476

J. Appl. Cryst. (1989). 22, 476-482

The Energy Dependence of the Neutron Coherent Scattering Length of Europium

By J. E. LYNN*

Los Alamos National Laboratory, New Mexico 87545, USA

(Received 23 November 1988; accepted 24 May 1989)

Abstract

General considerations are given for the importance of providing an energy/wavelength-dependent database of neutron coherent scattering lengths, and of the possibility of constructing such a base from the range of neutron cross-section data that are gen- erally available for a wide range of nuclides. The case of europium is considered in detail, partly because it is a good example of an element with strong resonance effects in the wavelength region of importance in crystallographic and condensed-matter studies, and partly because it is an exceptional case from the nuclear point of view, having multi-level interference effects. From the available cross-section data the com- plex coherent scattering length is constructed as a function of neutron energy up to 0.7 eV, for the isotopes and for the natural element. Some single- level approximations for the scattering-length func- tion of tS~Eu are given.

1. Introduction

The advent of powerful pulsed neutron sources has led to the use of neutrons in the energy range above thermal to - 0 . 5 eV in condensed-matter research. In spite of the rapidly increasing sophistication of instrumentation and data-analysis methods in pulsed- neutron condensed-matter studies, there appears to be no suitable nuclear database available for use in much of this work. The necessity for such a database is clear from the fact that very many heavy nuclides have strong resonances in their neutron cross sections at low neutron energies, thus strongly influencing the scattering characteristics over the energy ranges now being employed. The effects of such resonances, in Bragg scattering for example, have long been recog- nized (see, for example, Peterson & Smith, 1961, 1962; Nereson, Olsen & Arnold, 1962). The direct experi- mental measurements available for such a database are extremely limited, consisting of a few deter- minations of coherent scattering lengths at a few wavelengths for some nuclides. However, l~he indirect information is much greater; the energy dependence of total cross sections in the range from below thermal to above 1 eV has been measured for a large number

* Permanent address: UKAEA, Harwell, Oxon, England.

of nuclides, resonance parameters (often including the spin value) for all significant resonances in this low-energy region have been well determined, and for many nuclides there is a measurement of the total scattering cross section at thermal energy in addition to a spot value of coherent scattering. From this information it should be possible to construct the energy-dependent collision function of the neutron- nucleus scattering and hence the coherent scattering length for major isotopes and the natural element with a fair degree of confidence.

In this report I study the possibility of carrying out such a programme on europium and its isotopes. Europium is a rather typical rare-earth element from the nuclear point of view, in that it has closely spaced resonances (of the order of 1 eV), which are rather strong and less than 0.1 eV in width. The general principle I hoped to adopt in this work was to provide a parametrization for the coherent scattering length that is based on a generalization of the single-level Breit-Wigner formula, the generalization consisting of a representation of the effects of other levels by a term with only a slow energy variation. In the europium case it was not possible to do this over the full energy range of interest. Hence the data analysis was made with R-matrix many-level theory, and the resulting fully general coherent scattering-length function was refitted over more limited energy regions to an appropriate single-level formula.

In § 2 I review the formal reaction theory I employ to analyse the cross-section data. In § 3 1 describe the available data on europium. In § 4 the many-level analysis of the europium data is described. From this the coherent scattering length is generated over the energy range from near zero to 0-7 eV (§ 5) and single- level parametrizations over shorter energy ranges are deduced.

2. Formal reaction theory

The R-matrix nuclear reaction theory of Wigner & Eisenbud (1947) with further development by Lane & Thomas (1958) provides a fully general framework for carrying out analysis of neutron cross-section data. I limit the brief discussion here to the s-wave component of the neutron scattering wavefunction. A fuller account will appear in a forthcoming report.

The s-wave spherical component of a plane wave of neutrons impinging on a target nucleus situated at

0021-8898/89/050476-07503.00 © 1989 International Union of Crystallography

J. E. LYNN 477

the origin is

iTgl/2 j + l / 2 ~//plane __

k r v , / 2 [ I ( r ) - O ( r ) ] Y o o ( O , q ~ ) ~ ~', (1) s = j - 1/2

Here k is the wavenumber for the relative motion (with velocity v) of neutron and target nucleus, r is their relative radial separation, 0, q~ the spherical polar angles, Yoo is the zero-order spherical harmonic and ~', are channel spin functions for the coupled neutron spin (or = 1/2) and nuclear spin j. Finally, l ( r ) and O(r) are incoming and outgoing wave func- tions for the radial separation r of neutron and target.

