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THE CHEMICAL ROLE OF GRAINS IN THE INTERSTELLAR MEDIUM AND RELATED
PHYSICAL PROBLEMS
D. Williams
To cite this version:
D. Williams. THE CHEMICAL ROLE OF GRAINS IN THE INTERSTELLAR MEDIUM AND
RELATED PHYSICAL PROBLEMS. Journal de Physique Colloques, 1980, 41 (C3), pp.C3-225-C3-
232. �10.1051/jphyscol:1980335�. �jpa-00219854�
JOURNAL DE PHYSIQUE Coffoque C3, supplement au n o 4 , Tome 41, avri/ 1980, page C3-225
T H E C H E M I C A L ROLE OF G R A I N S I N THE I N T E R S T E L L A R MEDIUM AND R E L A T E D P H Y S I C A L PROBLEMS
D . A . Williams
Mathematics Department, U n i v e r s i t y o f Manchester, I n s t i t u t e o f Science and Technology, Manchtzster, Eng land.
1 . I n t r o d u c t i o n . - Over 80 m o l e c u l e s and i s o t o - p e s have now been i d e n t i f i e d i n t h e i n t e r - s t e l l a r medium ( s e e t a b l e I ) . Some o f t h e s e a r e found i n f a i r l y low d e n s i t y i n t e r s t e l l a r c l o u d s i n which t h e t o t a l g a s number d e n s i t y i s a b o u t 100cm-~ ; t h e s e a r e
H 2, CH, CH', CN, CO, and H2CO. The r e s t a r e normally found i n somewhat d e n s e r c l o u d s , - l o 3 t o - 1 0 ~ c m - ~ . The l a r g e s t m o l e c u l e s known a t p r e s e n t a r e q u i t e complex, and some have a q u i t e u n e x p e e t e d s t r u c t u r e , e . g . H-CEC-CiC-CEC-CEC-C-N.
A l l t h e m o l e c u l e s a r e d e s t r o y e d i n a f a i r l y s h o r t t i m e , by one o r more of a v a r i e t y o f p r o c e s s e s such a s p h o t o d i s s o c i a t i o n , o r che- m i c a l r e a c t i o n , e t c . , and f o r m a t i o n schemes a r e r e q u i r e d which can s u p p l y t h e s e molecu- l e s a t an a d e q u a t e r a t e . The q u e s t i o n we a r e concerned w l t h i n t h i s a r t i c l e i s t h i s : What c o n t r i b u t i o n t o t h i s a r r a y o f m o l e c u l e s
- and o t h e r s y e t t o be d i s c o v e r e d - i s made by c h e m i s t r y on t h e s u r f a c e s o f i n t e r s t e l l a r g r a i n s ? I n r e c e n t y e a r s , complicatedschemes i n v o l v i n g g a s p h a s e mechanisms have r e c e i v e d c o n s i d e r a b l e a t t e n t i o n and, i n d e e d , some workers have t e n d e d t o i g n o r e any p o s s i b l e c e ~ l t r i b u t i o n from s u r f a c e r e a c t i c n s . Is such a n a p p r o a c h j u s t i f i a b l e ? I s h a l l a r g u e t h a t it i s n o t , and i n t h i s a r t i c l e d e s c r i b e some r e c e n t work s u p p o r t i n g t h i s view.
