On the thermochemical action of ionizing radiation

International Journal of Applied Radiation and Isotopes, 1961, Vol. 11, pp. 1-9. Pergamon Fress Ltd. Printed m .Northern Ireland

On the Thermochemical Action of" Ionizing Radiation V. I. G O L D A N S K I I and Ytl. M. K A G A N Academy of Sciences of the U.S.S.R., Moscow, U.S.S.R.

(Received 10 October 1960) G e n e r a l characteristics of the t h e r m o c h e m i c a l action of ionizing r a d i a t i o n are considered, n a m e l y the initiation of" e n d o t h e r m i c reactions by local h e a t i n g u p of microvolumes along nuclear particle tracks, a n d also by the overall h e a t i n g u p of the system irradiated. T h e following problems are t r e a t e d : (1) d e t e r m i n a t i o n of the m a x i m u m (quasi-equilibrium) t h e r m o c h e m i c a l action of" r a d i a t i o n ; (2) d e t e r m i n a t i o n of the a m o u n t of h e a t spent in a n e n d o t h e r m i c reaction within a m i c r o v o l u m e along the ionizing particle track; (3) d e t e r m i n a t i o n of the total efficiency for a quasi-stationary case of averaged heating u p of a m i x t u r e ; (4) d e t e r m i n a t i o n of the energy i m P a r t e d to the reaction by a n a r r o w b e a m of c h a r g e d particles coaxial to a cylindrical volume. T h e results o b t a i n e d (especially in P a r t 2) m a y also be of some interest for discussing the problems of a n n e a l i n g a n d retention effects u p o n nuclear transformations in solids. SUR L'ACTION

THERMOCHIMIOUE

DU RAYONNEMENT

IONIZANT

O n consid~re les caract~ristiques g6n6rales de Faction t h e r m o c h i m i q u e d u r a y o n n e m e n t ionizant, soit l'initiation des rdactions e n d o t h e r m i q u e s p a r le chauffage local des micro-volumes le long d u c h e m i n des particules nucl~aires, ainsi que p a r le chauffage g6n6ral d u syst~me irradi6. O n traite des probl~mes suivants: (1) la mesure d u m a x i m u m (en quasi-6quilibre) de l'action t h e r m o c h i m i q u e d u r a y o n n e m e n t ; (2) ]a mesure de la q u a n t i t 6 de chaleur prise p a r une r6action e n d o t h e r m i q u e dans u n micro-volume le long d u c h e m i n d ' u n e particule ionizante; (3) la mesure de l'efficacit6 totale pour u n cas quasistationnaire d u chauffage moyenn6 d ' u n m61ange; (4) la mesure de l'6nergie r e n d u e /~ une r~action p a r u n faisceau mince de particules charg6es co-axial $ u n volume cylindrique. O TEH[IOXHMHqECHOM ,~EITICTBI/IH I4OHH3HP~VIOII~HX H 3 ~ ' H E H H 1 7 I PaccMaTprIBatOTC~ o S ~ n e xapat~Tep~cTHtCI4 TeII.1OXHMHqecHoFO 2XOI'~CTBHttnOHt~3npyromHx Ha,~yqeHHl~I, T.e. IIHHIXHHpOBaHH~I ~H~IOTepMHqecI~HX XHMHqeCHHX peaKaH~t BC.~Ie~CTBI4e .loHaJlbHOPO pa3orpeBa MHKpoo5%eMOB B~OorlbTpae~Top~fl n~epHb~X qacTgI~, a TaRnce o6n~ero p a s o r p e n a o6o~yqaeMott CHCTeMbI. Pema~oTcg caeRy~oi~ge r¢Ot~rcpeTHbie 3aRaqn: (1) onpe~ie:leHt~e MaHcHMa~3bHOFO (KBa314paBHOBeCHOFO) I~l~,~ TeII:IOX~MHqecKoFo Re~iCTn~g H3[tyqeHt4/~i (2) onpe~lesieH~e ~o~lqqecTBa Te~13a, pacxoRyeMoro Ha OHXOTepMnqecgy~o peamlnto B Mngpoo6~eMe B;I0ab TpaeHTopI~H ~OHHar~py~omefi qaCTHlIbI (3) ot~pe;Ie:ieH~e CyMMapHoro I~[I~ ;Iorl~ i~na.3Hcvarit~onapHoro c n y q a n ycpeAueuHoro 06nlero pa3orpeBa CMeCg (4) onpe~Ie~]eHne oueprrIH, 1]epe~aBaeMoi~ Ha XnMr~qecKyIo p e a m l n m ySt~HM nyq~OM 8apn~eHHHX qae, THl(, IIpOXO~l~lllI4M go OCH ~i4ylkIti;Ipnqec~oro cocy~la. IIoo~yqen~sm pe3y(ihTaTb~ (oc06eHtto B 8ao3aqe 2) MoryT Hpe~CTaB~Tb oHpeAe~rleHHbIl~ nuTepec i~ R:I~ o6cy~Ier~nn 3dp~eHTon OTht¢tira I~I y;IepmaH~n noc~e nnep~htx npenpa~r~ennit u TBep~IbiX Te~ax. UBER DIE THERMOCHEMISCHE WIRKUNG STRAHLUNG

