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THEORETICAL CALCULATIONS OF
RATE-DETERMINING STEPS FOR IGNITION OF SHOCKED CONDENSED NITROMETHANE AND
PETN
R. Bardo
To cite this version:
R. Bardo. THEORETICAL CALCULATIONS OF RATE-DETERMINING STEPS FOR IGNITION OF SHOCKED CONDENSED NITROMETHANE AND PETN. Journal de Physique Colloques, 1987, 48 (C4), pp.C4-265-C4-279. �10.1051/jphyscol:1987419�. �jpa-00226651�
JOURNAL DE PHYSIQUE
Colloque C4, suppl6ment au n"9, Tome 48, septembre 1987
THEORETICAL CALCULATIONS OF RATE-DETERMINING STEPS FOR IGNITION OF SHOCKED CONDENSED NITROMETHANE AND PETN
R.D. BARD0
Naval Surface Weapons Center, White Oak Laboratory, Silver Spring, PlD 20903-5000, U. S. A.
RBsumC:
Une analyse compl6te theorique de l'ignition par choc des liquides et des solides demande une connaissance des Btapes determinantes pour la vitesse, likes aux processus de transfert d'6nergie vibrationnelle et aux reactions chimiques. Malheureusement, les Btudes anterieures de ces &tapes ont kt6 entravees par l'emploi de cinetiques de &action non r6alistes et par la complexit6 des methodes utilisees pour le calcul des vitesses d'echange
d'6nergie. On d6crit ici une theorie plus simple de mecanique quantique que l'on applique au calcul des vitesses de transfert d18nergie de vibrations inter-intra molitculaires induit par choc dans le nitrom6thane en phase condens6e et dans le PETN. Une grandeur importante dans cette theorie est le temps tpv d'activation
des modes i n t e r n e s . Les v a l e u r s c a l c u l 6 e s de t p v < 1 p s s o n t i d e n t i f i e e s au
transfert le plus efficace de l'bnergie quand les mol6cules de nitromethane interagissent et rdagissent bimol6culairement et exothermiquement dans la configuration t&te b6che. Ces temps sont coherents avec ceux trouves dans les exp6riences d'initiation par choc. Une comparaison des t,, avec les temps de vie de reaction chimique indique que le transfert d'energie et la reaction
bimol6culaire se succbdent comme regulateurs de vitesse dans tout le domaine de pression de choc et de temperature respectivement : 10 d P d 80 kbar et 335 S.T d 850 K. A partir de ces resultats, on trouve qu'une pression minimale de P = 58 kbar est necessaire pour l'initiation d'une detonation ti grande vitesse du nitromdthane homogbne. Quant au PETN, la th6orie fournit aussi une explication pour interpreter la sensibilite directionnelle observee dans les monocristaux.
Abstract
A complete t h e o r e t i c a l a n a l y s i s o f t h e i g n i t i o n o f shocked l i q u i d s and s o l i d s r e q u i r e s a knowledge o f t h e r a t e - d e t e r m i n i ng s t e p s corresponding t o v i b r a t i o n a l energy t r a n s f e r processes and chemical r e a c t i o n s . U n f o r t u n a t e l y , p a s t - s t u d i e s o f t h e s e steps have been h i n d e r e d by t h e use o f u n r e a l i s t i c r e a c t i o n k i n e t i c s and by t h e c o m p l e x i t y o f t h e methods used f o r c a l c u l a t i n g r a t e s o f energy exchange. Here, a s i m p l e r quantum mechanical t h e o r y i s d e s c r i b e d and u t i l i z e d f o r t h e c a l c u l a t i o n o f r a t e s o f i n t e r m o l e c u l a r - t o - i n t r a m o l e c u l a r v i b r a t i o n a l energy t r a n s f e r i n shocked condensed nitromethane and PETN. An i m p o r t a n t q u a n t i t y i n t h i s t h e o r y i s t h e t i m e t r e q u i r e d f o r a c t i v a t i o n o f t h e i n t e r n a l modes. C a l c u l a t e d values o f t < 1 p!gc a r e i d e n t i f i e d w i t h t h e most e f f i c i e n t t r a n s f e r o f energy when n i t r o g g t h a n e molecules i n t e r a c t and r e a c t bimol e c u l a r l y and e x o t h e r m i c a l l y i n t h e h e a d - t o - t a i 1 c o n f i g u r a t i o n . These t i m e s a r e c o n s i s t e n t w i t h t h o s e found i n shock i n i t i a t i o n experiments. A comparison o f t w i t h chemical r e a c t i o n h a l f - l i v e s i n d i c a t e s t h a t energy t r a n s f e r and the.
bymolecular r e a c t i o n a l t e r n a t e as r a t e - d e t e r m i n i n g s t e p s over t h e shock pressure and temperature ranges o f 10 < P < 80 k b a r and 335 < T c 850 K. From t h e s e r e s u l t s , i t i s found t h a t a minimum pressure o f P = 58 kbar i s r e q u i r e d f a r t h e i n i t i a t i o n o f h i g h - v e l o c i t y d e t o n a t i o n o f homogeneous nitromethane. F o r PETN, t h e t h e o r y a l s o p r o v i d e s an e x p l a n a t i o n f o r t h e observed d i r e c t i o n a l s e n s i t i v i t y i n s i n g l e c r y s t a l s .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987419
JOURNAL DE PHYSIQUE
INTRODUCTION
Shock-induced v i b r a t i o n a l a c t i v a t i o n processes have been s t u d i e d much more e x t e n s i v e l y f o r gases t h a n f o r l i q u i d s and s o l i d s . F o r t h e low-1 i n g v i b r a t i o n a l s t a t e s o f d i a t o m i c and t r i a t o m i c molecules, l e l 'd o v i c h and R a i z e r have reviewed i n d e t a i l t h e t h e o r y o f r e l a x a t i o n o f v i b r a t i o n a l energy toward complete
thermodynamic e q u i l i b r i u m b e f o r e and d u r i n g m o l e c u l a r d i s s o c i a t i o n . They have d i s c u s s e d v i b r a t i o n a l re1 a x a t i on c o r r e s p o n d i n g t o v i b r a t i o n a l - t r a n s 1 a t i o n a l (V- T), v i b r a t i o n a l - v i b r a t i o n a l (V-V), and v i b r a t i o n a l - r o t a t i o n a l (V-R) energy exchange processes. F o r d i a t o m i c s , t h e s e a u t h o r s have d i s c u s s e d t h e r e s u l t s o f shock t u b e experiments which i n d i c a t e t h a t r e l a x a t i o n does n o t i n t e r f e r e w i t h r e a c t i o n , s i n c e e q u i l i b r i u m i n t h e v i b r a t i o n a l degrees o f freedom i s e s t a b l i s h e d w i t h i n a p e r i o d of t i m e which i s s h o r t compared t o t h a t f o r d i s s o c i a t i o n . I n g e n e r a l , t h e o r y and experiment have shown t h a t d i s s o c i a t i o n and r e l a x a t i o n r a t e s f o r t h e l o w - l y i n g v i b r a t i o n a l l e v e l s o f gases i n shock t u b e s f o l l o w t h e o r d e r , d i s s o c i a t i o n < V-R < V-T < V-V.
