Few-Body Systems9, 155-164 (1990)

ystcms sFgWa

9 by Springer-Verlag1990 Printedin Austria

Determination of the Reaction-Matrix Elements of the D(d, n)3He and D(d, p)3H Reactions for Ed 500 keV* S. Lemaltre and H. Paetz gen. Schieck Institut ffir Kernphysik, Universit~itK61n,ZfilpicherStrasse 77, D-5000 K61n41, Federal Republic of Germany Abstract. All available data of the mirror fusion reactions D(d, n)3He and D(d, p)3H have been subjected to a new analysis in order to extract the matrix elements of all 16 transitions necessary for inclusion of all 1 ~< 2 waves. Their energy dependence was assumed to be governed solely by Coulomb penetrabilities. The Levenberg-Marquardt algorithm was used to fit all experimental data. The experimental data are reproduced satisfactorily. The results compare well with an R-matrix analysis and with refined resonating group calculations. No suppression of quintet entrance-state transitions and therefore no neutron suppression in "polarized fusion" can be derived from this analysis.

1 Introduction The different observables of the D(d, n)3He and D(d, p)3H mirror fusion reactions at low energies (~< 500 keV) have been studied for many years. These studies included measurements of the total [1] and differential [2-4] cross sections, of the polarization of the outgoing nucleons (for a summary see ref. [5]) and measurements with a polarized incident beam [6-11]; however, no polarization-transfer measurements and no polarization-correlation experiments have been performed so far. The need for many different and mostly polarization measurements in these two reactions arises from at least two sources. One is the unusual complexity of these reactions even at every low energies, thus the lack of a workable theory or even of a simple model; to improve this situation requires more observables and more complex ones than just unpolarized cross sections. Another is the recent proposal of using polarized particles in future fusion reactors either to enhance the reaction rate of the desired reaction or to suppress the rate of unwanted reactions. As to the theoretical description of the D + D reactions, expansions of the observables in terms of combinations of contributing matrix elements have been

* This work was fundedby the German Federal Ministerfor Researchand Technology(BMFT)under the contract numbers 06-OK-153 and 06-OK-272

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given by several authors [12, 13]. The most comprehensive of these analyses was performed by Ad'yasevich et al. [14]. As new measurements at increasingly lower energies became available, it became also clear that the D + D reactions display unusual features: The anisotropy of the unpolarized cross sections even at energies as low as Ec.m. = 15 keV (see, e.g., the most recent results by Krauss et al. [4] and references therein) shows the contribution from P-waves (i.e. the coefficient of P2 (cos | in the Legendre expansion), and D-waves (coefficient of P4(cos | appear already significantly above 100 keV for the D(d, p) reaction and above 150 keV for the D(d, n) reaction. The vector-analyzing powers A r, which are mainly due to interference terms between different waves (predominantly S- and P-waves at the lower energies, i.e. terms such as Im(e0/?~l ) in the notation of ref. [14]), appeared significantly different from zero even at 30 keV [15]. Another interesting question, especially in connection with fusion-energy research, is the contribution (or suppression) of quintet states in the two reactions and possible channel-spin transitions with AS-- 1 (such as triplet-singlet transitions: matrix element B~, and quintet-triplet transitions: matrix elements 61 to 65) or AS = 2 (quintet-singlet transitions, matrix elements 7~, 72, and 73). Contributions from quintet states have in the past been excluded by an argument based on the Pauli principle [16-18]. This argument may be considered very weak because of the extremely large interaction radius of two deuterons. In fact, a careful analysis of the data of the D(d, p) reaction at 290 keV in terms of partial waves [14] showed a certain amount of quintet-state suppression. A later analysis of data by the same group [19] and comparison with all other available data in the energy range below 485 keV showed in addition that transitions from the quintet S state of the D(d, n) branch appear to be hindered relative to the D(d, p) branch. A renewed interest in the D + i~ reactions at low energies results from the proposal of an advanced "neutron-lean" fusion-reactor concept based on the 3He(d, p)4He reaction [20, 21]. Such a reactor will be aneutronic only if the D(d, n)3He reaction rate could be substantially suppressed. If quintet states were strongly suppressed in the D(d, n) reaction, then the use of deuterons polarized along the direction of the plasma-confining magnetic field would lead to a suppressed neutron-production rate and in conjunction with polarized 3He nuclei to a possible rate increase (or lower ignition limit) for the 3He(d, p)4He fusion in analogy to the 3H(d, n)4He case. Different approximative approaches to a theoretical description of the D + D reactions have been undertaken. These include a simple potential model [16-18], an R-matrix parametrization approach [22, 23], DWBA calculations [13, 24, 25], and resonating group (RRGM) calculations [26, 32]. Only recently microscopic 4-body (Faddeev) calculations have been performed for the four-nucleon system though with very limited complexity [27]. Especially on the question of quintetstate suppression these approximative approaches have produced vastly different predictions. In order to contribute not only to this question but more generally in an attempt to determine all transition matrix elements of both reactions a new straightforward analysis of all available data with E d ~< 500 keV was performed. The approach is similar to Ad'yasevich's [14] and the method and additional results will be published elsewhere in more detail. This approach was chosen as the least-model-dependent

