Liu, K.F.

Shuy, G.W.

Zhang, J.S.

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PFC/JA-86-47Fusion Reactions of Polarized DeuteronsJ.S. Zhangt. K.F. Liu t tand G.V. ShuyJuly 198Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139 USAt Present address: Institute of Atomic Energy, Beijing, Chinati Present address: Department of Physics k Astronomy. University ofKentucky, Lexington, KY 40506.This work was supported by the DOE Grant No. DE-F605-84ER40154 and a grantfrom Southern California Edison Company No. SCE-240302. Reproduction,translation, publication, use and disposal, in whole or part, by or for theUnited States government is permitted.Submitted for publication to and accepted by: Physics Review LettersFusion Reactions of Polarized DeuteronsJ.S. Zhang+ and K.F. LiuDepartment of Physics & AstronomyUniversity of KentuckyLexington, KY 40806G.W. ShuyPlasma Fusion Center, MITCambridge, MA 02139AbstractPolarized and unpolarized d+d -+ n + 3He fusion reaction cross sectionsin the center of mass energy region of 20-150 keV are calculated in adistorted wave Born approximation. The calculated unpolarized crosssections and the anisotropy of the angular distributions are within 20% ofthe experimental data. The polarized cross sections are found to be - 7.7%of the unpolarized ones despite the inclusion of the D-state component in3He. This shows that the idea of a "neutron lean" reactor may still befeasible. This result is consistent with the partial wave analysis of thetensor analyzing power in d(d.p)t reaction, but contrary to the findings ofa recent resonating group calculation.+ Present address: Institute of Atomic Energy, Beijing, China2It has been suggested1.2 that the spin degree of freedom in the plasmafusion reactions should be explored in light of the possibility that itcould be utilized to enhance the desired fusion reaction rates and/orsuppress the undesired reactions. Examples are given for the t(dn) He.3He(d,p) He reactions. 1 ,2 The cross sections of these reactions at lowenergies are largest at the J = 3/2+ resonances of the compound nuclei 5 Heand 5 Li. At such low energies (i.e. 107 keV for the deuteron in d+treaction and ~ 430 keV in the d+ 3He reaction), the reactions arepredominated by the relative S-wave in the entrance channel. Thus thecombined spins of the d+t and the d+3He system are equal to 3/2 in theseresonance reactions. Since the combined spins of d+t and d+3He can beeither a = 3/2 or a = 1/2, this shows that the a 1/2 channels do not takepart in these resonance reactions essentially. Hence, upon polarizing thespins of the incoming nuclei in the parallel direction i.e., s = 3/2, thesetwo reaction cross sections will be enhanced by - 50%, a substantial gain.It was further demonstrated that of the four mechanisms considered thedepolarization rate is sufficient slower than the fusion reaction rate sothat fusion reactors with polarized nuclei are feasible as far as thedepolarization is concerned.An example for the simultaneous suppression of the undesired reaction33is the polarized d(d,n) He (and d(d,p)t) reaction which could admit thepossibility of a "neutron-lean" fusion reactor . There is an experimental3evident of such suppression shown by the partial wave analysis of the tensoranalyzing power in d(d. p)t reaction7 . The suppression is understood tooccur when the deuterons are polarized in parallel (i.e., a - 2) so that atlow energies where the relative S-wave dominates, the Pauli principlesuppresses the two deuterons from approaching each other to initiate thereaction. In addition, since the total spin of the final state n + 3He canonly be s = 1 or 0, the conservation of angular momentum and parity dictatesthat the exit channel must be in the relative D-wave which leads toadditional suppression owing to the fact that this has to occur via thespin-orbit and tensor forces instead of the stronger central force.The latter example on the suppression of the polarized d+d reaction hasraised a controversial issue. The argument for the suppression is based onthe implicit assumption that the D-state probability in 3He and deuteron aresmall and they do not contribute in any significant way. However, it ispointed out recently by Fick and Hofmann4 that the inclusion of the D-stateprobability in 3He will allow the strong central force to contribute to thes=2 channel and could yield large cross section in the polarized d+dreaction. In their refinded resonating group calculation4 , they found thatthe polarized reaction is not suppressed at all. In fact, at E C.M.