒ ͑ ͒ fitted to the relation y=A exp −mvz 2kTe , where A and Te are the fitting parameters. a Tf =1170 K and Te =1010 K; b Tf =1220 K and Te =1100 K; c ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Tf =1260 K and Te =1300 K; and d Tf =1320 K and Te =2100 K. Plots a – c show a good agreement with the Maxwell distribution; plot d shows a case of poor agreement.

The two error functions in Eq. ͑13͒ differ from unity by The first experiment uses a typical commercial diode less than 1% ͑see Refs. 7 and 9͒ if ͑12AX3͒ consisting of a cylindrical tungsten cathode12 of / radius r =͑2.93Ϯ0.05͒ mm, enclosing an electrically heated ͓␩/͑R2 −1͔͒1 2 Ͼ 1.83. ͑16͒ c tungsten filament and a cylindrical anode of radius ra In this case, Eqs. ͑11͒ and ͑13͒ have the same shape. How- =͑4.39Ϯ0.05͒ mm. Thus, the anode-cathode distance is ever, Eqs. ͑11͒ and ͑13͒ are approximate because many ͑1.5Ϯ0.1͒ mm. Figure 2 shows the circuit diagram. A vari- 11 known factors, such as the Schottky effect, nonuniform ␾, able power supply is used to heat the filament and the two variation of ␾ with temperature, and the space charge pre- digital multimeters measure the filament current If and the vent the measured current density from obeying these equa- voltage Vf. The power supply provides the adjustable anode tions exactly. Because of the large number of these factors, retarding voltage Vs. A microammeter, Ia, with a resolution their role in explaining the differences between experiment of 0.1 ␮A, was used to measure the anode currents. Data at and theory is not easy. For these reasons, we look for tem- very low current intensity were controlled by using a mil- peratures where their influence can be neglected. limicroammeter.

III. THE EXPERIMENTS A. Calibration of cathode temperature We describe two experiments, starting from the simplest The cathode temperature was estimated by using the phe- from a conceptual and experimental point of view. The nomenological relation of the filament temperature Tf to the analysis of the results allows us to make its limitations evi- value of its resistance. This relation expresses Tf in terms of dent and to design a more complex experiment that verifies the ratio between the filament resistance and the resistance ͑ ⍀͒ the presence of a Maxwell distribution for a wider range of value at 300 K, Renv. For tungsten filaments Renv=3.2 , 13 temperatures. Tf can be written as

1304 Am. J. Phys., Vol. 78, No. 12, December 2010 Battaglia et al. 1304 R ͩ R ͪ2 Tf = 103.898 + 214.930 − 2.994 Renv Renv R 3 ͑ ͒ + 0.0433ͩ ͪ + ¯ . 17 Renv In our experiment the filament temperature is varied by changing the voltage Vf, and the filament resistance is obtained by measuring the corresponding current If. By taking into account the characteristics of the vacuum tube, the filament current and the voltage are varied in the ranges of 0.47–0.63 A and 8.00–13.50 V, respectively. The filament temperatures evaluated by using Eq. ͑17͒ are in good agreement with independent evaluations done by using the black body radiation of the tungsten filament and assuming that the power dissipated in the filament is mainly due to the radiation power according to I V Ϸ ␴S␥ T4, ͑18͒ Fig. 5. Plot of the experimental values of the anode current versus the anode f f w f retarding potential and theoretical curves given by Eqs. ͑11͒ and ͑13͒. The ␥ values of Tf are in the range of 1170–1420 K. where S is the filament surface, w is the tungsten emissivity, and ␴ is Stefan’s constant.

