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JOURNAL O F RE SEA RC H of the N ational Bu reau of Standards - A. Physics and Chemistry Vol. 72A, No. 1, January- February 1968 . Thermalization by Elastic : Positronium In a Rare Moderator

William C. Sauder* Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234

(July 20, 1967)

T he e ne rgy d eca y of particles mov ing through a moderatin.,: medium is discussed for the case in whic h only elas ti c co ll isio ns occur between these in coming particles and the mode rator ; the ra ndom the rmal motion of the moderato r atoms is taken into account. It is s hown that if the c ross section is inde pe nde nt of partic le ve locit y the equation for the e ne rgy d ecay e me rges in a rathe r simple fo r m involvin g the h yperbolic cotangent. Fina ll y, the theoreti cal de ve lopme nt is a pplied to estimating the thermalization tim e 'of positro nium in ra re gas mode rators, a nd is s ho wn to agree with the limited experi me ntal results presently a vail a bl e.

Key Wo rd s : E lastic colli sions, positronium , therma lization .

2. Average Energy Loss 1. Introduction Particles of mass m, velocit y VI, and mome ntu m In several areas of phys ics processes occur in PI = mVI e nter a moderating medium a nd each collides which e ne rgeti c particles enter a mod erating medium elasti call y with a mod erator of mass Inll/, velocity and s ubsequently are brought to thermal equilibrium V II/, and P II/ = mll/ vlI/ . It is straightforward to with the atoms of this medium by means of collisions. show that the momentum of the incoming particle in Examples are fo und in ne utron age theory [lJ I and in the center of mass syste m is related to these qua ntities the literature of positron annihilation in matter [21. as fo ll ows: Many pre vious theore ti cal treatm ents have included the approximation that the thermal e nergy of the mod­ _ mil/PI - mplII _ In (I a) erator atoms can be neglected; obviously, this is not PiC- M - PI - M Pc vali d if one is interested in the terminal portion of the thermaliza tion process. As will be shown below, it is where PiC is measured in center of mass coordinates, possible to carry ou t a simplified treatment of the Vc is the velocity of the center of mass, and thermali zation problem and yet to take th e moderator e ne rgy into account. p c=(m + 1n/l,)vc= Mvc. In the following, slowing of particles solely by means r of elastic collisions will be discussed; the thermal If, after the , an inco ming particle scatters with motion of the moderating atoms will appear ex plicitly velocity and momentum V2 and P2 respectively, then it in the treatme nt. The resulting re presentation of the is also true that e nergy decay of the particles as a function of time will m involve an integral over the particle velocities; for the (Ib) special case in which the collision is a P 2C= P2 - M p c· constant, the integration will be carried out in closed , form to yield a rather succinct equation for the ther­ S ubtracting eq (la) from eq (Ib) yields the result that C' I malization process. It will be demonstrated in the last the momentum transfer is the same in both coordinate sectio n of the paper th at results derived herein are systems: useful in considering positronium atoms in rare gas moderators. P 2 - P I = P2 C- PIC, (2)

Using eq (1) one can obtain *Associale Professor of Physics. Virginia Military Institute, Lexington , Va . 24450. ') Consulta nt to the Crys tal Che mi stry Section. Inorganic Materials Di vision. Nati onal Burea u m In - • of Stan dard s. Washington. D.C. 20234. ( ) ( ) P I' P 2= PIC' P-lC + M Pc ' (PIC + P2 C) + M P c· (3) I Figures in brackets indicate the lit.e rature refere nces at th e end of this paper.

91 Squaring both sides of eq (2), and employing the fact ot eq (7) is sufficiently small in magnitude so that one th at momentum and energy conservation in the col­ may write li sion imply that /lIe = P2C, yield the following expres­ sion for the change in momentum-squared of the (v~-v1)/v1 = d(!n v2 )_ incoming particle: Thus eq (7) can be written in differential form,

d(ln v)=-(1+m/mll/)- 2 {(~~J-(v~,'Y} (9) In order to derive an expression for the mean energy loss per collision, one must average eq (4) (which is which describes a continuous process rather than the equivalent to an expression for the energy loss in any discreet process indicated in (7)_ givt;n collision) over all possible angles of incidence We now take from kinetic theory [3] the result that between P I and pm, over all scattering angles, and over the mean collision rate of the incoming particles with the distribution (Maxwellian or otherwise) of moderator the moderator is velocities_ We shall carry out these averages under the assumptions that (1) the moderator motion is com­ v= PIIUV (10) ple tely random, and (2) the scattering is spherically symmetric in the center of mass system_ These where U = the elastic collision cross section and assumptions can be expressed mathematically as PII = the number of moderator atoms per cm3_ Com­ fo ll ows : bining (9) and (10) leads to the differential equation governing the thermalization process: random moderator: (11) (Sa) isotropic scatterin g: ( P IC - P-2c) scat = 0_ (5b) where

