the 3h(d, n)4he reaction below 1 mev

- - ~ Nuclear Physics A153 (1970) 193--210; ~ ) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE 8H(d, n)4He REACTION BELOW 1 MeV

F. SEILER and E. B A U M G A R T N E K

Physics Department, University of Basel, Switzerland

Received 8 May 1970

Abstract: The possibility of an analysis of the SH(d, n)4He reaction below I MeV is investigated. This is done in terms of a limited set of (l, s, J ) matrix elements, independent of specific models for the interaction. Sufficient experimental data for such an analysis can be obtained by supple- menting available information with measurements of the neutron polarization using polarized deuterons. Linear relations between redundant data are given, offering consistency checks between different experiments at the same energy.

1. Introduction

The 3H(d, n)4He reaction has been the object of numerous theoretical and experimental studies. At energies below 1 MeV it is a convenient high-yield source of fast neutrons, due to the ½+ resonance at 107 keV [refs. 1, 2)]. This isolated ~+ state in SHe formed by s-wave deuterons is of interest experimentally because it provides large isotropic cross sections as well as a pronounced sensitivity to tensor polarization of the incoming deuterons 3-6). Together with its mirror reaction 3He(d, p)4He, it has therefore been widely used as an analyser. From the theoretical point of view the reaction evokes interest by allowing the study of spin-dependent properties in the incoming channel, while the parameters of the n~ system in the outgoing channel are relatively well known 7, s). The process is thus well suited for calculations involving models, such as the coupled-channel Schr6dinger equation model used by De Facio et al. 9).

In using the 3H(d, n)4He reaction as an analyser for tensor polarization of deuterons, it is generally assumed that at 107 keV only the resonant channel formed by s-waves is open. To increase the efficiency of the analysing system, however, it is often desir- able to use deuterons of several hundred keV energy, incident on a thick tritium target. Unfortunately already at energies as low as 300--400 keV the cross sections for unpolarized deuterons 10-12) begin to show evidence of odd-parity contributions which are but poorly understood. Although it is improbable that these discernibly affect the analysing properties at the resonance energy s, 6), their influence should be taken into account when deuterons of higher energies are analysed.

In order to study these additional channels, cross-section measurements using polarized deuteron beams were undertaken 13,14). The results show that p-waves and at higher energies d-waves become significant. An analysis of the first set of data 13) was attempted by Huq and Goldfarb 15) in terms of the resonant ]+ matrix

193

194 r . SELLER AND E. BAUMGARTNER

element together with a contribution from a stripping mode. Subsequently Huq 16) analysed the same data in terms of interference between broad, unspecified, over- lapping states with the resonant 3 + state, taking into account four p- and two d- states. The difficulty in determining the resulting number of parameters from the experimental data then available is obvious. It is the purpose of this paper to show that with the results of several presently feasible experiments in addition to the most recent data 14), these difficulties could be overcome.

2. Formulae for reactions involving polarized particles

2.1. DESCRIPTION OF A SYSTEM OF POLARIZED PARTICLES

In a system of particles with spin s the polarization state is conveniently described by the expectation values tm.. of a set of irreducible tensor operators Tin,. [ref. 17)] which transform under rotations as the spherical harmonics of rank m. We shall adopt here the definition given by Lakin 18) and Satchler 19, 20) expressing the tensor moments in terms of the density matrix p,, ~,, where o" and a' are the projections of s along the z-axis.

tm,. = <Tin. n) = ~ ~. (--)~-¢(ssa'--almn)p¢.. , . . , . , (1)

Here ~ = (2s+ 1) ~, and (ab~#lcr) is the usual Wigner coefficient 21). The definition (1) differs from that used by Devons and Goldfarb 22) only by the insertion of the factor ~ in front of the sum. Welton 23) uses the factor ~/~.

In a system containing two kinds of particles with spin s and S the composite tensor moments are given by

tm, n,M,N -~ ~ Z ( - ) ' - ' + s - ~ ( s s a ' - a [ m, n) (SSY, ' - -ZIMN)p.x , . . r . (2) ~rj ~r"

If the polarization of one particle is not measured, eq. (2) reduces to (1) by setting

t m . , . o , o = tin. n or t O . O . M . N = t M . n . (3)

If no correlations between the two particles exist, as in the case of a polarized beam incident on a polarized target, the composite tensor moments are simply

tm, n,M, N = t,~,. " tu.~r. (4)

This factorization, however, is not valid for the outgoing channel of the reaction. Since the Tin,. are non-hermitian operators their expectation values are generally

complex numbers, subject to the conjugation property

: - ' : t* (5) t m , - n -~ ~ / m,n"

For a particle of spin s the number of parameters is thus (2s+ 1) 2, with to, o represent- ing the intensity normalized to 1. The inherent symmetries of polarized beams and

3H(d, n)4H¢ 195

targets, as well as the symmetry properties of nuclear reactions 22-24), impose selection rules and restrictions further reducing the number of parameters.

