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AN ANALYTICAL EXPRESSION FOR THE CURRENT-VOLTAGE CHARACTERISTICS OF CAPILLARY TYPE LIQUID METAL ION SOURCES
G. Mair
To cite this version:
G. Mair. AN ANALYTICAL EXPRESSION FOR THE CURRENT-VOLTAGE CHARACTERIS-
TICS OF CAPILLARY TYPE LIQUID METAL ION SOURCES. Journal de Physique Colloques,
1984, 45 (C9), pp.C9-173-C9-177. �10.1051/jphyscol:1984929�. �jpa-00224409�
AN ANALYTICAL EXPRESSION FOR THE CURRENT-VOLTAGE CHARACTERISTICS OF CAPILLARY TYPE LIQUID METAL ION SOURCES*
G.L.R. Mair
Department of Mathematics and Physics, The University of Aston in Birmingham, Gosta Green, Birmingham B4 7ET, U.K.
Résumé - A travers des considérations de charge d'espace, une expression analytique est obtenue pour les caractéristiques courant-tension des sources d'ions à métal liquide de type capillaire. Nous constatons une bonne concordance avec des résultats expérimentaux.
Abstract - An analytical expression, based on space-charge considerations, is derived for the current-voltage characteristics of capillary-type liquid metal ion sources. Good agreement is found with experiment.
1 - INTRODUCTION
An unresolved problem in liquid metal ion sources (LMIS) is the theoretical prediction of their current-voltage (i-V) characteristics; and whilst, in general, flow has a profound effect in this respect, it appears that in cases where the flow of liquid to the apex is not greatly impeded, the current level is controlled essentially by the effects of space-charges. The treatment presented below applies to the particular case of capillary-type LMIS; it is approximate, but leads to an analytical expression for the i-V characteristics, in terms of only measurable parameters, and agrees well with experiment. In the capillary-emitter
configuration, a cone forms at the nozzle of the capillary (inset, Fig. 1 ) , at a critical voltage, and emission occurs from the jet-like apex of the cone.
2 - THEORY AND COMPARISON WITH EXPERIMENT
The one dimensional space-charge analysis of Stern et al /l/ has been shown in the past to form quite an adequate approximation for the calculation of the surface field of non-planar charge-emitting electrodes /2,3/. It is briefly summarised below:
Consider two plane, parallel electrodes, with a potential difference VQ applied between them; assume that one of the electrodes emits in free space
(permittivity e0 ) ions of charge-to-mass ratio (e/m); let EQ be the electric field acting at the surface of the ion-emitting electrode, and let FQ be the electric field acting at the same surface in the absence of any emission (i.e. space- charge). After tedious manipulation of the original equations /1,4/ one can write for the current density jQ:
(1)
f o r t h e c a s e when t h e s p a c e - c h a r g e i s s m a l l (E ~ F ) . When t h e s p a c e - c h a r g e i s l a r g e (E << F ) one h a s :
o o
•Condensed v e r s i o n of a p a p e r i n p r e s s ( J . P h y s . D: A p p l . P h y s i c s )
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984929
JOURNAL DE PHYSIQUE
(or, since F, v0, jo OT
vO3l2
) Now, for Eo<<Fo eq. ( 1 ) reduces toand this differs from the limiting case of large space-charge (eq. (2)) by only
-
15%. For an approximate treatment, therefore, we can use (1) for any value of the space-charge, or jo.
Equation (1) now reveals that for an LMIS consistency can only be maintained if the apex of the liquid cone is assumed to elongate with current/5/: In low flow impedance LMIS the current(i) increases very rapidly with V (Fig.1); if the increase in j is comparably fast, then F has to be enhanced with a:ex elongation to keep E fromodecreasing. If, on the othe? hand, j and E do not vary much with i<eq.(l6) 2nd ref . / 6 / ) , E o / ~ constantiin this case Spex el8ngation with i is necessary to offset the increase ?n apex radius and keep F-.
constant.The hydrodynamics inside the cone-like protrusion, or cusp, is fairly straight- forward. Ignoring the pressure inside the capillary
-
something that would not be justifiable for high flow impedance "needle" ion sources-
one can integrate to obtain the total force acting at the lateral surface of the liquid cusp. Details of the treatment will be published elsewhere 171; but starting from the basic equation relating the net pressure at the surface of the cusp to flow, we obtainapproximately;
where PR designates the total force (consisting of oppositely acting electric and surface tension forces) acting at the lateral surface of the cusp; p is the fluid density, ro is the apex radius of the cusp, and v, the fluid velocity near the apex. For not too great an emission angle, one can assume that the force at the ion-emitting apex of the cusp is given simply by the product of the net outward stress at the apex and the apex area (Ao). Therefore, the total force Pt acting at the cusp-like liquid anode is given as:
T being the surface tension of the liquid. Also,
where (Pt)=is the total force acting at a liquid emitter of the same shape in the absence of space-charges. Assuming now £or simplicity that all material transported to the apex is emitted in the form of ions, once can relate v. to the current i;
with the further equation
and assuming that
where the parameter a, containing the flow effects, is given as:
Equation (10) includes the assumption that A
-
ar 2.
