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BEST IMAGE CONDITIONS IN FIELD ION MICROSCOPY
C. de Castilho, D. Kingham
To cite this version:
C. de Castilho, D. Kingham. BEST IMAGE CONDITIONS IN FIELD ION MICROSCOPY. Journal de Physique Colloques, 1986, 47 (C2), pp.C2-23-C2-29. �10.1051/jphyscol:1986204�. �jpa-00225634�
Résumé - La qualité de l^image produite par le Microscope Ionique de Champ dépend de certains paramètres opérationnels qui peuvent être ajustés afin d'améliorer résolution et contraste. Les conditions optimales d'opération sont étudiées en vue de les établir de fapon quantitative. A cette fin, la courbe de probabilité d'ionization est calculée à l'aide d'un modèle JWKB et une nouvelle interprétation des conditions optimales est suggérée. Ainsi sont calculées les valeurs approximatives du champ qui donnent lieu à la meilleure image pour les gaz inertes, de l'hélium au xénon.
BEST IMAGE CONDITIONS IN FIELD ION MICROSCOPY
C.M.C. DE CASTILHO AND D.R. KINGHAM*
Cavendish Laboratory, Madingley Road, GB-Cambrldge, CB3 OHE, Great Britain
+ VG Scientific Ltd, Imberhorne Lane, GB-East Grinstead, RH 19 1UB, Great Britain
Abstract - The Field Ion Microscope image quality is dependent on operational
?arameters which can be adjusted in order to improve resolution and contrast, he best image operation conditions are discussed with the aim to define them quantitatively. To do so, a JWKB calculation for the ionization distribution is used and a new interpretation of the best image conditions is proposed. An approximate range for the best image field values for the noble gases from He to Xe is established, based on this new interpretation.
I - INTRODUCTION
The Field Ion Microscope (FIM) /I/ has been able to give a wealth of
information about metal surfaces through direct imaging of individual atoms 111. The amount and precision of the data is then dependent on the images formed, for which contrast and resolution constitute the usual measure of quality. In the normal operating regime with the tip at temperatures in the range of liquid N, the "optimum imaging conditions" are those prevailing when "an image with a maximum of details and definition" HI is obtained. The best image conditions (BIC) vary according to the imaging gas used, with a crucial parameter being the voltage drop between tip and screen of the FIM - best image voltage (BIV). The field proportionality factor, 8, is defined as the ratio between the electric field strength at the tip surface, F , and the tip voltage, V, assuming a zero value for the screen potential. F is, in fact, not a constant over the tip surface due to the atomic scale roughness.
Nevertheless F can be interpreted as an "average" field over the tip surface. It will be referred to here as the tip field strength. The F value corresponding to the BIV is denominated the best image field (BIF). Experimentally, the BIV has been shown to be well reproducible 12/. However, the BIF has been found to be specimen dependent and the analytical expressions usually proposed for 8 are dependent on the tip shape model adopted. Hence, the quoted values of the best image field present some variation. For instance, quoted values for the BIF of helium on tungsten vary from 4.4 to 5.0 V/A /2-4/. A better characterization of the optimum imaging conditions requires a quantitative approach in order to define a parameter - or parameters - which express the otherwise subjective concept of best image. Also, it is necessary to consider the still not full clarified intricacies of the FIM imaging process 151. Ionization rates, accommodation processes and gas concentration are the main aspects to be taken into account /2,6-9/. So, despite the good reproducibility of the BIV, there is a lack of a quantitative description of the BIC.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986204
J O U R N A L D E PHYSIQUE
Our aim is then to find a quantity - or quantities - which by assuming some critical value, maximum or minimum, at the BIF (or BIV), could be used as a quantitative reference for the achievement of best image conditions.
11 - METHOD AND MODEL
We have adopted a hyperboloid of revolution model to represent the overall tip shape of the FIM. An atomic protrusion is simulated by a half sphere, of radius R, superimposed on the apex of the hyperboloid. Within this model, the electric field at a point along the tip axis and between tip and screen can be expressed as
3 3
F(z) = .F [ 1/(1 + 2 z/Rt).+ 2 R /z ] (1) where z is the distance along the symmetry axls of the tip, from the hyperboloidal apex and R is the hyperboloidal apex radius. Not very close to the tip apex
(z > 20 and typical values of R and Rt, say 2 and 600 A respectively, eq. 1 becomes the eq. for a smooth hyperboloid.