IONISIERENDER

Die allgemeine Charakteristik der t h e r m o c h e m i s c h e n W i r k u n g von ionisierender S t r a h l u n g wird untersucht, im H i n b l i c k a u f die Ausl6sung e n d o t h e r m e r R e a k t i o n e n d u r c h lokale

2

V.I. Goldanskii and Yu. M. Kagan

Erw~irmung von Mikrovolumen entlang der Bahnen nuklearer Teilchen, sowie durch die allgemeine Erw~irmungdes bestrahlten Systems. Die folgenden Probleme werden behandelt: (1) Bestimmung der maximalen thermochemischenWirkungderStrahlung (quasi Gleichgewicht); (2) Bestimmungder ftir endotherme Reaktionen innerhalb des Mikrovolumens entlang der Bahn des ionisierenden Teilchens verbrauchten W~rmeenergie; (3) Bestimmung der gesamten Ausbeute im quasi-statiomiren Fall der mittleren Erw~irmungeines Gemisches; (4) Bestimmungder durch einen ausgeblendeten Strahl geladener Teilchen entlang der Achse eines zylindrischenVolumens abgegebenen Reaktionsenergie. INTRODUCTION

THE initiation of chemical reactions by ionizing radiation is a problem obtaining an ever-increasingimportance both in theory and practice. O f especial interest from the standpoint of nuclear energy applied to chemistry would be the possibility of conducting energy-consuming syntheses i.e. of accomplishing radiation-induced initiation of endothermic chemical reactions. Such an application of the nuclear energy may in principle be directed along two paths. The first would involve individual hot atoms formed in the interaction between radiation and the substance at the primary or some other reaction stage. These atoms have an energy above the threshold energy of endothermic reactions and readily participate in the latter. The second path would be connected with a local heating up of microvolumes along nuclear particle tracks, as well as with the heating up of the overall system. A corresponding rise in temperature markedly increases the rates of endothermic reactions, especially in microvolumes along tracks of ionizing particles, Despite the amount of energy spent in chemical conversions, the temperature of heated microvolumes appears to be above that of the environment. O f great importance is the fact that, in the subsequent levelling out of temperature by diffusion and heat conduction, the rate of inverse decomCHARGED

position of reaction products may already be neglected (especially if the amount of these products is far from equilibrium even at a maximum initial heating up). Thus the products of conversion in heated up microvolumes are so-to-say " q u e n c h e d " and their yield may appear to be appreciably higher than that at equilibrium with uniform heating up of the overall system by ionizing radiation. However, the times of reactions within a a macrosystem are considerably longer than the cooling time of microvolumes along particle tracks, and the overall volume of the system considerably exceeds the sum of these microvolumes. Consequently, for a general case, the contribution of reactions in a system with averaged heating up may appear to be comparable to that of reactions in microvolumes. The action of ionized radiation, defined here as "thermochemical", is obviously nonequilibrium, i.e. of kinetic origin, and it will correspondingly depend on the reaction rates, diffusion and heat conduction, both in microvolumes along particle tracks and in the overall system. Unlike the mechanism of hot atom reactions, the thermochemical action of ionizing radiation is subject to ready investigation. The given paper will in fact be concerned with a mostly qualitative investigation of this kind.

