Barium depletion study on impregnated cathodes and lifetime prediction

Applied Surface Science 215 (2003) 5–17

Barium depletion study on impregnated cathodes and lifetime prediction J.M. Roquaisa,*, F. Poreta, R. le Dozea, J.L. Ricauda, A. Monterrinb, A. Steinbrunnb,1 a

Laboratoire d’Optique Electronique, Thomson/S.B.U. Displays and Components, Avenue du Ge´ne´ral de Gaulle, Genlis 21110, France b Laboratoire de Recherches sur la Re´activite´ des Solides, U.M.R. 5613 C.N.R.S./Universite´ de Bourgogne 9, Avenue A. Savary, BP 47 870, 21078 Dijon Cedex, France

Abstract In the thermionic cathodes used in cathode ray-tubes (CRTs), barium is the key element for the electronic emission. In the case of the dispenser cathodes made of a porous tungsten pellet impregnated with Ba, Ca aluminates, the evaporation of Ba determines the cathode lifetime with respect to emission performance in the CRT. The Ba evaporation results in progressive depletion of the impregnating material inside the pellet. In the present work, the Ba depletion with time has been extensively characterized over a large range of cathode temperature. Calculations using the depletion data allowed modeling of the depletion as a function of key parameters. The link between measured depletion and emission in tubes has been established, from which an end-of-life criterion was deduced. Taking modeling into account, predicting accelerated life-tests were performed using highdensity maximum emission current (MIK). # 2003 Elsevier Science B.V. All rights reserved. Keywords: Impregnated cathode; Barium evaporation; Tungsten pellet; Porosity; Diffusion in porous media; Accelerated life-tests

1. Introduction In the dispenser cathodes made of a porous tungsten pellet impregnated with Ba, Ca aluminates, the evaporation of Ba is the major factor determining the cathode lifetime. Models linking the Ba evaporation and the emission, allowing cathode lifetime prediction, have been proposed [1–4]. The scope of this work was to measure on our cathodes the depletion of barium resulting from evaporation as a function of *

Corresponding author. Tel.: þ33-3-80-47-63-00; fax: þ33-3-80-47-64-26. E-mail addresses: [email protected] (J.M. Roquais), [email protected] (A. Steinbrunn). 1 Tel.: þ33-3-80-39-61-30; fax: þ33-3-80-39-61-32.

key parameters. Based on existing models and additional calculations, a detailed equation to account for observed phenomena is proposed, including the effect of tungsten pellet porosity. The emission performance versus time in CRTs is evaluated at different temperatures and observed laws of variations are discussed.

2. Comparative measurements of Ba depletion and evaporation rate Alternatively to evaporation measurements to evaluate Ba loss in impregnated cathode, Shroff and coworkers [1,2] proposed to measure the depleted depth of Ba after a given time of cathode operation. The method was adapted to our equipment, consisting in

0169-4332/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(03)00318-0

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performing, in a scanning electron microscope (SEM), an energy dispersive X-ray spectroscopy (EDX) mapping of Ba on diametrical fractures of pellets after operation of the cathodes in the CRT. In the present study, the cathode pellets are discs of typical diameter close to 1.3 mm and thickness close to 0.4 mm. They are obtained by pressing a tungsten powder of 4–5 mm average particle size. They are impregnated with 4-1-1 aluminates and are subsequently submitted to a cleaning step of the surface that removes the aluminates from the pellet over a thickness of about 5 mm. After the surface cleaning, ˚ the deposition of an Os–Ru top coating of 5000 A thickness on emitting face is performed to lower the work function. The pellet is inserted in a Ta cup welded to a cylindrical Ta sleeve, this assembly being brought to the desired temperature by a heater (Fig. 1a). The Ba depleted depth e from cathode emitting face can be obtained from the EDX mapping, as shown in Fig. 1b, by setting a cursor on the estimated location of the depletion front on the EDX mapping. This method was satisfactorily correlated with the usual method [1,2] in which the location of the Ba depletion front is set at 50% of signal intensity on profiles of Ba concentration. Depletion occurs also from rear face of our pellet as seen in Fig. 1, but it is clearly restricted by the presence of the Ta cup. The Ba depletion versus time was measured on numerous cathodes operated at different temperatures