There a r e a number o f r e a s o n s why a s t r o p h y - s i c i s t s have t e n d e d t o r e g a r d g a s phase r e a c t i o n s more f a v o u r a b l y . P r i m a r i l y , t h e s u r f a c e problem i s i l l - d e f i n e d s i n c e t h e na- t u r e o f t h e g r a i n s i s a s y e t u n e x p l a i n e d , a l t h o u g h one may hope t h a t i n f r a r e d o b s e r v a - t i o n s w i l l s h o r t l y d e f i n e t h e g r a i n m a t e r i a l p r e c i s e l y . By c o n t r a s t , g a s phase r e a c t i o n r a t e c o e f f i c i e n t s may o f t e n be measured i n t h e l a b o r a t o r y , o r , even i f t h i s i s n o t pos- s i b l e , a r e l i a b l e t h e o r e t i c a l e s t i m a t e may o f t e n be made. However, it i s c l e a r t h a t g a s
phase s c h e m e s a r e by themselves i n a d e q u a t e t o e x p l a i n t h e o b s e r v a t i o n s of i n t e r s t e l l a r m o l e c u l e s . Most s i g n i f i c a n t l y , no g a s phase scheme can p r o v i d e a s u f f i c i e n t l y f a s t H z f o p mation r o u t e to a c c o u n t f o r t h e o b s e r v e d abun- d a n c e s . By d e f a u l t , t h e r e f o r e , , a s t r o p h y s i - c i s t s a r e f o r c e d t o invoke f o r m a t i o n o f H z on g r a i n s u r f a c e s . The r a p i d r e c o m b i n a t i o n o f atoms on s u r f a c e s i s , a f t e r a l l , a fami- l i a r p r o c e s s to p h y s i c i s t s even i f t h e d e t a i l s a r e u n c l e a r . S i n c e g g a s phase chemical sche- mes depend u l t i m a t e l y on t h e p r e s e n c e of HZ., t h e n a l l i n t e r s t e l l a r chemistry depends on g r a i n s . However, g r a i n s may be n e c e s s a r y i n o t h e r ways, t o o , f o r g a s phase schemes have been shown t o be i n a d e q u a t e i n r e s p e c t of o t h e r m o l e c u l e s . Formaldehyde ( H 2 C O ) i n low d e n s i t y c l o u d s , where i t i s r a p i d l y des- t r o y e d by p h o t o d i s s o c i a t i o n , c a n n o t be f o r - med s u f f i c i e n t l y r a p i d l y i n g a s phase r e a c t i o n s . However, b o t h 0- and C-atoms c o l l i d e w i t h g r a i n s a t a s u f f i c i e n t l y h i g h r a t e , i f a s u i t a b l e mechanism e x i s t s t o form H2CO (Barlow and S i l k 1977 ; M i l l a r , Duley and W i l l i a m s 1 9 7 9 ) . A s i m i l a r d i f f i - c u l t y may a r i s e w i t h ammonia ( N H 3 ) i n dense c l o u d s . Gas phase schemes do seem t o b e s u c c e s s f u l i n e x p l a i n i n g abundances of d i a - tomic m o l e c u l e s , b u t i t i s c e r t a i n l y n o t c l e a r how t h e y c a n l e a d t o l a r g e u n s a t u r a - t e d m o l e c u l e s which we s e e l i s t e d i n t a b l e I .
2 . Formation o f H 2 a t g r a i n s u r f a c e s . - We s h a l l b e g i n by d i s c u s s i n g t h e p r o t o t y p e r e a c t i o n 2H+H2 on s u r f a c e s , w h i c h h a s been i n v e s t i g a t e d by many a u t h o r s ( s e e r e f e r e n - c e s i n t h e a r t i c l e by Dalgarno and Black 1 9 7 6 ) . There a r e a c c e p t e d m o d e l s f o r t h i s p r o c e s s , e i t h e r f o r t h e c a s e o f weakly a b s o r b i n g ( p h y s i s o r b i n g ) g r a i n s o r s t r o n g l y a b s o r b i n g (chemisorbing) g r a i n s . F o r g r a i n s
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980335
C3-226 JOURNAL DE PHYSIQUE
D.i a t o m i cs
Hz, HD, CH', l3cI!', CH, OFI, 1701i, '80t(, C2, CEI,
CO, l 3 c 0 . ~ 1 7 0 , C ~ Q , !3c180, NO, CS, I ~ C S , C ~ S , ~ 3 4 s . SiO, 2 9 S i 0 , 3 0 S i C , SO, 2 4 5 3 , KS. S i S
T r i a:oni cs
H20, HDO. H2180, C2H. HCN, DCN, Hl?l,il, H15CN, HNC, DNC, HNI3C, Hl5IdC, HCO'. CCG', H ~ ~ c o ' , ~ ~ ' 8 0 ' . HCO, N~H', N ~ D + , H ~ S , HNO. OCS, so2
4 - a tomi cs
KH3. NH2D, C2H2, H2C0, HZ1?CO, HZClbO, HNCO, H2CS. C3N
5-aton:i cs
CHqt CH2NH. CH2C0, NH2CN, FCOOh, C4H. HC3N, HI3CCCN. !iC1=CCN, HCCI3CN
6 - a t o n i cs
CH30H, '3CH30H, CH30@. CH3CN. NHZCHO
7 - a t o m i cs
CH3NH2, CH3NHD, CH3C2H, CH3CH0, CH2CHCN. HC5N
8 - a t o m i cs HCOOCH3
9 - a t o m i cs
CH3CH20H, (CH3)*0, Cti3CH2CN, HC,N
11 - 6 t c m i cs HCgN
T a b l e I : L i s t of I n t e r s t e l l a r Molecules c u r r e n t l y known.
which physisorb H-atoms, the picture has been developed by Hollenbach and Salpeter (1970).