PARTICLE TRACKS IN A CONDENSED

Charged particle tracks may fall into three classes depending upon the localization of energy losses: (1) Tracks of the fragmentary type. These involve the liberation of very high energies within small track lengths (ionization density

MEDIUM

of the order of 40,000 1V[eV/g.cm 2, i.e. exceeding that of the relativistic ionization by a factor of almost 20,000). The zone of effective "thermal damage" of the substance will in this case be much larger than the volume of the primary ionization core along

On the thermochemicalaction of ionizing radiation the track. T h e p r i m a r y core m a y thus be considered as a thread-like source of energy located along the axis o f a h e a t e d cylinder, m

(2) Tracks of/astbutessentiallynon-relativistic heav}' particles. In this case the ionization zone m a y be modelled by a cylinder with a certain radial distribution of the ionization density, for instance o f the type described in c'~. T h e effective d i a m e t e r of the ionization zone decreases with increase in the particle velocity. Processes characterized by a certain energy threshold will ultimately occur, not t h r o u g h o u t the " a v e r a g e d " v o l u m e but along the p r i m a r y particle track and the tracks of individual a-electrons. At the limit the increase in velocities will obviously lead to a third type of track,

3

(3) Tracks of relativistic particles. Processes involving an energy threshold are concentrated in preference along discrete tracks o f d-electrons and, to a certain extent, in regions of elevated ionization density (of a tluctuation origin) along the p r i m a r y particle track. T h e models used in discussing tracks o f the first and second type can n a t u r a l l y be applied to conversions along tracks o f individual d-electrons as well. T h e given classification o f tracks is o f course largely conditional, a n y i n t e r m e d i a t e case m a y virtually take place. This p a p e r will be c o n c e r n e d only with the thermochemical action o f heavy ionizing particles corresponding to the first and second track models.

D E T E R M I N A T I O N OF T H E M A X I M U M ( Q U A S I - E Q U I L I B R I U M ) T H E R M O C H E M I C A L A C T I O N OF R A D I A T I O N

L e t us begin with a most simple c a s e - - t h e d e t e r m i n a t i o n of the t h e r m o c h e m i c a l action of radiation u n d e r conditions when all the energy input is spent in the h e a t i n g u p of the mixture and in the e n d o t h e r m i c reaction, while heat release is vanishingly low. Analysis of a quasi-stationary case of this kind is a strictly t h e r m o d y n a m i c p r o b l e m and no kinetic characteristics need be introduced, ttowever, as criterions should be established tbr the applicability o f the discussed variants, let us mention three time characteristics, taken as a t)asis for these criterions, n a m e l y : %, the p r o p e r time o f a reaction, i.e. the time necessary for attaining equilibrium (inverscly p r o p o r t i o n a l to the rate constant and consequently directly proportional to e u/k'r, E being the activation energy); me, the time o f efl'ective levelling out of the t e m p e r a t u r e microdistribution, a c c o u n t e d tbr by the heat release from track microvolumes; r, time o f the mixture exposure to radiation. In tim quasi-stationary case u n d e r discussion, % .~ r~. i.e. the reaction is cornpleted long betbre the termination o f heat release. T h e h e a t e d - u p microvolumes along tracks of h e a v y charged particles will be considered