ranging from 950 to 1100 8CB (Fig. 2). For this test on the influence of cathode temperature on depletion, the pellets had been selected so that their porosity P was in the range 0:185  0:005 (see Appendix A for definition of porosity). The brightness temperature was measured on the side of the cathode, on tantalum. From the ln/ln plot of Fig. 2, the equation describing the Ba depleted depth ‘‘e’’ from cathode emitting face, as a function of time is of the form: lnðeÞ ¼ 12 lnðtÞ þ lnðAÞ

(1)

or: e ¼ At1=2

(2)

where e is the depleted depth in mm, t the time of cathode operation in h, A is the depletion factor, dependent on temperature in mm h1/2. By means of a quadrupole mass spectrometer, Ba evaporation was quantified on cathodes from the same batches as the ones used for depletion study. For convenience, the relative ionic current was defined as being the ratio of the mass spectrometer ionic current corresponding to amu 69 (Ba2þ) at temperature T to the ionic current at a reference temperature T. The Arrhenius plot representing the relative ionic current in natural logarithm against reciprocal temperature is shown in Fig. 3. In the same manner, the square of the depletion factor, A2, is also plotted. The two ‘‘Least Mean Squares’’ linear regressions passing through the two series of points have the same slope.

Fig. 1. (a) Schematic cross-section of our M-type cathode illustrating the impregnant depletion after operation; (b) EDX mapping of Ba Ka line on fracture of pellet, showing 85 mm of depletion from electron emitting face of cathode after operation.

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Fig. 2. Depletion of Ba as a function of time in life-test for different cathode operating temperatures (8CB).

Fig. 3. Natural logarithm of relative intensity ln(iT/i1373) of amu 69 and ln(A2T ) vs. reciprocal brightness temperature. Here, 1373 KB (i.e. 1100 8CB) has been chosen for T .

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The ionic current corresponding to amu 69 being proportional to the Ba vapor pressure can be expressed as follows:

 Eevap 1 iT 1  ¼ exp  (3) T T iT  k where iT =iT  is the relative ionic current, iT and iT  being ionic currents measured by mass spectrometer at amu 69 (Ba2þ), respectively, at temperature T and at reference temperature T, T the temperature in K, Eevap the energy of activation of Ba evaporation from impregnated cathode in eV, k is the Boltzmann constant. The fact that A2 shows, in the Arrhenius plot, the same slope as iT =iT  , means that the limiting mechanism for the Ba evaporation and the Ba depletion is the same. Hence, the depletion measurements allow to determine Eevap, and A2 can be expressed as follows:
 Eevap A2 / exp (4) kT

3. General equation of depletion Writing the equations of transport of vapor by molecular flow as detailed in Appendix A enables to establish the relation between depletion factor A and the Ba vapor pressure:
1=2 2 CM elt PBa A¼ (5) N0 P Calculated expression (5) allows to predict a variation of A2 with reciprocal temperature as observed experimentally in Fig. 3. The activation energy for Ba evaporation, deduced from the slope of the two curves presented in the Arrhenius plot (Fig. 3) is Eevap ðeVÞ ¼ 3:23  0:15 This value is in excellent agreement with previous work [5–7,12]. Introducing the expression of A in Eq. (2), the following general equation for ‘‘e’’ as a function of T and t is obtained:
1=2 2 1=2 eðT; tÞ ¼ CM elt PBa=ref N0 P

