1 2.A.l 1 Nuclear Physics Al76 (1971) 261-272; @ North-Holland Publishing Co., Amsterdam

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NEUTRON-NEUTRON SCATTERING LENGTH

FROM A KINEMATICALLY COMPLETE MEASUREMENT

OF THE 3H(d, zn)n REACTION

R. GRGTZSCHEL, B. KUHN, K. MGLLER, J. MdSNER and G. SCHMIDT

Zentralinstitut fiir Kernforschung, Bereich Kernphysik, Rossendorf bei Dresden, DDR

Received 3 August 1971

Ah&act: Coincident nz spectra were measured from the reaction 3H(d, zn)n at Ed = 13.43 MeV. 8, = 29” and 6. = 65.4”. The Migdal-Watson approximation was used for data analysis. The value a., = - 16.0+1.0 fm for the ‘So neutron-neutron scattering length was deduced.

E NUCLEAR REACTIONS 3H(d, 3He, n)n; E = 13.43 MeV; measured u(&, E,,); nt coincidences; deduced scattering length. Ti(JH) target.

1. Introduction

For evidence of the charge symmetry of the nuclear interaction, the experimental determination of the neutron-neutron scattering length is indispensable. Scattering of neutrons on neutrons cannot yet be ascertained by direct scattering experiments. Therefore the n-n scattering length has to be determined from the interaction of two neutrons produced in the final state of a nuclear reaction. A review of the data from measurements obtained up to date is given in table 1. It must be pointed out, however, that most of these data result from kinematically incomplete experiments where only one of the three final particles was detected. The accuracy of the measured values not only depends on experimental error, but is also affected by the uncertainty introduced with approximations in the theoretical interpretation of the measured spectra. Appli- cations of the Migdal-Watson formula and other approximations to determine scatter- ing parameters from experimental spectra are discussed in a number of papers [refs. 19,21,31-35)]~

A kinematically complete measurement has several advantages. One is that the differential cross section along the kinematic locus is an unambiguous function of the relative energies of the three particle-pairs, enabling a more accurate identifica- tion and consideration of the influence of various reaction mechanisms. The other is to make use of the so-called lupe effect to improve energy resolution. The angles of the two detectors can be chosen in such a way that a little change in relative energy corresponds to a comparatively big change in the energy of one of the detected par- ticles.

The kinematically complete measurements undertaken up to now utilized the ‘H(n, p)2n reaction to determine a,,,, [refs. ‘, lo, 12, “)I. These measurements are rather

261

262 R. GRdTZSCHEL et al.

difficult to handle experimentally because neutrons are used as incident particles. The obtainable count rate is very low, making statistical errors important and background problems more difficult. For this reason, we decided on the reaction

TABLE 1

Measured ISo neutron-neutron scattering lengths

Reaction Energy (Me?

aAm) particles Type of Type of detected experiment analysis

Ref.

Wn, 2n)p 14.1 14.1 22 13.95

8. ‘28

14

-22 i2 P incomplete -23.6&1.8 P incomplete -15 to -18 P incomplete -14 33 P incomplete -16 f3 P incomplete -16.8il.O P incomplete -15.9il.l P incomplete -17 j;3 P incomplete

14.5 -16 to -24 11, n, P complete

14.1 -23.78 14 -19 r,“, 14.1 -23.231.8 18.5 -16.0&1.0

‘H(n, d)2n 20.8 15.1

3H(d, t)2n 32 and 40 30

-17 -17 12 -18 f3

-16.1&1.0 -16.5&-1.0

P n, u, P P n, ua P

d d d

t r

11 -18 h-3 to -19 12 --16.0&1.0

z incomplete Born appr.

*H(t, t)2n

3H(t, a&r

‘H(d, 2p)2n

‘H(n-, y)2n

13.43

22

22 1.5

16

-14to -22

-16 13 -15.0,iI.O

-15.5*1.1

-15.9 ‘“Z”

r, u

r

cc a, n

P, P

Y -12.6 to -24.7 y

*1 -16.4f1.3 n, n, lr *I -18.4115

-13 13 Y

incomplete complete incomplete complete

incomplete incomplete incomplete

incomplete incomplete

Born appr. diagr. meth. Born appr. Imp. appr. Born appr. Imp. appr. MW theor. MW theor. with

form factor diagr. meth. and

Faddeev eq. MW theor. Born appr. diagr. meth. MW theor.

Born appr. Born appr.

MW theor. MW theor. with

form factor _. ZO.2‘ 1

complete MW theor.

incomplete MW theor.

incomplete MW theor. complete MW theor.

incomplete

incomplete MW theor. incomplete complete complete incomplete

this paper

*) References ““) and z9) have theoretical analysis of the same data and differ mainly in the way background was treated.