F o r t h e h i g h l e v e l s o f v i b r a t i n a l e x c i t a t i o n c h a r a c t e r i s t i c o f
d i s s o c i a t i o n , Tardy and R a b i n o v i t c h g have reviewed t h e c o l l i s i o n a l a c t i v a t i o n and d e a c t i v a t i o n o f p o l y a t o m i c m o l e c u l e s as w e l l as t h e a p p l i c a b i l i t y o f RRKM ( R i c e - Ramsperger-Kassel -Marcus) t h e o r y t o t h e s t u d y o f unimol e c u l a r r e a c t i o n s . These a u t h o r s have i n d i c a t e d t h a t RRKM b e h a v i o r , ' w h i c h r e q u i r e s t h a t i n t r a m o l e c u l a r and i n t e r m o l e c u l a r V - V t r a n s f e r must be s u b s t a n t i a l l y f a s t e r t h a n decomposition, may be v i o l a t e d i n shock t u b e s a t v e r y h i g h temperatures. Here, t h e t i m e s c a l r a n d o m i z a t i o n o f v i b r a t i o n a l energy c o u l d be s i g n i f i c a n t l y 1 a r g e r t h a n 10-f2fz:c.
f o r c o l l i s i o n - a c t i v a t e d systems. I n t h i s case, t h e o r d e r g i v e n above f o r r e l a x a t i o n r a t e s would n o t be obeyed.
When t h e d i s s o c i a t i o n r a t e i s f a s t e r t h a n t h e r a t e o f e x c i t a t i o n o f t h e h i g h - l y i n g v i b r a t i o n a l l e v e l s , t h e k i n e t i c s o f b o t h t y p e s o f processes must be c o n s i d e r e d t o g e t h r I n o r d e r t o address t h i s s i t u a t i o n f o r p o l y a t o m i c
no1 e c u l es, E y r i n g g i n t r o d u c e d " s t a r v a t i o n k i n e t i c s " i n which he assumed t h a t t h e r a t e - d e t e r m i n i n g s t e p would be t h e f l o w o f energy from an e n e r g y - d e f i c i e n t r e s e r v o i r o f v i b r a t i o n a l degrees o f freedom i n t o a hidden bond t o be broken.
Here, he c o n s i d e r e d t h e b r e a k i n g bond t o be i n e q u i l i b r i u m w i t h t h e r e s e r v o i r , which i t s e l f i s n o t i n e q u i l i b r i u m w i t h t h e t r a n s l a t i o n a l degrees o f freedom o f t h e gas i# t h e shock tube. An e a r l i e r t h e o r y o f gas k i n e t i c s was developed by P r i t c h a r d who a1 so showed t h a t anomalously-1 ow A r r h e n i u s a c t i v a t i o n e n e r g i e s f o r gases i n shock t u b e s c o u l d be a c r i b e d t o n o n e q u i l i b r a t e d v i b r a t i o n a l degrees o f freedom. More r e c e n t l y , T a r v e r S has i n t r o d u c e d n o n e q u i l i b r i u m models f o r t h e one-dimensional , s t e a d y - s t a t e d e t o n a t i o n wave i n gases, 1 iq u i d s , and s o l i d s . I n t h e s e models, he has c o n s i d e r e d t h e d e t o n a t i o n wave t o c o n s i s t o f f o u r zones. I n one o f t h e zones f o r condensed systems, he a p p l i e d " s t a r v a t i o n k i n e t i c s " t o d e s c r i b e t h e slow, e q u i l i b r a t e d e x c i t a t i o n o f endothermic bond b r e a k i n g r e a c t i o n s .
Other methods, which a r e d i f f e r e n t from t h e ones mentioned above, have been u t i l i z e d t o analyze t h e a c t i v a t i o n and d e a c t i v a t i o n o f i n t r a m o l e c u l a r and i n t e r m o l e c u l a r i b r a t i o n a l degrees o f freedom. I n t h e case o f shock i n i t i a t i o n , P a s t i n e , e t al.' suggested t h a t t h e shock p u l s e d u r a t i o n r e q u i r e d f o r t h e growth t o d e t o n a t i o n may be i d e n t i f i e d w i t h t h e t i m e r e q u i r e d t o achieve a c r i t i c a l i n t e r n a l temperat r e o f t h e m o l e c u l e s c m p r i s i n g an e x p l o s i v e . Subsequent1 y ,
Z e r i l l i and ~ o t o n ~ and C o f f e y and Totons developed quantum mechanics?, methods t o c a l c u l a t e v i b r a t i o n a l e x c i t a t i o n r a t e s a r i s i n g from t h e coup1 i n g o f c o l d "
m o l e c u l a r and " h o t " l a t t i c e modes. I n t h e Z e r i l l i - T o t o n model, t h e s t u d y o f t h e e x c i t a t i o n o f t h e l o w - l y i n g s t a t e s o f shocked c r y s t a l 1 in e n i t r o m e t h a n e as based on a mathematical framework which i s s i m i l a r t o t h e one employed by L i n g t o d e s c r i b e v i bron-phonon re1 a x a t i o n i n condensed m a t e r i a l s . F o r n i t r o m e t h a n e , Z e r i l l i - T o t o n concluded t h a t t h e 0 - 1 t r a n s i t i o n r a t e was slower t h a n t h e h i g h e r e x c i t a t i o n s 1-2, 2 - 3 , . . . f o r t h e l o w e s t - f r e q u e n c y i n t e r n a l mode. I n t h e C o f f e y - Toton model , mu1 t i p h o n o n e x c i t a t i o n a r i s i n g from compressive waves was s t u d i e d .
T h i s model was a p p l i e d t o t h e e x c i t a t i o n o f t h e e n t i r e m a n i f o l d o f i n t e r n a l v i b r a t i o n a l l e v e l s l e a d i n g t o m o l e c u l a r d i s s o c i a t i o n . I n t h e f o l l o w i n g s e c t i o n o f t h e p r e s e n t paper, a n o t h e r quantum mechanical t h e o r y o f energy t r a n s f e r i s presented which i n c o r p o r a t e s some o f t h e f e a t u r e s from t h e Z e r i l l i - T o t o n and Coffey-Toton models.