The H-atom approaching the grain is attrac- ted by van der Waals forces, falls into a well of some -500K energy equivalent, and is bound to the surface with a probability -0.3. The bound H-atom is able to move over the surface by quantum mechanical diffusion, locates a site of enhanced binding caused by a dislocation, impurity, etc., and remains at this site long enough for the second H-atom to collide with the grain, stick, and locate the first. The two atoms interact, and use part of their energy of recombina-
tion to release themselves from the surface.
This mechanism requires that an H-atom finds an enhanced binding site before it evapora- tes, and that it remains there long enough for the second atom to find it. These requi- rements place restrictions on the grain temperature. Hollenbach and Salpeter conclu- de that if several enhanced binding sites exist per grain, and if the grain tempera- ture, T is less than --25K, then recornbi- nation is 100% efficient. Such a process, g' combined with the known photodestruction of H2, is sufficient to reproduce the observa- tions.
There is some indication that grains may be graphite, on which H-atoms are chemisorbed.
Barlow and Silk (1976) have shown that the recombination 2H+H2 proceeds efficiently if T 7 70K. The H2-molecule formed is not chemisorbed. Evaporation of the H-atoms is g in this case negligible, and the only re- quirements is that H-atom may "inspect" the whole surface. Recombination in the model of Barlow and Silk occurs via a rotational transition, followed by a rapid vibrational to kinetic energy transfer.
3.The formation of other molecules at the surfaces of grains.- The major uncertainty is whether such molecules will be released from the surface on formation from ahsorbed atoms. Some authors (e.g. Aannestad 1973, Watson and Salpeter 1972) assume that mole- cules containing C, N, and 0 will be retai- ned at the surface and released in regions of high radiation density or where sputte- ring by high speed atoms occurs. Others ta- ke the opposite view ; e.g. Barlow (1978) argues that for graphite or iron grains C, N, and 0 are rapidly converted to CH4, NH3, and H20, respectively, and in each case the energy release on formation is sufficient to eject the saturated molecule from the surface, but not the unsaturated radical.
The difficulties of two "heavy" atoms loca- ting each other on a surface have been em- phasized by Watson and Salpeter (1972).
Quantum mechanical tunnelling will c e r t a i ~ ly be less efficient than that for H-atoms.
Goldanskii (1979), however, considers that
mobility is significant even at very low
temperatures and even for polyatomic molecu- les. If "heavy" atoms have sufficient mobili- ty then reactions such as C + 0
+CO may oc- cur on the surface. If hot, then every
"heavy" atom will be visited by H-atoms and hydrides are formed. Whether such molecules are released from the surface is still uncer- tain ; Watson and Salpeter suggest that pho- toejection may be necessary. In dense clouds, therefore, molecules should all be locked up in grain mantles if the time available per- mits the accumulation of mantles. The release of molecules from grain surfaces may occur in other ways : a small grain absorbing a UV photon will have a rapid temperature fluctua- tion (Duley 1973) which may enhance the eva- poration of the mantle. Allen and Robinson
(1975) suggest that the energy release in forming an atom may disrupt the loosely bound material in a small grain.
4. - . ( The
probability of a gas atom or molecule sti- cking to a grain surface on collision is clearly a factor of crucial importance. It is frequently assumed that the probability for H-atomsticking is about G . 3 and for other
atoms and molecules it is about 1. There is some experimental evidence in support (Day 1972, Lee 1972). However, the difficulty of making such a measurement suggests that a ge- neral theory would be useful. Hollenbach and Salpeter (1970) were among the first to attempt such a calculation. Their model of the surface was a very simple one, a single classical oscillator, represented by a spring fixed at one end, with a mass at the other end. The spring constant was chosen to be Debye frequency of the solid.