The one electron potential is assumed uniform within the metal with a step of height Q above the Fermi level at the surface, CJ being the metal work function.
Outside the metal we assume a superposition of a hydrogenic atomic potential and the external electric field, as used by Haydock and Kingham / I l l . If a gas atom does exist at some point z, its ionization rate constant will be denoted by I(z) and numerically calculated from the one electron potential model mentioned. The velocity magnitude of the polarized atom is large enough that we can neglect any thermal velocity of the atom. Thus we can take:
v(z) = [ a/m l1l2 ~ ( z ) (2) where m is the atomic mass and a is the atomic polarizability. Assuming the
existence of an atom at a point z, its probability of ionization when it travels an infinitesimal distance dz is given by:
v(z) dz = [ I(z) /v(z) l dz (3) To calculate the probability of the atom ionizing at a certain point z, it is necessary to consider its survival probability up to this point. So, denoting P(z) as the probability of ionization of an atom travelling between a point far away from the tip (infinity) and the point z, and D(z) as the probability of ionization at the point z, we show that:
m
P(z) = 1 - exp[ -
I
v(zt) dz' and D(z) = v(z) [l - P(z) ] (4.a and b )z
The difference in energies at the screen arrival, between a hypothetical ion originated at the tip surface and a real one originated at z - an energy deficit, constitutes a measure of where the ionization has occurred and can be used instead of z. The ionization distribution as a function of position, D(z), is related to the ionization distribution as a function of the energy deficit, D (E), through:
D (E) = D(z) / F(z) (5)
The atomic ionization mechanismein the FIM /2/ occurs with the electron tunnelling to empty states at or above the Fermi level. Due to the Pauli exclusion principle, there is a region very close to the tip where ionization cannot occur, the forbidden zone, which width depends on the work function, tip field strength and atomic ionization potential. The minimum distance required for ionization to occur is known as the critical distance 1121, denoted here as z
.
Beyond this distance thetunnelling barrier for field ionization increaseg rapidly in such a way that for a certain range of tip field strengths the ionization is significant just in a small region, the ionization zone. We proceed numerically to calculate, from the critical distance and up to points far away from the tip, the values of I(z) and ~(z). By integration - eqs. 4 and 5 - we obtain P(z), D(z) and De(z) for several values of parameters like tip field strength, tip apex radius, etc and for various imaging gases. The maximum value of the ionization distribution D(z) will be denoted by D (z) and the corresponding z-coordinate by z For not too high values of the tip fTeld strength, the peak of the D(z) curve occurs at the critical distance. Hence m'
D (z) = D(z )
m ( 6 )
For this case a typical example of the curveC D(z) versus z is shown in fig. 1. TO characterize the position and width of the distribution, we define the position M as that corresponding to a value of D(z) equal to half of the distribution peak. Thus
D(M) = D (2) / 2 ( 7 ) From this we define the half width of themaXdistribution as being:
W = M - z (8)
equal to z . The peak value of D(z), its position and sharpness are dependent on tip field strekgth, imaging gas, tip apex radius and protrusion dimensions. Besides this, the atom velocity affects the ionization distribution as shown in eqs. 3 and 4. The velocity given by eq. 2 is due to the polarization energy on the first approach of the atom to the ionization zone. Nevertheless, it is known that, under or close the BIC, the atom suffers successive collisions and rebounds with the metal tip surface, so loosing kinetic energy at each collision 121. We did not consider the details of the thermal accommodation process of the gas atom. Its reduction of velocity increases the "residence time" in the ionization zone implying a higher ionization probability per unit of distance, p(z). It has been shown that, at BIC, the gas atoms are not fully accommodated to the tip temperature/9/. So, although the
"temperature" of the gas atom decreases by several thousand degrees during
accommodation, its equivalent temperature was calculated from the experimental data as being around 7 times greater then the tip temperature /g/. Forbes /5/ argues this value as an overestimation and suggests a possible value between 3 and 6. We impose
a reduction factor, 5 , on the velocity of eq. 2 in order to simulate the accommodation process. Thus,
ve,,(z) = 5 v(z) (9)
where v is the velocity due to the polarization energy as expressed in eq. 2. We should point out that, although the polarization energy is a consequence of the applied electric field, the insertion of the factor 5 in eq. 9 does not imply a reduction of the field strength. The field effect on the ionization rate constant, I(z), remains independent of 5. However, p(z) is reduced, as can be seen in eq. 3, since we reduce the atomic velocity by using v (z) given by eq. 9 instead of v(z), simulating the loss of kinetic energy ineffthe inelastic collision with the metal tip. This "ad hoc" imposed reduction on the velocity affects the ionization rate p(z) with subsequent changes on the values of M, D (z) and W.