 Eevap 1 1 

exp  (6) t1=2 2k T T 

PBa/ref is the reference pressure of Ba at temperature T . The above equation is similar to the one already proposed by Shroff [1]. Term A is proportional to the square root of the elementary molecular conductance CM elt which is a term depending on the geometrical characteristics of the W pellet. It was demonstrated that the pellet porosity P is one of these influencing geometrical characteristics [6,8,9]. To account for the latter, the elementary conductance will be referred more adequately as CM elt/P from this point and A as AP. Experimental data of Ba evaporation rate at 1050 8CB from a pellet impregnated with 4-1-1 aluminates as a function of pellet porosity are available from Dudley [8]. Other data on a ‘‘L’’ cathode are given by Rutledge and Rittner who measured the transmission factor for the gases of a W plug sealed to a calibrated volume filled with helium or nitrogen [9]. To study the effect of porosity on Ba depletion rate on our cathodes, three groups of pellets of very well-defined porosity were prepared, using the same tungsten powder of 4–5 mm average particle size for the three groups. The variation of porosity from one group to another was obtained by varying the quantity of pressed tungsten for a constant volume of pellet (constant diameter and thickness). The three groups were composed of pellets with measured porosity ranging in the three following intervals: 0:16  0:005, 0:187  0:005 and 0:223  0:009. It was found after impregnation that the weight of Ba, Ca aluminates introduced in the pellets was proportional to porosity. The results of the depletion at a cathode operating temperature of 1000 8CB as a function of the square root of time for the three tested porosities are given in Fig. 4. A trend of increasing depletion with porosity is clearly seen on our experimental data. One notices that this observed trend can be ascertained only when depletion exceeds 50 mm. From the time at which the depletion starts to be clearly proportional to t1/2, the evaporation rate of Ba in an impregnated pellet decreases with time in the following way (see Appendix A for details of calculation): dm A2 AP / P P / P 1=2 dt e t

(7)

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Fig. 4. Ba depletion as a function of the square root of time for three different porosities of tungsten pellet at a cathode operating temperature of 1000 8CB (experimental points and fitting calculated curves).

Remark: The law of depletion found in our experiments is a consequence of an evaporation rate decreasing as the reciprocal of t1/2. This dependence of evaporation rate of Ba from porous pellets versus time has been reproducibly observed by many authors [1,3,7,12]. It is sometimes clearly seen only after an early life period, for example a few hundred hours for Brodie et al. [7] or Mita [3]. The ratio of evaporation rates at time t for two different porosities P1 and P2 is derived from above equation:

Comparison of conductance of pellets as a function of porosity for Thomson, Dudley and Rutledge can be done by plotting on the same graph the transmission coefficient a as defined by Rutledge (a being proportional to the conductance, see Appendix A). The transmission coefficient, in the case of an impregnated pellet, decreases with increasing depleted depth e and can be written as a function of mean free path l and pellet geometrical factor GP (see Appendix A), as follows:

ðdm=dtÞP1 P1 AP1 ¼ ðdm=dtÞP2 P2 AP2

a ¼ aðeÞ ¼ (8)

To compare evaporation rates of our pellets with those of Dudley, relative evaporation or depletion rates were calculated. The relative rates are defined as being the ratio of rate at given porosity to rate at porosity 0.187. The relative rates of evaporation were calculated using term PAP for our cathodes (deduced from Fig. 4) to compare with curve of (dm/dt)P of Dudley showing average evaporation rates after 100 h of operation. The comparison displayed in Fig. 5 shows good consistency of relative evaporation rates between Dudley and us.

4 l P 3 GP e

(9)

As ðA2P =eÞ / ðl=GP Þð1=eÞ (see Appendix A), it is appropriate to compare between authors the following relative factors: P A2P aP ¼ 2 Pref APref aPref

(10)

The relative transmission coefficient aP =aPref of Eq. (10) is the ratio of the transmission coefficient at given porosity to coefficient at porosity PrefPref can be chosen arbitrarily. It has a value of 0.187 in the

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Fig. 5. Relative Ba evaporation rates for Thomson and Dudley’s cathodes as a function of pellet porosity.

proposed comparison graph of relative transmission coefficient of Fig. 6. From this plot, it appears that the curves of Dudley and Rutledge are very similar. It has to be noticed that the transmission coefficient a reaches a negligible value for a porosity Plim that seems to be between 0.05 and 0.12. One possible explanation is that interconnection of pores is very low when porosity falls below the value of Plim. One has also to keep in mind that residues of aluminates or reaction products of aluminates with W left in the pores decrease the porosity ‘‘seen’’ by Ba atoms, in other words the porosity between the depletion front and the pellet surface. The experimental points of relative values of a versus porosity are well fitted by the following function in the porosity range 0.12– 0.25:   P A2P aP P  Plim n ¼ ¼ (11) Pref A2Pref aPref Pref  Plim