2.2. DESCRIPTION OF NUCLEAR REACTIONS

The formulae for reactions involving polarized particles with arbitrary spin have been given by several authors 22-2,). Using the tensor moments as defined in subsect. 2.1 and Welton's formulae 23), the most general equation for the outgoing tensor moments can be written as

with

1 E ~ + ~-- i(n + N)~Am' , n', M', N'[O~ - - "m,n~M,N ~ "am, n , M , N w ) , (6) t m " n " M " N " - - ffp(0,~b) m,n

M , N

ffp(0, q~) = frO(0) E tm, n tM, N e - i ( n + N ) ¢ B m , n, M, N(0),

M, N

where ao(O) is the cross section for unpolarized beam and target

o(0) : ,o ,o , =

and the quantities B(O) are defined as

Bm, n,M, Iv(O ) - -

(7)

( 8 )

1 A 0, 0, 0, 0 (,0~ W'-m,n, M, iVkV ]. (9)

o(0)

The functions A(O) are given by

= d.+.,.,+N,(O ) • F(m, n, M, N; p, v; L) "Xm, n, M, N ~v] - ~ #, v

l "1 x F ' ( m ' , n', M', N';/~, v; L) i Im (R~R,,,) ,

[even/ (10) for K = m ' + M ' + m + M = (odd J

and have the symmetry property

Am',-.'.u',-wra~ ( ~r¢ +n + N +." + N'~.,', .', M'. n'ta~ (11) r a , - n , M , - N k v ) ~ k - - ] "tXm, n , M , N kwJ •

In the following the sub- or superscript indices m, n etc. are dropped if they are both zero and no ambiguity arises.

The coefficients F and F ' are defined as

F(m. n, M, N; 1, 2; L) = ( - ) " +LJ x a¢21112 ~t ~2 rfiJff

x X 7(I, I2 00ll0) Y', t(ItOn + NILn + N)(mmnN[tn + N) 1 t

x m M t , (12) s S s21\12 s2 J2/

196 F. SEILER AND E. BAUMGARTNER

where Rl = (ui, It, st, J~'lRIct;, l~, s;, Jr) is the matrix element of channel i, ut is the particle fragmentation in channel i, I t is the orbital angular momentum, s t is the channel spin, d~' is the total angular momentum and parity, dt,,L,),,,(O) are the reduced rotation matrices 21, 25), X = 1/k is the reduced wavelength of the incoming particles in the c.m. system, and

d e g h

is the X-coefficient or 9j-symbol 22, 23, 25). Incoming and outgoing channels are de- noted respectively by unprimed and primed quantities.

The coordinate system X used to describe the incoming beam and target has its z-axis aligned with k while the x- and y-axis are arbitrary. The outgoing beam is defined in a gravicentric coordinate-system X' rotated through the Euler angles (q~, 0, 0) [refs. 21, 25)], so that the z' axis lies along k', while the y ' axis is parallel to n = k x k ' / l k x k'[ according to the Basel convention 26). I f the outgoing beam is used in a further reaction, the tensor moments must be rotated to the lab system t 9) by (0, 0-0Xab, 0), SO that the incident z-axis for the second reaction lies parallel to the direction of motion in the lab as assumed in eqs. (6) to (10).

2.3. NUMERICAL EVALUATION

For the evaluation of these formulae a computer program was written which sim- plifies eq. (6) according to the symmetries of the problem. The numerical factors associated with a particular (R~ R*) are given for each combination of tensor moments and are grouped by L-values for convenient reference. The output is given in decimal fractions, although for the relatively simple model of the 3H(d, n)*He reaction considered here, the factors have been converted to a more convenient form.

3. Discussion of the approximations

3.1. SELECTION OF THE RELEVANT MATRIX ELEMENTS

At deuteron energies below 1 MeV penetrability arguments can be used to limit the number of matrix elements by allowing only s-, p- and d-waves in the incoming channel. The resulting set of 13 matrix elements is given in table 1.

Additional criteria helpful in eliminating some of the elements can be derived from an examination of the n~ system in the outgoing channel. The phase-shift analysis of Hoop and Barschall 8) is used, which is based on total cross sections, differential cross sections for elastic scattering and polarization measurements. In the energy region of interest (22-23 MeV excitation energy in 5He), the *I-Ie(n, d)T process is the only inelastic channel open. Partial waves up to l = 3 are sufficient to fit the data and only the p~ and d t phase shifts have significant imaginary parts.

"~H(d, n)41fe

TABLE 1 Matrix elements R(I, s, j~r, l ' , s')

i It st J~ 1~' s{

1 0 ] 3 + 2 ½ ") 2 0 ½ ½+ 0 ½ ") 3 1 ½ ½- 1 ½ ") 4 l ½ ~- 1 ½ ") 5 1 t ½- 1 ½ a) 6 1 t ~- 1 ½ ") 7 1 ] 3- 3 ½ 8 2 ½ 3 + 2 ½ ") 9 2 ½ 3 + 2 ½

10 2 ~ ½+ 0 ½ *) 11 2 ~- ~+ 2 ½ ") 12 2 ~r 3 + 2 ½ 13 2 -~ ~+ ~, ½

197

Unprimed quantities refer to the incoming channel.