3H(d, rn)n, which has the advantage that the incident particle and one of the detected particles are charged. In comparison to the deuteron break-up, the complexity of this reaction is a disadvantage because, for example, the final-state interaction between a neutron and a r-particle can become troublesome. In sect. 5 this aspect is further discussed.

3H(d, tn) 263

2. Experimental set-up

A scheme for the set-up is given in fig. 1. The titanium-tritium target - thickness

2 mg/cm2, atomic ratio 3H : Ti z 1 - was bombarded with 13.43 MeV deuterons

from the Rossendorf cyclotron. For beam adjustment a collimator was employed

with apertures of 1.5 x 3 mm2.

A p-silicon surface-barrier detector of 1 kO * cm served as a z-particle identifier.

It was placed at a distance of 58 mm from the target, at an angle of 29” to the incident

beam. To restrict the solid angle a slit of 1.6 x 3 mm2 was used. The detector voltage

Fig. 1. Experimental set-up.

was chosen so that the energy loss from elastically scattered deuterons, recoil tritons

and protons produced by Ti(d, p) reactions in the sensitive layer amounted to less

than 3.5 MeV. In this manner, only the z-particles were located in the analysed region

of 4 to 8 MeV. In the sensitive layer, the cl-particle energy loss from the 3H(d, a)n

and Ti(d, cc)Si reactions exceeded 9 MeV. For these reasons it was considered un-

necessary to use a dE/dx counter for particle identification.

The time-of-flight method was used to measure the neutron energy, utilizing a poly-

styrene scintillator of 10 cm diameter and 4 cm thickness. It was placed at a 65.4”

angle to the direction of the beam at a distance of 2.12 m from the target. An iron

and paraffin neutron collimator of 1 m thickness was interposed between the target

and the scintillator. It served to shield the detector from background neutrons and

y-rays. Detectors for the z-particle and the neutrons were located in one plane with

the incident beam (q = 180’).

The preamplifier was directly attached to the T-particle detector. It supplied pulses

proportional to energy of about 10 ns rise time. This was followed by the linear am-

plifier and the 3tchannel analog-to-digital converter for pulse processing. The pre-

amplifier also furnished the time signal, which was amplified and shaped in a fast

amplifier with two outputs. The output with the gain factor 100 served to start the

time-amplitude converter (TAC). To achieve a low dependency of the measured time-

264 R. GROTZSCHEL et al.

of-flight on the amplitude of the detector pulses, the threshold of discriminator I was

kept as low as possible. The other output, with the gain factor 10, supplied pulses with closely energy-proportional amplitudes. These were utilized to control the gating circuit. To fix the lower limit of the z-energy, the threshold of discriminator II was adjusted to about 3.7 MeV.

A K14 FS50 photomultiplier supplied the stop pulses. To determine the lower limit Eb of the proton energy, we employed two methods for the calibration of the threshold of discriminator III. One made use of the Compton edge of the p-radiation from “Na, 203Hg and 13’Cs, in the other we utilized the a-radiation from ThC and ThC’ at 6.047 and 8.778 MeV to calibrate the threshold setting scale. Conversion calculations of the fi- and cc-energies to proton energy were taken from the papers 36, 3 ‘). Within the error limits, both methods yielded results in good agreement. In the first run E,, was 450 + 50 keV and in the second 500 f 50 keV. The TAC covered a range of 100 ns. It was followed by the 64-channel analog-to-digital converter for sorting the time pulses. The coincident events were collected in the two-dimensional store with 32 x 64 channels. The width of the time channels was fixed with calibrated cables and amount- ed to 1.91 ns.

The natural pulsation of the cyclotron beam produced narrow peaks spaced 90 ns apart from the random y-ray coincidences. Through proper flight path selection, one of the peaks was located at the edge of the evaluated spectrum range and also served to fix zero on the time scale.

With a mean deuteron current of 5nA, the count rate at the TAC start input was about 500 pps, with some lo4 pps in the stop channel. Thisyieldedabout200registered events per hour in the storage unit. Two runs of 80 h each, were made. Experimental conditions were the same, except for a slightly altered setting of the electronic system. From the first run, we received 16 000 registered events. After selecting the region and subtracting the background, 12 600 events remained for the calculation. The second run yielded 17 500 events, leaving 13 800 for the calculation of the scattering length.