Experimental points are well fitted using, respectively, for Dudley’s curve and Rutledge’s curve, 0.08 and 0.11 for the value of Plim. With these values for Plim, the power factor n is in a range of 1.5–2 for both authors. More generally, it seems to have a value between 1.5 and 3 if Thomson experimental data are also taken into account. Introducing the dependence of transmission coefficient versus porosity, one can give for our pellets the following general expression for the Ba depletion e, valid in the tested porosity range 0.15–0.25 typical of cathodes for CRTs, and in the temperature range 950– 1100 8CB:
1=2   2 P  Plim n=2 CM elt=ref eðP; T; tÞ ¼ N0 P Pref  Plim

 E 1 evap 1 1=2  

PBa=ref exp  t1=2 2k T T

where Plim is the value of porosity such that transmission coefficient aPlim ¼ 0, n is the power factor (1:5 n 3).

where CM elt/ref is the molecular conductance of pellet of porosity Pref, for emitting face of surface unity and per unit thickness of pellet.

(12)

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Fig. 6. Relative transmission coefficient a as a function of pellet porosity.

The conditions of validity of expression (12) in our model are the following:

CM elt/ref, P and N0 are constant along z-axis over pellet thickness,

PBa is constant over time at the depletion front,

initial depletion e0 due to cleaning step is negligible (5 mm in this work),

e < ðepellet =2Þ (epellet being the pellet thickness). PBa/ref can be calculated from Lipeles and Kan [11] and factors CM elt/ref/N0, Plim and n, depending on pellet technology, are obtained from experiment. Expression (12) has been established based on numerous experimental results on pellets impregnated with 4-1-1 aluminates. It is also valid for other aluminates commonly used in cathodes for CRTs, i.e. 3-1-1 and 5-3-2 aluminates, taking into account that PBa/ref and N0 are dependent on the aluminates composition. On our pellets, the validity of expression (12) was verified up to a value of e equal to half of the pellet thickness

epellet. Departure from the proposed law of variation for e beyond this limit may occur, especially when backside depletion is not negligible with respect to depletion from the electron emitting face. In any case, gaining data beyond 200 mm of depletion on our cathode was estimated unnecessary to build a satisfactory model of lifetime prediction. It is also possible to observe a deviation from (12) in the early stage of the cathode life. A possible explanation is that in this transient early regime, the Ba partial vapor pressure appearing in (12) has not reached a stabilized value. It is also possible that the transient regime is the time necessary for the depletion e to satisfy Eq. (A.11) given in Appendix A. Our proposed model for depletion is applicable to both S-type and M-type cathodes. For M-type cathode, it may be useful to consider CM elt/ref varying along z-axis to account for the presence of a top coating. In the case of our M-type cathodes studied here, this refinement was not necessary.

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4. Relation between depletion and emission on life Life-tests of impregnated cathodes were performed in CRTs in order to evaluate the cathode lifetime. Cathodes featuring a pellet porosity of  0.185 were run at two different temperatures, namely 980 and 1050 8CB. The tubes were operated in real TV mode with TV off-the-air signal. The average cathode current was close to 0.5 mA. The cut-off voltage (difference of voltage between cathode and G1 at beam cutoff) was set at 150 V. The G1 hole diameter was 530 mm. Periodically, cathode emission at both the average (avgMIK) and maximum (maxMIK) current required from cathode was measured at 965 8CB. When cathode is in space charge limited regime, the maximum current density in maxMIK condition at the center of the emitting area is close to 7.6 A cm2 [15]. Table 1 gives the respective conditions for both MIK measurements. Fig. 7 shows the MIKs readings versus time. The fact that the maxMIK variation is far greater than the avgMIK one is due to the higher requested current