") Only these elements were considered in this analysis.

For incoming s-waves there are two matrix elements. Element 1 forms the resonant 3 + state and decays with l ' = 2, giving the main contribution to the cross section. Element 2 with J = ½+ and l ' = 0 is of special interest in view of the suggestion by Mclntyre and Haeberli 27) that it may contribute appreciably near the 3 + state at 430 keV in the mirror reaction SHe(d, p)4He. Measurements of that process at the resonance energy using a polarized beam and target 28) have shown that its influence there is small. Nevertheless it should be investigated in the all(d, n)4He reaction as well.

Five matrix elements (3 to 7) can be formed with incoming p-waves. From cross- section and polarization measurements it is known that they are relatively small [refs. 1o. 13.14)]. Element 7 can be eliminated since the f~ phase shift in 4Fie(n, n)4He is less than 4 ° , which should make it negligible compared to the others. The uncer- tainty in the determination and interpretation of the imaginary parts of the P~r and p~ phase shifts precludes any further simplification.

The six matrix elements formed by d-waves are expected to be very small. Element 13 is eliminated because of the absence of g-waves in 4He(n, n)4He. Matrix elements 9 and 12 are omitted, since in addition to their formation by d-waves, they should have small amplitudes in the exit channel. This is indicated by the d~ phase shift in 4He(n, n)4FIe which is for both real and imaginary parts considerably smaller than the p~ phase shift. Elements 8 and 11, in which the resonant 3 + state is formed, are retained together with element 10 which decays by an l ' = 0 channel.

Thus 9 matrix elements remain, giving rise to 17 parameters. In order to determine these quantities from experimental data a considerable number of different measurements are needed, even if experimental errors are very small. By making use of the


possibilities opened by sources of polarized deuterons, more than sufficient independent information could be accumulated to determine the reduced set of parameters above.

3.2. APPROXIMATIONS IN THE FORMULAE

The experimental data available show that below 1 MeV the dominant contribution to the reaction stems from the resonant matrix element R1, all others being small by comparison. We have therefore ignored all terms in the formulae not involving element Rt , retaining only interference terms of the type Uv = Re(RiR*) and Vv = Im(Ri R*).

In this approximation, independent systems of equations emerge for both U~ and V~ with even and odd incoming angular momentum I, thereby simplifying the analysis.

The experimental quantities t" n,(tm,~), where t~,.. denotes the tensor moment of the outgoing neutron and tin, ~ that of the incoming deuteron, are given in tables 2 to 5. Experiments using a polarized target or a polarized tritium-beam have not been included in the present work. The relevant formulae will be published in a sub- sequent paper on the mirror reaction 3He(d, p)4He for which polarized aHe targets are available.

4. Discussion of experiments

4.1. GENERAL REMARKS

The theoretical description of the 3H(d, n)4He reaction is considerably simplified by the presence of a spin-0 and a spin-½ particle in the exit channel. Experimentally there are thus only four observable outgoing tensor moments: the cross sections o-p(0, ~b) and the neutron polarizations t ' 1, o and tl, ± i, which can be measured for a variety of incoming polarizations. Such experiments use both polarization insensitive and polarization sensitive detectors. The latter can be quite complex, involving for instance the use of a magnetic field as well as a nuclear reaction.

The description of the polarization of an incoming deuteron beam usually involves more than one tensor moment. As a result several terms contribute to the right-hand side of eq. (6). In order to separate the unknown quantities A(O), sources of polarized deuterons are particularly useful since most of them allow the elimination or sign reversal of one or more of the tensor moments. Further separation can be achieved by making proper use of the known qS-dependence of the outgoing moments. For measurements of the neutron polarization, the orientation of the reaction plane of the analyser provides an additional degree of freedom. Even so, difference measurements are often required, making such experiments more difficult. The cross sections ap(0, tk) for polarized neutrons are usually measured relative to the cross section tr0(0 ) for unpolarized deuterons in order to avoid an absolute measurement. The experiments thus determine the quantities B(O) given in eq. (9) and the extraction of the corresponding A(O) requires knowledge of the differential cross section go(0 ) at

all(d, n)'*He 199

the same energy. Similarly for a measurement of the neutron polarization, the cross section ar,(O, q~) appropriate to the polarization of the incoming beam should be available.

Measurements of the neutron polarization fall into three classes with regard to the structural symmetry of the formulae:

(i) Measurements of the outgoing transverse polarization using longitudinally polarized incoming deuterons. These relatively easy experiments can be performed with existing equipment.

(ii) Measurements of the longitudinal neutron polarization. These involve the use of magnetic fields to rotate the polarization vector of the neutrons perpendicular to the direction of motion. Experimentally this introduces considerable difficulties, but where polarized ion sources with/~A beams are available, even such demanding measurements of the all(d, n)4He reaction should be feasible.