3. Results

The two-dimensional spectrum of the first run is given in fig. 2, with the solid line representing the kinematically allowed curve. In calculating the curve, mean values have been used with respect to incident energy, angle, time scale zero, z-particle energy loss in the target and time-of-flight. The measured distribution smears in the direction of the z-particle energy axis, which can be principally attributed to the energy loss in the target and the angular resolution in the z-channel, with the latter depending on the geometry set-up and Coulomb scattering in the target. As revealed in the lower region of the z-channels, representation smears in the direction of the time axis, which is mainly caused by the time-resolution inherent in the whole experimental set-up. Jn sect. 4 these questions are discussed in more detail.

3H(d, zn) 265

For the neutron-neutron scattering length analysis the channel contents were pro- jected on the time axis. The background was identified from the density of the random

&.: ::.:::. 2 ::::;:;;:::

. ..0.......

i- / w ‘-L-z?---

! . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 ;;J-.-.. “. .

..................

......................

........................ ........................

. . . . . . ... . . . . ......... ....................... ........................ ......... ........... ...................... . ...................... ............... . .............. ............

’ FIRST KI_W .

. . . . . . . . . . . . .

Fig. 2. Map display of the measured two-dimensional spectrum (first run) from the reaction 3H(d, z, n)n. Areas of the points are roughly proportional to the number of events, with five sizes signifying 4-9, 10-19, 20-39, 40-69 and 70-100 events, respectively. The solid line is the kinematically allowed

curve.

FIRST RUN -7

SEQND RUN

arm- -16fm ct = Lens Eb- 503W

Fig. 3. Representation of the data projected on the time-of-flight axis (background subtracted). The solid curves are the Migdal-Watson fits yielding the given values for arm, ct and E,,. For com-

parison, the Migdal-Watson curve for a,,. = -23 fm is shown as a dashed line.

coincidence events outside of the kinematic locus and subtracted. The resulting spectra are shown in figs. 3a and 3b. From Migdal-Watson theory the following formula can

266 R. GRiiTZSCHEL

be derived for the differential cross section 38):

-- ___ (x)

dt,dQ,dQ,

[( - -..!-.

2

-k+-K’ +K2 t,” a ml ) 1

where E;;, is the energy and f, the tame-of-fight of the registered neutrons, Eb the proton threshold of discriminator III, p(E,) the phase-space factor, r the effective radius, K the wave number of the relative movement of the two neutrons and an,, the scattering

Fig. 4. Kinematics of the reaction 3H(d, t, n)n. The a-energy, the three relative energies, the neutron counter efficiency 11 and the phase-space factor p are shown as functions of the neutron energy. The dashed Et curve is for a mean energy loss in the target. Theregion used for anaIysis is defined by the dashedvertical lines. Arrows indicate the position of the 21.4 MeY (GO-) and 22.4 MeV (Z-) ‘He levels.

length. To calculate the neutron efficiency q of the plastic scintillator we utilized a code of Kumpf 39). Fig. 4 shows the three relative energies, the z-energy, the effi- ciency q and the phase-space factor p as functions of the neutron energy E,,. The dashed curve is for a mean energy loss of the r-particle in the target. Curve g is calcu- lated for the threshold value Eh = 450 keV_ The region used for the determ~na~on of the scattering length is defined by the two vertical dashed lines on the En axis. To fit the theoretical curve to the experimental points of fig. 3, eq. (1) was folded. We proceeded on the assumption that time and incident energy distribution and angular dispersion satisfies Gaussian distribu~ons. For the calculation of the integral, we used the Gauss-Nermite formula 40) with three points. To fold eq. (1) we used the follow- ing values: time resolution, At = 3.5 ns, angular dispersion, AS, = 1.2”, (composed

3H(d, rn) 267

of the geometric resolution and the z-particle Coulomb scattering in the target),

AS, = 0.75”, Acp = 2.4”, incident energy width, AE, = 80 keV, (from beam width

and energy loss in the target), with A representing the HWHM of the Gaussian dis-

tribution, i.e. A = 1.176a. The values At and AE, were experimentally found and

the other values were calculated. To determine the angular dispersion from Coulomb

scattering, we used the Moliere equation 41). This gave the angular dispersion

AS,(Coulomb) = 0.7”, averaged across the target thickness, which we quadratically

added to the geometrical angular dispersion.

x2; FIRST FUN

so/-

55-

50.

i -i6 %n [fm

Fig. 5. The x2 curves represent x2 as a function of arm. The number of degrees of freedom is 38. The influenceofctand E,isshown. Foreachcurvefixedvalues were used for ct and Eb, onlythe proportional factor from eq. (1) was fitted. For the solid curves Eb is 450 keV. The statistical error for the

ct = 44 ns curve is represented by the dashed line.