Reaction-Matrix Elements of the D(d, n)3He and D(d, p)3H Reactions

157

approach succeeding in extracting significant numbers for real and imaginary parts of the transition amplitudes. As in ref. 1-14] and older references the energy dependence of the matrix elements was assumed to be solely determined by Coulomb penetrabilities. The analysis includes all 16 transitions necessary when waves up to and including D-waves are considered. A new analysis such as this is prompted by the availability of more recent sets of new data and also of new means of performing the necessary extensive computations. 2 Theoretical Framework

2.1 Relevant Transitions of the D + D Reactions Below 500 keV Let us now consider, which transitions must be allowed for the D + D reactions. In spite of the very low energies relative P- and even D-waves influence the observables strongly besides the incoming S-waves. This is probably due to the extraordinary size of the deuteron (the interaction radius of two deuterons is about 7 fro) and the low reduced mass of the two deuterons. This last feature leads to a centrifugal barrier which is higher than the Coulomb barrier even for P-waves [18]. With respect to total-angular-momentum and parity conservation and to the symmetries that result from the identity of the particles in the entrance channel one obtains 16 relevant transition amplitudes for incoming S-, P-, and D-waves (according to the notation (2S~+l I~s~Ij.12sp+l lps/, \ with ~ for the entrance channel,/3 for the exit channel), which ~ are shown in Table 1. There are three incoming S-waves, five P-waves, and eight D-waves which undergo a different repulsion from the Coulomb and centrifugal potentials. Because of the large Q-value of the reactions (3.27 and 4.03 MeV for D(d, n)3He and D(d, p)3H, respectively) the outgoing S-, P-, and D-waves are influenced little by their respective barriers and the F-wave will only change little with energy 1-18]. For the same reason the Coulomb barrier for the protons may be neglected. The two deuterons may be in singlet, triplet or quintet configurations corresponding to the channel spins S~ = 0, 1 or 2. The maximum channel spin in the exit channel is Sr = 1. Therefore, when S, = 2, either a second-order spin-flip transition must occur or an admixture of higher internal-angular-momentum states to the ground state of the nuclei render possible a first-order transition.

Table 1. List of the 16 matrix elements of the two D(d, n) and D(d, p) reactions including angular m o m e n t a up to L = 2 % = (1So]0+[1So)

71 = (5S212+[1D2)

~o = (3poI0-I3Po)

~2 = (SOoIO+l~So)

311 = ( 3 p i l l - l I P 1 ) ~11 = ( 3 p l I l - I 3 P 1 )

Y3 = (SD212+I1D2) dl = (5S212+13D2)

~12 = (3P212-13P2)

~2 = (50111+13Si)

c~u = (1D212+I1D2) 32 = (1D212+I302) % = (3puI2-[3F=)

~3 = (5D1]l+[3D1) 64 = (50313+1303) fi5 = (SOz12+[3De)

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S. Lernaitreand H. Paetz gen. Schieck

The contributions from the quintet configuration were often considered negligible because of the action of the Pauli principle [18]. However, as our calculations show, quintet transitions must not be neglected. Especially the incoming quintet S-waves (Ss212+llD2) and (Ss212+ 13D2) are of great importance down to the lowest energies.

