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TOWARDS A UNIFIED THEORY OF HELIUM FIELD IONIZATION PHENOMENA
R. Forbes
To cite this version:
R. Forbes. TOWARDS A UNIFIED THEORY OF HELIUM FIELD IONIZATION PHENOMENA.
Journal de Physique Colloques, 1986, 47 (C2), pp.C2-3-C2-10. �10.1051/jphyscol:1986201�. �jpa-
00225631�
Colloque C2, supplément au n°3, Tome 47, mars 1986 page C2-3
TOWARDS A UNIFIED THEORY OF HELIUM FIELD IONIZATION PHENOMENA
R.G. FORBES
University of Surrey, Department of Electronic and Electrical Engineering, GB-Guildford, GU2 5KH (Surrey), Great Britain
Résumé" - La prédiction de la largeur à mi-hauteur de la distribution d'énergie des ions d'hélium est re-examinée à la lumière de progrés récents en la théorie d'ionisation de champ. La possibilité de faire cette prédiction sur la base de formalismes de type JWKB est discutée; il est besoin d'utiliser de meilleurs modèles de barrière électronique à la surface d'un émetteur.
Abstract - The prediction of the half-width of the helium ion energy distribution is re-examined, in the light of recent progress in imaging theory. It is argued that prediction on the basis of JWKB-type formalisms should be possible: the need is for more realistic models of the electron barrier at a real emitter surface.
1. Introduction
Recent developments in the theory of local-contrast formation in field-ion images have at long last provided it with a more-or-less satisfactory numerical basis.
These developments, along with the past history of the local-contrast problem, have recently been reviewed elsewhere [1]. The stimulus for the recent burst of progress was some work of Homeier and Kingham [2]. But from the present author's point of view, the key to improved understanding has been the realisation that the across-surface variations in the average field Fav between the critical surface and the emitter surface could be very much greater (as much as 100 times) than the corresponding variations in critical-surface field per. This turns out to be true even for a relatively close-packed emitter plane, such as tungsten (111). In consequence, it now seems certain that - at imaging temperatures near 80K - the local ionization density variations in the critical surface (and hence the observed local contrast) are primarily due to across-surface variations in the ionization (i.e. the electron transition) rate-constant P
e.
In contrast theory it is more important to know the relative values of rate-constant at different surface locations than to know the absolute values well.
Appropriate relative values can be obtained by the following series of modelling steps:
(1) Establishment of a charged-surface model, and calculation of the potential structure above it, - in particular the "one-dimensional potential sections" along lines normal to the surface, passing through critical-surface positions of interest.
(2) Fitting, for each such section, between the model surface and the critical surface, an equivalent trapezoidal barrier. This gives the parameters f, B and h (or, equivalently, f, B and t) of the model barrier illustrated in Fig.1(a).
(3) Use of some rate-constant formula dependent on these parameters.
(4) Normalisation of the resulting rate-constant value, using a divisor given by applying the same rate-constant formula to a reference barrier.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986201
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Figure 1. Models f o r t h e e l e c t r o n b a r r i e r . ( a ) Simple t r a p e z o i d a l b a r r i e r , w i t h mode1 parameters shown ( s o l i d l i n e ) . ( a ) and ( b ) Reduced p o t e n t i a l b a r r i e r when coulombic and e l e c t r o n - s u r f a c e i n t e r a c t i o n terms a r e added t o t h e simple b a r r i e r ( b r o k e n l i n e ) . ( b ) T r i a n g u l a r b a r r i e r used i n t h e CBM approach t o modelling t h e reduced b a r r i e r . These diagrams and t h e o t h e r s i n t h i s paper a r e schematic.