(iii) Measurements of the outgoing transverse polarization using incoming deuterons with transverse polarization, Such experiments are considerably less difficult than those of type (ii) but, because of the more complicated angular dependence, they demand more careful measurements at more angles than experiments of type (i).

4.2. CROSS-SECTION MEASUREMENTS

For one polarized particle in the incoming channel, eq. (7) reduces to

,,,._o im(tm,.e-~"¢) , for m = [ o d d J '

where the real quantity C,., .(0) is defined by

Cm,.(O) = (2-6.,o)tro(O) {li} Br~,.(O)

{1} ovont = ( 2 - 6. o) no, o, o, otto for m = (14) , "1,.,., o, o v'J, t odd J

and can be expanded in terms of Legendre polynomial t coefficients am,.(L):

C,n,.(O) = ~2 Z ar'..(L)PL,.(COS 0). (15) L

For the all(d, n)gHe reaction the am,.(L) are given in table 2 in terms of the appropriate combinations of matrix elements. For tro(0 ) and Ca, 1(0) coefficients with L up to 2 appear while for the Cz, .(0) the L = 3 term is also non zero. The expressions for tro(O ) and C2..(0) provide six equations for the five real parts U, with even l ' (v = 1, 2, 8, 10, 11), including the linear dependence:

½~/6a2, o(2) + 2a2, t(2) + a2, 2(2) = 0. (16)

* The Legendre polynomials used here are given by Jahnke and Emde, Tables of functions.


This is sufficient for the calculation of this set of U,. For the quantities U, with odd 1 (v = 3, 4, 5, 6) there are also six equations, subject to the conditions:

and

a2,o(1) = -- ½x/6a2,1(1) (17)

a2.o(3) = -¼x/6a2,1(3) -- 346a2, 2(3), (18)

which leave the set underdetermined. The imaginary parts V~ appear only in the two coefficients of C1, l(0), necessitating

the use of data obtained from measurements of the outgoing polarization. Measurements of go(0 ) below 1 MeV have been performed by several groups

[refs. 1 o-i2)]. For an analysis, precise data at the energies of the polarization measurements would be of great value. The advent of sources of polarized deuterons has made the determination of Cm, n(0) a relatively simple experiment, so that such information is available at several energies on and above resonance s, in, 1,). The recent experiment of Grunder et al. 14) determined all C2,,(0) at eight angles at 0.2, 0.6, 0.8 and 1.0 MeV. The resulting Legendre polynomial coefficients fulfill conditions (16) and (17) well within the error, while relation (18) diverges slightly toward 1 MeV. This is discussed in subsect. 5.2.

4.3. T R A N S V E R S E N E U T R O N P O L A R I Z A T I O N P R O D U C E D BY A L O N G I T U D I N A L D E U T E R O N P O L A R I Z A T I O N

A longitudinally polarized particle in the incident channel leads to the following transverse polarization of one of the outgoing particles:

/',/ x ovon: t ' 1 tra, 0 m, = , - - - C l' io(0) for m (19) 1.1 S ( 0 , ¢) / o d d j

where the real quantity

,1 {:) . , , oo,o, oven/ C.,o(O) = f o r . , = , (20) --~=, o. o, o~"j [ odd J

can be expressed as 1,1 c., o(0) = ~2 X ak ~(L)eL, 1(cos 0). (21)

L

With incident spin-1 particles, experiments using to, o, tt, o and t2, o can be performed. For the 3H(d, n)4He reaction the formulae for the contribution of a particular tin, o to the total polarization are given in table 3. For an unpolarized incident channel terms up to L = 2 appear, while with an incident polarization L = 3 terms are also non-zero. For even rank incoming tensor moments five equations involving the imaginary parts Vv result, while the odd rank tensor tl, o gives three additional equa-

_i 1/2~ (formula 7.2) completes a linearly tions for the quantities Uv. The coefficient al; ot )

TAB

LE 2

Lege

ndre

poly

nom

ial c

oeff

icie

nts a

~,, n

(L) f

or th

e de

uter

on p

olar

izat

ion

effic

ienc

ies C

m, n(

O)

Cm" n

(O) =

~

2 E

am

,,,(L

)PL.

,(cos

0),

am,,.

(L) =

•

og P

,, L

,,,

Exp

erim

ent

p~

eq.

L

ui

~2

0~3

0~4

0~5

~6

0~11

(Z

8 ~1

0

I un

pola

rized

cros

s sec

tion

ao(0

)

coef

ficie

nts a

o, o(

L)

u~

(1.1

) 0

(1.2

) 1

-,X

/2

~X

/5

(1.3

) 2

2eff

icie

ncyf

ortx

.1

V v

(2.1

) 1

-i!~

8X/3

---i

!-gs

x/3-

-AX

/6

AX

/15

coef

ficie

nts a

l.~(L

) (2

.2)

2 --

~X/3

-i~

8X/3

--~

2-2X

/6

3 ef

ficie

ncy f

or t2

. o

coef

ficie

nts a

2, o(

L)