Table 1 MIK test conditions

Cut-off voltage (V) Cathode load

avgMIK

maxMIK

50 Single pulse for 2 s

150 Single pulse for 2 s

density of the former (cf. Table 1). Consequently, the maxMIK is used to have an early diagnosis of the cathode emission performances, while the avgMIK is more representative of the mean cathode current requested during a regular TV operation mode. As the latter is far less sensitive to cathode evolution than the former, we focused on maxMIK for the present study. The experimental data of emission decay presented in Fig. 8 are well fitted by the following equation proposed by Higuchi et al. [13]: maxMIK ¼ K  K 0 t2=3 0

(13)

where K and K are positive constants. It is noteworthy that Eq. (13) was obtained from comparable life-tests performed on impregnated cathode in a CRT electron gun. But, the fact that cathode

Fig. 7. avgMIK and maxMIK evolution vs. time for 980 and 1050 8CB operating temperatures.

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Fig. 8. maxMIK vs. time for 980 and 1050 8CB.

electron emission, or ratio of emission decay [13], is proportional to t2/3 is apparently not coherent with results obtained by Shroff and co-workers [1,14], where
 DI Jsat0 / log (14) / et / t1=2 I0 Jsatt DI/I0 being the cathode emission decay ratio, Jsat0 and Jsatt are the cathode saturated emission density, respectively, at t ¼ 0 and at a time t, and et is the depleted depth at a time t. As the proportionality between t1/2, depletion and emission density cannot be questionable, to solve such a discrepancy we have to reconsider the relation between emission current and current density in the case of electron guns for CRTs. Once noted that Eq. (13) is obtained in triode configuration while Eq. (14) for a diode or diode-like configuration of the beam-forming region, the most probable hypothesis is that in triode configuration: logðJsat Þ ¼ C þ C 0 I 3=4 0

where C and C are constants.

(15)

In order to validate Eq. (15), the current density versus maxMIK was calculated using the model proposed by Itazu [16] of current density distribution for saturated cathode emission. The results displayed in Fig. 9 confirm that this hypothesis is valid for the maxMIK range 6.6–2 mA, corresponding to a Jsat range of 6.2–1 A/cm2. From exponential regression equation deduced from the Itazu model, the log(Jsat) corresponding to maxMIK decay region of Fig. 8 versus time was calculated. Doing so, good consistency with Shroff et al. results was found, where logðJsat Þ ¼ KJ  KJ 0 t1=2

(16)

0

where KJ and KJ are positive constants. According to Shroff [1] and Dieumegard et al. [17], activation potential Vs of log(Jsat) equals Vs ¼ 12 Ve þ VLd where Ve is the depletion activation energy established above (Ve  3:23 V), and VLd the activation energy for the Ba surface migration (VLd  0:7 V). Hence, the expected Vs value for our cathode is 2.3 V. Fig. 10

Fig. 9. Log(Jsat) vs. maxMIK3/4 calculated using Itazu [16].

Fig. 10. Log(Jsat) vs. square root of time.

J.M. Roquais et al. / Applied Surface Science 215 (2003) 5–17

shows the variation of log(Jsat) as a function of time for the two tested life-test temperatures. From this plot, we confirm:

the expected decay of log(Jsat) as per relation (16),

a value of 2.3 eV for Vs. Finally, it appears that for a dispenser cathode in saturated mode placed in a triode configuration of the beam-forming region, e.g. CRT gun, a linear relationship exists between the four following parameters: MIK3=4 ; logðJsat Þ; e; t1=2

(17)

Consequently, the depletion mechanism can be considered as the limiting mechanism regarding emission decay, for the temperature range here considered (980–1050 8CB). The practical condition is actually to have a stable top-layer composition, without tungsten concentration increase that would impede the electron emission level [3]. Based on Eq. (17), controlled accelerated life-test can be defined on the basis of depletion. The acceleration factor is defined as being the ratio of times to reach the same decay of emission, respectively, at the operating temperature and at the temperature of the accelerated life-test. The acceleration factor was found about 3 between 980 and 1050 8CB for the tested cathodes. In the same manner, the cathode end-of-life can be determined for a given depleted depth corresponding to the maximum tolerable emission decay.