Leitch-Devlin and Williams (1979) have a more sophisticated quantum-mechanical model of the process. The system considered is the atom + solid, and the model permits a calculation of the transition probability from an initial state, in which the atom is free and no pho- nons exist, to one in which the atom is bourd and simuLtaneously a phonon is excited. The potential energy for the atom-surface inter- action is formed by summing pairwise inter- actions between the incident atom and atoms of the lattice. The mathematical treatment
is similar to that now found in standard works (e.g. Goodman and Wachman 1976) al-
though these calculations are normally con- cerned with scattering of atoms, whereas Leitch-Devlin and Williams are concerned with sticking. The nature of the calcula- tion is a first order distorted wave Born approximation, but a higher order approxi- mation is also being calculated.
* s u r + o c e s t o t a ~ Srr wage
E N E R G V I k E N E R G Y I k
To.$ MI
Fig. 1. The sticking coefficient for the collision of H-atoms with graphite. The contributions from the surface modes, and the surface + bulk modes are shown, and the total averaged for a Maxwellian dis- tribution of gas velocities.
E N E R G Y / k E N E R G Y / k Tsar(K)
Fig. 2. Same as figure 1 , for H2.
Fig. 3. Same as figure 1, for Nz.
Figures 1, 2, and 3 show results for H, HZ,
and N 2 on graphite ; the contributions of
C3-228 JOURNAL DE PHYSIQUE
surface modes, surface + bulk modes, and for the total averaged over a Maxwellian distri- bution of gas velocities are given as a function of temperature. The influence of the bound levels within the potential well is clearly seen in the curves, but is washed out in the Maxwellian averaged curve. The molecu- lar nitrogen case has many bound levels and the effects are not so prominent. The H-gra- phite case represents chemisorption, and the H z and N 2 cases represent physisorption. As expected, where the binding is strong, as in chemisorption, the sticking probability is high. The sticking of HZ on graphite, for a gas tentperature of loOK, is quite small, and even for a molecule as massive as N z , with a large number of bound levels the sticking probability is only on the order of 0.1 (but increases rapidly to unity as the temperatu- re falls). The accretion of mantles on grains by the sticking of molecules from the sur- rounding gas will therefore take considera- bly longer than was previously thought to be the case. For clouds of total density 2 lo3
the accretion time is probably longer than the free-fall time of the cloud, and significant accretion of mantles does not occur (Duley and Williams 1979). Enhanced binding regions will tend to ahsorbmolecules more readily, but when this has occurred, their effect will be much reduced.
Mobility is the other very important parame- ter, and has been calculated by various au- thors, either by a semiclassical barrier pe- netration expression, or more properly by a quantum mechanical calculation of the width of bound energy levels. So far, only very crude estimates of mobility have been made.
More accurate quantum mechanical calculations are being carried out (Leitch-Devlin and Williams 1979). The question of mobility is considered again in section 5.
The nature of the interaction between ahsor- bed atoms and their ejection from the surfa- ce has not received detailed attention, al- thoughvarious suggestions have been made
(Williams 1968, Willis and Fitton 1975, Barlow and Silk 1976). Much more work needs to be done in this area.
5. A simple quantitative model of surface reactions.- It is obviously important to describe the parameters of sticking, mobi- lity, and interaction as accurately as pos- sible. Since mobility is a very uncertain parameter, it is a useful approach to test the sensitivity of molecule formation to variations in this parameter. This has been done in some detail by Pickles and Williams (1977a, b ; 1979). The model quantified by Pickles and Williams is essentially that of the physisorption model described in sec- tions 2 and 3 :
(i) atoms from the interstellar gas strike the surface, and remain bound to it, with a certain probability ;
(ii) absorbed atoms migrate over the surfa- ce with a given mobility ;
(iii) absorbed atoms interact to form mole- cules or radicals, the latter remaiaing bound and ultimately forming molecules ;
(iv) molecules are ejected from the surface either on formation or later.
Treating the surface population of atom XI Sx, as a continuous variable, we may write an equation for Sx
The first term on the right-hand-side re- presents the rate of collisions on the grain of atoms X in the gas where their number density is nx. The second term shows that atoms X are lost from the surface only in reaction with atoms Y ; the coefficient Y x ~ represents an interaction coefficient for these atoms, and contains an inter- action length and the surface mobility.
