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Predicted Methods of Changing the Muon Catalized Fusion Cycling Rate E Predicted methods of changing the muon catalized fusion cycling rate E. V. Sheely, S. E. Jones, L. M. Rees, J. K. Shurtleff, S. F. Taylor, and J. M. Thorne Citation: AIP Conference Proceedings 181, 79 (1988); doi: 10.1063/1.37891 View online: http://dx.doi.org/10.1063/1.37891 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/181?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of nuclear interaction on muon sticking to helium in muon catalyzed dt fusion AIP Conf. Proc. 181, 330 (1988); 10.1063/1.37919 Muon losses in deuteriumtritium muoncatalyzed fusion due to fast transfer reactions to helium nuclei AIP Conf. Proc. 181, 161 (1988); 10.1063/1.37913 Muonic molecular formation under laser irradiation and in the clustered ion molecule (The effect of protonium additive on the muon catalyzed fusion cycle) AIP Conf. Proc. 181, 185 (1988); 10.1063/1.37895 Progress report on muon catalyzed fusion studies in H2+D2 and HD gaseous targets AIP Conf. Proc. 181, 68 (1988); 10.1063/1.37890 Density dependent stopping power and muon sticking in muon catalyzed DT fusion AIP Conf. Proc. 181, 355 (1988); 10.1063/1.37876 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 79 PREDICTED METHODS OF CHANGING THE MUON CATALIZED FUSION CYCLING RATE E.V. Sheely, S.E. Jones, L.M. Rees, J.K. Shurtleff, S.F. Taylor, J.M. Thorne Brigham Young University, Provo, Utah, ABSTRACT The following paper uses an easily modifiable computerized simulation proce- dure to predict reaction yields of muo-fusion. Reaction yields are predicted under varying conditions without the explicit solution of rate equations. Condititions and methods of optimizing reaction conditions by controlling the concentrations of hydrogen isotopes are discussed. The effects of non-equilibrated manipulation of reaction components and intermediates are predicted. I. INTRODUCTION There are three steps to solving muo-fusion rate equations: the determination of the Gibb's free energy, the solution of equilibrium equations, and the solution of differential kinetic equations. An easily modifiable method of predicting reaction yields under varying conditions without the explicit solution of rate equations will be presented. The method enables the effects of non-equilibrated reaction components and intermediates to be quickly and accurately predicted. The number of fusions observed during muon-catalyzed fusion is dependant upon many factors. Among the most important of these is the concentration of the various isotopes. For temperatures _< 500 K optimum yields during muon catalyzed D-T fusion are obtained when the concentration of DT is small in comparison to the concentration of D2 and T2. Recent analyses indicate that at high temperatures the cycling rate increases as the concentration of DT increases. Recent predictions also indicate that the presence of free T atoms in a reaction chamber will increase the cycling rate. Methods and effects of manipulating these reaction concentrations are disscussed. II. DERIVATION OF EQUATIONS In order to derive muon-catalyzed fusion kinotic equations we make several simplifing assumptions. There is experimental evidence to support some, but not all of these assumptions. Assumptions: 1) All reactions are either first order or pseudo-first order (they may be accu- rately considered first order in kinetic equations). © 1989 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 80 This assumption has been shown to be accurate in all cases in which it has been tested, and is in agreement with current theory on muon reaction mechanics in the rest of the cases, l There is no proof that this assumption will hold for all conditions. 2) The transfer reaction t# ~ dp may be considered negligible. 3) The tt# muo-molecules do not form via resonance. 