as the reaction zone, since at n o r m a l flow densities these tracks do not overlap. T h e v a l u e of r will t h e n be d e t e r m i n e d by the rate of difl'usion and will obviously be close tO r~. Thus, as r r < r, it m a y be disregarded. It will be assumed in qualitative terms that chemical equilibrium corresponding to the new t e m p e r a t u r e will be established within a time, t, m u c h shorter than that necessary for heat release (r~) and diffusion (r). Cooling is a c c o u n t e d for solely by the e n d o t h e r m i c reaction, and consequently the final tempera.ture, 7), x.~ill tail to attain its m a x i m u m possible value, 7",~ (in the absence o f endothermic processes). It will also be assumed that the initial system was also at equilibrium, as consistent with its initial t e m p e r a t u r e , T o. T h e microvolumes will be considered as axially symmetrical. T e m p e r a t u r e changes along their axes will be neglected. T h e n , after heating, the t e m p e r a t u r e becomes T,, = T,,,(r), where r is the distance from the track axis. This results in an e n d o t h e r m i c reaction, the t e m p e r a t u r e falling off in every point. A new equilibrium d e p e n d i n g upon r will be attained. T h e condition T O < Tt(r ) < 7;,(r) for the final t e m p e r a t u r e Ti(r ) will be valid at every point. Let us estimate the a m o u n t o f substance

4

V. 1. Goldanskii and Yu. M. l(agan

having reacted within a microvolume before the new equilibrium set in or, in other words, the amount of energy spent in the chemical reaction, For the sake of definiteness the reaction will be considered as occurring in a condensed phase, and possible phase transitions will be neglected, The chemical equilibrium before heating is r[C,0", = K(To) and the final stage

(1) (2)

I I C , 7 , ( r ) = K[T:(r)]

where C~ = n~ and is the concentration of n reagents and reaction products; n = Y~n, the mixture density, and v~ are the corresponding stoichiometric coefficients (positive for reaction products and negative for initial substances), The amount of heat absorbed per unit volume at the expense of the reaction is q(r) -- n , k ( r ) - nio qo = N(r)qo

(3)

where nik and nio correspond to final and initial conditions respectively, q0 is the heat of reaction, and N(r) is the number of elementary reactions per unit volume. Let us express C,:(r) through Cio" C,~

Nv~ 1 nio + Nvi n,o -- n o ÷ N ~ v , = C'° 1 + - - N X v "

(4)

r/0

Substituting (4) into (2) we obtain after transformation /][i/11[ + Nv, ~v, nm | K[Tt(r)] + NYy,I -- K(To) (5) no / The final temperature distribution after completion of the reaction in the given case (r~
where C is the specific heat and p the density. In a general case the product Cp, equal to the specific heat per unit volume, will be related to temperature, first of all due to the temperature dependence of the density. However, it may be assumed here that no essential change in the substance density succeeds in taking place within a microvolume during t. On the other hand, if one allows for the fact that the heat capacity increases with temperature, it would be possible to assume, disregarding the slight decrease in density, that Cp = const. Once the Tm(r ) function is known, equations (5) and (6) provide an unambiguous description for N(r) and T:(r). The amount of product i formed per unit path length will, obviously, be

w,

(rm~x. ),ml.. N(r)v, 2rrr dr

(7) '

where the limits rmi,. and r,,,ax, will be selected according to the given mode of Tm(r ) determination. The amount of heat absorbed in the chemical reaction is Q ....

:

rm~x.

N(r)q o 2rrrdr. :3 rm~,, Since we wish to establish what am,)unt

of the nuclear radiation energy is imparted to the chemical reaction, of main interest would be the efficiency value ( .... qOJrmln. ~ ( r ) 27rr dr ('} ~

---

{,rmax"

J,

'' )

[ T , , ( r ) - 7o]Cp27rrdr ""R e a c t i o n A " B -~" D t-F (where,}fv,, 0) will be discussed for the sake of simplicity, assuming thatna0 = nB0andnD0 = nv0. The equation N(r) exp I-~kl 1 1 }] = ~_.., T o T:(r),~ -- 1 n~ o 1 [qo{ 1 1 }] %/k--~0 + exp ~-~ 7~0 Ty(r ) (tO) will be readily obtainable.