Averaging eq (4) overall scattering angles according R II ,= In/ mlll _ to eq (5b) results in Separation of variables leads to the following integral: (6) ( p Dscat- P T=-2 (~) pc - PIC - v dv J (12) "0 u(v)[R lllv2 - VTIIJ (1 + RI/I)2 Since Pc = P I + Pili, then performing the averaging of eq (6) over moderator velocities [eq (Sa) I yields the where Vo is the initial velocity of the incoming particle following expression for the fractional energy change: as it enters the moderating medium. Equation (12) cannot be integrated unless the explicit dependence of the cross section upon velocity is known. - 2(1 + m/ m lll) - 2 { (.!!!.-) _ (v;~) } mill VI For the case in which u is a constant the results are (7) particularly concise. If the quantities a and f3 are defined as follows: where (VYII) is the mean-squared moderator velocity_ Having obtained this expression for mean energy loss, (13) we will (for convenience) drop the average bracket notation in all further work_ T he effect of taking the thermal energy of the Eo moderator into account can be identified with the term ,B=coth- I ~- (14) EIII (VII,/V I)2_ As the velocity of the incoming particle is reduced through elastic collisions to a magnitude where Eo and E,,, are the kinetic energies correspond­ comparable to that of the moderator atoms, the energy ing to Vo and Vm, respectively, then the integration of lost in successive collisions becomes smaller. Finally, eq (12) for (J constant yields when the particle reaches thermal equilibrium, that is, when E /EI/I=coth2 (,B+at) (8) (15) where E is the of the incoming particle. then no further decrease in energy occurs_ Note that a can be related to the collision frequency of the incoming particle at thermal equilibrium by 3. Thermalization employing eq (8): mmm mm", Usuall y for the physical situations mentioned in a= 2 PIIUVT = .,VT (16) the introductory paragraphs the left hand member (m+ milt) (m+.mlll )- 92

I I~_- wh ere 11'/' and VI' are the velocity and the collision Equation (15) of the preceeding secti on i s uitabl e fr equency of the ther malized particle , respectively. for making conservative estim ates of the e nergy of The form of th e thermali zati on equation, eq (15), positroni um at the time of its annihila ti on if one use is de monstrated in fi gure 1 in whi ~ h is pl otted the for (J the geometrical cross section, whi ch, as stated [uncti on coth t (a t ). One can see from eq (15) that (3 above, is a lower limit for the true elasti c colli sion indicates the point of the curve a t which the thermaliza· cross secti on. S uch estimates are of interest since the ti on process is initiated. Note that when (3 is s ma ll , for energy of the annihilating positronium atom is related a]] intents and purposes thermal equilibrium is to the width of the angular di stribution of the photons establi shed after the characteristi c time in te rval a - I. emitted in the annihilatio n process. Consider the thermalization of positronium in argon 10·r------, gas at 300 OK and 2 atm pressure. Using for the radius of the argon atom a value of 1.5 A in order to determine the geometrical cross section, a nd making use of the fact that argon under the stated conditions be haves a pproximately as an ideal gas, we have computed the value of a to be 10' y = C oth2at 0'= 3.9 X 106 sec-I (P s in argon).

Thus the estima ted th e rm alizatio n tim e for P s in argon is a - I = 2.6 X 10- 7 sec. S in ce the Ore gap for argon lies be tw een 9.0 a nd 11.6 e V, the n the ra ti o Eo/ EII/ is approximately 100. The onl y existing measureme nts of angul a r di s­ tributions in a rare gas that are of sufficient accurac y to test this result we re performed by Heinberg a nd Page [6]. In tha t experime nt, argon at room te mpera­ ture a nd 2 a tm pressure was e mployed as a modera tin g 10 medium ; the expe rim e nt al sensitivity was suc h th at the a uthors were a ble only to state tha t the Lim e re­ quired for the positronium ki neti c e nergy to fall to 0. 5 e V is greate r than 9 X 1O - ~) sec. Using the values of a a nd Eo /EII/ de te rmined above, eq (15) indicates that the tim e fo r the P s e ne rgy to I reach 0.5 eV is 5 X lO- H sec, whi c h agrees nicely with o 2 3 the expe riment al result s. t in units of a-I

FIGU RE 1. The energy decay fllllction for velocity·independent cross section. I tha nk R. D. Deslattes of the Nati onal Bureau of Standards for many helpful disc ussions during the time that the above results we re be in g de ri ved. I a m par­ 4 . Thermalization of Positronium ti c ularly grateful to R. C. Case ll a of the Nati onal Bureau of S tandards for the derivati on of the e ne rgy loss equati on [eq. (7)] in terms of expressions in mo­ W he n positrons enter matter, positronium atoms me ntum space; this is fa r more elegant th an the a re fo rm ed only after the kineti c e nergy of the posi· geo metri cal derivati on that had been done ori gin all y. tro ns has fall e n to a value lyi ng in the so-called Ore gap [4]; this region li es below the ionization po te ntial of th e moderator a toms, and typica lly is several elec­ 5. References tron volts wide; therefore the Ore gap occupies the region 1- 20 eV. If the moderator is composed of rare [1] Samu el c lasstone a nd Milt on C. Edlund , The Ele ment s of Nuclear Reaclor T heory, chp. VI (D. Va n Noslra nd Co. , In c., gas a toms, then th e elastic colli sion is th e only e nergy New York , 1952). loss mechani s m available to the positronium. Massey [2] P. R. Wallace, Phys. Rev. 100,738 (1955). and Mo hr [5] indica te that the elastic coll ision cross [3] E. H. Kenna rd , Kin etic Theory of Cases, p. 104 (l\ lcc raw·Hill secti on fo r positronium in the Ore gap is of the orde r Boo k Co., Inc., New York , 1938). [4] M. Deui sch, Prog. Nuc. Phys. 3, 131 (1953). of magnitude of the geome trical cross section, a nd [5] H. S. W. Massey and C. B. O. Moh r, Proc. Phys. Soc. (Lo nd on) th a t furthermore, thi s cross secti on increases at lowe r A67, 695 (1954). ene rgies. [6] Milt on Heinberg and Lome A. Page, Phys. Rev. 107, 1589 (1 957). (P a per 72AI- 485)

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