The i m p l i c a t i o n i s o f t e n made t h a t , d u r i n g shock i n i t i a t i o n and d e t o n a t i o n , m o l e c u l a r d i s s o c i a t i o n proceeds as a s i m p l e unimol e c u l a r r e a c t i o n . However, a t t h e h i g h shock p r e s s u r e s which may be generated i n l i q u i d s and c r y s t a l s ,
u n i m o l e c u l a r bond s c i s s ' n r e a c t i o n s a r e o f t e n so s l o w t h a t o t h e r , f a s t e r r e a c t i o n s a r e dominant.I8 The r a t e s o f t h e s e r e a c t i o n s and t h e i n t e r m o l e c u l a r - t o - i n t r a m o l e c u l a r energy t r a n s f e r processes g i v i n g r i s e t o them must t h e n be compared i n o r d e r t o p e r m i t i d e n t i f i c a t i o n o f t h e r a t e - d e t e r m i n i n g step. I n t h e s e c t i o n on c a l c u l a t i o n o f r a t e s o f energy t r a n s f e r and r e a c t i o n , comparisons w i l l be made between r e a c t i o n h a l f - l i v e s o f a new i n i t i a l r e a c t i o n f o r n i t r o m e t h a n e and t h e t i m e s o b t a i n e d from t h e t h e o r y d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n .
THEORY OF VIBRON-PHONON ENERGY TRANSFER R e l a x a t i o n Processes and t h e Master E q u a t i o n Approach
Quantum r e 1 a x a t i o n processess i n atoms, molecules, o r condensed m a t e r i a l s a r e g e n e r a l l y t r e a t e d by c o n s i d e r i n g t h e i n t e r a c t i o n between two o r more zero- o r d e r subsystems which comprise t h e complete p h y s i c a l system. F o r i s o l a t e d atoms and molecules, t h e re1 a x a t i o n o r r a d i a t i o n l e s s processes a r e a u t o i o n i z a t i ons, unimol ecul a r chemical d i s s o c i a t i o n r e a c t i o n s as t r e a t e d by RRKM t h e o r y , m o l e c u l a r p r e d i s s o c i a t i o n , and i n t r a m o l e c u l a r r a d i a t i o n l e s s t r a n s i t i o n s whi h i n v o l v e a change i n t h e e l e c t r o n i c and v i b r o n i c s t a t e s o f a l a r g e molecule." A11 o f t h e s e processes a r e i n t e r p r e t e d i n terms o f energy t r a n s f e r between two subsystems o r s e t s o f l e v e l s w i t h t h e f a s t e s t f l o w o c c u r r i n g i n t o t h e denser set. F o r
m o l e c u l a r l i q u i d s and s o l i d s , t h e s e subsystems may be t a k e n t o be t h e m a n i f o l d o f i n t r a m o l e c u l a r v i b r a t i o n a l l e v e l s , and a denser s e t o f i n t e r m o l e c u l a r l e v e l s which, i n t h e case o f c r y s t a l s , corresponds t o l a t t i c e modes. F o r s t r o n g l y - i n t e r a c t i n g molecules i n shocked, condensed systems such as t h e n i t r i c o x i d e s , n i t r o a l ip h a t i c s , and n i t r o a r o m a t i c s , i t has been suggested t h a t i n t r a m o l e c u l a r r a d i a t i o n l e s s t r a n s i t i o n s can a1 so occurl$s a consequence o f l a r g e r e d s h i f t s o f t h e l o w - l y i n g e x c i t e d e l e c t r o n i c s t a t e s .
For t h e s t u d y o f v i b r a t i o n a l r e l a x a t i o n i n condensed systems, ~ i n ' ~ has employed t h e g e n e r a l i z e d master e q u a t i o n i n t h e form
where p e t c . a r e t h e d i a g o n a l elements o f t h e d e n s i t y m a t r i x , Kmmnn(t) i s a memory F t r n e l , and k,,,,(t) i s a r a t e c o n s t a n t d e f i n e d by
By u s i n g p e r t u r b a t i o n t h e o r y , he has expanded Eq. ( l a ) t o i n c l u d e c o n t r i b u t i o n s t h r o u g h f o u r t h o r d e r . W i t h t h i s approach, he has shown t h a t t h e f i r s t - o r d e r c o n t r i b u t i o n vanishes and t h a t t h e h i g h e r o r d e r s may be i d e n t i f i e d w i t h Paul i -
t y p e master e q u a t i o n s where t h e r a t e c o e f f i c i e n t s a r e expressed i n t h e "golden r u l e " form, and where t h e i n i t i a l and f i n a l m o l e c u l a r o s c i l l a t o r s t a t e s a r e p e r t u r b e d . S t i l l h i g h e r o r d e r a p p r o x i m a t i o n s i n c l u d e t h e non-Markovian o r memory e f f e c t s .
JOURNAL DE PHYSIQUE
A l l o f t h e s e o r d e r s o f a p p r o x i m a t i o n address t h e t o t a l i t y o f energy exchange events f o r s t r o n g c o u p l i n g o f t h e m o l e c u l a r o s c i l l a t o r s t o t h e l a t t i c e modes.
These e v e n t s i n c l u d e b o t h s e q u e n t i a l and d i r e c t processes. S e q u e n t i a l processes i n v o l v e a t l e a s t two m o l e c u l a r o r v i b r o n i c modes which a r e coupled t o t h e l a t t i c e o r phonon modes. I n t h e case o f v i b r o n d e a c t i v a t i o n , i n t r a m o l e c u l a r and v i b r o n - phonon c o u p l i n g r e s u l t i n c o n v e r s i o n o f h i g h t o l o w e r frequency v i b r o n modes f o l l o w e d by vibron-phonon induced r e l a x a t i o n o f t h e l o w e r frequency v i b r o n modes and e x c i t a t i o n o f phonons. D i r e c t processes, on t h e o t h e r hand, i n v o l v e o n l y one v i b r o n mode which i s c o u p l e d t o t h e phonon modes. T h i s s i t u a t i o n o c c u r s f o r v i b r a t i o n a l r e l a x a t i o n o f d i a t o m i c s and f o r t h e l o w e s t frequency modes o f l a r g e r p o l y a t o m i c s . F o r a c t i v a t i o n o f t h e v i b r o n modes t h r o u g h vibron-phonon c o u p l i n g , t h e r e v e r s e o f t h e above two-stage process occurs i n which phonon modes a r e d e a c t i v a t e d f l l o w e d by v i b r o n i g a c t i v a t i o n . 9 T h i s i s t h e s i t u a t i o n d e s c r i b e d by Z e r i l l i - T o t o n and Coffey-Toton , who used methods which may be shown t o be e q u i v a l e n t t o t h e v a r i o u s t y p e s o f P a u l i master equations.