The imaging process and contrast in the FIM has peen object of some dispute with considerations about which aspect should be dominant for the contrast:
ionization rate constant or gas concentration. Some recent papers have raised again the question, adding some new aspects /13/ and reviewing the general problem /5/.
Although some influence has been reported from the gas concentration, mostly at very low tip temperatures, we did not take into account this in our calculations.
For much higher tip field strengths, compared to the case where most of the ionization occurs close to z , the peak of the distribution occurs away from the metal - free space field ionTzation /3,10/. In this case, for increasing values of F , the peak of the distribution moves away from the tip. However, for a certain rgnge of the tip field strength, a two-peak structure develops in the ionization distribution curve, with one local maximum at the critical distance and another one a few tens of %ngstr&!ms away from the tip. This is due to the local enhancement of the electric field close to the tip, as a consequence of the protrusion, and to the sharp cut-off in the potential at the metal surface. It should be mentioned that the value of D(z) corresponds to the product of the ionization rate and the survival probability - eq. 4.b - as will be discussed later and the sharp rise of p(z) close to z _ makes this two peak structure appear in the calculated results.
111 -"RESULTS
We performed a JWKB calculation of p(z) and subsequent numerical integration gave us P(z), D(z) and D (E). Fig. 2 shows the curves D (E) versus energy deficit for He on W. For the pargmeters specified in the figureecaption and tip fields equal to or less than 5.2 V/.&, the peak of the ionization distribution occurs at the critical distance. In reducing the tip field strength, the D (E) curve becomes sharper while the critical distance increases. However, two &antities do not vary monotonically with the tip field strength: the characteristic position M of the
distribution and the maximum value of D (E) or, alternatively, of D(z). At the range of tip fields for which the ionization sistribution peaks at the critical distance -
below 5.2 V/A in our example with He-W - there is a value of the tip field strength for which the value of M is a minimum and a value of F for which the peak value of D(z), Dm(z), is a maximum. So, D (z) and M are depe$dent on the value of F , i.e.
D (z) = f(Fm) and M = g(Fo) (lO.a and by and, at conditions of eq.m6 - ionizztion close to the ~ r i t i c a l ~ d i s t a n c e - there is
J O U R N A L DE PHYSIQUE
a v a l u e of F which maximises D ( z ) , which w i l l b e d e n o t e d by F . S i m i l a r l y t h e t i p f i e l d s t r g n g t h c o r r e s p o n d i n g m t o t h e minimum v a l u e o f M w i l l be d e n o t e d by F mM .
Thus, i s t h e v a l u e o f F c o r r e s p o n d i n g t o t h e maximum o f t h e f u n c t i o n f ( F ) and, a n a l o g o u s l y , i s t h e F 'value c o r r e s p o n d i n g t o t h e minimum of t h e f u n c t i o g g ( F ) . F i g . 3 shows t h e v a r i a t i o n O o f M and of t h e c r i t i c a l d i s t a n c e w i t h t h e t i p f i e l d O s t r e n g t h . A l s o shown a r e t h e v a l u e s o f W , a s d e f i n e d i n e q . 8.