The interaction of the target and neutron modifies the amplitude of the outgoing component of the plane wave. This modification represents elastic scattering. In addition, reaction will normally take place result- ing in waves appearing in particle and photon chan- nels that are physically distinct from the outgoing waves o f ( l ) ; I shall not explicitly use these. The wave function for only the incident and elastically scattered neutron components of the system is now

i77.1/2 j + 1 / 2

[ ,~, . - u.~e~], (2) O - krvl/2 .~=lj ~/21

in which the angle and spin functions have been incorporated with the radial functions into the gen- eralized incoming and outgoing channel functions 5~ and C, for the incident neutron channel. Subtraction of (1) from (2) gives the scattering wave function

i7r~/2 j+l/2 0 ~ ' - ~ [ 1 - U~]C,. (3)

krv~/2 s = l / 1 / 2 i

This can be written in the usual form for the s-wave elastic scattering wavefunction:

where

j + l / 2

s - I J-- 1/21

Z [ e x p ( i k r ) / r ] ~ . ~ / v 1/2 (4)

A = ( i / 2 k ) ( 1 - U~). (5)

The form of U~. is determined by nuclear reaction theory. From the general R-matrix theory an impor- tant result that is particularly useful for the treatment of slow neutron reactions is the reduced R-matrix formulation of the collision function U. The collision function is expressed as

U = e x p ( - 2 i k a ) ( 1 - i P R ) - ~ ( l + i P R ) , (6)

where a is a channel radius (in the construction of the theories this formally divides the 'internal' region of configuration space, in which nuclear forces are operative, from the external 'channel' region), and P - - k a is the s-wave penetration factor. R is the reduced R matrix

R E 2 = 7A(,,/[EA -- E - i1"~,,~,/2]. (7) A

channel spin s, and the sum over levels in (7) is limited to those levels with total angular-momentum quantum number J = s. Here 3'~,) is the reduced neutron width of the compound nucleus level A with energy EA and 1\~,~) is the sum of partial widths for all reaction channels; it is thus approximately the total resonance width 1\ minus the neutron width FA~,) when reson- ances are narrow compared with their spacing.

For the limited energy range used in condensed- matter work, a single resonance will usually dominate the reaction. In this case a single-level treatment will suffice in which this one level is treated exactly, but the background effects due to other levels are properly included. The reduced R function is split into an explicit term for the single level of interest and an additional term, which is complex, to represent the remaining levels. This latter term is written as a simple polynomial. Thus

R 2 = 3,~,(,ff[Ea - E - i F a ~ , , f f 2 ] + R e ~ + i S e x (8)

R ~ × = A ~ + B f i E - E o ) + C ~ ( E - E o ) 2 + . . . (9)

Sex= A, + B,( E - Eo) + C,( E - Eo)2 +. . . , (10)

where Eo is an approximate median point in the energy range in question. If parameters are known for neighbouring levels, approximate evaluations can be made for the polynomial coefficients in (9) and (10). From the expressions for these coefficients in terms of other levels it is also found that B, and Ai, C, cannot be negative.

From (8) the collision function of (6) can be arranged into the single-level form

U = exp [ -2 i (ka - a r g X )]{ 1 -2PS~x/X

iF ;,I,,~ + 2P( A A~,,) + iFA~.t/2)Se~}

+ E where

X = 1 + iPRe~+ PS~x, (12)

F~(.) 2 2 (13) = PTa~.~(l+PSe~)/lXl 2

is an energy dependent neutron width, and

= 2 2 (14) A, = A~(, , - P R.3 ,~ , . 41XI =

is an energy-dependent level shift. The substitution of (11) into (5) results in a generalized single-level exoression for the scattering-amplitude behaviour as a function of energy:

f = - [ a ( 1 - Rex)]+2ka2Sex(1- Rex)

aT~¢.~[(Ea + A, - E ) + k a l ~ ( 1 - Rex)] (Ea + A a - E ) 2 + F ~ / 4

+ i[aSex+ k a 2 ( 1 - Rex)2-ka2S2~x]

aT~(,)[ l \ / 2 - 2 k a ( E~ + A~ - E)(1 - Rex)] + i

( EA + AA -- E )2 + 1-'~/4 The functions U and R are implicitly labelled by (15)