R e l a x a t i o n Processes and t h e D e n s i t y o f S t a t e s Approach
Bardo and w o l f s b e r g 1 4 have shown t h a t t h e n o n r e l a t i v i s t i c Schr'ddinger wave e q u a t i o n f o r a s e t o f c o o r d i n a t e s f i x e d i n a t r a n s l a t i n g and r o t a t i n g m o l e c u l a r system can be w r i t t e n i n t h e form
I n t h i s s e t o f (25 + 1) coupled e q u a t i o n s , J and s a r e t h e quantum numbers o f t h e t o t a l a n g u l a r momentum and t h e p r o j e c t i o n o f t h e t o t a l a n g u l a r momentum o n t o t h e z - a x i s o f a r o t a t i n g c o o r d i n a t e system, r e s p e c t i v e l y . T, T+, T-, T++, and T-- a r e k i n e t i c energy o p e r a t o r s , t h e forms o f which depend on t h e c h o i c e o f c o o r d i n a t e system r o t a t i n g w i t h t h e m o l e c u l a r system. T+, T-, T++, T-- a r i s e f r o m t h e C o r i o l i s - 1 i k e i n t e r a c t i o n s between t h e r o t a t i o n a l , e l e c t r o n i c , and v i b r a t i o n a l motions. V i s t h e usual e x p r e s s i o n f o r t h e p o t e n t i a l energy which i n c l u d e s t h e e l e c t r o s t a t i c p o t e n t i a l energy terms among e l e c t r o n s and t h e N n u c l e i . E i s t h e t o t a l energy ( e x c l u d i n g t h e energy o f t h e center-of-mass m o t i o n ) o f t h e r o t a t i o n a l and i n t e r n a l motions o f t h e system.
When J=s=O i n Eq. (2a), t h e e l e c t r o n i c s t a t e o f t h e m o l e c u l a r system i s nondegenerate, and a l l m o l e c u l a r r o t a t i o n s w i t h i n t h e system may be assumed t o be quenched so t h a t Eq. (2a) reduces t o
T h i s a p p r o x i m a t i o n , which w i l l be used i n t h e sequel, i s an e x c e l l e n t one even a t h i g h temperatures i n condensed systems which have been s u b j e c t e d t o h i g h
pressures.
E q u a t i o n ( 2 b ) i s a p p l i c a b l e t o an e n t i r e assemblage o f m o l e c u l e s i n condensed systems. F o r t h e s e systems, each w a v e f u n c t i o n Y may be expanded i n terms o f a complete s e t o f orthonormal e l e c t r o n i c w a v e f u n c t i o n s aK,
where FK i s a f u n c t i o n , o f t h e 3N-6 c o o r d i n a t e s o f t h e n u c l e i , and OK i s a f u n c t i o n o f t h e s e n u c l e a r c o o r d i n a t e s and t h e c o o r d i n a t e s o f t h e e l e c t r o n s . Here, t h e s e t o f w a v e f u n c t i o n s {QK) determines t h e ground and e x c i t e d s t a t e
e l e c t r o n i c b e h a v i o r f o r t h e complete system. I n m o l e c u l a r s o l i d s , f o r example, l a K } d e f i n e s t h e band s t r u c t u r e . The s e t o f w a v e f u n c t i o n s {F 1 i n Eq.
( 3 ) determines t h e v i b r a t i o n a l b e h a v i o r f o r t h e complete system and, f o r s o l i d s , i n c l u d e s a l l v i b r o n and phonon c o n t r i b u t i o n s . Thus, when Eq. (3) i s a s o l u t i o n o f Eq. ( 2 b ) , t h e e l e c t r o n i c - v i b r a t i o n a l d e s c r i p t i o n o f t h e condensed system i s compl ete.
The s o l u t i o n o f Eq. (2b) i s g r e a t l y s i m p l i f i e d i f {aK} and {FK} a r e chosen i n such a way t h a t t h e number o f terms i n Eq. ( 3 ) i s as small as p o s s i b l e . I n t h e s o - c a l l e d a d i a b a t i c s o l u t i o n , o f which t h e Born-Oppenheim r (BO)
a p p r o x i m a t i o n and t h e a d i a b a t i c c o r r e c t i o n a r e s p e c i a l cases ,e4 Eq. ( 3 ) i n c l u d e s a s i n g l e t e r m
where FKr i d e n t i f i e s t h e r - t h v i b r a t i o n a l s t a t e i n t h e K - t h e l e c t r o n i c s t a t e
". These a p p r o x i m a t i o n s a r e e x c e l l e n t ones when QK v a r i e s s l o w l y w i t h n u c l e a r displacement. I n t h e s o - c a l l e d n o n a d i a b a t i c s o l u t i o n , t h e v a r i a t i o n o f QK i s l a r g e and a t l e a s t two terms a r e needed i n Eq. ( 3 ) t o d e s c r i b e v i b r o n i c c o u p l i n g i n i n t r a m o l e c u l a r r a d i a t i o n l e s s t r a n s i t i o n s , and i n t h e o t h e r e l e c t r o n i c
r e l a x a t i o n processes mentioned p r e v i o u s l y . I n o r d e r t o simp1 i f y t h e subsequent a n a l y s i s , t h e p r e s s u r e - i nduced e l e c t r o n i c n o n a d i a b a t i c processes d e s c r i b e d i n Ref. 12 w i l l be i g n o r e d so t h a t Eq. ( 4 ) a p p l i e s . Also, i t w i l l be assumed t h a t t h e shocked m o l e c u l a r system e x i s t s i n i t s ground s t a t e o n l y . I n t h e sequel, t h e s u b s c r i p t K i n Eq. ( 4 ) w i l l be dropped.
The l a r g e r a t i o R o f t h e e l e c t r o n i c " f r e q u e n c y " t o v i b r a t i o n a l frequency, 100<R<1000, a l l o w s t h e s e p a r a t i o n o f e l e c t r o n i c and v i b r a t i o n a l m o t i o n s t o b e made i n t h e a d i a b a t i c a p p r o x i m a t i o n , Eq; (4). Analogously, t h e l a r g e r a t i o R' o f v i b r o n f r e q u e n c y t o phonon frequency, R =lo, p e r m i t s t h e a d i a b a t i c s e p a r a t i o n o f t h e i n t r a m o l e c u l a r and i n t e r m o e u a r v i b r a t i o n a l m o t i o n s t o be made i n t h e l o w e s t o r d e r o f approximation.by5*4 i n t h i s a p p r o x i m a t i o n , t h e w a v e f u n c t i o n F,.
i n Eq. ( 4 ) becomes t h e s i m p l e p r o d u c t
where $(q,Q) and h r ( Q ) a r e v i b r o n and phonon w a v e f u n c t i o n s , r e s p e c t i v e l y . The s e t s o f c o o r d i n a t e s f o r t h e i n t r a m o l e c u l a r and i n t e r m o l e c u l a r c o n f i g u r a t i o n s o f t h e n u c l e i a r e g i v e n by q and Q, r e s p e c t i v e l y . Here, t h e f u n c t i o n s $,,(q,Q) a r e t a k e n t o be s o l u t i o n s o f t h e SchrBdinger e q u a t i o n f o r t h e v i b r o n subsystem.