The i n c l u s i o n of t h e v e l o c i t y r e d u c t i o n f a c t o r 5, r e f e r r e d t o i n eq. 9 , a f f e c t s t h e v a l u e s o f M , i n c r e a s i n g s l i g h t l y t h e i r a b s o l u t e v a l u e s f o r l o w e r v a l u e s of 5 , and s h i f t i n g t h e p o s i t i o n of t h e minimum o f t h e c u r v e M v e r s u s F t o l o w e r v a l u e s o f t h e t i p f i e l d , a s shown i n f i g . 4.a. The v a r i a t i o n of D ( z ) , ' t h e maximum
m
v a l u e o f D ( z ) , a s a f u n c t i o n o f t h e t i p f i e l d i s shown i n f i g . 4 . b , f o r v a r i o u s v a l u e s of t h e f a c t o r 5. With t h i s imposed r e d u c t i o n o f t h e v e l o c i t y we h a v e a n i n c r e a s e o f e a c h D ( z ) v a l u e and a r e d u c t i o n o f t h e v a l u e o f F c o r r e s p o n d i n g t o t h e maximum o f t h E D ( z ) , i . e . , i s r e d u c e d a s 5 d e c r e a s e s . O c o n s i d e r i n g t h e c a s e He-W and t h e values of 5 e q u a l t o 1.0 (no accommodation) a n d , s a y , 0.10, t h e maximum v a l u e s o f D ( z ) i n c r e a s e s by 22% and a s h a r p e r peak on t h e n u m e r i c a l l y c a l c u l a t e d c u r v e s of m f i g . 4.b c a n b e o b s e r v e d . However, t h e v a l u e s o f M i n c r e a s e s t o o , MDthough j u s t by 13% i n t h e same example, w h i l e t h e s h i f t on t h e v a l u e s o f F~
and F a r e a b o u t t h e same. T a b l e 1 shows t h e v a l u e s o h t h e t i p f i e l d s c o r r e s p o n d i n g t o minimum of M , and t & t h e ma&mum of D ( z ) , F , f o r v a r i o u s g a s e s on t u n g s t e n . F o r e a c h v a l u e of F and F t h e e q u T v a l e n t g a s t e m p e r a t u r e a t t h e c r i t i c a l d i s t a n c e was c a l c u l a t e d . T a b l e 2 c o n t a i n s t h e v a l u e s o f M, z and W f o r a F v a l u e a t which i o n i z a t i o n , i n t h e mean, o c c u r s c l o s e s t t o t h e tipc- minimum v z l u e of M - f o r t h e v a r i o u s n o b l e g a s e s and 5 e q u a l t o 1.0.
The i n c r e a s e o f t h e l o c a l e l e c t r i c f i e l d a t p o i n t s c l o s e t o t h e t i p , d u e t o t h e i n f l u e n c e of t h e p r o t r u s i o n c a u s e s a s h a r p r i s e i n t h e i o n i z a t i o n r a t e c o n s t a n t
~ ( z ) . As c a n b e s e e n from e q . 4 . b , t h i s c a n l e a d t o a n i n c r e a s e i n t h e v a l u e s of D(z) a t s u c h p o i n t s . So, i n a c e r t a i n r a n g e of t h e t i p f i e l d s t r e n g t h - between 4.5 and 4.15 V I A i n t h e c a s e of Ne-W ( f i g . 5) - a two peak s t r u c t u r e d e v e l o p s on t h e c u r v e s of D ( E ) v e r s u s e n e r g y d e f i c i t o r , a l t e r n a t i v e l y , o n t h e c u r v e D ( z ) v e r s u s z . The p e a k e c l o s e r t h e t i p i n c r e a s e s f o r d e c r e a s i n g v a l u e s o f F and becomes t h e dominant one. F o r F = 4.05 V I A , i n o u r r e f e r r e d example, i s no@ p o s s i b l e any more t o n o t e t h e d o u b l e i e a k s t r u c t u r e . T h i s e x t r a peak d o e s n o t c o r r e s p o n d t o a J a s o n peak 1141. T u n n e l l i n g t r a n s m i s s i o n r e s o n a n c e s w e r e n o t i n c l u d e d i n o u r c a l c u l a t i o n s . IV - D I ~ C U S S I O N AND C ~ N C L U ~ I ~ N ~
A v i t a l r e q u i r e m e n t f o r good imaginn i n t h e FIM i s t h a t i o n i z a t i o n s h o u l d -
o c c u r c l o s e t o t h e t i p . The p a r a m e t e r M h a s b e e n i n t r o d u c e d t o h e l p c h a r a c t e r i s e b e s t image c o n d i t i o n s , a s i t e x p r e s s e s how c l o s e t o t h e t i p t h e i o n i z a t i o n o c c u r s . The c r i t i c a l d i s t a n c e may b e r e d u c e d by i n c r e a s i n g t h e f i e l d s t r e n g t h , b u t i f t h e f i e l d i s t o o s t r o n g t h e n f r e e s p a c e f i e l d i o n i z a t i o n w i l l o c c u r . So, i t i s n e c e s s a r y t o s t r i k e a b a l a n c e between a narrow i o n i z a t i o n zone and a s m a l l c r i t i c a l d i s t a n c e . The m i n i m i z a t i o n o f M r e p r e s e n t s t h i s b a l a n c e .