Tf =1420 K, can be ascribed to the use of our simplified planar model and can be eliminated by the more accurate B. Measurement of the anode current and evaluation of cylindrical model. the electron velocity distribution Figure 5 shows the plots of ln͑I͒ versus the anode retarding potential V and the fits given by Eqs. ͑11͒ and ͑13͒. The A set of experiments was performed by fixing the filament a experimental points for the lower three temperature values temperature and varying the retarding anode potential from are well fitted by the simple exponential in Eq. ͑11͒. In these zero to values that allow a current in the range of the sensi- three cases the condition ͑16͒ is satisfied at very low retard- tivity of our microammeter. Figure 3 shows a plot of the ratio ing anode potentials and Eqs. ͑11͒ and ͑13͒ show the same of the anode current I at a given anode potential Va to the shape. The data at the highest temperature values ͑T ء value I ͑V =0͒ against the anode retarding voltage V for f a a =1320 K, T =1360 K, and T =1420 K͒ are not in agree- different filament temperatures. f f ͑ ͒ ment with the theoretical curves. We conclude that electrons To investigate the shape of the n vz distribution function, exhibit a Maxwellian behavior only for values of Tf in the we numerically differentiated the data in Fig. 3 by applying range of 1170–1260 K, corresponding to T in the range of ͑ ͒ ͑ ͒ ͑ ͒ e Eqs. 2 , 4 , and 7 and fitting the results to a half Max- 960–1055 K. The deviation of the experimental values of the wellian. As shown in Eqs. ͑2͒, ͑4͒, and ͑7͒, the electron ve- ͑ ͒ ͑ ͒ anode current from Eqs. 11 and 13 for high values of Tf locity distribution function is directly proportional to the first suggests that these data are strongly affected by the space derivative of the Va-I relation. We performed the numerical charge ͑mutual repulsion of electrons in the space between differentiation of our experimental data by using the central the electrodes͒ for high charge densities due to low current difference method. values for high Va or high electron emission for high Tf. Figure 4 plots the derivative versus the electron velocity Moreover, high values of the retarding potential can modify ͱ / component vz = 2eVa m and the fits at four temperatures. the cathode work function barrier ͑the Schottky effect͒. Figures 4͑a͒–4͑c͒ show good agreement between the theory For these reasons, we designed a new set of experiments and the experimental data. The electron temperature Te is in using a triode with the grid polarized at a small positive good agreement ͑the differences are less than 15%͒ with the ͑ ͒ filament temperature Tf evaluated according to Eq. 17 . Fig- ure 4͑d͒ is an example of a bad fit referring to experiment at Tf =1320 K. The same behavior is displayed by experiments at Tf =1360 K and Tf =1420 K. For these experiments, Te is greater than Tf. As has been shown in Ref. 14 for indirectly ͑ ͒ heated cathodes, the value of Tf obtained from Eq. 17 can be considered as an upper limit of the possible electron temperature Te. To verify the validity of the assumption of the Maxwellian distribution for experiments at Tf =1170 K, Tf =1220 K, and Tf =1260 K, we fitted the experimental data normalized by the anodic current I to Eqs. ͑11͒ and ͑13͒, which hold only if the electron velocity distribution is Maxwellian. This fit allows us to analyze the experimental data by avoiding errors introduced by numerical differentiation. Moreover, we can Fig. 6. The triode circuit diagram. The two dc power supplies V1 and V2 are ͑ ͒ ⍀ test if the disagreement between our data and the theoretical at 1 V. Rp is a potentiometer giving 0–470 . V is a variable power ͑ ͒ ⍀ ͑ ͒ curves, for experiments at Tf =1320 K, Tf =1360 K, and supply giving 0–30 V. Renv =3.1 at Tenv =300 K .