where T(q), V(q,Q), and Em(Q) are, r e s p e c t i v e l y , t h e k i n e t i c energy o p e r a t o r f o r t h e m o l e c u l a r o s c i l l a t o r s , t h e t o t a l p o t e n t i a l energy f o r e l e c t r o n i c and n u c l e a r m o t i o n s , and t h e t o t a l v i b r o n energy a t t h e n u c l e a r c o n f i g u r a t i o n Q. The
p o t e n t i a l energy V(q,Q) i n c l u d e s t h e e l e c t r o n i c ground s t a t e energy which may be o b t a i n e d b s o l v i n g t h e s t a n d a r d BO e l e c t r o n i c problem f o r t h e e n t i r e system o f m01eCUleS!4 The c o n d i t i o n s under which t h e a d i a b a t ' c a p p r o x i m a t i o n f o r v i b r o n - phonon modes i s v a l i d have a1 so been examined by L i n 4 who used a s i m i l a r analogy t o e l e c t r o n i c - n u c l e a r s e p a r a b i l i t y . F o r coupled v i b r o n and phonon modes,
~ e r i l l i - ~ o t o n ' analyzed t h e breakdown o f t h e a d i a b a t i c a p p r o x i m a t i o n i n shocked n i t r o m e t h a n e by means o f a n o n a d i a b a t i c o p e r a t o r formalism.
The n o n a d i a b a t i c s o l u t i o n f o r c o u p l e d vibron-phonon o s c i l l a t o r s i s now o b t a i n e d by expanding t h e wavefunction Fr i n Eq. ( 4 ) i n terms o f t h e complete s e t
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o f e i g e n f u n c t i o n s @,,,(q,Q) o f Eq. ( 6 ) ,
From Eqs. (2b), ( 4 ) , and ( 7 ) , i t may be shown t h a t t h e phonon w a v e f u n c t i o n s +,
,,
.(Q) a r e s o l u t i o n s o f t h e s e t o f coupled d i f f e r e n t i a l e q u a t i o n s ,
I n Eq. ( 8 ) , t h e b r a - k e t n o t a t i o n i n d i c a t e s i n t e g r a t i o n o v e r t h e v i b r o n i c c o o r d i n a t e s q, and T ( Q ) denotes t h e k i n e t i c energy o p e r a t o r f o r t h e pnonons,
E q u a t i o n ( 8 ) has t h e same f o f l as t h a t f o r v i b r o n - v i b r o n c o u p l i n g between d i f f e r e n t e l e c t r o n i c s t a t e s .
The magnitude o f t h e r i g h t hand s i d e (RHS) o f Eq. ( 8 ) determines t h e v a l i d i t y o f t h e a d i a b a t i c approximation. I f t h e c o u p l i n g m a t r i x elements on t h e RHS o f Eq. ( 8 ) a r e s m a l l , t h e a d i a b a t i c s o l u t i o n i s a good one. On t h e o t h e r hand, i f t h e c o u p l i n g i s l a r g e , t h a t a p p r o x i m a t i o n breaks down. I n e i t h e r case, a p e r t u r b a t i o n approach t o t h e s o l u t i o n o f Eq. ( 8 ) w i l l r e q u i r e t h e f o l l o w i n g m a t r i x elements between t h e a d i a b a t i c w a v e f u n c t i o n s of Eq. ( 5 ) ,
where
and
E q u a t i o n (10c) i s o b t a i n e d from t h e d i f f e r e n t i a t i o n o f t h e S c h r s d i n g e r e q u a t i o n s ( 6 ) c o r r e s p o n d i n g t o t h e orthonormal s o l u t i o n s @m and $,. It may a l s o be shown from Eq. ( 9 ) t h a t T Eq. ( l o b ) , reduces t o a sum o f terms c o n t a i n i n g t h e m a t r i x elements en, Eq. (Pa;). According t o Eq. ( l a c ) , t h e a d i a b a t i c a p p r o x i m a t i o n a p p l i e s when t h e energy d i f f e r e n c e between t h e v i b r o n s t a t s i s l a r g e r e l a t i v e t o t h e phonon m a t r i x elements c o n n e c t i n g t h e s e s t a t e s . I f t h e energy d i f f e r e n c e i n Eq. (10c) i s s m a l l , t h e a d i a b a t i c a p p r o x i m a t i o n c o m p l e t e l y f a i l s and
r a d i a t i o n l e s s t r a n s i t i o n s occur.
The models of Z e r i l l i - T o t o n 7 and in^ r e q u i r e t h a t t h e a d i a b a t i c
a p p r o x i m a t i o n i s v a l i d i n t h e z e r o t h o r d e r , and t h a t t h e nonadia a i c c o r r e c t i o n s 9,b.
g i v e energy t r a n s f e r r a t e s w i t h i n t c o n t e x t o f t h e golden r u l e and t h e c o n v e n t i o n a l P a u l i master equation." W i t h t h e l a t t e r method, n in'^ c o n s i d e r e d v i b r a t i o n a l a c t i v a t i o n and d e a c t i v a t i o n t o c o n s i s t o f s i n g l e - s t e p o r d i r e c t processes f o r one v i b r o n mode, and s e q u e n t i a l processes i n o l v i n g t w o v i b r o n Y
modes. I n t h e case o f nitromethane, CH3N02, Z e r i l l i - T o t o n c o n s i d e r e d d i r e c t processes i n w h i c h t h e a c t i v a t i o n o f NO2 r o c k i n g p a r a l l e l t o t h e NO2 p l a n e was chosen t o be t h e f a s t s t e p . Another approach which i c l u d e s t h e a c t i v a t i o n and 4
d e a c t i v a t i o n s t e p s d e s c r i b e d by ~ e r i l l i-Toton' and L i n b u t which a1 so i n c l u d e s memory o r feedback e f f e c t s c o r r e s p o n d i n g h i g h o r d e r s o f a p p r o x i m a t i o n o f master e q u a t i o n , w i l l now be considered.
Shocking a m o l e c u l a r system mixes t h e s t a t i o n a r y v i b r a t i o n a l s t a t e s Fr o f Eq. ( 7 ) so t h a t
where t h e e v o l u t i o n o f t h e system among t h e b a s i s s t a t e s $,,hs i s c o n t a i n e d i n t h e time-dependent o e f f i c i e n t s a r s ( t ) . The t i m e d e r i v a t i v e s o f t h e
5 .
p r o b a b i l i t i e s (ar ( g i v e t h e r a t e s o f a c t i v a t i o n and d e a c t i v a t i o n . T h i s
f o r m u l a t i o n o f v i g r a t i o n a l r e l a x a t i o n which i n v o l v e s t h e e x p l i c i t t i m e dependence o f t h e w a v e f u n c t i o n i n Eq. ( 1 1 ) c o n s t i t u t e s t h e Schriidinger p i c t u r e of quantum dynamics. F o r t h e purposes o f t h e p r e s e n t f o r m u l a t i o n o f r e l a x a t i o n , an
e q u i v a l e n t , b u t more c o n v e n i e n t approach i s p r e s e n t e d n e x t which i s based on t h e Heisenberg p i c t u r e . P a r t o f t h e f o r m a l i s m t o b used h e r e has a l r e a d y been developed and u t i l i z e d by J o r t n e r and coworkersf5 i n c o n n e c t i o n w i t h t h e problem of v i b r a t i o n a l r e l a x a t i o n o f a guest m o l e c u l e i n a h o s t m a t r i x c o n s i s t i n g o f e i t h e r atoms o r p o l y a t o m i c molecules.