= 20 keV.the ratio of the polarized to unpolarized cross sections is around 1. Thisseems to concur with the findings of the R-matrix analysis. 5Despite these results from the R-matrix analysis and the resonatinggroup calculation, we think the problem deserves further critical4scrutiny 6 because these results contradict the partial wave analysis of thetensor analyzing power for d(dp)t at Ed = 290 keV.7 Furthermore, thecentral force contribution through the D-state probability in 3He that Fickand Hofmann argued for is certainly allowed, but the magnitude of the matrixelement in their calculation is inscrutably large. The reason that thematrix element <3 D2 , D2 Vc S 2 > for the central force Vc between the 5S2 d-d channel and the 3 D2 , D2 n 3He channel is smaller than the matrix element<1 S V | S > is related to the smaller internucleon D-wave component in theincoming d-d system in the relative S-wave6 ; a smaller D-wave projection for3 8the central potential onto the relative coordinate between n and He , and anumber of cancellations upon antisymmetrization of the total wavefunction.The cancellation arises from the permutation symmetry uponantisymmetrization of the spin, isospin, and the spatial coordinates of thenucleons. This is quite transparent in the DWBA formalism where theantisymmetrization of the total wavefunction is treated analytically. Ontop of the above-mentioned reduction factors, the D-state probability in 3Heis merely ~ 4%. Therefore, based on the above reasonings, we estimatedethat the ratio of the polarized to the unpolarized cross section due to the-3 -4central force should be 10 10-. This is 3 to 4 orders of magnitudesmaller than that obtained in the recent resonating group calculation. 45In order to verify our point quantitatively, we carried out a distortedwave Born approximation (DWBA) calculation. Since the d+d reaction crosssections are of the order of ~ 10 mb , they are more than two orders ofmagnitude smaller than the elastic d-d10 and n 3He11 cross sections. Thisshows that the off-diagonal matrix elements in the coupled d-d and n 3Hechannels are at least an order of magnitude smaller than the diagonal matrixelements. Therefore, the perturbative calculation such as DWBA should bereasonably good.The DWBA formalism has been adopted by Boersma 1 2 to study the ddreactions at these low energies. Contrary to what is implied in the Reply13of Fick and Hofmann , there is no criticism to the suitability of DWBA forthese reactions in Ref. 12. Instead, the success on the energy dependenceof the cross sections is stressed and a more realistic dd wavefunction(which was not available at that time) beyond the Coulomb wavefunction usedis called for improvement. The present work represents an attempt to answerthis call for improvement.Effective potentials (including the folded Coulomb potential in theentrance channel) are used to generate the distorted waves between the d andd in the entrance channel and between the n and 3He in the exit channel.They are fitted to the d-d elastic scattering phase shifts given by D.R.Thompson 4 and the n 3He elastic cross sections. The internal deuteronwave function (2G) is taken from the work of F.S. Chwieroth, Y.C. Tang andD.R. Thompson and the 3He wave function from the work of B.R. Bransden.6H.H. Robertson. and P. Swan. 16  The same nucleon-nucleon interaction asused in the recent resonating group calculation4 (called S in Ref. 4) isemployed to calculate the matrix elements in the Born approximation so thatthe results can be compared with those in Ref. 4.The unpolarized reaction cross section for the DWBA is written as:2 [ 0 e2 3 3 2 Dkw (2Sd+1)t p c+ (2s+) }s1 (with the polarized cross section Oao written as2sl+ D< 3D2. D2 Ic+VLS+VT 5 2 2D (2)where k is the relative wave number in the entrance channel in the C.M.system. PD is the D-state probability in 3He which we take to be 4%. Vc'VL9S, and VT denote the central, the spin-orbit, and the tensor interactionsrespectively. The subscript D in Eq. (2) denotes the matrix elements forthe D-state