I n c o n n e c t i o n w i t h Fig.1, we cal1 f t h e "model-field", h t h e "step-height" and B t h e " o u t e r height"; t h e t r a p e z o i d a l barrier can be f i t t e d t o s u r f a c e mode1 p o t e n t i a l s i n more t h a n one way, s o a c l e a r d i s t i n c t i o n must be made between s t e p - h e i g h t h and s u r f a c e work-function 6 , and between model-field f and t h e s u r f a c e f i e l d s FaV and F c r . I n t h e c a s e o f i n e r t - g a s f i e l d i o n i z a t i o n t h e parameter B can be equated t o t h e i o n i z a t i o n e n e r g y 1, though s t r i c t l y some small i n t e r a c t i o n terms ought a l s o t o b e included. For helium i o n i z a t i o n on t u n g s t e n at best image v o l t a g e the r e f e r e n c e barrier u s e s t h e parameters B = 1 = 24.59 eV;
h
=4.5 ev; f
=45 V/nm.)
For t h e f i r s t t h r e e s t e p s l i s t e d above, v a r i o u s a l t e r n a t i v e c h o i c e s a r e a v a i l a b l e ; o t h e r modelling procedures are a l s o p o s s i b l e , f o r example t h a t o f Bon0 and
G o o d[3]. Some c h o i c e s o f charged-surface d e l , b a r r i e r f i t t i n g procedure, and r a t e - c o n s t a n t formula, a r e d i s c u s s e d q u a n t i t a t i v e l y i n Ref. C l ] . I t is found t h a t t h e formula most commonly used i n f i e l d - i o n l i t e r a t u r e (eq.
( 3 )below) is not f u l l y r e l i a b l e . b u t s e v e r a l c h o i c e combinations u s i n g o t h e r formulae g i v e f a i r l y s i m i l a r r e s u l t s as r e g a r d s r a t e - c o n s t a n t v a r i a t i o n .
Some c o n c l u s i o n s o f R e f . [ l ] r e l e v a n t t o t h e p r e s e n t d i s c u s s i o n a r e :
(
1 ) Q u a s i c l a s s i c a l t h e o r y should be s u f f i c i e n t t o d e s c r i b e imaging.
( 2 )
I n t h e t h e o r e t i c a l d e r i v a t i o n o f e l e c t r o n t r a n s i t i o n r a t e - c o n s t a n t , u s e o f a JWKB o r e q u i v a l e n t method is t o be p r e f e r e d , r a t h e r t h a n a method based on o v e r l a p i n t e g r a l s . T h i s is because it is important t o g e t t h e wave-functions i n t h e barrier r e g i o n as n e a r c o r r e c t as p o s s i b l e [ 4 , 5 ] , and t h i s seems t o be e a s i e r w i t h a JWKE-type formalism t h a n w i t h a n o v e r l a p i n t e g r a l formalism. ( P a s t a t t e m p t s [6,71 t o u s e o v e r l a p i n t e g r a l formalisms t o e x p l a i n l o c a l image c o n t r a s t have been s i n g u l a r l y u n s u c c e s s f u l .
)( 3 )
I n t h e d i s c u s s i o n o f l o c a l c o n t r a s t , where r e l a t i v e r a t e - c o n s t a n t v a l u e s are t h e important f a c t o r , three-dimensional formalisms do n o t g i v e s i g n i f i c a n t l y b e t t e r r e s u l t s t h a n one-dimensional fonnalisms, and t h e formula - e q . ( l ) below - d e r i v e d
£rom a simple t r a p e z o i d a l b a r r i e r w i l l o f t e n y i e l d r e s u l t s o f adequate q u a l i t y .
This c o n c l u s i o n , about t h e pragmatic s u f f i c i e n c y o f a one-dimensional JWKB
approximation, is a t v a r i a n c e w i t h some p a s t t h i n k i n g . I n t h e m i d - s i x t i e s , it Was
t h e a p p a r e n t i n a b i l i t y o f one-dimensional JWKB formalisms t o e x p l a i n t h e observed
s p r e a d s f o r helium f i e l d i o n i z a t i o n t h a t l e d t o t h e i n t r o d u c t i o n o f o v e r l a p
i n t e g r a l formalisms, s u c h as t h o s e o f Refs.[8.9]; t h e s e methods o f r a t e - c o n s t a n t
c a l c u l a t i o n were a b l e t o e x p l a i n observed half-widths.