1305 Am. J. Phys., Vol. 78, No. 12, December 2010 Battaglia et al. 1305 Fig. 7. Plot of the experimental values of the anode current versus the anode Fig. 9. Plot of the experimental values of the anode current versus the anode retarding potential and theoretical fitting curves described by Eq. ͑11͒.The retarding potential for the triode-diode assembly and theoretical curves values of T are in the range of 1120–1540 K. f given by Eq. ͑11͒. potential with respect to the cathode to reduce the space charge effect and to shield the cathode from the variation of Although no systematic investigations were made on the the anode potential. effect of different grid voltages on the I-V curves, we verified that grid voltages in the range of ͑1Ϯ0.5͒ V did not modify the anode current shapes, in agreement with Ref. 10. C. Experiments using a triode ͑ ͒ Figure 7 shows ln I versus Va and the fitted curves using A commercial triode ͑6J5͒ was assembled according to the Eq. ͑11͒. The electron velocity distribution can be considered circuit diagram in Fig. 6. The cathode-anode spacing is the Maxwellian for values of Te in the range of 950–1370 K, same order of magnitude as the diode and a polarizing grid is corresponding to Tf in the range of 1120–1480 K. positioned between them. Figure 8 shows the values of the derivatives found from ͱ / The grid was positively polarized by using a constant volt- the data of Fig. 7 versus the electron velocity vz = 2eVa m age Vg =1 V and a retarding potential Va was applied be- for the two experimental temperatures that exhibit good fit- ͑ ͒ tween the cathode and the anode. The grid directs the elec- ted results experiments at Tf =1120 K and Tf =1480 K . trons emitted from the cathode to the anode and shields the The data were fitted by the positive half of a Maxwellian cathode from the anode voltage. As a consequence, the cath- distribution. The fits to the data and the characteristics of the ode is not greatly affected by the variation in the anode volt- Maxwellian distribution at different temperatures ͑differ- age during the measurements and the effects of space charge ences in the width and peak height͒ are evident. are partially reduced. The first effect represents an advantage To show that the different shapes of the curves in Fig. 7 of a triode over a diode in reducing the Schottky effect, can be ascribed to the grid, the triode was mounted as a especially for oxide coated cathodes in which this effect is diode by connecting the anode and the grid ͑so that the cath- more relevant.8 ode is the emitter and anode + grid is the collector of emitted

Fig. 8. Calculated derivatives of the data in Fig. 7 for experiments at Tf =1120 K and Tf =1480 K. The derivatives are ﬁtted as in Fig. 4.