2.2 Energy Dependence of the Transitions The experimental data show smooth excitation functions for all observables with deuteron energies below Ea = 500 keV. This implies the absence of nearby resonances of the nuclear potential [28]. (A resonance-like behaviour appearing in some measurements near Ed = 100 keV turned out to be an artefact [-28, 29].) For energies lower than E a = 500 keV this means that the nuclear potential will not contribute significantly to the energy dependence of the reaction matrix elements. The energy dependence is then mainly due to the Coulomb and centrifugal barriers in the entrance channel. Therefore like many recent approaches [-13, 14, 16, 18] we assume the transition amplitudes T~,(E) to factorize like

T~(E)= Q(E)Tt~,,

(1)

Q(E) = x/P~I,(E) exp(i(6,, + qoto)),

(2)

with

1

P,o(E) = F,~(E) + 6,~(E)"

(3)

Here Ft=and G~=are the regular and irregular solutions of the Schr6dinger equation for the scattering of two charged particles approaching each other to a distance of R = 7 fro. The phases 6~o,~0~oare well known, 61, = -- arctan Flo/Gt~, q~z== F ( / + 1 + it/).

(4)

The 2?p~are constant in the energy range of interest. 3 Method

Because of the lack of experiments that involve the polarization of at least two particles simultaneously, like the measurement of polarization transfer, it is not possible to determine the transition amplitudes in a direct way from experiment without additional information (compare ref. [30]). However, if we consider the energy dependence to behave according to Eq. (1), we find the available experimental data to be sufficient for determination of the reaction matrix elements. Since observables are always written in terms of Ti Tj* (here T / - Te~),in the present case only 31 real parameters can be deduced from experiment, while one common phase will remain unknown. Here we are concerned with 31 real parameters from the ]P~,in the energy range investigated. The data base for the present analysis consisted of all data for both reactions available in numerical form below 500 keV including total and differential cross sections, vector- and tensor-analyzing powers and po-

Reaction-Matrix Elements of the D(d, n)3Heand D(d, p)3H Reactions

159

larizations of the outgoing nucleons. Only data which were manifestly inconsistent with more than one data set of other authors or had significantly larger errors have been omitted in the analysis (for a discussion of the experimental situation see, e.g., ref. [31]). Except for the total cross sections the experimental angular distributions of all observables have been parametrized by coefficients of Legendre polynomials or associated Legendre functions expansions. These expansion coefficients are bilinear expressions of the transition amplitudes and depend on energy. Therefore all total cross sections and each expansion coefficient can also be factorized in terms of the energy-independent amplitudes ~ . For the D(d, p)3H-reaction the calculations were performed with 153 equations (data points) and for D(d, n)3He with 131 equations. They were carried out (separately for both cases) by minimizing the z2-function,

)f2 ;~ 2 (Yi -- Y(Xi, a))2/a2.

(5)

The Yi, o-~in Eq. (5) are the experimental data and their statistical errors, respectively. The model function y(xi, a) depends on the independent variables xi (the energy of the incoming deuterons, their relative angular momenta and the tensor moments of the beam) and the dependent variables a, which actually are the real and imaginary parts of the transition amplitudes. In order to find the absolute minimum of the z2-function a derivative-free form of the Levenberg-Marquardt algorithm was used, a "+1 = a" - [~,D, + J~,J,J-lJ~f(a').

(6)

Here J, is the numerical Jacobian matrix evaluated at a", D, is a diagonal matrix equal to the diagonal of J',J,, and ~, is a positive scaling constant, the so-called Marquardt parameter. The procedure starts with an arbitrary value a s for the amplitudes and then continues either with a Newtonian step or a "steepest-descent" step. Near the minimum at am~. a Newtonian step will be most effective, because linearization of the model function is allowed. In this case the Marquardt parameter ~, is small (<< 1). If, on the other hand, a Newtonian step does not succeed in reducing the )~2 by reason of nonlinearity of the model function the iteration will proceed in the direction of steepest descent by increasing the Marquardt parameter. To ensure that the solution ar.i. is unique and that it belongs to the absolute minimum of the xZ-function the procedure was repeated many times with initial values chosen randomly out of the interval [ - 1. . . . ,1]. For the reduced Z2 the fit yields 5.1 in case of the D(d, p) reaction and 3.4 for D(d, n). 4 Results

4.1 Matrix Elements The best-fit results for the absolute values and phases of the 16 energy-independent transition amplitudes ~ for both reactions are shown in Table 2 along with their errors. Because a common phase remains undeterminable the phase of~ o was chosen to be zero. From Table 2 we learn that the two quintet S-wave transitions are of the same order of magnitude as the singlet S-wave transition for both reactions. Furthermore