As w e l l a s i o n i z a t i o n o c c u r r i n g c l o s e t o t h e t i p , i t i s a l s o i m p o r t a n t t h a t t h e p r o b a b i l i t y o f i o n i z a t i o n a t t h e c r i t i c a l d i s t a n c e i s h i g h b e c a u s e t h i s i s r e l a t e d t o t h e image i n t e n s i t y c o r r e s p o n d i n g t o t h a t p o i n t . T h i s i s r e p r e s e n t e d by t h e m a x i m i z a t i o n of D ( z ) a s a f u n c t i o n of F .
The v a l u e s o f m t i $ f i e l d s t r e n g t h a t wgich M and D ( z ) assume t h e i r r e s p e c t i v e s t a t i o n a r y v a l u e s d i f f e r by o n l y 4% i n t h e c a s z 0% He-W n e g l e c t i n g any t h e r m a l accommodation ( 5 = 1 . 0 ) , however, t h e v a u e s a r h t o o h i g h . The i n c l u s i o n o f accommodation e f f e c t s r e d u c e s t h e s e v a l u e s o f F' and F by a b o u t t h e same amount e a c h . Accommodation i n c r e a s e s t h e maximum v a l u e of D ( z ) , b u t i t a l s o i n c r e a s e s t h e minimum v a l u e of M w h i c h , a c c o r d i n g t o o u r r e a s o n i n g , m t g h t b e e x p e c t e d t o r e d u c e m t h e image q u a l i t y . However, t h i s must be outweighed by r e d u c e d b l u r r i n g of t h e image due t o t h e d e c r e a s e d t h e r m a l v e l o c i t y of t h e imaging g a s .
The b e s t image q u a l i t i e s of d i f f e r e n t imaging g a s e s c a n be r e l a t e d t o t h e minimum v a l u e of M a s i n t a b l e 2 where t h e r e i s a s y s t e m a t i c b e h a v i o u r w i t h h e l i u m c o r r e s p o n d i n g t o t h e c a s e where i o n i z a t i o n o c c u r s c l o s e s t t o t h e t i p . I n c o n c l u s i o n , we b e l i e v e t h a t , w i t h o u t f u l l y c o n s i d e r i n g g a s c o n c e n t r a t i o n e f f e c t s and t r e a t i n g accommodation i n a n "ad hoc" manner, we h a v e d e m o n s t r a t e d some p h y s i c a l f a c t o r s a f f e c t i n g b e t i m a g e M D f i e l d and b e s t image q u a l i t y . F o r v a r i o u s g a s e s o u r c a l c u l a t e d v a l u e s o f FA and F a r e i n good agreement w i t h q u o t e d v a l u e s o f b e s t image f i e l d . One of u s (CMC d e C) acknowledges f i n a n c i a l s u p p o r t from CAPES-UFBa, B r a z i l .
/ 2 / Muller, E. W. and Tsong, T. T., in: Field Ion Microscopy, Principles and Applications, Elsevier, New York ( 1 9 6 9 ) .
/ 3 / Sakurai, T. and Muller, E. W., Phys. Rev. Lett. 30 (1973) 532.
/ 4 / Chamberlain, P., thesis, Univ. of Cambridge ( 1 9 7 1 ) . / 5 / Forbes, R. G., J. Phys. D: Appl. Phys. 18 (1985) 973.
/ 6 / Forbes, R., J. Microscopy 96 pt.1 (1972) 5 7 . / 7 / Forbes, R., J. Microscopy 96 pt.1 (1972) 63.
/ 8 / Van Eekelen, H. A. M., Surface Sci. 1 (1970) 21.
/ g / Chen, Y. C. and Seidman, D. N., Surface Sci. 6 (1971) 61.
/ 1 0 / Sakurai, T. and Muller, E. W., J. Appl. Phys. 5 (1977) 2618.
/ 1 1 / Haydock, R. and Kingham, D., Surface Sci. 103 (1981) 239.
1121 Inghram, M. G. and Gomer, R., J. Chem. Phys. 2 (1954) 1279.
/ 1 3 / Homeier, H. H. H. and Kingham, D. R., J. Phys. D: Appl. Phys., 2 (1983) L115.
1 1 4 1 Jason, A. J . , Phys. Rev. 156 (1967) 266.
TABLE 1 Ti Fields for Maximum D ( z ) and Minimum M
vaEues in parenthesis correspond to equivalent temperatures ( O K ) .