r a t e - c o n s t a n t v a r i a t i o n s a c r o s s t h e s u r f a c e o r normal t o t h e s u r f a c e . n e c e s s a r y o b j e c t i v e is t h u s t o look a g a i n a t al1 t h e r e l e v a n t helium f i e l d i o n i z a t i o n phenomena, i n t h e hope o f e v e n t u a l l y d e v e l o p i n g a c o h e r e n t t h e o r e t i c a l s t o r y . I n t h i s c o n t e x t , t h e e x p e r i m e n t a l phenomena t h a t d e s e r v e o u r a t t e n t i o n are l o c a l c o n t r a s t , e n e r g y - d i s t r i b u t i o n half-widths, and c u r r e n t / v o l t a g e c h a r a c t e r i s t i c s . The first o f t h e s e is d e a l t w i t h i n R e f . [ l ] and 1 c o n c e n t r a t e h e r e on e n e r g y d i s t r i b u t i o n p r e d i c t i o n . T h i s p a p e r should be understood as a n i n t e r i m d i s c u s s i o n a b o u t what c a n and cannot be achieved w i t h r e l a t i v e l y simple-minded arguments.
2. Rate-constant fonnulae
A s a preliminary, 1 want t o b r i e f l y d i s c u s s the main r a t e - c o n s t a n t formulae s o f a r d e r i v e d , i n t h e c o n t e x t o f f i e l d - i o n imaging, from JWKB-type b a r r i e r a n a l y s e s . The STB form. The s i m p l e s t formula i s obtained by e v a l u a t i n g a JWKB i n t e g r a l a c r o s s t h e s i m p l e t r a p e z o i d a l b a r r i e r shown i n F i g . 1 . T h i s g i v e s :
Pe
=Ve e ~ p [ - ( b / f ) ( 8 3 / 2 - h3/2)] ( S m form) ( 1 ) where ve is t h e "frequency o f a t t e m p t s b y t h e e l e c t r o n t o p e n e t r a t e t h e barrier", and b is a u n i v e r s a l c o n s t a n t r e l a t e d t o a n o t h e r u n i v e r s a l c o n s t a n t c l
by:where me is t h e e l e c t r o n mass, e the elementary c h a r g e , and 6 P l a n c k ' s c o n s t a n t .
The GBM form. E q . ( l ) i g n o r e s b o t h t h e atomic p o t e n t i a l at t h e "vacuum" end o f t h e b a r r i e r and s u r f a c e e f f e c t s a t t h e "emitter" end. A n a l y s i s o f t h e t r i a n g u l a r b a r r i e r o f F i g . l ( b ) a t t e m p t s t o t a k e t h e s e i n t o account, and l e a d s t o r e s u l t ( 3
)below, which is a t t r i b u t e d t o G o m e r [ l O ] by Brandon [ll] ( a l t h o u g h it d o e s n ' t i n fact appear i n Ref.
[ I O ] ),and is r e d e r i v e d by Müller and Tsong [12]. T h i s
"Gomer-Brandon-Müller" form, widely used i n f i e l d - i o n l i t e r a t u r e , is:
where : A
=2 ( c 2 f )1/2
(3 b )
and c2 is another u n i v e r s a l c o n s t a n t , d e f i n e d by:
The FHG form.
Afosm p r e v i o u s l y used 1131 by t h e p r e s e n t a u t h o r is:
where a and a 2 were presumed t o be weakly f i e l d - d e p e n d e n t p a r a m e t e r s ( w i t h a 2 of o r d e r u n i t y ) , and ag is t h e c o r r e c t i o n f a c t o r proposed b y Burgess e t a l . 1143 t o t h e Fauler-Nordheim t h e o r y o f f i e l d e l e c t r o n emission. Values o f o ; ~ a r e t a b u l a t e d i n R e f , [ 1 4 ] as a f u n c t i o n o f t h e parameter y given i n t h e p r e s e n t c o n t e x t by:
Eq.
(1 2 ) was developed from t h e e q u i v a l e n t free-space formula g i v e n b y Gomer
[ l O ] ;a b r o a d l y s i m i l a r formula was used by Halpern and Gomer 1151 i n t h e c o n t e x t o f
f i e l d i o n i z a t i o n i n t o l i q u i d g a s e s .