160

S. Lema~tre and H. Paetz gen. Schieck

Table 2. The constant "internal" transition amplitudes ~ , that minimize the x2-function for D(d, n)3He and D(d, p)aH, respectively, in the energy range 0 < Elab < 500 keV. The normalization of the absolute value (upper number) is such that the total cross sections are given in millibarns. The phases (lower number) are in degrees relative to the phase of %. The errors are specified in parentheses 7~

D(d, p)



3,81 0

(0.15) (2)

(3po[0- L3po)

4.02 80

(0.43) (9)

(3pill-lIP1>

4.39 14

(0.12) (2)

(3pill-laP1)

7.78 -67

( 3p212-13P2)

1.17 227

(10212+1102)

5.7 -- 131

(1D212+IaD2) (3P212-I3F2)

~,

D(d, n) 1,38 (0,11) 134 (5)

15.4 -244

(1.3) (4)

19.1 (3,7) 128 (11)

4.33 (0.20) (50212+[102> 30 (3)

6.46 (0.77) 8 (4)

7.5 (1.9) - 10 (12)

(0.20) (1)

8.90 (0.37) (sS212+13D2) -92 (3)

1.37 (0.05) -111 (2)

1.71 (0.09) -138 (3)

(0.18) (9)

2.89 (0.38) (5Dl11+1351) - 83 (4)

(1.2) (15)

(0.21) (4)

4.04 (0.26) (55212+11D2) 0 (5)

D(d, p) 1.85 (0.07) -241 (2)

0.2 (1.3) 64 (151) 2.23 -67

D(d, n)

5.5 ( 1 . 1 ) -49 (12)

7.6 - 142

(2.3)

(SDo[0+ 115o>

(59111+1301)

(17)

8.0 46 3.4 56

(1.1) (8) (1.2) (27)

5.30 (0.95) (SD313+I3D3) 9 (23)

8.03 (0.75) -- 155 (5)

0.59 (0.32) (SD212+1302) 72 (36)

3.25 (0.94) --93 (8)

1.8 (1.9) 35 (57) 11.2 25

6.9 172

(3.4) (15)

(1.4) (9)

3.4 (1.1) 174 (21)

the triplet-singlet transition fll 1 is stronger than the singlet-triplet transition f12 for both reactions, especially when considering that fiE is connected with an incoming relative D-wave as compared to an incoming P-wave in fll 1. Within the error fi2 is even negligible for the D(d, p) reaction in contrast to the D(d, n) case. Finally there are remarkable contributions from the incoming quintet D-waves, especially the ( 5D010+ 11So) transition. As mentioned above much theoretical work on the D + D reactions in the past was done under the assumption that quintet transitions can be ignored [12, 16, 18]. As one can see, this is in no way justified. Moreover, flll is neither forbidden as expected by Rook and Goldfarb [12-1 nor negligible as supposed in the recent RRGM calculations [26] and in the R-matrix analysis [22]. Incoming quintet D-waves must also be taken into account, which were neglected in the framework of RRGM and R-matrix analyses so far. In order to explain the occurrence of transitions with S~ = 2 either internal D-state components in the deuteron together with a central potential or a tensor potential are needed. The experimentally observed polarization Pr(| [28] also suggests that non-central forces are of importance. 4.2 Polarized Fusion

With the results from Table 2 all observables of the D + D reactions can be calculated. Of particular interest are those that are not yet determined by experiment

Reaction-Matrix Elements of the D(d, n)3He and D(d, p)3H Reactions

161

like polarization-transfer and polarization-correlation observables. Because of the multitude of such observables and the uncertainty whether they will be accessible to experiment, it is helpful to calculate them in order to support the planning of new measurements. Here we shall discuss only the cross sections for polarized fusion reactions. The sum rule for the total cross section in terms of the total polarizationcorrelation cross sections (where both deuterons are polarized in different relative orientations along a common quantization axis) is given by [21] ao = ~(2al,a + 4~

+ 2o-a,-1 + o'o,o).

(7)

The indices (m, n) of the cross sections o-,,,, denote the projections of the deuteron spins on the z-axis, which (in the helicity frame) is in the direction of the initial momentum k'. In our calculations, which make use of the channel-spin representation, no distinction is made between ~ and o'o,o, since in both cases the channel spin is S= = 0. In Fig. 1 our results for the D(d, n)3He reaction are compared to an R-matrix analysis by Hale [21, p. 11] and especially for o-a,a to results from a DWBA calculation [24]. The D(d, p)3H reaction behaves very similarly. The RRGM results of Hofmann and Fick for o-1,a/O-o,o [26] are in close agreement with the R-matrix results of Hale and are therefore not shown separately. Our results agree reasonably

3.0

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~

2.8 2.6 2.4

g 1,i / 0

2.2

0

2.0 1.8 1.6 1.4 1.2 i.0 .5 .5 .4 .2 0.