On the thermochemical action of ionizing radiation

Substituting then into (10) the expression (6) for T1(r), and assuming a definite form of Tin(r), we obtain a simple transcendental equation for N(r) and consequently for the values Wi, Q and ~ under question (see equations 7-9).

5

If the distribution of Tin(r) may be approximated over a H-shaped function, a quite obvious relation for ~ will be obtained: qoN rI -- Cp( T m _ To ) .

(11)

THE KINETIC PROBLEM OF REACTIONS IN MICROVOLUMES A L O N G T R A C K S OF I O N I Z I N G P A R T I C L E S

Let a large volume be exposed to radiation. An endothermic reaction will consequently take place within this volume. Heat excess will be withdrawn via the system surface. Due to the continuous removal of reaction products their concentration remains unchanged. An averaged steady-state picture with a temperature distribution of T~(r) will thus be attained. However, an investigation of this picture will be insuffÉcient for our purpose, since a large amount of energy spent in chemical reactions is due to non-stationary processes connected with individual particle tracks. Assuming that there is no overlapping of tracks, we may estimatethe total contribution of these local non-stationary processes for one track. When the time of the mixture exposure is sufficiently short, conditions may be obtained under which the inverse reaction in the averaged steady state will be negligible, These would apparently be (a) 7 <; %(T~, Ci s) and (b) ~'l -~-~% (7"tr~¢,k,C J). The first condition corresponds to a short time of exposure as compared with the time required for the attainment of equilibrium at a constant temperature, Ts, and steady-state concentrations C~'. The second corresponds to conditions under which the heat release and diffusion from microvolumes is sufficientlv fast and consequently the chemical equ{librium, similar to that described in the preceding section of this paper and corresponding to elevated temperatures, d o e s n o t succeed in becoming established. Thus the concentration of end-products is supposed to be always considerably below that at equilibrium. Let us now consider tracks of the fragmentary type. As stated before, they may be considered as thread-like heat sources,

In the absence temperature A T moment, t, upon be approximately

of heat release, the rise in within the volume at a passage of a particle, may written as Qo A T -- 4rrCpz t (r2 < 4zt) (12)

AT ~ 0

( r 2 > 4zt )

where Z is the thermal diffusivity coeËficient (cm2/sec),). and is the heat conducCPZ tion coefficient (cal/cm sec grad), Q0 is the amount of heat liberated per unit track length (cal/cm), C is the specific heat, and p is the density. When the reaction is endothermic, a part of the energy will be absorbed, the rate of absorption per unit volume being A e -EIkT (cal/cm a sec) (where E is the activation energy, and A is the product of the reaction heat by the reaction rate pre-exponential). On the assumption that the heat propagation rate remains unchanged in the course of the reaction, the latter's progress may be considered as an effective decrease in Qo. Then, instead of equation (12) we obtain Q(t) A T ( t ) - 4wCpzt (r 2 < 4zt ). (13) The overall integral relation for O(t) will be Q(t) = Qo - AQ(t) = Q0 - 4~rA f0t :~

{ zt'dt' exp

E

l_Q(t,) i). (14)

-- k T~ + 4~rCpz t }

For a general case, equation (14) may be solved only by the method of successive approximations. In practice, however, the first approximation, with replacing of Q(t')

6

V.I.