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The IiK form. Defects of the above approaches are their one-dimensionality and, more ïmportantly, their lack of attention to the pre-exponential factor. It is relatively straightforward to derive three-dimensional versions of the above formulae (see Ref. Cl]), but derivation of an expression for the pre-exponential requires attention to the problems of wave-matching at the atomic end of the barrier. A formula was derived by Goldenfeld [4], but is somewhat unwieldy. More recently Haydock and Kingham [16], using an approximate wave-matching treatment, have produced a result that has the form:
Pe
=[a3 £1-C exp( C
)/(~ 3 / 2 - h3/2)] exp[-(b/f)(~3/2 - h3/2]
(HK form)
(7
)where C is a constant, given in terms of the universal constants cl and c2 defined earlier and an adjustable parameter Z ("the effective charge on the ion") by:
a3 is a parameter that depends on B and 2, but not on f or h, and incorporates a nunierical adjustment factor; this factor has been so chosen that eq.(7) leads to the known correct (low-field) result for the free-space field ionization of a hydrogen a t m . Kaydock and Kingham do give a specific form for aj, but it is not clear to the present author that their form is in fact correctly derived, even in the case of inert-gas field ionization, m y reservations being about the way the parameter v is introduced into their argument. (This difficulty is also present in the derivation of the revised formula, appropriate for metal-ion post-ionisation, given by Kingham in Ref. [17].
).However, since a3 does not significantly affect the sensitivity of the rate-constant to field and step-height variations, we need not conaider these problems of fine detail here.
Comparisons. A convenient way to compare the performance of the various formulae is to compare values of the partial derivatives of and
phdefined by:
Some results derived for the rate-constant formulae discussed here are shown in Table 1 below. ReSultS are alS0 shown for the three-dimensional version of the FHG formula, derived in Ref. [1] .
Table 1. Comparison of values of and ph for various different rate-constant formulae (and of for three different physical situations). For helium and helium/tungsten we use the parameter values B=24.59 eV, f=45 V / m . For hydrogen we use an f-value such that ~3/2/f is the same as for helium.
rate-constant formula
STB 1D GBM 1D FHGa 1D FHGa 3D HKb 3D true 3D
value of of for: value of
phfor:
hydrogen free-space ionization
helium free-space ionization
heliwtungsten critical-surface ionization
helium/tungsten critical-surface ionizat ion
aTaking a2=l and da~/df=O. Making Z=1.
=lD
=one-dimensional; 3D = three-dimensional
a l t h o u g h t h e 3D formalisms do g i v e b e t t e r agreement w i t h the t r u e r e s u l t i n t h e o n l y c a s e which t h e formulae can b e t e s t e d , t h e r e is no major d i f f e r e n c e between t h e r e s u l t s o f 1D and 3D formalisms. The GBM formula perhaps l o o k s s l i g h t l y l e s s r e l i a b l e t h a n t h e o t h e r s .
3. The h a l f - w i d t h o f t h e energy d i s t r i b u t i o n
Helium i o n e n e r g y d i s t r i b u t i o n w i d t h s were measured many y e a r s ago b y Tsong and Muller [ l a ] , and more r e c e n t l y b y Hanson and Inghram [19]. From t h e s e d a t a 1 t a k e a t y p i c a l helium half-width (AE) v a l u e t o b e 0.7 eV, which i n t h e c o n t e x t o f a n assumed c r i t i c a l - s u r f a c e f i e l d o f
45V/nm i m p l i e s a s p a c e half-width
( A x )o f
1 6p.
The t a s k is t o e x p l a i n t h e s e f i g u r e s : s e v e r a l approximations and c o r r e c t i o n s are now i n v e s t i g a t e d .