1306 Am. J. Phys., Vol. 78, No. 12, December 2010 Battaglia et al. 1306 electrons͒. Figure 9 shows plots of ln͑I͒ versus V for the ͑ 1 2 1 2͒ ͑ 1 Ј2 1 Ј2͒ ͑ ͒ a 2 mu + 2 mv − 2 mu + 2 mv = eVr. A2 same filament temperatures as the data shown in Fig. 7.By comparing Figs. 5 and 9, we can see their similarity. We From angular momentum conservation, it follows that found that the exponential fits are adequate only for values of Tf in the range of 1120–1205 K. We conclude that the posi- ͑ ͒ tive grid reduces the spatial charge and shields the cathode ravЈ = rcv, A3 and, thus, allows us to investigate the velocity distribution of the emitted electrons in a wider temperature range. ͑ ͒ where rc and ra see Fig. 1 denote the cathode and the anode radii, respectively, and uЈ and vЈ are the radial and the tangential velocity components, respectively, of an electron that IV. DISCUSSION has reached the anode surface. From Eqs. ͑A2͒ and ͑A3͒,we have We used commercial vacuum tubes and easily available measurement devices. Experiments using a diode show that 1 the Maxwellian distribution can be inferred only in a re- u2 = uЈ2 +2␥V − ͩ1− ͪv2, ͑A4͒ stricted range of temperatures ͑no more than 100 K͒. To in- r R2 crease the range would involve decreasing the temperature of the filament ͑and, consequently, the current density͒ and ␥ / / where =e m and R=ra rc denotes the cathode to the anode would require a more accurate microammeter than those usu- ratio. ally available in undergraduate laboratories. Experiments us- For an electron to reach the anode, it is sufficient that its ing a triode allow us to widen the range for which electrons planar velocity at the cathode is such that it can arrive at the exhibit a Maxwellian distribution. Consequently, students anode with vanishing radial velocity. This requirement leads can use a wide range of temperatures to compare standard to two conditions on the minimum value umin of the radial deviations, percentages of electrons at high or low velocities, velocity of the electron emitted by the cathode and all the characteristics that differentiate velocity distributions at different temperatures. In the range of temperatures we used and for the characteristics of our triode, a simple 1 ͱ ␥ ͩ ͪ 2 2 Ͻ ␥␭ 2 V − 1− v for v 2 Vr model of planar geometry is sufficient and more complicated u Ͼ Ά r R2 · ͑A5͒ calculations required by the cylindrical approximation are Ͼ 2 Ͼ ␥␭ not necessary. 0 for v 2 Vr, The experiments were performed in a physics laboratory for undergraduate engineering students. Some students only where ␭=R2 /͑R2 −1͒. analyzed their exponential fits of current versus anode poten- If we assume that electrons are distributed according to tial data. Other students used numerical differentiation pro- Maxwell–Boltzmann statistics so that cedures and some were interested in trying fits with a geo- metrical model closer to the actual vacuum tube geometry. −Km͑u2+v2͒ All students showed an understanding of the need for an f͑u,v͒ = Ce , ͑A6͒ accurate experimental design and models able to describe the experimental data. where K=1/2kT and k and T are the Boltzmann’s constant and the electron gas temperature, respectively, the current density can be written as APPENDIX: THEORETICAL DERIVATION OF ϱ ϱ SCHOTTKY EQUATION 2 2 J = C͵ dv͵ e−Km͑u +v ͒udu, ͑A7͒ ϱ The structure of a vacuum tube affects the anode current. − umin When a retarding potential Vr is applied between the cathode and the anode, the form of the current as a function of the where the minimum value of u is given by Eq. ͑A5͒. From 7 ϱ ϱ retarding potential was found by Schottky for a cylindrical considerations of symmetry, we have ͐ dv=2͐ dv so that geometry assuming that electrons in the metal obey the Max- −ϱ 0 well distribution and do not interact with each other. ϱ ϱ In cylindrical coordinates, the velocity vector of an elec- 2 2 ͵ ͵ −Km͑u +v ͒ tron emitted by the cathode can be split into a radial compo- J =2C dv e udu 0 u nent u, a tangential component v, and an axial one w.To min ϱ ϱ evaluate the electron current collected by the anode, the axial 2 2 −Kmv −Kmu ͑ ͒ component w is irrelevant because it is assumed to be paral- =2C͵ e dv͵ e udu A8a lel to the cylindrical surface of both the cathode and anode. 0 umin Hence, if we denote the velocity distribution in a plane perpendicular to both the electrodes as f͑u,v͒, the current den- ϱ C ͑ 2 2 ͒ sity collected by the anode surface is given by = ͵ e−Km v +umin dv. ͑A8b͒ Km 0 J = C͵ f͑u,v͒ududv. ͑A1͒ To account for the restrictions imposed by Eq. ͑A5͒, we need From energy conservation, we can write to split the integral over the variable v as

1307 Am. J. Phys., Vol. 78, No. 12, December 2010 Battaglia et al. 1307 ͱ ␥␭ 2 Vr when the distance of the anode surface from the cathode C −Km͑v2+2␥V −͑1−1/R2͒v2͒ J = ͫ͵ e r dv surface is negligible with respect to the cathode radius. Km 0 Moreover, for ͱ␩/͑R2 −1͒Ͼ1.83, the two error functions in ϱ 2 Eq. ͑A10͒ differ from unity by less than 1%, and Eq. ͑A10͒ + ͵ e−Kmv dvͬ ͑A9a͒ ͑ ͒ ͱ2␥␭␾ can be approximated by Eq. A12 .