Goldanskiiand Yu. M.

b y Q0 in the e x p o n e n t o f the integral, already appears to be a d e q u a t e . Strictly speaking, this corresponds to absorption of a small p a r t only o f the liberated energy. T h e main c o n t r i b u t i o n to the integral of e q u a t i o n (14) is m a d e by t e m p e r a t u r e s above T~ / o f the \ order ~1

kj •

Assuming a power d e p e n d e n c e of

X on T a n d taking into a c c o u n t that within the t e m p e r a t u r e range u n d e r question A T > 7",, we m a y write Z ~ fl(AT)'. W e obtain, then, from e q u a t i o n (13) { Q0 ~ 1/1 + , A T = \4--~pflt] (15) T h e heat absorbed by the chemical reaction will be a c c o u n t e d for by the approximate expression [ Q0 ~ , n + , AQ(t) ,~ 4trAil \4--~pfl]

It × ~o t m+'dt

{ exp

Etm+* --~- [

}

Q0 ~m +* . (16)

\ ~ ] O f interest is the overall value of heat absorbed d u r i n g the time taken by the track t o b e c o m e diffuse. Since with the rise in t the integral in equation (16) rapidly converges, it m a y be assumed that at the u p p e r limit t = 0o and then, as s is nearly always positive,

1 AQo~

4~r Z

A

[ Oo~]2

/[ T = w,~ ~pCE] k] \

Kagan

o f fairly wide practical i m p o r t a n c e , it would be feasible to make the most general qualitative estimates p e r t i n e n t to equations (12) and (17). Condensedphase pC ~ l cal/cm~grad z ~ 10 ' c m ' l s e c Q0 ~ 10 ' cal]cm (for fissionfragments) Ifr* < 5 x 10 3tcm*, 6x10 s A T ~ -----~--- grad

Gases

pC ~ 10 4 cal/cma grad Z ~ 1 cm'/sec Qo~ 10 '~ cal/cm (for fission fragments) Ifr* .t tcm*, 2×10' AT~ t grad

It will be assumed that the effective course of e n d o t h e r m i c reactions would require a heating up of no less t h a n A T ~ 1000'C. As m a y be seen from the above figures such a heating up will occur both in the condensed and in the gas phase within a cylinder along a fission fragment track o f the radius r 10 6 cm. However, in the condensed phase the heating up time tbr such a cylinder will be t ~'~ 10-x° sec which is about three orders higher than the characteristic time interval, r, between molecular collisions in a liquid (r ~

1 /6-M 10 -aasec), whereas in the rrd2-----n~/k-T ~" gas phase t ~-~ 10 ~2 sec which is two orders lower than the characteristic r value ('or gases. T h u s the reaction in microvolumes along fission fragment tracks m a y be considered as taking place effectively in ttw condensed phase, while the xcrv us(" of equation (12) for the gas phase is possible only u n d e r definite restrictions. It will b e s e e n i n passing t o e q u a t i o n ,: 17) that o

x (1 + s ) P ( 2 } s). (17) where P is the g a m m a - f u n c t i o n . By allowing for the p a t h length of the fragments a n d their total n u m b e r per unit v o l u m e in unit time, it is possible to determine from e q u a t i o n (I7) the specific absorption o f their energy in an energy-consuming chemical synthesis. It will be noted that the conversion on nuclear into chemical energy is more effective with processes of a lower activation energy ( A Q ~ , ~ E-(2+'))(assuming, however, as before that E > kT,). Since the above considerations seem to be

the f u n d a m e n t a l factor AQ0 k characZ ' teristic of the efficiency given by the thermochemical utilization of the particle energy AQ~j Q0 ' is also different for the condensed and the gas phase (the numerical factor (1 } s)F(2 ~ s) 4 changes from 0.32 to 7.6 with changing of 1 to 3, i.e. tends to unity). U n d e r the assumption that E = 2q and q ~ 2"5 eV, the

s fl'om

On the thermochemicalactionof ionizing radiation value of---~-0 will be about unity for fission ti'agments in the condensed phase and about 10 3 in the gas phase. This provides another confirmation for the efficiency of thermochemical utilization of fission fragment THE

QUASI-STATIONARY

energies in the condensed phase. It will be noted, moreover, that the additional thermochemical effect connected with molecular thermal dissociation and the consequent reactions between free atoms and radicals was not accounted for in any section ofthispaper.