Assuming after Gomer 161, that
we have a value of k
-
8 X 1011 ~ /,
mfor ~ KO 20 and a near onset current of -1 uA. Inserting (11) into (10), and evaluating the result for CS+ ions, with V.
-
3 kV,
one has a-
1.3 X 1 0 - ~ . It follows, therefore, that a can be ignored from (9) with small error; at high currents (i-100 M ) , however, the error may increase, in view of the substantial neutral component of the emission which has been omitted in deriving (10).We shall now proceed to calculate (Pt)= :
If we designate by g the angle that the liquid makes with the axis of the capillary (radius R) at the point of contact with the nozzle, then,
The first term on the r.h.s of (12) is the total electric force acting on the liquid, and is derived using the slender body approximation /8,9/; in this
approximation the total force acting at the surface of a drop protruding through the nozzle of a capillary does not depend on the exact shape of the drop. k is a numerical constant /g/, given by:
h being the nozzle-to-screen distance.
Combining (9) and (12), and ignoring a(<<l), one has:
The extinction voltage, Vox
,
is the voltage for which i+o; hence,JOURNAL DE PHYSIQUE
From ( 14) and ( 15), and assuming that $ does not vary with Vo,
The best experimental data available with capillary type LMIS are those of Culham Laboratory, obtained using CS (ref /l11 and private communication by K L Aitken).
However, even in these experiments the geometry was not exactly the same as that assumed by the theory. For example, the electric term in (12) assumes an infinitely large, plane counter-electrode; in the Culham experiments the counter-electrode was in the form of an apertured electrode placed near the nozzle. Also, the outside of the rim of the nozzle was slighly rounded for better operation
-
although, this should not affect the comparison with theory greatly, since the rounded rim would normally be embedded inside the liquid cone (K L Aitken, private communication).For an effective nozzle-to-screen distance, h, in the range of a few to several mm, and with 41 = 49.3O/10/, eq. (15) predicts, within f10% values of Vox of 3.5 kV, 3.03 kV and 2.43 kV, for R = 400 p, 225 vm and 125 vm respectively; these values compare satisfactorily with observations (Table 1).
Table 1
-
Values of the extinction potential Vox(V) for the three capillaries used in the experiments of Culham Laboratory /11/ (symbols as in Fig 1).In evaluating (16) an average mass has been assumed for the CS ions, based on the mass-spectroscopic observations of Clampitt and Jefferies 1121; these authors found that the CS ion beam consisted of 83% CS+, 12.5% and 4.5% (CS)~'.
Fig. 1 shows a comparison of (16) with the Culham experiments. It is seen that the agreement is encouraging. In spite of the approximate nature of the theory and non- ideality of the experimental geometry, the agreement found strongly suggests that space-charges might indeed control the current level in capillary type LMIS (and, generally, in LMIS of low flow impedance).
Acknowledgements
The author is indebted to Mr K L Aitken for supplying the experimental data, and many useful discussions.
4. MAIR G.L.R., J. Phys. D: Appl. Phys.
15
(1982) 2523.5. GAUBI H., SUDRAUD P., TENCE N., VAN DE WALLE J., Proc. 29th Int. Field Emission Symp., Goteburg (Sweden), H-0 Andren and H Norden (eds), Almqvist and Wiksell (Stockholm, 1982) 357.
6. GOMER R., Appl. Phys.
9
(1979) 365.7. MAIR G.L.R., J. Phys. D: Appl. Phys. (in press).
8. TAYLOR G.I., Proc. Roy. Soc. (1966) 145.
9. TAYLOR G.I., Proc. Roy. Soc.,
m
(1969) 453.10. TAYLOR G.I., Proc. Roy. Soc. (1964) 383.
11. AITKEN K.L. Proc. Field Emission Day (Nordwijk, ~olland) (ESTEC, 1976) 23.
12. CLAMPITT R. and JEFFERIES D.K., Nucl. Instrum. & Methods
149
(1978) 739.Fig. 1