478 ENERGY DEPENDENCE OF NEUTRON SCA'Iq'ERING LENGTH OF Eu

For a given nuclide the coherent scattering length is the spin-weighted sum over the two channel spins (or one channel spin, s =j, for an even nucleus) of minus one times the scattering amplitude:

a¢oh = - E gsfs, (16) s

where g~ is the spin-weight factor (2s+ 1)/[2(2j + 1)]. For a given element the sum must also be taken over isotopes:

acohx, = --E W, ~, g,f.~, (17) i s

where wi is the fractional abundance of the isotope i in the element. Most of our information on the parametrization of the scattering length comes from the cross sections. The total scattering cross section of a single isotope is

trsc= ( rr/ k2) E g, I1- U,I 2 (18) s

and that for the element is

o .... l = Z wio~c.i. (19) i

The total cross section for a single isotope is

o , = ( 2 7 r / k 2 ) Z g ~ [ 1 - R e ( U , ) ] (20) $

and that for the element is weighted as in (19). The absorption cross section is the difference between (20) and (18).

3. Nuclear cross-section data for europium

A few direct measurements of the coherent scattering length of europium have been made over a very limited energy range. Cable & Koehler (1977) give values of acoh. E . . . . t at 0.75A (=0.145eV) and at 1.067,~ (=0.0719eV); they are 5.97 (2) fm and 6.69 (3)fm, respectively. It is not indicated if these values represent the real part or the modulus of acoh. I assume in my analysis that they are Re (acoh); it turns out that l acohl is only a few percent different at these energies. Nereson et al. (1962) give acoh,E .. . . t = 6"3 (3) fm at 1.391 A (=0.0423 eV); I have not used this datum in view of the much greater accuracy of the former work, with which it is in statistical agree- ment. Als-Nielsen, Dietrich, Kunnmann & Passell (1971) have published a coherent scattering length for 153Eu; acoh.Eu.153 - - 8"3 (3) fm.

A value of the total scattering cross section of natural europium has been reported by Brockhouse (1953); it is trsc.E .. . . t=630(40) fm 2. The thermal capture cross sections have been measured (Poortmans, Girlea & Fabry, 1971) for Eu(nat) [=0.4565(100) pm2], 15~Eu [=0.9184 (140) pm ~] and ~53Eu [=0.0312 (7) pro2].

The total cross section has been measured in detail as a function of energy from 0.001 eV to well above

1 eV by a number of workers; the data and references can be found in the compilation of cross-section data by the Brookhaven compilation centre (see, for example, Goldberg, Mughabghab, Malgurno & May, 1966; Mughabghab, 1984; Garber & Kinsey, 1976). Some of these data and other measurements have been used to determine the parameters of the reson- ances found at 0.32 and 0.46 eV. I have not attempted to redetermine these parameters, but rather have used the evaluated parameters given in the compilation of Mughabghab (1984). These are Et = 0.321 eV, fit(,,)= 0-0714 meV, F~(a) = 0-08 eV; E2 = 0.46 eV, F2(,) = 0.665 meV, F2(a)= 0.087 eV. Both these resonances are attributed to the target nucleus ~5~Eu and have total angular momentum J = s =3 (Stolovy, 1964). No resonances below 1 eV have been found in 153Eu. The isotopic abundances of europium are w~5~ = 0.4786 and wt53=0.5214. Both isotopes have spin j = 5 / 2 .

4. Many-level analysis of data

The two data points for 153Eu were first used to give simple constant values for Rex and Sex on the assump- tion that local level effects were significant for only one channel spin; this channel spin was taken to be s = 2. The channel radius a was assigned the value of the nuclear potential radius

a =.1.16A~/3 +0.6 fm. (21)

This is a conventional choice, although not formally correct under the basic tenets of R-matrix theory (Wigner & Eisenbud, 1947; Lane & Thomas, 1958); for s-wave neutrons it does not affect the analysis, and it does have the advantage of giving the potential scattering cross section at the right order of magnitude in the absence of an R term. With these assumptions I find

Re . . . . 2=-0"49(5 )

S . . . . . 2 = 0.031 (1).