N i t z a n , e t a1 .I5 have s t u d i e d t h e v i b r a t i o n a l r e l a x a t i o n o f a m o l e c u l e i n a dense medium ( o r phonon b a t h ) by employing a model which i s based on t h e
assumptions t h a t ( 1 ) t h e m o l e c u l a r v i b r a t i o n s a r e harmonic, ( 2 ) t h e m o l e c u l a r v i b r o n modes a r e n o t coupled t o each o t h e r by t h e vibron-phonon i n t e r a c t i o n , ( 3 ) t h e vibron-phonon i n t e r a c t i o n i s l i n e a r i n t h e i n t r a m o l e c u l a r d i s p l a c e m e n t s , and (4) d i s s i p a t i v e processes i n v o l v i n g t h e phonon b a t h a r e f a s t on t h e t i m e s c a l e of v i b r a t i o n a l r e l a x a t i o n so t h a t t h e ba h i s i n t ermal e q u i l i b r i u m , whfch i s an a s s u m p ~ ~ o n a1 so made by Z e r i 1 1 i -Toto> and L i n
.'
U i Th t h e s e assunpf ions, Ni t z a n ,e t a1 . have shown t h a t t h e m u l t i p h o n o n v i b r a t i o n a l r e l a x a t i o n r a t e i s
where G ( v ) i s a vibron-phonon c o u p l i n g term, n ( v ) depends on t h e t e m p e r a t u r e - dependent phonon o c c u p a t i o n number <nv>T so t h a t
and ~ ( v ) i s t h e compound many-phonon densWity o f s t a t e s
p(v) = $ dc, J dc, . . . J dc,_,p,(w - cI)pZicI - 6,) . . . P ~ ~ E , - , ) > (1 4 ) which g i v e s t h e c o n v o l u t i o n o f s i n g l e phonon d e n s i t i e s of s t a t e s {p,(~)). The c o l l e c t i o n o f phonon s t a t e s s a t i s f i e s t h e energy c o n s e r v a t i o n l a w f o r a p o l y a t o m i c m o l e c u l a r s o l i d ,
where ma and wg correspond t o c o l l e c t i o n s o f phonon modes ( a ) and o t h e r v i b r o n modes ( B ) , r e s p e c t i v e l y , where u < W. Each o f t h e terms i n Eq. ( 1 2 ) corresponds t o a g i v e n o r d e r o f m u l t i - p h o n o n processes.
The ( G ( v ) l2 terms r a p i d l y decrease w i t h i n c r e a s i n g o r d e r o f t h e m u l t i p h o n o n processes s i n c e G(v) may be i d e n t i f i e d w i t h Eq. (10a). Consequently, t h e dominant c o n t r i b u t i o n t o t h e r a t e i n Eq. (12) i s t h e t e r m c o r r e s p o n d i n g t o t h e s e t o f phonon s t a t e s ( = 1,2,...,N where N i s t h e s m a l l e s t number o f phonons N E N( u) which can r e s u l t i n a v i b r a t , i o n a l r e l a x a t i o n process c o n s t r a i n e d by energy c o n s e r v a t i o n , Eq. (15). E q u a t i o n ( 1 2 ) may t h e n be w r i t t e n as
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which i s a form o f t h e golden r u l e . For t h e p o l y a t o ~ i c m o l e c u l a r s o l i d , Eq. ( 1 6 ) i s s a t i s f i e d when t h e minimum number o f phonons Na( v ) i s
where ,w, i s o f t h e o r d e r o f t h e "Debye" frequency, twg> i s t h e average o f v i b r o n f r e q u e n c i e s o t h e r t h a n w, and Ng, t h e most p r o b a b l e number o f v i b r o n s , l i e s i n t h e range
The i n c l u s i o n o f t h e v i b r o n f r e q u e n c i e s w p r o v i d e s a d e s c r i p t i o n o f v i b r a t i o n a l r e l a x a t i o n i n nonhomogeneous o r impure s o f i d s . I n t h e sequel, i t w i l l be assumed t h a t n i t r o m e t h a n e i s pure, so t h a t wg = 0.
I f t h e exchange o f energy among t h e d i f f e r e n t v i b r o n modes i s much more r a p i d t h a n vibron-phonon energy t r a n s f e r , t h e t o t a l r a t e WT f o r a l l o f t h e i n t r a m o l e c u l a r modes i s o b t a i n e d from t h e sum
where W corresponds t o Eq. ( 1 6 ) , and V may be t a k e n t o be t h e r a t e a s s o c i a t e d w i t h t h e l a r g e s t terms, which correspond t o t h e most e f f i c i e n t l o w frequency modes. I n t h i s regard, t h e assumption t h a t t h e vibron-phonon i n e r a c t i o n i s l i n e a r i n t h e i n t r a m o l e c u l a r d i s p l a c e m e n t s has been shown by L i n 4 t o be a good one f o r m u l t i p h o n o n r e l a x a t i o n processes which i n v o l v e o n l y t h r e e o r fewer phonons. T h i s w i l l be t h e case f o r r e l a x a t i o n o f l o w frequency v i b r o n modes where N(, u) < 3, i f w/ 3 i s s u b s t i t u t e d i n t o Eq. (17) I n n i t r o m e t h a n e under p r e s s u r e s o f 5-l8?.b;r, f o r example, B = w = 485 cm-I f o r t h e l o w frequency mode w i t h NO r o c k i n g p a r a l l e l t o t h e NO2 plane. At t h e shock p r e s s u r e o f 7 8 0 kbar, say, t e frequency w~ may be e s t i m a t e d from a power-law r e l a t i o n s h i p t o be u,, = 193 cm-' so t h a t u/% = 3, where i t has been p s u m e d t h a t w i s e s s e n t i a l l y unchanged from i t s l o w e r p r e s s u r e v a l u e o f 485 cm- . Here, then, t h e above assumption c o n c e r n i n g l i n e a r i t y i n t h e vibron-phonon i n t e r a c t i o n p r o v i d e s a r e a s o n a b l y good a p p r o x i m a t i o n i n l o w e r - o r d e r p e r t u r b a t i o n t h e o r y .