V~

(3.1

) 0

~X/2

--

~X/2

(3.2

) 1

--~

/2--

~o

x/2

~

@X

/10

(3.3

) 2

--lx

/2

--~X

/2

--~X

/2

(3.4

) 3

3ox/

2 3o

x/10

¢

4 ef

ficie

ncy f

or ta

, 1

coef

ficie

nts a

2.1 (

L)

U v

(4

.1)

1 ~4

3 ~0

X/3

-~

2x/6

-@x/

15

(4.2)

2

3~84

3 14

3 -1

~843

-A

x/6

(4.3)

3

~X/3

@

415

5 ef

ficie

ncy f

or t2

.2

coef

ficie

nts a

2.2(

L)

V~

(5.1

) 2

--~

x/3

-AX

/3

~x

/3

--~

X/6

(5.2)

3

3-1o4

3 -,

~ox/

lS

1,4

TABL

E 3

Lege

ndre

pol

ynom

ial c

oeffi

cien

ts alm

~ I(L

) fo

r th

e ne

utro

n po

lariz

atio

ns t

'l, 1

(tin, O

) Cm

, o(O

)1

,1

= ~2

2 a

l'm, ao

(L)P

L, l( c

°s 0

), a i

'm, ~

(L)

= ~

8vPv

L

v

Expe

rimen

t Pv

eq

. L

81

82

83

f14

flS

86

811

8s

8~o

6 po

lariz

atio

n t'l

, 1(to

, o)

coef

ficie

nts

a I' I

(L)

O,

i

41o

V v

(6

.1)

1 12

(6.2

) 2

1 12

7 po

lariz

atio

n t't

, t (t

l, o)

coef

ficie

nts

al: ~

(L)

Uv

(7.1

) 1

--~

43

~43

(7.2

) 2

~66~

/3

~24

6

(7.3

) 3

4x/3

~4

3

-~o x/

3 3-

~x/3

&46

8 po

lariz

atio

n t'l

, x(tz

. o)

1 1

coef

ficie

nts

a2: o

(L)

v~

(8,1

) 1

112

151

~4~/

2 _

7_~/

5

1 (8

.2)

2 a-

~

(8,2

) 3

110

5--~

4 5

~4~/

2

aH(d,n)*He 203

independent system for the set U 3, U 4, U s and U6. Thus these quantities could in principle be determined once sufficient data for t ' 1, l(tl, o) become available. Experi- ments using unpolarized or aligned * deuterons yield two equations for the Vv in the system with even-/terms and three for the system with odd-/terms, leaving both sets still underdetermined.

Experimental data for the neutron polarization using unpolarized beams and targets are available at several energies 29-31) but for one angle only. In the forward direction very small negative polarizations are found. These increase slowly with energy reflecting the growing influence of p-waves. Precise measurements at several angles and energies below 1 MeV would be of considerable importance both for an analysis and as a basis for more complex experiments.

For incident beams with tl, 0 only or with ta, 0 and t2, o, existing polarized deuteron sources using high-frequency transitions deliver enough beam to make experiments with good precision possible. An earlier measurement of t~, x(q, o) [ref. 6)] near the 3 + resonance could readily be expanded to other angles and higher energies. The separation of effects from tl, 0 and t2, 0 occurs automatically since the resulting transverse polarizations lie perpendicular to each other, while the effects of t2, 0 and to, 0 can be separated by changing the sign of t2, 0 at the source. The reaction mode t t a, t(q, o) has been proposed as a source of highly polarized 14 MeV neutrons 32)

4.4. M E A S U R E M E N T OF T H E L O N G I T U D I N A L N E U T R O N P O L A R I Z A T I O N

The formula for the outgoing longitudinal polarization is

1 E a,o / I m . _,.#./ /even / (22) ' -- - - C=,.(0) IR e( t= , .e )j for m = [ o d d J ' h,o p(0,

where the real quantities

o c.,.(o) =(2-a. ,o) 1.o.0.o,m "am, n, O, 0 kvI

can be expanded as

/even/ (23) m = todd 1

1,0 1,0 C=,.(0) ~;2 y, 0). (24) = am, n(L)Pr~, . (cos L

The coefficients 1, o a = , . ( L ) are given in table 4.

For longitudinal incoming polarization, only tl, o gives an effect, the influence of to, 0 and t2, o being zero because of parity conservation. This experiment and one using a pure ta, 1 beam provide seven equations involving the Uv. For each one there exists a linear relation, within the already complete set of equations.

= --3x-~/2al: 0(3) = a 1' ota~ = ¼~/2bl: xx(3) (25) 2al;oa(3) 1 - - 1 0 l. lk "~1

* The term aligned is used to describe the axially symmetric case of tensor polarization where only t2, 0 is non-zero.