Pickles and Williams (1977a) have written
equations similar to (1) for H I C, N, 0
and various radicals formed from them, and
assuming chemical equilibrium have calcula-
ted the formation rate R ( ~ m - ~ s-') of mo-
lecules XY ejected into the gas phase. The XY
coefficients y have been varied indepen-
dently by very large factors (lo7). In low XY
density interstellar clouds (n = 100cm-~)
it is found that calculated rates of forma-
tion of HZ and hydrides are independent of
Table I1 : Formation rates s-') predicted for some molecules in low and high density clouds.
the surface properties as contained in y.
Rates of formation of XY are sensitive to y r but the rate is small in real terms ; see table 11. In high density clouds (n = lo4
~ m - ~ ) , H-atoms are a minor constituent and so the formation rate for H E is sensitive to all y terms. The rate at which all molecu-
XY
les other than H2 form again turns out to be independent of y.
These molecular formation rates (table 11) are significant in astrophysical terms. Sup- pose, for example, that all CHI, emerging from the grains is photodissociated to CH, and then to C + H. In a thin cloud, this oc- curs at a rate s-'. Then n(CH)/n(H)
and a column density of CH in the ran- ge 1012 to 1013 is contributed from this source. This is typical of the observed column density.
In fact, atoms feeding the grain reactions will also have gas phase reaction paths to follow, and the products of surface reac- tions will enter a complex network of gas phase reactions. However,although a number of simplifying assumptions have been made in this theory and specific reaction routes (as opposed to general) are excluded (see sec- tion 7), it is interesting that a crucial part of this theory is 'parameter independent.
6. The contribution of surface reactions to total interstellar chemistry.- It is now possible to test the importance of grain surface chemistry based on physisorption leze ding to indiscriminate molecule production.
The work is simpler if one restricts the chemistry to that relevant to a related set of atoms and molecules. We shall consider here the reactions occurring in low and me- dium density interstellar clouds which dis- tribute carbon among the three important
species C + , C, and CO, see figure 4. Grains may affect this chemistry by supplying CO directly, or by catalysing the formation of other molecules which by subsequent gas phase reactions lead to the formation of cO.
For example, grains may yield C H 4 , which photodissociates to give CH, which then reacts with 0-atoms, forming CO. Included in the model are the possible products of surface reactions C2, CO, CHu. In figure 5 are curves showing the variation of CO at the centre of a spherical 500M0 cloud of uniform density 100 < n ( ~ m - ~ ) < 2000, as a function of temperature. Grains may contri- bute at the rate calculated as described in the previous section, or their contribution may be excluded. The curves show clearly where grain chemistry may be important. ~t low densities the abundance of CO is unaf- fected by grain chemistry, but at higher densities grain chemistry dominates. For example, in a cloud of 1000cm-~ at a tempe- rature of 40K, the model of grain chemistry described here puts almost an order of ma- gnitude more carbon in CO than gas phase schemes. Clearly, such a marked change should provide a test for these models.
However, it is not yet possible to apply such a test, for - as the curves indicate -
the total chemistry is a very sensitive function of temperature and density in exactly the region of interest. Observations have not yet defined n and T sufficiently well to be able to discriminate between mo- dels with and without grain contributions.
The reason that CO is formed mainly by gas phase reactions in low density clouds is because of formation schemes such as :
-
depending on the abundance of OH which ari-
C3-230 J O U R N A L DE PHYSIQUE
ses as a result of the cosmic ray ionization of H-atoms, thus :
Most cosmic ray ionizatio~s of H lead to OH and hence to CO. Therefore CO is formed at a rate proportional to n, whereas grains form CO either directly or indirectly at a rate proportional to n2, and - though slower ini- tially - soon dominate as n is increased.
The nitrogen situation is different. No che- mical routes exist which lead to N-bearing molecules in a way comparable to those lea- ding to CO. Indeed, gas phase routes to amrno- nia are generally slow. If grain surface che- mistry plays a role, then it dominates the chemistry of N-bearing molecules, and mole-
cular abundances even in low density clouds 2 0 40 T 60 80 are larger by one or two orders of magnitude.
Nevertheless, there is no conflict with ob-
servations. F i g . 5. The number d e n s i t y of CO a t the c e n t e r of a 500M0 c l o u d , a s a f u n c t i o n of t e m p e r a t u r e , f o r d i f f e r e n t cloud d e n s i t i e s . S o l i d l i n e s
:g r a i n chemistry excluded ; dashed l i n e s : g r a i n chemis-
h3 try i n c l i d e d .