component in 3He. The normalization of the relative wavefunctions in the entrance and the exit channels include the_-" h 1/2 ad hjs )1/24w 2h2 412h27factors respectively (p/p' and k/k' are the reduced mass and the relativewave number in the entrance/exit channel), so that the matrix elements aredimensionless. These matrix elements are calculated for the relative S and Pwaves in the entrance channel. The relative D wave is very small at theseenergies and is therefore neglected.The unpolarized cross sections and the anisotropy C2/C0 in the angulardistribution (dc/dO - C0+ C2 P2(coso) + ... ) at 25 to 160 keV C.M. energiesare plotted in Figs. 1 and 2 respectively together with the experimentaldata and the results from Fick and Hoffman.4 It is seen that our calculatedunpolarized cross section and the angular anisotropy are generally within20% of the experimental data in this energy region which justifies ourapproximation based on the DWBA calculation. It is worthwhile to point outthat our results on the unpolarized cross sections represents an improvementover those in the resonating group calculation4 which are roughly a factorof 2 smaller than the measured cross sections.The results of the matrix elements (dimensionless) in differentchannels due to the various parts of the NN interaction are tabulated inTable 1 for the center of mass energies of the deuterons at 50, 100, and 150keV. From Table 1 we have learned that the P-wave contribution constitutes~ 4% of the total cross sections. Thus, it verifies that in this energyrange the cross sections are indeed dominated by the S wave in the entrancechannel.8The presence of the D-state component in 3He allows an a a 3/2 wavefunction in He which gives rise to an s a 2 matrix element by the centralforce. According to Table 1, its magnitude turns out to be an order ofmagnitude smaller than that of the S-wave, s = 0 matrix element due to thecentral force. This is attributed to a number of cancellations due to thepermutation symmetry among the interacting nucleons upon antisymmetrizationwhich involves large numbers and reflects the fact that the D-wavecomposition between nucleons in different incoming deuterons is smaller thanthe corresponding S-wave composition at these low energies. Multiplied by ~4% for the D-state component in 3 He, we predict that the central forceyields polarized cross sections which are ~ 1.3 x 10 times of theunpolarized cross sections. in agreement with our earlier estimate. This iscontradictory to the findings of the resonating group calculation whichpurports that the central force is responsible for their - 1:1 ratio of thepolarized to the unpolarized cross sections.The spin-orbit and tensor forces, on the other hand, yield largercontributions to the polarized cross sections via the D-state component in3He (Table 1). Especially, the tensor force gives the largest contributionsince it could connect a D-wave to an S-wave. However, according to ourcalculation, the combined central and these spin-dependent forces yieldpolarized cross sections which are - 7.7% of the unpolarized ones in thislow energy region. This is certainly consistant with Ad'yasevich andFomenko's analysis which predicted the polarized cross section to be about95% of the unpolarized one at Ed = 290 keV. However, this is much smallerthan those predicted by the resonating group calculation4 and the R-matrix5analysis. The ratios of the polarized to unpolarized cross sections at lowenergies are plotted in Fig. 3 together with those given by Fick andHofmann.4To further unravel the situation, we turn to the tensor analyzingpowers which have certain bearings on the issue of the s=2 channel. It waspointed out in Ref. 4 that, without the s=2 channel contributing, theangular distributions of T20 and T2 2 (Azz and A ) should be symmetricwith respect to 900 while those of T2 1 (Az) antisymmetric with respect to900. The experimental results' 8 on the angular distributions of the tensoranalyzing powers for the reaction d(a,n)3He at Ed - 320 keV (160 keV in thecenter of the mass system) reveals that the angular distributions of T2 0 andT22 are rather symmetric with respect to 900 and those of T2 1 are quiteantisymmetric with respect to 900. These are certainly contradictory to theimplications of Fick and Hofmann in their article. 