204 r. SEILER AND E. BAUMGARTNER

and

a I: oO(2) = - x/2a ~: o(2) (26)

may be of use as consistency checks. (For the last term in eq. (25) see subsect. 4.5.) Experiments using incident rank 2 polarization give three relations for the imagi-

nary parts V~ with even l. These together with formulae (6.2) and (8.2) from table 3 and formula (2.2) from table 2 form six equations in the four quantities V2, Vs, Vx o and V11. The linear dependencies (see also eq. (40)) are

1,0 a2. x(3) = -2a~:°(3) (27)

and

0 , 0 1,0 1,0 -- 1, = 3az, ~(2)+ 3a2, 1(1)-2a2,1(3)-61/6ao, o1(2) 0. (28)

From other Legendre polynomial coefficients of the same experiments, two equations for the Vv with odd l can be extracted. Together with formulae (1.1), (8.1) and (8.3) (table 3) and formula (2.1) (table 2) six equations for the quantities V 3, V4, V5 and V6 are obtained. One linear dependence involves 5 terms

5x/3a°: °(1) - 14~/2a~: o1(1) -- 16a~: ~(1) + 24a~: 1(3) + 3x/3a ~2: 2°(2) = 0. (29)

The other one is

a~: °(2) = -- a~:°(2). (30)

Inspection of relations (27), (28) and (30) shows that a measurement of h', o(t2,1), which would be very difficult, does not yield any independent information. In fact, none of the measurements of the longitudinal polarization of the neutrons are needed to arrive at a complete system of equations.

4.5. M E A S U R E M E N T O F T H E T R A N S V E R S E P O L A R I Z A T I O N P R O D U C E D BY A T R A N S - V E R S E D E U T E R O N P O L A R I Z A T I O N

The evaluation of formula (6) for this case is not straightforward since the quantities "'m,'~a" ~,o,:t:., o,° owjta'~ involve the functions a(L)_±., 1(0) . The real and imaginary parts of the contribution to the outgoing moment of rank 1 can be written in the following form

Qm, n} 1,1 re+n+1 1, 1 , ~-. 1 Z Pro,. [Cra'n(O)'~(-) CI'-n(O)]' Re (t,.x)

Im (t~, 1) - 1 [ - P . , . / 1,x . ÷ . 1 , 1

trp(O, ~),,,,~>o. Q.,,., [Cm,.(O)+(--) Cm,-.(0)] tevenl

for m = [odd J '

(31)

(32)

TA

BL

E 4

Lege

ndre

poly

nom

ial c

oeffi

cien

ts a t,

°(L

) for

the

neut

ron p

olar

izat

ions

t'l,o

(tm, n

) ra

in

1,0

1

,0

c,.,

(o)

o),

,.o

=

am, n

(L)P

L, n

(C°s

am

. n(L

) =

E 7v

Pv

L

v

Expe

rimen

t Pv

eq

. L

72

72

73

74

75

76

7tl

7a

71o

9 po

lariz

atio

n t'x

, o(tl

, o)

U v

(9.1

) 0

-- 1-

i~46

-i~

8 430

coef

ficie

nts a

~:°(

r)

(9.2

) 1

~/6

--

~8~/

6 ~-

s~/6

--

~ox

/6--

~X

/3

(9.3

) 2

1846

--

~/6

--

~/3

--

~rV

/30

(9.4

) 3

AX

/6--

~4

6

10 p

olar

izat

ion t

'l.o

(t,. i

) U

~ (1

0.1)

1

--~X

/3

~/3

2,

,/3

--~o

43

--~

46

"~.

o

coef

ficie

nts a

l: °(

L)

(10.

2)

2 ~X

/3

~X/3

-~

/6

24

15

(10.

3)

3 A

43

~-

s V/3

11 p

olar

izat

ion t

',.o(

t2.,)

V

v (1

1.1)

1 ~-

s~/3

-~

-~/3

-~

/6

coef

ficie

nts a

l: I°(

L)

(11.2

) 2

(11.

3)

3

A4

3

~ax/

3 --

~/6

-4--

~-~

/15

Ax/

3 -

~-s~

/3

12 p

olar

izat

ion t

'l.o(

t~. ,

) V~

(1

2.1)

2

-~ax

/3

--~a

~/3

~/6

~-

s~/1

5

coet

ticie

nts a

~: °(

L)

(12.

2)

3 ~o

~/3

~oV

/3

0

TABL

E 5

Coe

ffic

ient

s fo

r th

e ne

utro

n po

lari

zati

ons

t'1,

~ (t

in,

n ~

o)

1, 1

,~

2 a

1, 1

/I

~/(

L)

1, 1

C

m, "

l'n(O

) =

~ m

, 1(

0),

= +

nk~

iW+

n,

am, +

.(L)

~, c

5{ p,

L

v

Exp

erim

ent

p, eq

. L

c5~

~ c5

~ 6~

c5

~ c~

tS

fl tf~

5~

o .~

13 p

olar

izat

ion

tq,

l(h,

l)

coem

cien

ts a

1' ~

l(L

) 1,

14 p

olar

izat

ion

t'i,

l(t2

, x)

(13.2

) 2

~/6

~/3

___{

~/3

(13.3

) 3

~-5 ~

/6

~-~5 ~/

6 ~,

T1,

/3

~,/3

V

v (1

4.1)

1

coef

fici

ents

a 1

' ~ l(

L)

2,

(14.