Velocity Reduction Factor (5)
G A S 1.00 0.70 0 . 5 0 0 . 2 0 0.10 0 . 0 5 B I F / 2 , 4 /
FmM 5 . 1 0 5.00 4 . 9 0 4.70 4.50 4 . 3 0
(5530) (2560) (1240)
Helium ( 1 7 6 ) (39) ( 9 )
FMD 4.90 4.80 4.70 4.50 4.30 4 - 1 5 4 . 4 - 5 . 0
(4940) (2290) (1100) (157) ( 3 5 ) ( 8 )
FmM 4 . 1 0 4.00 3.90 3.70 3.60 3.40
(6610) (3030) (1440) ( 2 0 0 ) ( 4 7 ) ( 1 0 )
Neon
FMD 3.90 3.80 3.70 3.50 3.40 3 . 3 0 3 . 5
(5780) (2640) (1260) (175) (41) ( 9 )
Argon
FMD 2.25 2.20 2.15 2.05 2.00 l.9O 1 . 9 - 2.20
(7500) (3480) (1680) (240) (56) (13)
FmM 1 . 8 5 1 . 8 0 1 . 8 0 1.70 1.65 1.60
(7600) (3480) (1780) (248) ( 5 8 ) ( 1 3 )
Krypton MD
F 1.80 1.75 1.70 1 . 6 5 1.60 1 . 4 5 - 1.50
(7110) (3250) (1550) (230) ( 5 4 ) ( 1 2 )
FmM 1 . 4 0 1 . 3 8 1 . 3 6 1 . 2 8 1.24 1 . 2 0
(6910) (3270) (1610)
Xenon (220) ( 5 2 ) ( 1 2 )
FMD 1 . 3 6 1 . 3 4 1 . 3 2 1.26 1.22 1.16 1.10 - 1.30
(6440) (3050) (1500) (210) ( 5 0 ) ( 1 1 )
TABLE 2
Values of position M, critical distance, zc, ionisation zone width, W, and tip field strength at minima values of M for noble gases on tungsten. The values of Rt and R are, respectively, 600 and 1 . 5 8 A.
VALUES AT MINIMUM M
Gas M(A) z p ( A ) w(A) F_ (VIA)
Helium 4.24 3.76 0 . 4 8 5 . 1 Neon 4.54 3 . 9 5 0 . 5 9 4.1 Argon 5 . 2 1 4.58 0 . 6 3 2.3 Krypton 5.42 4.75 0.67 1.85 Xenon 5 . 6 6 4.96 0 . 7 0 1 . 4 0
JOURNAL DE PHYSIQUE
Figure 1 - A schematic diagram of the ionisation distribution versus position when it occurs close to the critical distance z The position M corresponds to the z-csordinate for which D(z) is half of the maximum value D (2). The distribution half width, w , ~ is the difference between M and zc.
Figure 2 - Ionization distribution curves versus ener y deficit, for ti field values for which tte ionization is cyose to the critical distance in the case of He - W.
The tip apex radius is 600 A and the half sphere protrusion radius is 1.58 A.
4.0 4.5 5.0 5.5
T I P FIELD ( V / A )
Figure 3 - Position M (full line) and critical distance z (dotted line) versus tip field stEength. The variation of the peak half width, W, is represented by the dashed line. M and z should be read on left scale and W oncright. Both vertical scales are in 3ngstr8ms.
4.0 4.5 5.0
TIP FIELD ( v / A )
b 4.0 4.5 5.0
TIP FIELD ( V / &
F i g u r e 4 - F o r t h e i n d i c a t e d v a l u e s of t h e v e l o c i t y r e d u c t i o n f a c t o r 5 , t h e v a r i a t i o n w i t h t h e t i ~ f i e l d s t r e n g t h i s shown f o r :
a The p o s i t i o n M.
b]The maxima v a l u e s of t h e i o n i z a t i o n d i s t r i b u t i o n c u r v e s .
ENERGY DEFICIT (eV)
F i g u r e 5 - A " d o u b l e peak" s t r u c t u r e d e v e l o p s f o r t i p f i e l d s t r e n g t h s between t h o s e f o r which we h a v e e i t h e r f r e e s p a c e f i e l d i o n i z a t i o n o r
i o n i s a t i o n c l o s e t o t h e c r i t i c a l d i s t a n c e .