The basic " u n i f o n n - f i e l d " d e l . The s i m p l e s t approach is t o assume a f l a t s u r f a c e e m i t t e r d e l , w i t h a unifonn f i e l d ( ~ e x t ) above it. I n t h i s case, a displacement o f t h e atomic nucleus outwards b y a d i s t a n c e 6X causes t h e step-height o f t h e e l e c t r o n barrier t o be reduced by a n amount 6h
=e ~ e x t ô x , as i l l u s t r a t e d i n F i g . 2;
t h e energy o f the d e p a r t i n g i o n i s less by a n amount 6E a l s o e q u a ï t o e ~ e x t 6 x . F o r c r i t i c a l - s u r f a c e i o n i z a t i o n t h e s t e p h e i g h t is e q u a l t o the e m i t t e r work-function, assumed
4.5eV f o r t u n g s t e n . So, fram eq.(lO), s i n c e 6 Q n ( P e ) =
ph6h, t h e r a t e - c o n s t a n t falls t o h a l f i t s o r i g i n a l v a l u e i n an energy i n t e r v a l :
Using the STB v a l u e f o r
ph,we o b t a i n AE
;- 1.43eV. The HIC v a l u e is s l i g h t l y less, at 1.35 eV. A s i m i l a r f i g u r e ( 1 . 3 3 eV) was d e r i v e d i n Tsong and M u l l e r ' s o r i g i n a l c a l c u l a t i o n s [ l a ] . It was t h e d i s c r e p a n c y between t h i s f i g u r e and t h e observed h a l f - w i d t h s t h a t , i n t h e m i d - s i x t i e s , l e d t o t h e i n t r o d u c t i o n o f o v e r l a p i n t e g r a l formalisms .
Figure 2. B a r r i e r f i t t i n g wikh t h e elementary f l a t - s u r f a c e e m i t t e r d e l . B a r r i e r s 1 and 2 a r e t h e b a s i c t r a p e z o i d a l ' b a r r i e r s a p p l i c a b l e f o r imaging-gas atom n u c l e i a t p o s i t i o n s ( a ) and ( b ) r e s p e c t i v e l y , w i t h p o s i t i o n ( a ) b e i n g i n t h e c r i t i c a l s u r f a c e . FL denoLes t h e f e r m i l e v e l .
The e f f e c t o f s u r f a c e i n t e r a c t i o n s . A d e f e c t w i t h t h e STB, FFiG and EIK formalisms is t h a t t h e y d o not t a k e i n t o account t h e e l e c t r o n / s u r f a c e i n t e r a c t i o n terms i n t h e e l e c t r o n p o t e n t i a l energy. The most important of t h e s e is t h e image i n t e r a c t i o n , and R e f . [ l ] a r g u e s t h a t it can be s i m u l a t e d by u s i n g a t r a p e z o i d a l barrier of s l i g h t l y g r e a t e r step-height. I n effect, one m e s t h e s t e p o u t a b i t . R e f . [ l ] estimates t h a t , f o r c r i t i c a l - s u r f a c e i o n i z a t i o n i n t h e &
=4.5eV case, a n i n c r e a s e i n s t e p - h e i g h t o f between 3 and
4e V might be a p p r o p r i a t e .
F o r argument, l e t u s t a k e t h e e f f e c t i v e step-height a s
8eV. The corresponding Values o f
phare shown i n Table 1. (The GBM v a l u e is n o t shown because t h e
GBMformula a l r e a d y i n c o r p o r a t e s a s i m u l a t i o n o f e l e c t r o n - s u r f a c e e f f e c t s . ) The
SigJOURNAL DE PHYSIQUE
and HK formulae now lead t o a v a l u e s o f 1.08 e V and 1.01 e V r e s p e c t i v e l y ; t h e s e v a l u e s a r e c l o s e r t o t h e observed ones, b u t a r e still not s m a l l enough.
Obviously, t h i s approach is f a i r l y crude, b u t two p o i n t s deserve note:
( 1 ) With t h e STB, FHG and HK fonnulae, t h e e f f e c t o f t h i s simulation is t o i n c r e a s e t h e ph v a l u e . T h i s is i n marked c o n t r a s t t o t h e GBH. formula, where t h e attempt t o simulate e l e c t r o n / s u r f a c e i n t e r a c t i o n s ha8 l e d t o a reduction i n ph as compared w i t h t h e basic (h-4.5eV) STB case. mis c o n t r a s t i n c r e a s e s my r e s e r v a t i o n s about t h e u s e f u l n e s s o f t h e GBM formula.