2)

2 ~z

~/6

-T- ~

/6

(14.

3)

3 ~/

6

15 p

olar

izat

ion

t'l,

l(t2

, 2)

coef

fici

ents

al:

1 2(

L)

V v

(15.

1)

2 ~-

2~/6

-T-

~-z~

/6

T 1-

~/3

~2~/

3

(15.2

) 3

k415

1~

43 ~

A41

5 ~

41

5

3H(d,n)4He 207

where

and

em, n = Re (tin, n) C O S (n~b)+Im (tin,.) sin (n¢),

Qrn, n = Im (t,...) cos ( n ¢ ) - R e ( t . , . ) sin (n~b)

1,1 {i} 11 = a" Ant: 5:n(O) Z m, 1 ( l r~t t (L) c~, ~.(0) = 1 ~ ± .~: -± . ,1(0) .

From the set of coefficients am,X' a+. given in table 5, the quantities

1,1 1,1 1 ,1 am, - .(L) am, n(L)-I- bm, n(L) =

and

(33)

(34)

(35)

(36)

1, 1 1, 1 1, 1 c=,.(L) = a m , . ( L ) - a . , _.(L) (37)

can be formed, which again belong to one of the two systems of equations with even or odd L For all combinations, the highest non-zero term is one with L = 3.

For incident rank 1 polarization, three additional equations for each system of equations of the U v result. Useful relationships are given by eqs. (25), (38) and (39):

3~/2aI: ~(2) = b~: I(2), (38)

c I, 1:aa = O. (39) 1, l k " /

For incident rank 2 polarization t2,1 and t2, 2, five equations for each Vv system are found. With these eq. (27) is augmented by

a~]°(3) = ¼~/2c~: ~(3) = tao-~5c.~: ~(3) (40)

and in addition, the expressions

6~/6a~' o1(2) + 4b~: 1 1,1 (41) • 1(2)- b~, 5(2) = O,

2~/6aI: ~(2)- 4~/3a~: ~(2) + b~; ~(2) = O, (42)

x/:6a1: ~(1) = bl'2, *[lXl~. 1, (43)

c I, It'2"~ 2 4 : I (2 ) = ,_, 2, : , (44)

-- a~: ~(3) = ~ b ~ : I(3) = ~ o - ~ b ~ ; ~(3), (45)

can be used to test the validity of the approximation. From these relations it is also apparent that the measurement of the t;, 1(t2. i) contribution gives redundant data as in the case of the measurements of the longitudinal polarization.

2 0 8 F. SEILER AND E. BAUMGARTNER

5. Discussion

5.1. DETERMINATION OF THE MATRIX ELEMENTS

As shown in the last section, measurements of the polarization of outgoing neutrons, in addition to the data now available, give more than enough equations to determine the matrix elements. Many of these experiments are very difficult however, so that in practice a set of experimental data with its relatively large errors will not overdeter- mine the problem.

A possible complete set could involve measurements of the angular distributions of

%(o,,/,) t ,l(to, o) t ,l(tl,o) t ,l(t2,o)

for h,1 and t2,. (n = 0, 1, 2)

longitudinal polarization of the incident beam

Re [t~, l (q, ~ 1)] / transverse polarization of the incident beam. J

These data contain more than sufficient linearly independent information, and additional measurements could be undertaken on the basis of these results to eliminate spurious solutions.

The analysis should be performed using least-square methods to fit directly all available data points at one energy rather than the quantities C"" "'(0~ m,. , z or the corresponding Legendre polynomial coefficients. This procedure would not be overly biased by the Legendre polynomial fit or the measurements of the appropriate cross section (see subsect. 4.1). Furthermore the redundancies in the measurements would be used to fullest advantage. The equations for the Legendre coefficients, apart from their application as consistency checks, could be used to determine the starting values for U~ and V,.

5.2. VALIDITY OF THE APPROXIMATION

The approximation used here should be valid as long as the consistency relations are not violated and no additional terms have to be used in the expansions. This will certainly occur with increasing energy, where contributions from matrix elements not considered in the analysis become appreciable in their interference with R1 or where the nonresonant elements taken into account become large enough to give noticeable terms not involving R1.

The data for the C2,,,(0 ) [ref. 14)] show that very small L = 4 terms are possible at 0.8 and 1.0 MeV. However the consistency relations (16) and (17) still hold, while (18) shows a deviation outside the error limits. This could be caused either by terms neglected in the calculation or by the fact that the eight data points available had to be used to determine four and in one case even five coefficients and their errors.