For the determination of the time of flight from the channel number we not only

needed the channel width, which was very accurately fixed by calibrated cables,

but also the time for the channel “0” which is designated as “ct” in the following.

On the one hand, the value of ct is derived from the section of the two-dimensional

spectrum (fig. 2) where the kinematic curve closely parallels the z-energy axis. Here,

neutron energy is 4.8 MeV (fig. 4). Taking the z-particle time-of-flight into considera-

tion, ct can be found. On the other hand, we utilized the y-ray background peak, as

already mentioned. Within the error limit of f 1 ns, both calibrating points gave

values for ct that were in good agreement.

To fit the theoretical curves to the experimental points in fig. 3 we presumed that

a was negative and that the effective radius was r = 2.67 fm. For given values eq.

(;j was folded and only the proportional factor fitted so that x2, the sum of the

quadratic deviation weighted with statistical errors, became a minimum. For other

values of arm we used the same calculation. In this manner we obtained the x2

curves in fig. 5 by varying an,,. To identify the influence of the parameters ct, Eb and

the above mentioned dispersions A we altered the respective parameter. In this

manner the four different x2 curves in fig. 5 were obtained, with the solid lines

Eb = 450 keV at various values of ct, while the dashed line stands for Eb = 400

268 R. GRdTZSCHEL et al.

keV and ct = 45 ns. A discussion illustrating the influence of these parameters on a,, determination is given in the next section.

We obtained a_ from the minimum of the x2 curves (fig. 5) with the first run re- sulting in

arm = - 15.9 fm, while the second run yielded

a nn = - 16.0 fin.

The curves entered in fig. 3 result from the calculation of the folded equation (1). For the parameters et, Eb, Y and the dispersions A the mentioned values were inserted.

4. Error analysis

Only the experimental errors are dealt with in this section; uncertainties caused by the use of the Migdal-Watson theory are discussed in sect. 5. In fitting, we assumed that the sign of arm (negative) and for the present also the effective radius I are known.

Fig. 6. The nn-scattering length an,, obtained from the fit as a function radius r. AIong the line x2 is nearly constant.

of the assumed effective

Analysis of the errors affecting the a,, determination centered on (i) statistical errors of the channel contents, (ii) uncertainty in ct determination, (iii) time-scale linearity errors, for instance, due to the time-amplitude dependency in the neutron channel, (iv) uncertainty of the dispersions A required in folding eq. (l), (v) inaccuracy of the proton energy threshold Eb, (vi) correction of q in the region of low relative energy of the two neutrons. This increases the re~stration probability of an event because both neutrons pass through the scintillator.

3H(d, m) 269

The statistical error (i) can be determined from the x2 curve. Under specific con- ditions 42. 43) ‘t I can be demonstrated that an,, is within the region defined by xi = Xii, + 1 with a 67 % probability.

In fig. 5, this region is given by the intersections of the dashed line with the x2 curve for ct = 44 ns, which corresponds to an error of +0.4 fm.

The uncertainty in the determination of ct (ii) contributes the largest error com- ponent to a,,. It was pointed out that an accuracy of only + 1 ns could be achieved for cl. As the location of the x2 curves in fig. 5 shows, this uncertainty results in an error of -0.9 fm and f0.7 fm in unn. Time scale linearity (iii) was checked with the aid of the reaction 3H(p, n)3He, with the set-up as in fig. 1. The only difference was that instead of the z-energy, the photomultiplier pulse amplitude was registered in the store. For proton energies of iess than 1 MeV a slight time-amplitude dependency was detected which resulted in a negligible error. Uncertainty of the dispersions d (iv)con- tributes an error of 0.15 fm to arm; a value which was obtained in the same manner as for the error in (ii).

The contribution from the inaccuracy of the threshold & (v) was determined in a similar way. The threshold uncertainty of + 50 keV caused an error of 40.2 fm in arm. One of the curves used for the error determination is the broken-line curve in fig. 5. To assess the error (vi), the Monte Carlo method was employed. ft could be shown that the change in the curve for q, caused by this error, did not exceed 3 % and has no influence on the a,,” determination.

Obtained error values were quadratically added and the scattering length then was

a “II = -16.O+l.Ofm.

As mentioned above, in fitting we assumed a fixed value for the effective radius r. In a two-parameter fit r and a,,,, correlate very strongly since the relative energy of the two neutrons in the region used for analysis is rather small. The straight line in fig. 6 gives the calculated a,,” as a function of r. Along this line, between r = 2 and 3.5 fm, x2 does not change its value practically. Therefore m our measurement a real two- parameter fit is meaningless.