( 2 ) Haydock and KinQham argue [16] t h a t t h e image i n t e r a c t i o n c m
besimulated i n t h e i r formalism by i n c r e a s i n g t h e s i z e of t h e i r " e f f e c t i v e chargew Z . This w i l l undoubtedly i n c r e a s e t h e value o f Pe i n a q u a l i t a t i v e l y c o r r e c t fashion. But the ph-formula derived from t h e i r formalism i s not Z-dependent, s o t h e i r proposa1 may not be f u l l y s a t i s f a c t o r y .
F i e l d v a r i a t i o n s near t h e c r i t i c a l s u r f a c e . An a l t e r n a t i v e explanation o f t h e small observed half-widths h a s been p u t forward i n Ref. [16]. Haydock and Kingham i n e f f e c t argue t h a t t h e uniform-field model does not adequately d e s c r i b e t h e f i e l d v a r i a t i o n above a p r o t r u d i n g s u r f a c e atom. I f , o u t s i d e t h e c r i t i c a l s u r f a c e , t h e l o c a l f i e l d F falls o f f r a p i d l y w i t h d i s t a n c e , then s m a l l e r Ax (and hence AE) v a l u e s might be e-cted. m e y assume f i e l d f a l l - o f f i n t h e fonn:
where t h e R is t r e a t e d as an adjustment parameter. I n terms of t h e p r e s e n t n o t a t i o n , t h e y i d e n t i f y t h e model-field f with t h i s f i e l d F and t a k e h as given by:
where x c r is t h e critical d i s t a n c e between t h e model s u r f a c e and t h e c r i t i c a l s u r f a c e . They can achieve a Ax v a l u e o f 16 p m i f R is taken as about 1 . 5 mn.
Now t h i s argument h a s some merit f o r i o n i z a t i o n above a protruding kink-site atom, b u t f o r i o n i z a t i o n about a f l a t ( 1 1 1 ) s u r f a c e 1 can s e e two d i f f i c u l t i e s with it.
F i r s t , Haydock and Kingham a r e i n e f f e c t f i t t i n g t r a p e z o i d a l b a r r i e r s by identi-ing t h e model-field f w i t h t h e l o c a l f i e l d F a t t h e p o s i t i o n o f t h e imaging-gas atom nucleus, as i l l u s t r a t e d i n Fig.B(A). This t y p e o f f i t t i n g procedure is lilrely t o overestimate the rate at which t h e barrier i n c r e a s e s with d i s t a n c e , and hence may overestimate t h e r a t e at which t h e rate-constant f a l l s o f f . ( T h i s would a l s o be t r u e f o r i o n i z a t i o n above a k i n k - s i t e atom.
)Second, f o r i o n i z a t i o n above t h e
(111) plane, an R-value as s m a l l as 1.5 nm does not s e e m e n t i r e l y p l a u s i b l e . C a l c u l a t l o n s c a r r i e d o u t using t h e Forbes and Wafi [20] charge-array model suggest t h a t it is d i f f i c u l t t o deduce a p l a u s i b l e R-value less than 10 nm. and t h a t t h e best v a l u e might b e somewhat higher.
I t is t h u s seems t h a t t h e Haydock and Kingham c a l c u l a t i o n s do not provide a s a t i s f a c t o r y explanation of observed half-widths, c e r t a i n l y not f o r i o n i z a t i o n above f a c e t s , and p o s s i b l y not f o r i o n i z a t i o n above k i n k - s i t e s .
Average-field v a r i a t i o n s . An a l t e r n a t i v e f i t t i n g procedure, i n which t h e mode1
f i e l d i s taken e q u a l t o t h e average f i e l d between t h e s t e p p o s i t i o n and t h e
gas-atom-nucleus p o s i t i o n , is proposed i n R e f . Cl]. T h i s procedure i s i l l u s t r a t e d
i n Fig.B(B). The model-field again f a l l s o f f as t h e gas-atom-nucleus p o s i t i o n m e s
outwards, b u t t h e f a l l - o f f rate is less than with t h e f i t t i n g method described i n
t h e preceding paragraphs. Preliminary c a l c u l a t i o n s , based on t h e Forbes-Wafi
s u r f a c e model and t h e STB formula ( b u t ignoring e l e c t r o n / s u r f a c e i n t e r a c t i o n s )
again produce energy h a l f - w i d t h e s t i m a t e s o f around 1 . 0 eV. I n o t h e r words, t h i s
approach, t o o , does n o t by i t s e l f provide a s a t i s f a c t o r y explanation o f observed
h a l f -widths .
