Volume 56B, number 1 PHYSICS LETTERS 31 March 1975

T H E I N C O H E R E N T N E U T R O N S C A T T E R I N G L E N G T H O F 3 He

V.F. SEARS and F.C. KHANNA Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada, KOJ 1.10

Received 4 February 1975

The complex incoherent neutron scattering length of 3He is estimated on the basis of effective range theory and also by means of a Breit-Wigner analysis. Both methods indicate that the incoherent scattering is a factor 4 weaker than the coherent scattering.

The interpretation of thermal neutron scattering data [1 ] for liquid 4He is facilitated by the fact that 4He is a coherent scatterer with negligible absorption so that the scattering data are directly proportional to the atomic number-density fluctuation spectrum [2]. The situation with regard to 3He is, however, compli- cated both by the large absorption cross section (5327b) and by the presence of incoherent effects as a result of which the scattering data contain contribu- tions from both number-density and spin-density fluc- tuations.

The imaginary parts of the coherent and incoherent scattering lengths are accurately determined by exist- ing absorption data as will be shown below. In addition, the real part of the coherent scattering length has re- cently been obtained from reflectivity measurements [3]. However, the real part of the incoherent scatter- ing length has not previously been determined. Thus it was not known to what extent recent thermal-neutron inelastic scattering data [4] for liquid 3He reflect num- ber-density or spin-density fluctuations. Also, the inter- proration of earlier experiments [5] on 3 H e - 4He mixtures is to some extent uncertain for the same rea- son.

In this article we present theoretical estimates of the complex incoherent scattering length of 3 He based on both effective range theory and on a Breit-Wigner analysis. Both methods lead to the conclusion that the incoherent scattering cross section is a factor 4 smaller than the coherent cross section. Thus the scattering is predominantly coherent, as tacitly assumed in [4] and [5], but incoherent effects are not entirely negligible.

We begin by examining the available neutron data for 3He gas. It is conventional to express cross sections in terms of the "bound scattering length"

in which A denotes the nucleus/neutron mass ratio and a = a' - ia" is the scattering length. In particular, the bound coherent and incoherent scattering lengths are given by

3 1 b c =~b t +-~bs, b i = ¼x/r3 (bt bs) , (2)

where b t and b s are the bound scattering lengths for the triplet (1 +) and singlet (0 +) states respectively. In terms of the quantifies (2) the scattering, absorption and total cross sections per atom are given for unpo- larized neutrons by [6]

[ ~ A 2 es = (~-~--~) 47r(lbc12 + Ibil2)r~ + ...,

4 I t , , ( A ) 2 Oa = bc - 8.(b 2 + + . . . , ( 3 )

at = 4rtb" +(AA~_r)24rr(b'c2_h"2 +b~ 2 _o i -"2- )~7 +.." k o c -r 1 / ~ c '

in which r/= 1 + k B T/2AE o where k B is Boltzmann's constant, T the temperature and E o = h2k2o/2mn the incident-neutron energy in the laboratory system, m n being the neutron mass. The next higher-order terms in (3) contain the effective range correction for the s- wave phase shift and (in o a and o t but not Os) a p- wave contribution as well. As a rule only the ko 1 term in o a is significant for thermal neutrons thereby giving the familiar 1/o-law.

For center-of-mass energies below 3.27 MeV, the threshold for the (n, d) pick-up reaction, the only available absorption channels are the 3 He(n, p)T and 3He(n, ~/)4He reactions for which the cross sections

Volume 56B, number 1 PHYSICS LETTERS 31 March 1975

2n + 2 p 7.72

D + n ÷ p 5.49

D + d 3.27

0.00 n -I- 3He - 0 . 7 6

T + p - o . 5 ~ ~ o"

- 2 0 . 5 8 ' O* 4He

Fig. 1. Thresholds and 0 + levels for 4He.

Table 1 Bound scattering lengths (b = b' - ib") and scattering cross sections (o) for SHe: s = singlet state, t = triplet state, c = co- herent, i = incoherent, f = free atom

Quantity Experiment* Effective range Breit-Wigner approximation analysis

b' s (fm) 8.1 b s (fm) 5.9264 ± 0.0012 b[ (fm) 5.5 b[ (fm) (2.3 ± 1.1)× 10 -s

b' s (fm) 6.1 + 0.6 b e (fro) 1.4816 ± 0.0003 b[ (fm) 0 < Ib~l < 6;3 -1.1 b~' (fm) -2.5662 ± 0.0005

o c (b) 4.9 ± 0.9 o i (b) 0.9 < o i < 5.9 1.0

of (b) 2.8 < of ~< 6.6 3.3

9.9

4.9

-2.2

1.5 3.6

* t The value ofb c is from ref. [3] and the others are deter- mined in this article from available absorption data.

for 2200 m/sec (0.0253 eV) neutrons are Op = 5327 + 10b [7] and o~ = 60 + 30/ab [8] . The origin of the factor 108 disparity in these cross sections can be seen by examining the energy levels of the com- pound nucleus [9]. At thermal neutron energies we can confine our at tent ion to the s-wave component of the n + 3He channel for which J ~ = 1 + or 0 +. Hence, only 1 + and 0 + states of the compound nucleus occur in the collision. 4He has no well-identified 1 + states be- low at least 12 MeV and the only 0 + states are (fig. 1) the ground state at - 2 0 . 5 8 MeV and a resonance at - 0 . 5 MeV with a proton width F = 1.2 MeV [10].