5. Conclusion The interpretation of the different measurements of Ba depletion on our impregnated cathodes has allowed to confirm the law of Ba evaporation from the porous pellets previously described by other authors. The dependence of the evaporation versus time and temperature was found as expected and the activation energy for Ba evaporation deduced from the measurements of depletion is equal to the one found by the more classical mass spectrometry measurements. Moreover, the depletion rate as a function of the pellet porosity was shown and calculations based on the assumption of a flow of Ba in a Knudsen flow regime appear really adequate to describe the observed phenomena. The effect of the porosity was introduced in a

15

general equation describing the evolution of the depletion as a function of the main parameters. It was confirmed that the Ba depletion is the major factor explaining the emission decay versus time observed in cathode ray-tubes. From the knowledge of the law of decay versus time and temperature deduced from experiments, appropriate accelerated life-test can be proposed with satisfactory control of the accelerating factor.

Acknowledgements We wish to thank M. Paul, S. Thoral, J.C. Pruvost and J.R. Adamski (LOE/Thomson Genlis) for samples preparation and characterization. We are grateful to D. Dieumegard, D. Brion and J.C. Tonnerre from Thale`s Ve´ lizy, France, for fruitful discussions about impregnated cathodes and associated characterizations and more specifically the method for measuring the depletion.

Appendix A

A.1. General equation of flux of particles of a perfect gas The flux of particles of gas diffusing parallel to the z-axis (main axis) of a container, i.e. the number of particles of gas passing through unit surface perpendicular to z-axis for unit time is given by [10]:
 1 dN J ¼ cl 3 dz where J is the flux of particles in cm2 s1 (for simplicity, J is an absolute value), c the mean speed of particles in cm s1 (c ¼ ð8kT=pm0 Þ1=2 ), T the temperature in K, k the Boltzmann constant, m0 the mass of particle in g, N the density of particles in cm3, and l is the mean free path of particles in cm. In agreement with Fick’s law, the flux is proportional to the gradient of the concentration of particles. For a perfect gas, substitution of N ¼ P=kT gives
 1 cl dP J¼ 3 kT dz

16

J¼

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 4 l dP 3 ð2pm0 kTÞ1=2 dz

(A.1)

where P is the pressure in g cm1 s2. The flux of particles is proportional to the pressure gradient along the z-axis. A.2. Equation of flux of barium atoms through the cathode pellet Interpretation of the observed Ba evaporation rate was proposed by Rittner et al. [12] who considered that the Ba vapor transport from the depletion front to the cathode surface (i.e. into vacuum) takes place via Knudsen flow through the network of pores of the pellet. The difference of partial vapor pressure of Ba between the depletion front at pressure PBa and the cathode surface at pressure P0, is the driving force for the Ba atoms flow. As P0 is negligible compared to PBa, one has PBa  P0  PBa In the impregnated cathode, the mass rate of Knudsen flow of Ba is proportional to the pressure gradient between depletion front and surface, this pressure gradient being equal to PBa/e [12]. Introducing a geometrical factor GP, the flux of Ba atoms through unit surface of pellet perpendicular to z-axis, can be expressed as
 4 lP PBa J¼ 1=2 3 GP ð2pm0 kTÞ e
 4 leff PBa ¼ (A.2) 3 ð2pm0 kTÞ1=2 e where l is the mean free path of Ba atoms in independent parallel straight pipes of radius r p in cm, leff is the mean free path of Ba atoms in network of pores of pellet in cm, GP the non-dimensional geometrical factor depending on pellet (GP > 1), P the porosity of pellet ¼ ratio of the volume of pores to the total volume of the pellet, e is the Ba depleted depth in cm. The introduction of factor GP in the equation of the flux of Ba atoms can be justified as follows. The condition for molecular flow regime is that the gas pressure is so low that the mean free path of a gas molecule is not limited by its collisions with other