I n o r d e r t o d e s c r i b e t h e h i g h - o r d e r m u l t i p h o n o n a c t i v a t i o n and d e a c t i v a t i o n o f t h e l o w frequency v i b r o n modes l e a d i n g t o chemical r e a c t i o n , i t i s c o n v e n i e n t t o express Eq. ( 1 9 ) i n terms o f t h e t o t a l r e l a x a t i o n t i m e tnr,
t,, = - h
2~u*i5n(i4 '
where V = t-l, v2 = fi2(6(?) l2 and p 5 p(<)/ll i s a m u l t i p h o n o n , rnultimode d e n s i t y o f s t a t e s . YR Eq. ( 2 0 ) , r v Z F n ( 7 ) i s t h e v i b r a t i o n a l r e l a x a t i o n h a l f - w i d t h y.
E q u a t i o n (20) may be r e w r i t t e n as
where ii = p y i s t h e average number o f phonon s t a t e s e x c i t e d w i t h i n t h e h a l f - w i d t h y. I n Eq. ( 2 1 ) , 5i;; may be c o n s i d e r e d t o be a measure o f t h e t i m e tp, r e q u i r e d f o r energy t r a n s f e r from t h e denser s e t o f phonon l e v e l s i n t o t h e v i b r o n s t a t e s so t h a t
S i n c e t - tv t h e t i m e f o r vibron-to-phonon energy f l o w , i t i s seen from Eqs.
( 2 1 ) anar(22) !;at tv < t which i m p l i e s t h a t ( v p ) t r a n s f e r i s more p r o b a b l e t h a n (pv). It i s o f Pnterg:; t o n o t e t h a t Eq. ( f 2 ) i s analogous t o t h e
r e c u r r e n c e t i m e i n t r o d u c e d by Bixon and J o r t n e r f o r t h e n o n r a d i a t i v e decay o f a v i b r o n f o r an e x c i t e d e l e c t r o n i c s t a t e i n t o a quasicontinuum o f v i b r o n i c l e v e l s c o r r e s p o n d i n g t o a l o w e r e l e c t r o n i c s t a t e . It w i l l be shown i n a n o t h e r paper t h a t t h e i n t r o d u c t i o n o f Eq. (22) i n t o t h e p r e s e n t a n a l y s i s i s e q u i v a l e n t t o t h e i n c l u s i o n o f h i g h o r d e r s o f a p p r o x i m a t i o n i n t h e p e r t u r b a t i o n expansion o f t h e g e n e r a l i z e d master e q u a t i o n .
There a r e f o u r i m p o r t a n t reasons f o r w r i t i n g t i n t h e f o r m o f Eq. (21). -
F i r s t , t h e o n l y q u a n t i t i e s needed f o r t h e d e t e r m i n a v l o n o f tnr a r e p and ii. Second, t h e e v a l u a t i o n o f p, which w i l l be c o n s i d e r e d below, i s s t r a i g h t f o r w a r d when t h e phonon mode f r e q u e n c i e s a r e known as-functions o f p r e s s u r e , which i s t h e case f o r nitromethane. T h i r d , t h e q u a n t i t y n , which i s a measure o f t h e minimum number o f phonons a v a i l a b l e f o r e x c i t a t i o n , may a l s o be e s t i m a t e d f r o m t a b u l a t i o n s o f shock p r e s s u r e d a t a f o r nitromethane. A d e t e r m i n a t i o n o f ii i n t h i s manner, a v o i d s t h e v e r y d i f f i c u l t e v a l u a t i o n o f t h e Nth o r d e r c o u p l i n g t e r m s v. Fourth, Eqs. (20) and ( 2 1 ) a p p l y t o t h e h i g h e r - o r d e r problem o f simultaneous m u l t i p h o n o n a b s o r p t i o n where t h e energy c o n s e r v a t i o n l a w o f Eq. (15) must b e g e n e r a l i z e d t o d e s c r i b e energy jumps between n o n n e i g h b o r i n g d i s c r e t e s t a t e s . I n t h i s case, a d e t a i l e d t h e o r e t i c a l d e r i v a t i o n o f Eq. (21) would r e q u i r e a more general H a m i l t o n i a n w i t h n o n l i n e a r , h i g h e r - o r d e r terms i n t h e anharmonic,
i n t r a m o l e c u l a r displacements. A d i f f e r e n t approach was t a k e n by c o f f e y - ~ o t o n ~ , who observed t h a t t h e f o r m a l i s m f o r t h e simultaneous a b s o r p t i o n o f many phonons i s i d e n t i c a l t o t h a t used i n many-photon e x c i t a t i o n s when d e n s i t i e s o f quanta a r e s u f f i c i e n t l y high.
F o r t h e c a l c u l a t i o n o f tnr and t i s Eqs. ( 2 1 ) and ( 2 2 ) , t h e compound, m u l t i p h o n o n d e n s i t y o f s t a t e s , which 8; a g e n e r a l i z a t i o n o f Eq. (14) and which corresponds t o t h o s e s t a t e s c l u s t e r e d around t h e l g x c i t a t i o n energy E, may be approximated w i t h an e x p r e s s i o n due t o H a a r h o f f ,
where n i s t h e number o f v i b r a t i o n a l degrees o f freedom, <v> i s t h e average o f t h e n f r e q u e n c i e s vi , and A, 6, and n a r e d e f i n e d by t h e e q u a t i o n s
I n Eq. (23e), E i s t h e i n t e r n a l energy i n t e r v a l and Eo i s t h e z e r o - p o i n t energy.
JOURNAL DE PHYSIQUE
CALCULATIONS OF RATES OF ENERGY TRANSFER AND REACTION FOR NITROMETHANE Energy T r a n s f e r
The known l a t t i c e f r e q u e n c i e s r e q u i r e d f o r use i n Eq. (23a) p e r t a i n t o Raman and n e u t r o n - s c a t t i ng s p e c t r a which have been o b t a i n e d f o r s o l i d n i t r o m e t h a n e a t ambient pressure.e5 I n Ref. 17, t h e bands assigned t o t h e 8 t r a n s l a t i o n a l modes ( a t o t a l o f 9 e x i s t f o r t h e u n i i c e l l ) a r e l o c a t e d a t 45.0, 69.0, 72.0, 77.0, 85.5, 95.5, 99.5, and 114.0 cm- . As a r e a s o n a b l e a p p r o x i m a t i o n a t h i g h shock p r e s s u r e P , i t may be assumed t h a t t h e s e f r e q u e n c i e s a r e i n c r e a s e d by t h e same f a c t o r so ghat Eqs. (23b) and (23d) remain unchanged. Then, Eq. (23a) w i l l depend p r i m a r i l y on t h e v a r i a t i o n 05 Eq. (23e). For example, a t PS = 80 kbar, t h e power-law p r e s s u r e r e l a t i o n s h i p f o r t h e "Debye" frequency YD, which i s t a k e n t o be t h e average o f t h e 8 f r e q u e n c i e s , g i v e s % ( 8 0 ) = 143 cm- so t h a t
o ( 8 0 ) / w ( 1 atm) = 1.8. As a r e s u l t , t h e l a t t i c e f r e q u e n c i e s a r e i n c r e a s e d t o 8y.0, 129.2, 129.6, 138.6, 153.9, 171.9, 179.1, and 205.2 cm-l.