ͱ2Ke␭V C r ͑ / 2͒ 2 a͒ = ͫe−2KeVr͵ e− 1 R x dx Electronic mail: [email protected] ͑ ͒3/2 1 Km 0 J. C. Maxwell, “Illustrations of the dynamical theory of gases,” Philos. Mag. 19, 19–32 ͑1860͒. ϱ 2 2 R. C. Miller and P. Kush, “Velocity distributions in potassium and thal- + ͵ e−x dxͬ, ͑A9b͒ lium beams,” Phys. Rev. 99, 1314–1321 ͑1955͒. ͱ2Ke␭V r 3 O. R. Frisch, “Molecular beams,” Sci. Am. 212͑5͒, 58–74 ͑1965͒. where Kmv2 =x2. 4 L. H. Germer, “The distribution of initial velocities among thermionic Equation ͑A9͒ can be written in the more convenient form electrons,” Phys. Rev. 25, 795–807 ͑1925͒. ͱ 5 O. W. Richardson, The Emission of Electricity From Hot Bodies, 2nd ed. ␲ 1 ͑Longmans Green, London, 1921͒, ͗www.archive.org/details/ J = C ͫRe−2KeVr erfͩ ͱ2Ke␭V ͪ +1 2͑Km͒3/2 R r emissionelectri00richgoog͘. 6 F. C. Brown, “The kinetic energy of the positive ions emitted by hot ͑ ͒ ͑ ͒ ͑ͱ ␭ ͒ͬ ͑ ͒ platinum,” Phil. Mag. Series 6, 17 99 , 355–361 1909 . − erf 2Ke Vr A10a 7 Theoretical calculations for the cylindrical diode are reported in Walter Schottky, “Uber den austritt von elektronern aus qluhdrahten bei verzog- erden potentialen,” Ann. Phys. 44, 1011–1032 ͑1914͒;P.A.Lindsay,in 1 R2 ͫ −␩ ͩͱ␩ ͪ ͩͱ␩ ͪͬ Advances in Electronics and Electron Physics, edited by L. Marton ͑Aca- =J0 Re erf +1−erf , R2 −1 R2 −1 demic, New York, 1960͒, Vol. XIII, pp. 250–277. 8 ͑A10b͒ See, for example, Ref. 4 and C. S. Hung, “Thermionic emission from oxide cathods: Retarding and accelerating ﬁelds,” J. Appl. Phys. 21, where J =Cͱ␲/2͑Km͒3/2 is the current density when no re- 37–44 ͑1950͒. 0 9 tarding potential is applied to the electrodes. For a wire- J. G. Dodd, “An experiment on electron emission,” Am. J. Phys. 39, ͑ ͒ shaped cathode ͑R→ϱ͒, Eq. ͑A10͒ reduces to 1159–1163 1971 . 10 K. L. Luke, “An experiment on thermionic emission using a nuvistor triode,” Am. J. Phys. 42, 847–856 ͑1974͒. −␩ 2 ͱ ͱ J = J ͫe ␩ +1−erf͑ ␩͒ͬ 11 ͑ 0 ͱ␲ S. M. Sze, Physics of Semiconductor Devices, 2nd ed. Wiley, New York, 1981͒, p. 250. ϱ 12 The cathode surface is coated by a layer of a mixture of barium and 2 −␩ͱ −x2 = J ͫe ␩ + ͵ e dxͬ. ͑A11͒ strontium oxides. This layer enhances the electron emission process from 0 ͱ ␲ ͱ␩ the surface due to its small work function ͑usually 1–1.9 eV͒ compared to ͑ → ͒ ͑ ͒ that of tungsten, which is about 5 eV. See K. R. Spangenberg, Vacuum For a planar geometry R 1 , Eq. A10 reduces to Tubes ͑McGraw-Hill, New York, 1948͒, pp. 29 and 44. −␩ ͑ ͒ 13 H. A. Jones and I. Langmuir, “The characteristics of tungsten ﬁlaments as J = J0e . A12 functions of temperature,” Gen. Electr. Rev. 30, 310–319 ͑1927͒. Even if Eq. ͑A12͒ rigorously holds only for a planar geom- 14 C. Wagner and H. H. Soonpaa, “A simple picoammeter for thermionic etry, it can be considered a good approximation to Eq. ͑A10͒ emission measurements,” Am. J. Phys. 62͑5͒, 473–474 ͑1994͒.

1308 Am. J. Phys., Vol. 78, No. 12, December 2010 Battaglia et al. 1308