Since the n + 3He and T + p channels both lie well inside the 0 + resonance it is natural to assume that the large observed value of OpiS due to s-wave absorption in the singlet state. This assumption is confirmed by experiments [ 11] with polarized neutrons and oriented 3He nuclei which show that within the experimental error (3%) the (n, p) reaction is due entirely to singlet- state collisions. Thus, with b" = b"(p) + b"(7), we can put b t (P) = 0 in which case the transmission data of ref. [7] give

I~' 1 IJ' be(P) =- ibs (P) = 1.4816 +- 0.0003 fm. (4)

Radiative capture can occur in the triplet state by an M1 transition but in the singlet state only indirectly via electron-positron production. Since the former

mechanism is dominant we can put bs(7 ) = 0 in which case the observed value [8] o f o~ gives

~r 3 t t bc(7) =-~bt (7) = (1.7 -+ 0.8) X 10 - 8 fm. (5)

Hence we obtain the imaginary parts o f the bound PI

scattering lengths listed in table 1. The values of b e and b~' enable us to compute the second term in the expression (3) for %. Wi thA = 2.98903 and '7 = 1 (i.e. E o >> k B T / 2 A "" 0.004 eV) we find the value - 1 . 2 3 9 0 +- 0.0005 b which agrees with the observed value - 1 . 1 -+ 0 .2b [12] at energies in the range

1 0 < E o < 30 000 eV. At present the real parts of the bound scattering

lengths cannot be determined as accurately as the imaginary parts. The value o f b ' c in table 1 was ob- tained in recent reflectivity measurements [3]. An up- per limit on Ib~l can be obtained from the observed "value", - 0 . 8 + 5.1 b [7], o f the second term in the expression (3) for o t which gives b[ 2 = - 0 . 3 9 + 0.79 b or, since b I is real, 0 ~< b~ 2 ~ 0.40 b. This enables us to obtain upper and lower limits on the cross sections in table 1. Here o c and o i denote the bound coherent and incoherent scattering cross sections,

Oe = 47rlbe12 , o i = 4~rlbil2 , (6)

and of the scattering cross for a free a tom which is

initially at rest,

PHYSICS LETTERS 31 March 1975 Volume 56B, number 1

A 2 of = (A----~) (Oc + oi). (7)

" I

A more precise estimate o f b i can be obtained from effective range theory [13 -16 ] by regarding the 0 + resonance as a shallow, quasi-stationary bound state o f the neutron with a complex energy

h2k2b . P

Eb - 2U - - 8 - 1 3 . (8)

Here B = 0.5 MeV, F = 1.2 MeV [10] and t~ is the n - 3 He reduced mass so that k b = 0.07 - 0.15 i f m - 1 . The singlet scattering length has an expansion

1 O(k4), (9) 1 /a s = i k b +~re k 2 +

in which r e denotes the effective range. Although t h e above expansion is derived in the above references for a stationary bound state (e.g. the deuteron) one can easily verify, within the framework of a complex opti- cal model, that it is equally valid for quasi-stationary bound states. For neutron-proton scattering the kb 4- term is approximately 0.03 r3e k4 [13]. I f this is also true in the present case, where we shall fred that Irekb I = 0.53, then the k4-term will contr ibute less than 0.5% to the value o f a s. Neglecting this term, a s is then uniquely determined b y the complex effective

f • I? range r e = r e - lre. We expect that , as in the case of a stat ionary bound state [ 16], r ' e is approximately equal to the radius of the potent ial well. Thus we assume

I t h a t r e = 1.7 fro, the charge radius of 4He [17]. The

I f value o f r e is then determined by requiring that (9) yield the correct value for a s , i.e.

f? I? a s = 3b c = 4.4448 + 0.0009 fm. (10)

• ? l

Hence we find that r e = 2.7 fm whereupon a ' s = 6.1 fm t

and b s = 8.1 fm. It then follows with the help of the t

value o f b ' c in table 1 that b t = 5.5 fm and, finally, that f

b i = - 1 . 1 fm. Since the collision in the singlet state is dominated

by a single resonance of the compound nucleus, we ex- pect that a s can also be est imated with the help of a single-level Breit-Wigner formula. Since the collision is non-resonant in the triplet state, we therefore assume

that [10]

a s = r - - S / E b , a t = r , (11)

in which the potential scattering length r is taken to be spin-independent (as is usually the case) and r and s

f

are bo th real. It then follows from the values of b c and be that r = 3.7 fm and s = 4.5 MeV fm whereupon we obtain the results listed in table 1. The approxi- mate agreement of the effective range and Breit- Wigner results indicates that our basic assumptions, i.e. that r ' e equals the 4He charge radius and that r is spin-independent, are substantially correct. The above results suggest that it is reasonable to adopt the value b~ = - 1 . 5 + 1.0 fm in which case o i = 1.2 +0.3 b and of = 3.4 + 0.7 b.

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