PROBLEM

Let us now consider the chemical reaction vield in terms of an averaged stationary problem, i.e. the yield of a macrosystem heated up uniformly under the action of ionizing radiation. A problem of this kind may be encountered, for instance, in a case when a mixture of substances capable of undergoing an endothermic reaction is utilized for the removal of encrgy in w~rious reactors. The efficiency of utilizing the energy liberated in the averaged heating up of a system is, as a rule, considerably lower th;m ihat of the local heating up of microvolumes along tracks of heavy ionizing particles with subsequent " q u e n cnlng ~" ~ . [towever, the energy concentrated in microvolumes may be a very insignificant part only of the total energy of nuclear particles emitted by a radiation source. Besides, practical accomplishment of the removal of "averaged" heat is often obtained by more simple means than the utilization of heat from 1he microvolumes. Consequently' the averaged stationary problem would also be of interest. The part played by the time criterions cited above should then also be taken into account, assuming again that the time of reaction, r~, in an "averaged" system considerably exceeds that of the removal of reagents (r). I f the heat sources are distributed uniformly throughout the volume, the equation tbr heat conduction in the given case will be

7

OF A V E R A G E D

HEATING

UP

Obviously

W nl ( Q o - AQ.~) (19) where n(cm a sec 1)is the densityofappearing tracks, and l their track length. For certainty, let us consider a reaction proceeding in a parallel plate vessel at a fixed wall temperature T o. Assuming that 7~(r) -- T o = A T is relatively low compared with To, we may use the approximate transformation of the exponential term in equation (18). Then neglecting the temperature dependence of 2, we get, instead of cquation (18), 2 - -

~ W -- A* exp

A

(20) (

~T0)

where A* = A exp

--

conditions being {dAT'I = 0

and

~ dZ ]z=o

, the boundary

(AT)z=a0 = O. (21)

Integrating equation (20) and allowing for (21,) we get a solution in an implicit form ((±~.)~ d),~/2 I

'v/ A*kT"2 (k~o 2 ) 2~ exp y -- 2Wy + F = Z0 -- Z

div(2 grad 7;(r)) + W

.-lexp[..~E. ~ = 0 Lkl ~(r) J (18)

where W(cal/cm a sec) is the amount of heat liberated per unit volume in unit time (subtracting the energy spent for chemical reactions in regions adjacent to tracks),

= 0

dZ"

(22)

where

F = 2W(:XT)z =0 > exp

(E ~

2A*kT°2E (AT)z=0

)

.

(23)

The value of (A T)z ~ o is obtained by solving equation (22) at Z 0.

8

V.I. Goldanskiiand Yu. M. Kagan

T h e energy absorbed by a chemical reaction per unit time inside a c o l u m n 2 Z 0 long with a unit base area (s :- 1 cm 2) will be d e t e r m i n e d by the relation U

:: 2ZoW -- 22 ~--)-Z-]z = z.

As follows from e q u a t i o n (20) ---~--] z- z _ = ~

--

=

U + nl2ZoAO®

nlQo being the overall nuclear radiation ÷ F (25)

--

Uto t

(24)

so that =

u n d e r question (2Z 0 × 1 cm a) per unit time will be

+ F

• (24')

T h e total a m o u n t o f n u c l e a r energy absorbed by a chemical reaction inside the c o l u m n

energy liberated per unit time in unit volume. T h u s the total part o f the utilized nuclear energy will be ~ = 1 -- ~/2 ~ / A*kTo 2 It will again be emphasized that the efficiency of the utilization o f energy liberated in the averaged heating up o f the system is, as a rule, less with the heating up of microvolumes along particle tracks.

THE THERMOCHEMICAL E F F E C T OF A N A R R O W OF C H A R G E D P A R T I C L E S

N a r r o w beams of accelerated charged particles are often used in practice. Analysis of reactions along individual tracks of the b e a m reveals no specific characteristics provided the tracks do not overlap. In o r d e r to describe stationary conditions outside the b e a m , it would be feasible to consider the

where ~ (cal/cm sec) is the energy lost every second per unit path length. A transformation o f the e x p o n e n t similar to transition from equation (18) to (20) is assumed to hold within the range r > r 0. Substituting the variables we get

andge°metrYhaving°faaradiusCylinderR0.c°axialwith the b e a m M a k i n g use of e q u a t i o n (18) and assuming that 2 -- const, we have

td2(Ar) 1 dAT 1 2[ dr~ + r dr ~ 0.