gas molecules but by collisions with the walls of the container. The pellet could be modeled, in rough approximation, as Np parallel independent straight pipes of length ‘‘e’’ and radius equal to the mean pore radius r p , l being considered as the mean free path in those pipes. In this model, l depends on r p which may vary with porosity in the range studied. This model does not describe the real pellet and, as a refinement, it is appropriate to introduce factor GP to account for an effective length of the pipes superior to ‘‘e’’. The path to the surface being tortuous and the pipes more or less interconnected, the effective length is eeff ¼ GP e. Factor GP depends on pellet porosity, morphology and size of particles of tungsten. This network of pipes is equivalent to a single pipe of length e with effective mean free path of Ba atoms leff ¼ lP/GP. The lower the tungsten pellet porosity, the higher the value of GP. A higher value of GP translates itself into:

an increase of eeff in the tortuous multi-pipe model,

a decrease of leff in the straight single-pipe model. The relation between the evaporated Ba mass rate through the pellet top face dm/dt and the depletion rate de/dt is given by [14]: dm de ¼ trbSP dt dt

(A.3)

where S is the surface of pellet emitting face (perpendicular to z-axis) in cm2, t the fraction of pore volume filled with aluminates, r the density of the Ba, Ca aluminates in g cm3, b is the mass fraction of Ba, Ca aluminates liberating Ba. The flux of Ba atoms through unit surface of pellet perpendicular to main axis is J¼

trb de P m0 dt

Equalizing this equation of flux with our previous one brings
 trb de 4 l PBa ¼ (A.4) m0 dt 3 GP ð2pm0 kTÞ1=2 e From (2), one can calculate first derivative of e with respect to t: de A2 ¼ dt 2e

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Introducing A2/2e in above equality of fluxes leads to
 trb A2 4 l PBa ¼ (A.5) m0 2e 3 GP ð2pm0 kTÞ1=2 e

Comparing with our proposed equation leads to
 4 l 1 4 1 ¼ leff (A.9) P a ¼ aðeÞ ¼ 3 GP e 3 e

The volumetric flow dVBa/dt through surface unity associated to the flow of Ba atoms is  
 dVBa trb kT A2 4 l kT 1=2 1 ¼ P ¼ P m0 PBa 2e 3 GP 2pm0 e dt

and

One can define the conductance of a section of pellet of emitting face of surface unity with respect to gas flow in several manners—Cv: volumetric conductance in cm2 s g1, Cm: molecular conductance in g1 cm1 s, CM: molecular conductance in cm2 s1, where  
 4 l ðkTÞ 1=2 1 P  Cv ¼ Cv ðeÞ ¼   P1 Ba 3 GP ð2pm0 Þ e
 4 l h m0 i1=2 1 Cm ðeÞ ¼ P 3 GP e 2pkT  1=2
 4 l 1 1 CM ðeÞ ¼ P 3 GP 2pm0 kT e Then, it is convenient to write the equation of the flux of Ba atoms through unit surface of pellet as follows: trb A2 CM elt PBa P ¼ CM ðeÞPBa ¼ m0 2e e

ð2pkT=m0 Þ

P 1=2 Ba

[1] [2] [3] [4]

[7] [8]

[9] [10] [11] [12] [13]

[15]

The flux of Ba atoms derived from above equation is J¼

a ð2pm0 kTÞ

1=2

PBa

(A.8)

(A.10)

(A.11)

References

[14]

a

aðeÞ

It is reasonable to assume that the minimum value of e satisfying (A.11) is of the order of a few layers of tungsten grains.

[6]

where N0 ¼ trb=m0 in cm and N0, P, CM elt are considered constant along the z-axis in the present model. In Rutledge’s equation for Knudsen mass flow rate, Jm, through tungsten pellet [9], a transmission coefficient a is introduced so that

ð2pm0 kTÞ1=2

e  43 leff

[5]

3

1

As transmission factor a is 1, this sets the minimum value of depletion e from which our equation of flux of Ba is valid:

(A.6)

where CM elt is the conductance of a pellet of emitting face of surface unity, per unit of pellet thickness. This elementary conductance is useful for the expression of coefficient A:
1=2 2 CM elt PBa A¼ (A.7) N0 P

Jm ¼

CM ðeÞ ¼

[16] [17]

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