4 Results plotted on thecontour diagrams at lower energies (Ed = 60, 80, 105, and 205 keV) 1 8 showthat the asymmetries from the expected symmetries without the s=2 channelcontribution are even smaller. This suggests that the s=2 channel isrelatively unimportant at such low energies.10Combined with the experimental results on the tensor analyzing powersand our DWBA calculation, we feel that the polarized (s=2) cross sectionsmay well be strongly supressed (at the 8-15% level) despite the presence ofthe D-state component in 3 He. Thus a "neutron lean" fusion reactor based onthe suppression of the polarized d-d reactions may be attainable. However,more experimental data on the tensor analyzing powers and theoretical worksinvolving the D-state component in the deuteron are still desired.This work is supported in part by a DOE grant No. DE-FGO5-84ER40154 anda grant from Southern California Edison Company No. SCE-240302.11REFERENCES1. R.M. Kulsrud. H.P. Furth. E.V. Valeo and M. Goldhaber. Phys. Rev.Lett. 42. 1248 (1982).2. S. Tamor, G.W. Shuy, and K.F. Liu, Bull. Am. Phys. Soc. 2i, 922(1982).3. G.W. Shuy. A.E. Dabiri. H. Gurol. Accepted for publication inFusion Technology.4. H.M. Hofmann and D. Fick, Phys. Rev. Lett, 52. 2038 (1984)5. G.M. Hale, Proceedings of the Polarized-Fuel Reactor Workshop,Madison, Wisconsin, 1983, p. 11; G.M. Hale and G.D. Doolen, LosAlamos preprint LA-9971-MS.6. K.F. Liu, J.S. Zhang, and G.W. Shuy, Phys. Rev. Lett. 5, 1649(1985).7. B.P. Adyasevich and D.E. Fomenko, Sov J. Nucl. Phys. 2. 167(1969).8. The D-state probability in the deuteron is neglected in thepresent work. It is only ~ 7% in the deuteron and has beenneglected in the calculations of Ref. 3. For the configurationthat all the internucleon coordinates are S-waves in the ddsystem, three pairs of the central NN potential drops out due toorthogonality which leads to an additional reduction factor of 2for the matrix involving this configuration.9. H. Liskien and A. Paulsen, Nucl. Data Table 11, 569 (1973).10. Balashko and Barit, JETP Z. 715 (1958).11. A.R. Sayres, K.W. Jones. and C.S. Wu, Phys. Rev. 122. 1853 (1961).12. H.J. Boersma, Nucl. Phys. AM. 609 (1969).13. D. Fick and H.M. Hofmann. Phys. Rev. Lett. fi, 1650 (1985).14. D.R. Thompson, Nucl. Phys. LAW, 304 (1970).15. F.S. Chwieroth, Y.C. Tang, and D.R. Thompson, Nucl. Phys. AIM , 1(1972).16. B.H. Bransden, H.M. Robertson, and P. Swan, Proc. Phys. Soc. LXIX,12-A. 877 (1956).1217. H. Eikemsier and H.H. Hackenbroich, Nucl. Phys. LMI.2, 407 (1971).18. B.P. Ad'yasevich, V.G. Antonenko. and D.E. Fomenko, Sov. J. Nucl.Phys. M, 313 (1981).Table 1The DWBA matrix elements (dimensionless) in different channels atthree different enegies due to the various parts of the NNinteraction. Vc stands for the central force, VL.S thespin-orbit force and VT the tensor force. Matrix elements withsubscript D are those with the D-state component in 3 He (the 4%D-state probability is not multiplied in these matrix elements).C.M. Energy (keV)50 100 150<1 SOIVcI1 SO) 0.29x10-1 0.58x1&l0 0.86x10-1<3plVcl 3 P> 0.31x10-2  0.63x10-2  0.94x10- 2<3D2IVL.Sl 5S2> 0.25x10-2  0.49x10-2  0.74x10-2<3D2,lD2VTI'5S2> 0.33x10-2  0.66x10-2  0.99x10-2D' 3D2,1 D2IVcI5 S2> -0.12x10-2  -0.24x10-2  -0.36x10-2D< 3D2,lD2IVL-Sl5S2> 0.61x10-2 0.12x10-1  0.18x10-1D< 3D2,1 D2 IVTI5 S2> -0.15x10- 1  -0.30x10-1  -0.45x10l-Figure CaptionsFig. 1. The d(d,n)3 He reaction cross sections as a function ofthe center of mass energy in the entrance channel.The dots are the experimental data. The solid lineis from our DWBA calculation and the dashed line isfrom the resonating group calculation in Ref. 3.Fig. 2. The anisotropy C2 /CO vs. the C.M. energy of thedeuterons. The dots represent the experimental data.The solid line is our prediction and the dashed lineis from Ref. 3.Fig. 3. The ratio of the polarized to unpolarized crosssections. The solid line is our prediction and thedashed line is from Ref. 3.iI I II I I-o0~~010I jIla)c-o0000I I I I ~LL1 i I I I I I I0(qLu)~oI Iali~00I 0dI I I I I I l III I I I I I I I- 01 -- 0~ -0' -00to0ci0LoI I I I00I 1\ I I I00OD/ Zo-'00IO 00_I 0o~

Fusion reactions of polarized deuterons

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Fusion reactions of polarized deuterons

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