:~e:

(AT)r=R0 = 0

AT

-- Z2.

(30)

E q u a t i o n (28) m a y then be rewritten as

{dZiZ 1 zdZ \ d r ] ; r- T ~ B :~ 0 (31)

,128)where

Let us i n t r o d u c e an effective b e a m radius, r 0. Neglecting the energy absorption within a range r < r 0 and assuming that the energy liberated per unit b e a m length considerably exceeds that absorbed in a similar v o l u m e at a wall t e m p e r a t u r e , To, we obtain for b o u n d a r y conditions

dAT I --]t--~r27rr

exp , ~

d2Z Z~--

- - A * exP(k~02 A T ) =

BEAM

B

A*E 22kToZ.

(32)

E q u a t i o n (31) will be solved as Z -- %r" + :~,,r°.

(33)

Direct substitution readily shows that the solution o f equation (33) satisfies (31) if B

(34)

On the thermochemical action of ionizing radiation

Making use of the first boundary condition of equation (29), we obtain ~2 (2

-

a) r20(x -a)

-

a + "--xx

1 + otj r20(l_a)

Ee a - - 4 r r 2 k T ~ -- 9,0

= 9'0 - - 4rrZkTo ~ •

(35) The second boundary condition of equation (29) will be, allowing for equation (30). OClRoa _}_ ezR,~- a = 1.

(36)

Multiplying both parts of equation (36) by el and making use of equation (34) we get

1 ~1 - - 2 R0 a

source. The value of a (lower index) will then be readily obtained:

Ee

°~1

9

(38)

Besides, the coefficient of the term with a lower index will be positive in this case, and consequently at r0 ~ 0 the root of el in equation (37) will also be positive. The energy imparted by the beam to the chemical reaction per unit path length may now be determined : [ \ W = ~ -- 22rrR0 I\-W-r dAT| I,=R.

l+

(39) "

Determining 0c2 from equations (37)and (34), we readily obtain the ratio of ~[a~. The value of a may then be obtained from equation (35). If 9,° < 1' the limit r° -~ 0 can be used in equation (35). The first boundary condition of equation (29) will then be represented by a conventional condition for a threadlike

a

Making use of equation (36) we obtain the following expression for the efficiency of the thermochemical utilization of the beam energy. W l,_l { } ~7 . . . . e a (2 -- a) -- 2~lR0a(1 -- a) (40)

CONCLUSIONS

The given work is concerned with general characteristics of the thermochemical effect of ionizing radiation, no quantitative illustrations being given, It will be emphasized that a more explicit discussion of the problems treated above would involve data on radial temperature distribution along tracks of ionizing particles, as well as allowance for the fact that many parameters obtained for macroscopic systems under ordinary conditions may be essentially different under irradiation, especially at a high density of ionization. Indetermined phase transitions in the zone affected by radiation may also be of importance. All this makes the given work essentially quail-

tative. Nevertheless it may be anticipated that comparison with more definite results would permit more precise definition and development of the given calculations. The latter would also be of assistance in finding out whether the main part in the chemical action of radiation is played by the thermochemical action of ionizing radiation (either localized in microvolames or averaged) or by the effects due to hot atoms and radicals. In addition, the given calculations may be of use in discussing the possible application of energy-consuming chemical systems for the utilization of heat energy emitted by various radiation sources.

REFERENCES

1. LIFSHITZI. M., KAGANOVM. I. and TANATAROV 2. KAGANYU. M. Dokl. Akad. Nauk SSSR 119, 247 L. V. Atomnaya Energiya 6, 391 (1959). (1958).