A t Ps = 80 k b a r and t h e shock t e m p e r a t u r e T = 850 K a l o n g t h e shock Hugoniot o f n o n r e a c t i n g nitromethane, t h e i n i & i a f s t a t e o f which i s below t h e m e l t i n g p o i n t o f 244.6 K Hardesty and Lysne have determined t h e i n t e r n a l energy EH t o beld.2 x 10:' erg/!, and t h e c o r r e s p o n d i n g energy d e n s i t y EH/V = 2.2 x 10 erg/cm . I f i t now i s assumed t h a t t h e l o c a l o r d e r i n g o f molecules a t 80 k b a r i s such t h a t t h e r e a r e s t i 11 mo e c u l e s p e r u n i t c e l l , i t
i s found from t h e volume o f t h e c e l l o f 1.9 x loez4 cm' t h a t t h e t o t a l energy p e r m o l e c u l e cH = 14.8 k c a l . The t h e r m a l energy c(T,V) i s t h e n o b t a i n e d b y
s u b t r a c t i n g from E t h e i n t e r m o l e c u l a r r e p u l s i o n energy p e r molecule, e(V). From Ref. ( 1 9 ) , E(V) k c a l a t 80 k b a r so t h a t c(T,V) = 10 k c a l .
The q u e s t i o n now a r i s e s as t o whether o r n o t t h e t h e r m a l i z e d phonon b a t h p r o v i d e s a l a r g e enough d e n s i t y o f phonons p e r m o l e c u l e so t h a t a g i v e n m o l e c u l e i s a b l e t o absorb a s i g n i f i c a n t amount o f t h e energy E(T V) = 10 k c a l . From t h e average number o f phonons p e r molecule, [ e x p ( h w v / k ~ - l ) ] - l , f o r each o f t h e 7 l a r g e s t f r e q u e n c i e s g i v e n above, t h e t o t a l average number o f a v a i l a b l e phonons p e r m o l e c u l e i s 22. Since a l l o f t h e s e phonons a r e i n i t i a l l y a v a i l a b l e i n t h e phonon b a t h a t t h e s u s t a i n e d p r e s s u r e o f 80 kbar, c(T,V) may be equated w i t h E i n Eq. (23e). From Eq. 23a), then, o n e o b t a i n s 7 = 1.5 x 1019/erg and, from Eq.
(221, tTy = l.BOx lo-' sec. Here, n = 20 so t h a t , from Eq. ( 2 1 ) ,
tnr = x 10- sec.
The a n a l y s i s g i v e n above f o r Ps = 80 k b a r may be a p p l i e d t o t h e e x t e n s i v e d a t a g i v e n i n Refs. 18 and 19. Graphs o f l o g t and t a r e d i s p l a y e d i n F i g u r e 1 f o r - t h e range o f shock p r e s s u r e s 10 r pS-r 8oP!bar. ?he c o r r e s p o ~ d i n g ranges f o r p , n , t , and t n age 5.5 x l o 1 9 > P r 1.5 x 1 0 ' ~ / e r ~ , 1 r n r 22.
5.5 x , FVv, 1.5 X!O- sec,and 2.6 x l o m 9 > tn , 7.0 x 10-'0 sec. Since
t h e s e t i m e s i r e p e r t o t h e simultaneous a b s o r p t i o n of 15 t o 22 phonons between nonneighboring i n t r a m o l e c u l a r s t a t e s , t h e y are, as a consequence o f p e r t u r b a t i o n t h e o r y , l a r g e r t h a n r e l a x a t i o n t i m e s p e r t a i n i n g t o e x c i t a t i o n s i n more l i m i t e d r e g i o n s o f t h e m a n i f o l d o f l e v e l s .
Chemical R e a c t i o n s
The energy t r a n s f e r t i m e s t p l o t t e d i n Fig. 1 w i l l now be compared w i t h r e a c t i o n ha1 f - 1 i v e s . I n Ref. 20Yvan i n i t i a1 , h i g h l y - e x o t h e r m i c ( - 4 8 k c a l / m o l ) r e a c t i o n s t e p f o r shocked n i t r o m e t h a n e i s presented which i s b i m o l e c u l a r and which i s , t h e r e f o r e , s t r o n g l y a c c e l e r a t e d by pressure. T h i s step, which i s t h e f a s t e s t p o s s i b l e b i m o l e c u l a r r e a c t i o n f o r n i t r o m e t h a n e , i s r e p r e s e n t e d by t h e h e a d - t o - t a i l r e a c t i o n
YH
where t h e p r o d u c t i s a r e a r r a n g e d dimer. At P = 80 k b a r and Ts = 850 K, f o r example, t h e e n t r o p y , volume, and e n e r g r o f a c 2 i v a t i o n f o r t h i s s t e p a r e g i v e n i n Ref. 20 as Sa = -10 eu/mol, V = -12 cm /mol, and Ea = 22.5 k c a l / m o l
r e s p e c t i v e l y . I n o r d e r t o o b t a i n t h e r e a c t i o n ha1 f - 1 i f e t b ( 1 / 2 ) , t h e s e
parameters a r e s u b s t i t u t e d i n t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e r a t e c o e f f i c i e n t k,
where v = kBT/h. From E q . (25), t b ( 1 / 2 ) = t ( 1 / 2 ) = 5 x 1 0 - ' ~ s c f o r t h e r e a c t i o n i n Eq. ( 2 4 ) . A comparison o f t h i s v a l u e w i t h t = 1.5 x ID-' sec c a l c u l a t e d p r e v i o u s l y shows t h a t energy t r a n s f e r f r o m t h e /%onon b a t h i n t o t h e v i b r o n s t a t e s i s t h e s l o w o r r a t e - d e t e r m i n i n g s t e p a t Ps = 80 k b a r and TS = 850 K. These r e s u l t s a r e d i s p l a y e d i n F i g . 1. Also p l o t t e d i n F i g . 1 a r e l o g t b ( 1 / 2 ) f o r 1
atm < Ps < 80 k b a r and 244.6 < TS G 850 K. The ranges of a c t i v a t i o n parameters
which a r e u t i i z e d f o r t h e s e p r e s s u r e s and t e m p e r a t u r e s a r e 1' -32< Sac -10 eu/mol, -30<Vac-12 cm /mol, and 32.5>Eab22.5 k c a l / m o l . These ranges were c a l c u l a t e d from t h e e s t i m a t e d pressure-dependent i n t e r m o l e c u l a r d i s t a n c e s and t h e e q u a t i o n s g i v e n i n Ref. 20.
X
t = t , (1121
F i g u r e 1. Shock p r e s s u r e dependence o f e n e r g y t r a n s f e r t i m e s tpv, t n r , and r e a c t i o n h a l f - l i v e s t b ( 1 / 2 ) and t u ( 1 / 2 ) f o r b i m o l e c u l a r and u n i m o l e c u l a r p r o c e s s e s .