4) There are no significant reactions taking place other than those represented in the following diagrams. (Epithermal and hyperfine effects have not been in- cluded.) Reaction Diagrams and Equations The following diagrams and equations represent reactions which are predicted to occur during muon-catalyzed-fusion:2 parallel arrows (T]') represent the reaction path of molecules with parallel nuclear spin, anti-parallel arrows (TJ.) represent the reaction path of molecules with anti-parallel nuclear spin, and Lambda (~) represent rate constants. Not included in the diagrams or equations is the effect of muon scavenging by impurities in reaction chambers. T x. --- t ~t* ~'o D2 ...J.~_x,.~ dd~td D 2 + /.t + DT x,~ = dd~tt T2 ),6 ~t~ I Gershtein, Soy. Phys. JETP, 51(6), pp. 1053-1058. 2 Jones, Nature,Vol 321, pp. 127-133. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 81 d[du] dt = x,[t:rr]O,-I + [O,l{~-] + x,dOlb,-i- Xolat,l - .h,l~,llr] - (.~, + ;~,)[a~,l[O:l -x,,[at, lItrrl - ;~,{a~,llr, l ~ tt ~td + DT ~ dutt T2 t~t+ + D2 . ~ -- dBtd T +g-~ g'+ DT~ 1.24 + T2 tt0.t ,/It,,,] = X, la~,)[r2) + .~.,[a~)[rl + ,Xslfllr21 + .xuO,-llrl + x,,~-)(zrrl - (x. + x.,)lo,lltrrl dt -(,h, + x,t)[t~l[n,)- ,h,[t~l[T,i L ~LI8 t~t +DT .~ dt tt / ~,, ~-- -"""M n2 + 12- t~t + ~.o This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 82 d[dtpt] = A~dt/~l[Dr] + A.[T][dtM] + A,.{T=][dt#d] - A4~[Dl{dtpt] dt -L,,{O,l[d~,q - (~o + ~.)[d0,l t + t~ T2 ,,~. "x tt gt .~~ 4 He + 2n3 + ~ ~'-~l~4He + 2n 3 tt Bd ~ Y tB+ TD t d[,,q dt = A24[T2][tp] - ( Ao 4- ,~,e)[ttpt] d[upd] = ;hd~r.][t. ] _ (;~o + ~.)[upd] dt d[dd,,1 = (~r + ~9){D2l[dp] + Aas[Dl[ddpt] + )qt[D2l[ddpt] - ('~o + ~s)[ddpd] dt -~a4[~[ddpd]- ~-tl[T~][ddpd] d[ddpt] dt = ~,2[DT~[dp] + ~a4{~[ddpa~ + ~_tt[T2][ddpd] - (,~o + ~to)[ddpt]- ,~[ D][ddpt] - )q l [D~][ddpt] d{p-] ! = ~{~s[dd.dl + ~o{dd,~t}}(2 - ~ - ,.'3) + {~.[dt.q + ~.[d0,d])(l - ,~) dt +{~26[ttpt] + .hT[tt.d]}(l - ,'4) - [p-]{.Xo + At[E/'] + Az[D2] + A~dD] +~.[T,] + ~.6[T] + ~dDTJ} This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 83 ~'o d~ +D2 /,.',,/~,, ~-"~'t~. + P d~ + DT ddlat ~ / ~laHe+ t ! ! ! ~.o d~ tt 4 = ~ { ~.[d@d] + )~to[ddl~t])(l - ~) d[4//e] dt It = dt dt dt = + + din'-] dt This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 84 III. DETERMINATION OF EQUILIBRIUM CONSTANTS In order to determine the relative amounts of various hydrogen molecules which will be present in a reaction chamber at a given temperature and pressure, it is nec- essary to know the equilibrium constants (Keq) of the gases involved. Equilibrium constants are most often calculated from the Gibb's free energy (AG) by the equa- tion: 3 A G = - RT in Keq R = 8.3144 T is the temperature in degrees Kelvin Values of AG for non-tritium containing hydrogen molecules can be obtained from J. phys. chem R~.f. Data, Vol 14, suppl. 1, 1985 pg. 991,1211. For tritium containing molecules, the Gibb's free energy and equilibrium constants must be determined from the partition function. Due to the radioactivity of tritium, the standard form of the partition can not be used. Corrections need to be made. 4 In order to expand known values to all possible temperatures, it is necessary to determine an equation that fits the data or to perform interpolation and extrapo- lation. Iterative non-linear interpolation can give greater accuracy, however, it has the disadvantage of taking longer to calculate. The iterative technique which I have used is Nevil's non-lincar interpolation method, s IV. EQUILIBRIUM EQUATIONS In order to determine the equilibrium concentration of hydrogen isotopes, it is necessary to consider the reactions: 1 1 I I 1 1 DT ~ ~D~ + ~T~ ltt 1 D 3 Levine, Physical Chemistry P. 166. 4 Souers,Hydrogen Properties for Fusion Energy, Pg 20-24,160,295 s Burden, Numerical Analysis, P. 97 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.187.97.22 On: Tue, 01 Apr 2014 20:40:57 85 1T•--~ T Using the above reaction equations it is possible to derive the following equi- librium equations KX,H =~ ~ KX,HD ~ XHD Kx,D : ~ Kx,Hr =~ XHT KX,T =-- ~ KX,DT ~-~ XDT These reaction diagrams and equations do not include all possible components, however, they do include all components which occur in quantities which can effect normal muo-fusion reaction conditions. 8 It is necessary to evaluate the initial concentrations of the isotopes.
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