"an(d, n) ' tHe 209

5.3. A P O S S I B L E S I M P L I F I C A T I O N

Cross-section measurements with tensor-polarized deuterons i4) indicate that both Us = Re(R1R*) and Ulo = Re(R~R*o) are very small up to energies of 1MeV, while Ull = Re(RIR~I) is relatively large. If this behaviour is explained by penetrability considerations, assigning the U~, term to the formation of the resonant ½+ state with s = ½, the argument could be extended to the corresponding imaginary parts Vv. Elements 8 and 10 could then be omitted altogether, which would lead to a set of seven contributing matrix elements and thus thirteen parameters. The simplification is felt only in the sets of equations for terms with even l where new linear dependencies emerge. Instead of relation (16) we now have

a2,o(2) = - - ½ 4 6 a 2 , 1(2) = ~/6a2, 2(2), (46)

while condition (25) is augmented to

ao, o(2) = - x / 2 a 2 , o ( 0 ) = lo--~ 1, 5 - 1 , 0 -- ~ q a1,1(3) = ~x/6a,,o(3 ) 5 -- 1 , 0 --~x/3a1, 1(3) = -~2x/6b1:~(3). (47)

In the equations for Vv relations (27), (28) and (40) become

3al, 1(2 ) 1,o 1,o = -a2,1(1) = a2,1(3) = -2a i : ° (3 ) = -¼~/?.c~: ~(1)

-- ¼ ~ / 2 c I ; I ( 3 ) = 1 - 1. T~x/5c2, I(3). (48)

Furthermore eqs. (41) and (42) yield the conditions

2~/6a2~; ~(2) = - b~; I(2) = - b~2; 2 ~ (2) (49)

and

1 ,1 ao, 0(2) = o. (50)

Condition (46) is fulfilled well within the experimental error by the data of Grunder et al. 14). The first equality of eq. (47) can be checked using the interpolated data of Bame and Perry 1 o) for ao(O). In view of the uncertainties inherent in the interpola- tion for ao(O ) and in the determination of 4 coefficients from eight data points for (?2, o(0), it is not surprising that deviations as large as twice the combined error are found at one energy. If this set of seven matrix elements is adopted, a preliminary estimate of Vii can be made from formula (2.2) and the measured C1, l(0). For terms with odd l the existing experimental data are too limited to give any indication of possible simplifications. Thus the suggested set of experiments remains unaltered.

The authors would like to express their gratitude to Prof. P. Huber for his continued interest and support and to the Schweizerischer Nationalfonds for financial assistance.

2 1 o r. SEILER AND E. BAUMGARTNER

R e f e r e n c e s

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385 7) H. H. Barschall, Proc. 2nd Int. Syrup. on polarization phenomena of nucleons, 1965 (Birkhauser,

Basel, 1966) p. 393 8) B. Hoop, Jr. and H. H. Barschall, Nucl. Phys. 83 (1966) 65 9) B. de Facio, R. K. Umerjee and J. L. Gammel, Phys. Rev. 164 (1967) 1566

10) S. J. Bame and J. E. Perry, Phys. Rev. 107 (1957) 1616 11) H. V. Argo, R. F. Taschek, H. M. Agnew, A. Hemmendinger and W. T. Leland, Phys. Rev. 87

(1952) 612 12) A. Paulsen and H. Liskien, Nucl. Phys. 56 (1964) 349 13) W. Tr~chslin, H. B~rgisser, P. Huber, G. Michel and H. R. Striebel, Helv. Phys. Acta 38 (1965)

523 14) H. Grunder, R. Gleyvod, J. Lietz, G. Morgan, H. Rudin, F. Seiler and A. Stricker, Helv. Phys.

Acta, to be published 15) L. J. B. Goldfarb and A. Huq, Hdv. Phys. Acta 38 (1965) 541 16) A. Huq, Helv. Phys. Acta 39 (1966) 507 17) G. Racah, Phys. Rev. 61 (1942) 186 18) W. Lakin, Phys. Rev. 98 (1955) 139 19) G. R. Satchler, The scattering and polarization of spin one particles, Oak Ridge Nat. Lab. Rep.

ORNL-2861 (1960) 20) G. R. Satchler, Nucl. Phys. 8 (1958) 65 21) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) 22) S. Devons and L. J. B. Goldfarb, Handbuch der Physik vol. 42 (1957) 362 23) T. A. Welton, Fast neutron physics, vol. II (Interscience, New York, 1963) chap. V.F 24) A. Simon and T. A. Welton, Phys. Rev. 90 (1953) 1036 25) D. M. Brink and G. R. Satchler, Angular momentum (Oxford University Press, Oxford, 1962) 26) Sign convention for particle polarization, Nucl. Phys. 21 (1960) 696 27) L. C. McIntyre and W. H~berli, Nucl. Phys. A91 (1967) 369 28) Ch. Leemann, H. B~irgisser, P. Huber, H. Schieck and F. Seiler, Helv. Phys. Acta 41 (1968) 438 29) P. Pasma, Nucl. Phys. 6 (1958) 141 30) R. Perkins and J. Simmons, Phys. Rev. 124 (1961) 1153 31) F. Borelli, V. Lazarevic and N. Radisi¢, Nucl. Phys. 66 (1965) 301 32) G. G. Ohlsen, Phys. Rev. 164 (1967) 1268

the 3h(d, n)4he reaction below 1 mev

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