Figure 3. Methods o f E l t t i n g t r a p e z o i d a l : m b t s
HHbarriers t o a non-linear model p o t e n t i a l ( r e p r e s e n t e d by t h e broken l i n e ) . I t is assumed t h a t t h e p o s i t i o n o f t h e s t e p h a s been chosen is some w e l l - d e f i n e d way. As
b e f o r e , b a r r i e r s 1 and 2 a r e t h e basic / t r a p e z o i d a l b a r r i e r s a p p l i c a b l e for /
imaging-gas atom n u c l e i a t p o s i t i o n s ( a ) FL---i- 1 and ( b ) , w i t h p o s i t i o n ( a ) b e i n g i n t h e
c r i t i c a l s u r f a c e .
A) F i t t i n g b y t a k i n g t h e d e l - f i e l d t o ( B)
b e e q u a l t o t h e l o c a l f i e l d a t t h e g a s atom nucleus.
B) F i t t i n g b y t a k i n g t h e model- f i e l d t o be e q u a l t o t h e average f i e l d between t h e p o s i t i o n o f t h e g a s atom n c u l e u s and t h e p o s i t i o n o f t h e step.
4 . Discussion
The f o l l o w i n g c o n c l u s i o n s c a n b e drawn from t h e above a n a l y s i s :
1 ) The "dimensionality" o f t h e JWKB formalism is not a p a r t i c u l a r l y important f a c t o r i n t h e c o n t e x t o f energy-half-width p r e d i c t i o n . I n f a c t , al1 t h e JWKB formalisms examined h e r e ( w i t h t h e p a r t i a l e x c e p t i o n o f t h e GBM formula) g i v e s i m i l a r r e s u l t s .
2 ) It seems more important ( i n t h e c o n t e x t o f half-width p r e d i c t i o n ) t o have a good model o f t h e e l e c t r o n b a r r i e r t h a n t o have a quantum-mechanically "good"
formalism.
3 ) A s compared with t h e s i m p l e "uniform f i e l d " model, p r e d i c t e d half-width v a l u e s c a n be reduced b o t h b y u s i n g a b e t t e r model o f t h e f i e l d v a r i a t i o n above a real s u r f a c e , and by i n c o r p o r a t i n g e l e c t r o n / s u r f a c e i n t e r a c t i o n s i n t o t h e t u n n e l l i n g b a r r i e r .
4 ) Half-width v a l u e s as low as t h o s e observed have n o t been s a t i s f a c t o r i l y p r e d i c t e d by the r a t h e r s i m p l e a n a l y t i c a l methods d e s c r i b e d h e r e .
N e x t s t e p s should probably be t o u s e more comprehensive b a r r i e r models, and t o employ numerical methods t o e v a l u a t e t h e corresponding JWKB i n t e g r a l s . A t p r e s e n t , 1 s e e no reason t o doubt t h a t s a t i s f a c t o r y e x p l a n a t i o n s o f half-widths c a n
befound i n t h e c o n t e x t o f one-dimensional JWKB formalisms.
There is a l s o . s o m e s u g g e s t i o n i n t h e above a n a l y s i s t h a t e n e r g y half-widths might be d i f f e r e n t f o r i o n i z a t i o n above a k i n k - s i t e and above a n imaged c r y s t a l f a c e t . W h i l s t t h e r e is l i t t l e i n d i c a t i o n o f t h i s i n r e s u l t s s o far p u b l i s h e d , it M g h t be o f i n t e r e s t f o r experiments t o look e x p l i c i t l y f o r such a n e f f e c t .
c l ] FORBES, R.G., J. Phys. D: Appl. Phys.10 (1985) 973.
[ 2 ]