0
( < 100A) particles of diatomic oxides such
as MgO, FeO, SiO (Duley 1975, 1976, 1977).
Duley and Millar (1978) have shownthat such particles can provide a natural explanation for the selective depletion of elements in the interstellar medium. Such particles would also give rise to the so-called loand 20 pm features in the infrared (Millar and Duley 1978) .
Alkaline earth oxides are wellknown chemical
Fig. 4. The C + - C - CO network of chemical r e a c t i o n s .
7. A specific grain model.- The discussion so far has been limited to grain models in which the grain material is unimportant ;
the grain surface merely provides a weakly or strongly binding substrate on which reac- tions may occur. We now consider a different situation in which the nature of the grains is specified in a detailed way.
It has been proposed that interstellar grains or a component of them may consist of small
catalysts and have been the subject of many studies. Their chemical reactivity is asso- ciated with particular surface sites whose nature has been elucidated by IR and ESR spectroscopy. The simplest model of a per- fect diatomic oxide crystal isa cubic form, similar to NaQ, in which the oxygen is 02- and the metal M'+. All crystals contain de- fects, however, and these form the reactive sites in metal oxides. Figure 6 (from Duley, Millar and Williams 1978)shows some defect centers known or believed to exist in me- tal oxides. The notation used is standard.
The net charge at a centre is found by
summing charges on the centre and others
associated with it. reactions at sites on interstellar oxide grains. These are summarized in table 111.
II
From the table, we can immediately identify
r o n o n (011-)
+3 0
O Yvs7-
a catalytic site at which recombination of
o n o n n O P O P I O
H-atoms occurs :
n-
H+Here, we may be describing the enhanced
n o o n n O H e n Fst(" binding site arbitrarily introduced by
o a o t ~ o o t ~ o n o Hollenbach and Salpeter.
Oxygen atoms may be incorporated into the
II
0- o n v(o-)s- n o n
E C \IFs grain to replace those lost in chemical
o t l o n o o n o n o
reaction ; e.g. :
Fig. 6. Defect centres in metal oxlde grains
Alternatively, 0-atoms may be incorporated in gas molecules in reactions such as The absence of M ~ + thus yields a net negati-
ve charge of 2 units, and the absence of 0 ' - X+ + '(OH-); - v2- s F : + + ( x o H ) ~ ~ ~ . yields a net positive charge of 2 units. The
sites (0-1; and "(0-)- are known to be Carbon ions will react readily at negative s
reactive, and in the presence of hydrogen sites :
(as in the interstellar medium) will be con-
verted to (OH-): and "(OH-);. By appealing C+ + '(OH-); - v 2 - s P : + + (HCO) gas
to the literature we can predict the likely
I
F ~ + (HI (H-) + F ~ + + HZ (gas) -- - -
FS +s
to) (0-1 s+ (OH-) S+ O. . . (OH-) sf 0.. . v ~ ~ ~ - ) S- F, + + OH(gas)
"
-- - - - - - - VS '- FS '+ +HCO (gas
2 +
+ + + v
N.'. .FS +NH(gas) N..-(oH-)~ N...
N.. .FS F ~ - +
( N H 1, 1 I
P Hz CO
Table 111.- Summary of possible surface reactions at various defect sites in interstellar oxide grains.
-- -- - - --
vs 2- FS2++ H~O(gas)
2 + + + + -
Hz.. .FS Hz.. .FS Hz.. . (H-) Hz.. . (OH-) Hz.. OH-)^
+ + +
C0...FS2+ CO. . . FS CO.. . (H-1 CO.. . (OH-) CO.. .V(~~') S-
- Fs + + HCO(gaS)
c c...F~~+ c.. . F ~ + F ~ + + CH (gas) c.. . (OH-) + c.. OH-)^-
2 +
pS (c) (c-1 (CH-) s+
C3-232 JOURNAL DE PHYSIQUE
and the product may be formaldehyde (H2CO) Hollenhach, D. and Salpeter, E.E., J. Chem.
if hydrogen is also weakly bound at the va- Phys. 53 (1970) 79.
cancv. Lee, T.J., Nature Phys. Sci. 237 (1972) 9 9
~.d-
