THE BREATHER STATISTICS IN CRYSTALS FAR AWAY FROM EQUILIBRIUM

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Dale Terry
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1 TH RATHR STATISTICS IN CRYSTALS FAR AWAY FROM QUILIRIUM Vldimir Dubinko NSC Khrkov Institute of Physics nd Technology, Khrkov, Ukrine

2 ASTRACT The sttistics of discrete brethes (Ds) formed by therml spikes (TSs) in solids under irrdition with swift prticles is considered, nd the corresponding rection rte mplifiction fctors re derived. Rdition-induced formtion of Ds is shown to chnge mechnicl properties of mterils under irrdition, which is confirmed by experimentl dt. The rection rte mplifiction fctor increses strongly with incresing D lifetime nd the mximum D energy tht depend on the system The present model shows tht the effects due to the crystl nhrmonicity re of both fundmentl significnce nd of considerble technologicl importnce in the fields of nucler engineering nd rdition effects.

3 Relxtion of n initilly strongly heted lttice prt (therml spike) Ivnchenko, Knkov,.Shlfeev nd S. Flch Discrete brethers in trnsient processes nd therml equilibrium Physic D 198, 1 (4).

4 Rdition-induced therml spikes (TS) [Lifshits, Kgnov, Tntrov, 1959] T c = κ ΔT α T T t ( ) e e e e e p T c T T T t ( ) p p = κ pδ p + α e p xtremely short lifetimes ~ 1-1 ps

5 TS -induced rection rtes (LKT-theory) 1 ( ) R& ( T) = R& T + T( W) dw, T WT ( ) T T R& = R exp ( k T ) b 1 ε ep, ϕ, ( Tep) 16π L χ ( cep) ep,, - dibtic regime [LKT, 1959] ( T ) 1 ε 16π L ϕ χ c -symptotic regime [Dubinko, -Klepikov, Novikov, et l, 7] kbt RT & ( ) Rexp R T + >> 1 kt kt b b

6 TS -induced formtion of Ds D CONCNTRATION TMPRATUR (K) Therml equilibrium Therml spikes z T ( ) = τ 1 TS T min f K f ( ) = K exp( ) τ 1, min z Temperture dependence of the men concentrtion of Ds formed t therml equilibrium nd in therml spikes 14 ϕ = 1 ion/cm s K τ = 1 min =. ev ε + ε = 1 MeV e p z = 9 mx = 1 ev nergy distribution of Ds under irrdition nergy distribution of Ds under therml equilibrium

7 D -induced rection rtes RACTION RAT (1/s) mx ( ) ( ) ( ) TS, RT & = R & f I d= A R & K TS, TS K min RACTION RAT (1/s) b TMPRATUR ( K) Therml equilibrium Ds Arrhenius' low 13-1 R = 1 s ; mx = =1 ev TMPRATUR ( K) Rdition-induced Ds Rdition-induced TSs Arrhenius' low

8 D -induced mplifiction fctor ( ) = &( ) RT & A RT, TS TS mod min mx min z z A = Kτ exp( ) y min ( y 1) I( ymin) dy y ( y 1) I( r ) dy + k T b 1 mod min D AMPLIFICATION FACTOR mx = =1 ev 5 1 D DCAY XPONNT, z Therml equilibrium Therml spikes D AMPLIFICATION FACTOR b z = D MAXIMUM NRGY (ev) Therml equilibrium Therml spikes

9 Rdition-Induced Softening: experiment

10 ε T Rdition-Induced Softening: theory ( σ ) exp = ε kt b Therml kt & ε = + H & ε ex & & σ ( T, ε ) σ σ 1 b ln & T ex in c ε TS kt b ε ( σ ) TS & & ( & ) σ ϕ, ε σ σ 1 & ε TS ex = in + c & εex 14 kt b H ( ϕ ) & = A & ε, TS TS ( & ) ε D ( ϕ ) & ε kt b σ D ϕ, εex = σ in + σ c 1 A & εex H A A 3 ( ϕ ) 14 & ε kt b TS (, & ϕ ex ) = M c d & ε ex H δσ ϕ ε σ ρ ϕ T ( ϕ ) 1 ε = 16π L ϕ χ c ( ϕ ) & ε kt b δσϕ D ( ϕ, & εex ) = Mσ c A ρdϕ & ε ex H

11 Rdition-Induced Softening: theory vs. experiment

12 Outstnding problems 1) Time evolution of brethers (, ) f t d = K f t S t dt ( ) (, ) ( ) d dt =? ) Quodons - high energy mobile longitudinl opticl mode Ds tht cn trnsfer energy over gret distnces in tomic-chin directions [Russell, ilbeck, 7] ( ) r r 1 p( ) d = exp d πσ σ r r r probbility distribution of the rection potentil brrier + ( ) &( ) ( ) R& r = R r p r dr = R r r σ exp exp kt kt σ = - dispersion = function (?) of the quodon sttistics r

13 Summry Rection rte theory in solids hs been modified tking into ccount intrinsic loclized modes or discrete brethers tht cn pper in crystls with sufficient nhrmonicity resulting in violtion of Arrhenius lw. The rection rte verged over lrge mcroscopic volumes nd times including lot of Ds cn be incresed by mny orders of mgnitude depending on the D sttistics. The brether sttistics in therml equilibrium nd under irrdition with swift prticles hs been considered, nd the corresponding rection rte mplifiction fctors hve been derived. Rdition-induced formtion of Ds chnges mechnicl properties of mterils under rector conditions s compred to the surveillnce smples in out-rector tests fter equivlent irrdition dose, which should be tken into ccount in forecsting the mteril service life.

CHEMICAL KINETICS

CHEMICAL KINETICS Long Answer Questions: 1. Explin the following terms with suitble exmples ) Averge rte of Rection b) Slow nd Fst Rections c) Order of Rection d) Moleculrity of Rection e) Activtion Energy