FFCCCC,-CCCCCI-~-I-I-rS~3355~ ] 7 ~ 7 Y T ? ? r F Fi-FFI-i-CCi-Ci-CCC

~o

~oo

~5o

~o0

~5o

3oo

3~o

4oo

4so

E lab [ k e V ]

soo

(a)

Fig. 1. Integrated polarization-correlation cross sections, divided by the integrated unpolarized cross section ao, of the D(d, n)3He reaction for different relative spin orientation of the two deuterons. The results of the present analysis are shown as solid lines and are compared to the R-matrix [21] (and R R G M [26]) results (dots) and for al, 1 also to the DWBA prediction by Zhang et al. [24] (dashed line)

162

S, Lema~'tre and H. Paetz gen. Schieck

3,0

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2.8 2.6 2.4 2.2

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2,0 1,8 1.B 1,4 1.2 1.0

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k -j

.8 .6 .4 .2 I

O.

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100

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~

150

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300

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lttlllllIlIlIII

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3.0

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4-50

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[keY]

(b)

Illllll/llllllt~lt~tlttll'~f~f~

2,8 2,B 2,4 2,2

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+o

1,-I

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150

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400

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(c)

Reaction-Matrix Elements of the D(d, n)3He and D(d, p)3H Reactions

163

Table 3. Matrix-element ratios of the quintet-state transitions 31 and 71 of the D(d, n)aHe reaction at E~ b = 80 keV

l(sSz]2+ laO2) t I(lso[0+llso)l

t(sSzt2+13D2) I I('Sol0+lls0)l

0.12 cited from ref. [323 0.34 this work

0.24 cited from ref. [32] 0.43 this work

well with the R-matrix analysis, especially with respect to the energy dependencies of the polarized cross sections, but both are in marked contrast to the DWBA results. This confirms that at these low energies and over the small energy range investigated the use of penetrability functions as the only energy-dependent factor of the transitions amplitudes is reasonable. The cross section o-1,1 is enhanced over the unpolarized one by more than a factor of two for Ela b < 40 keV and still more than a factor of one at the resonance energy of 430 keV of the 3He(d, p) He reaction. These findings contradict strongly the DWBA calculations of Zhang, Liu, and Shuy r24], who expect o-i,1 to be only 7.7~ of the unpolarized cross section, and they are also at variance with the findings of Ad'yasevich et al. [,-14]. For the transitions 71 and 61, which are essential in this context, we find the following ratios, which in Table 3 are compared to RRGM results at Ela b -- 80 keV [-23]. Our results are, of course, independent of energy in the energy interval examined.

5 Summary The results obtained show that the existing data set for the two D + D reactions at energies below 500 keV is sufficient to determine all relevant matrix elements of these reactions with a minimally model-dependent approach and using a suitable algorithm, in this case the one of Levenberg-Marquardt. As is suggested by chargesymmetry arguments the strong similarity between both reactions is again borne out by the near-equality of the leading matrix elements. However, some of the matrix elements with higher partial waves, the effects of which are strongly suppressed by the centrifugal barrier, show marked differences, which must be responsible for the differences seen in the observables. The comparison with results from an R-matrix analysis of the 4-nucleon system and with RRGM calculations indicates that, though differences occur in the absolute magnitudes of the spin-correlation cross sections, the trend with energy is quite similar and, like in these analyse~ rules out quintet-state suppression and therefore neutron suppression in the D(d, n)SHe reaction for fusion-energy purposes. On the contrary, for the relevant cross section o-1,1 is even enhanced over the unpolarized case. Besides the strong contributions from entrance quintet states in both reactions we find also non-negligible tripletsinglet transition strength.

Acknowledgment. The authors thank R. Reckenfelderb/iumer, K. R. Nyga, P. Niessen, G. Rauprich, and L. Sydow for many helpful discussions. The authors are grateful to Dr. E. Pfaff, University of Giegen, for making available the data of his thesis before publication.

164

Reaction-Matrix Elements of the D(d, n)3He and D(d, p)3H Reactions

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25.

26. 27.

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Received January 5, 1990; revised April 27, 1990; accepted for publication May 23, 1990