5. Discussion and conclusion

In the error assessment of the foregoing section we did not touch upon the theo- retical un~rtainty in the determined a,, value introduced by using the Migdal-Watson approximation. Until it is possible to accurately calculate multipfe nucleon systems, it will be difficult to define this uncertainty. For relative energies below 1 MeV, how- ever, the Migdal-Watson theory appears to give an acceptable approximation, pro- vided reaction mechanisms other than the considered final-state interaction do not distort the spectrum in this region.

In the present case, there is a possibility of the spectrum being distorted through the influence of the n-r final-state interaction. Two of the excited states of 4He at

270 R. GRijTZSCHEL et al.

21.4 MeV (O-) and at 22.4 MeV (2-) should appear at the points marked by the

arrow in fig. 4 - cf. Meyerhof 44) and Haase et al. 45). When looking at the spec-

trum in fig. 3, however, no indication of peaking is seen at the point where the 2-

level should occur (arrow). The O- level located outside of the interpreted region at

E,, = 4.3 MeV also does not appear in the spectrum (see fig. 2). Assimacopoulos et al. 46) mentioned the O- level in the mirror reaction ‘H(r, tp)p, and Poppe et al. 47) observed the same phenomenon in the reaction 3H(d, n)3H. Hitherto, no reference

has been made of the appearance of this energy level in the reaction 3H(d, Tn)n. This

is because kinematically complete measurements had not been undertaken and only

fm n-n scattering length

Fig. 7. Distribution of the arm values from the measurements compiled in table 1.

the z-particle was detected in the kinematically incomplete measurements by Baum-

gartner et al. I*) or by Larson et al. ‘l). As demonstrated by the already cited work

of Poppe et al., which covered a wide range of angle and energy, the reaction mecha-

nism changes noticeably with the incident energy and the neutron detection angle.

In the region of higher incident energy (> 12 MeV) and larger angles (a,,, > SO’)

the peaks of the 4He levels disappeared. In the study of the reaction 3He(d, PP)~H

Jarmie et al. 48) reported the same effect. These results suggest that these levels exert

only little influence on our measured spectrum because of the kinematic conditions

Ed = 13.43 MeV, 9, = 65.4”. In other words, the final-state interaction between the

neutron and the t-particle did not supply a marked contribution to the cross section.

The same applies to the spectator mechanism. This produces peaking in the spec-

trum only when the neutron constituent of the incident deuteron, continues its flight

(particle spectator) or when one of the two neutrons from the triton remains in its

place (target spectator). Across the investigated range of the spectrum, kinematic

conditions have been selected in such a manner that both neutrons fly in the direction

of the neutron detector (9, = 65.4”) with nearly the same energy.

For these reasons, we consider the Migdal-Watson formula satisfactory for a close

approximation in the interpretation of the measured spectra. In support of this postu-

late, the measurement of the mirror reaction 3He(d, tp)p under similar kinematic

3H(d, tn) 271

conditions would furnish a further argument. It can be supposed that both reactions

will demonstrate analogous mechanisms and it should be possible with the aid of the

well-known p-p scattering length to demonstrate the good fit of the spectrum obtain-

able with the Migdal-Watson theory. This is being investigated.

Fig. 7 gives a graphic presentation of the n-n scattering lengths compiled in table 1.

Although the various results are shown with widely-varying error limits and should

be differently weighted, fig. 7 demonstrates a clear accumulation of values between

- 16 fm and - 17 fm and between - 23 fm and -24 fm. The values at -23 fm have

all been obtained from kinematically incomplete measurements of the ‘H(n, 2n)p

reaction. In all probability the value of the n-n scattering length is between - 16 fm

and - 17 fm, which would coincide with the value an,, = - 16.0+ 1.0 fm we obtained

from our measurementsi.

With regard to the problem of charge symmetry of nuclear interaction this result

has to be compared with the p-p scattering lengths. This value is known with high

accuracy: app = -7.826 kO.005 fm [ref. “‘)I. Sher et al. 50) considered the electric

and electromagnetic contributions to the p-p interaction and derived under the as-

sumption of charge symmetry ano = - 17.06 fm for the n-n scattering length. The

uncertainty of this value due to different realistic potentials used in the calculation

is smaller then 0.1 fm. It turns out, that within present accuracy charge symmetry of

nuclear interaction is realized in nature.

On this occasion, we want to express our appreciation to Prof. Dr. J. Schintlmeister

for his support. We also thank Dr. Uhlenhut and Mr. H. Biihme for making the target

as well as Dr. H. Kumpf for his helpful discussions and for letting us have the code

for the calculation of neutron efficiency. We also appreciate the help we received from

our cyclotron staff.

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