On the assumption that only one level is dominant in this energy region, these values imply that this level is bound at about 1 eV below zero. The energy depen- dence of the total cross section of "~'Eu indicates that there is a bound level very close to zero neutron energy, and the relative magnitudes of the thermal cross sections of the two isotopes show that this level is in the ~5~Eu+n system. Although parameters are suggested for this level in Mughabghab's (1984) com- pilation and in the work of Poortmans et al. (1971), I redetermine them in this investigation so that they are consistent with the determination of Rex and Sex parameters. The spin of this level has been measured by polarization techniques (Stolovy, 1964); it is s = 3 as for the other two resonances. Radiation widths are normally very constant from resonance to resonance of a given nuclide, so I assume a value of this

J. E. L Y N N 479

1 0 5 , , , , , , , , | , , , , , , , , I , , , , , , , ,

.~_ 1 0 4

ca')

o

t~ 10 3

1-

10 2 , , , , , , , , = , , , , , J t t f , , , , , , , ,

10-3 10 .2 10-1 10 0

N e u t r o n e n e r g y

Fig. 1. Three-level fit to total cross section of natEu. The full curve is obtained under the assumption that all these levels have the same spin s = 3.

25

2 0

1 . 5

~ 1 0 E

~ 05

g 0 g c)

-0.5

I I I I

I I I I ]

0.1 0.2 0.3 0 4 0.5

N e u t r o n e n e r g y

-10 I 0 0.6 0.7

Fig. 2. The real part of the coherent scattering length of tStEu calculated from the parameters of the three-level fit of Fig. 1 (full curve). The broken curve is calculated from the two s = 3 levels and one s = 2 level fit.

I I I

- 1 . 0

"C'-

(D - 1 5

~ -20

0 -2.5

-0.5

-3.0 J t t I .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

N e u t r o n e n e r g y

Fig. 3. The imaginary part of the coherent scattering length of tS~Eu. The curve has the same significance as in Fig. 2.

Table 1. The calculated coherent scattering length of 15~Eu

units are 10 -12 cm ( - 10 fm).

E Re (acoh) Im (acoh) (eV) (10 -12 cm) (10 -12 cm)

0-0141 0.658 -0.308 0.0281 0.632 -0"239 0.0421 0.598 -0 .192 0.0561 0.563 -0 .160 0.0701 0.528 -0.139 0.0841 0.493 -0.125 0.0981 0.461 -0 .116 0.1121 0-428 -0.111 0-1261 0.396 -0.109 0.1401 0.364 -0 .110 0.1541 0.330 -0.113 0.1681 0.296 -0-119 0.1821 0.259 -0.127 0.1961 0.219 -0 .139 0-2101 0-176 -0-155 0-2241 0.129 -0 .176 0.2381 0.077 -0 .204 0.2521 0.020 -0.243 0-2661 -0-040 -0 .299 0-2801 -0.098 -0.378 0.2941 -0.136 -0.486 0.3081 -0.127 -0 .608 0.3221 -0.065 -0 .689 0-3361 -0-020 -0.695 0.3500 -0-052 -0-681 0.3640 -0.146 -0-707 0.3780 -0.273 -0.799 0.3920 -0.410 -0 .974 0.4060 -0.531 -1 .26 0.4200 -0.575 - 1.68 0.4340 -0.415 -2 .25 0.4480 0-118 -2-80 0.4620 0-997 -3 .00 0.4760 1.81 -2-65 0.4900 2.23 -2 .05 0-5040 2.32 -1-51 0.5180 2-26 -1 .10 0-5320 2.14 -0.825 0.5460 2.02 -0.633 0-5600 1.91 -0.497 0.5740 1-81 -0.400 0.5880 1.72 -0-327 0.6020 1.64 -0.273 0.6160 1.57 -0.231 0.6300 1.52 -0-198 0-6440 1.46 -0.171 0.6580 1.42 -0 .150 0.6720 1.38 -0.132 0-6860 1.34 -0.118 0.7000 1.31 -0.105

p a r a m e t e r f o r t h e b o u n d l eve l t h a t is v e r y c l o s e to

t h e m e a n v a l u e m e a s u r e d f o r t h e first f e w r e s o n a n c e s

o f 15~Eu, i.e. 90 m e V . In t h e d a t a a n a l y s i s o f n~tEu I u s e t h e t w o v a l u e s

o f t h e c o h e r e n t s c a t t e r i n g l e n g t h , t h e t h e r m a l c a p t u r e

c r o s s s e c t i o n a n d a s e l e c t i o n o f t o t a l c r o s s - s e c t i o n

v a l u e s . I u s e d a s t r a i g h t f o r w a r d l e a s t - s q u a r e s - f i t t i n g p r o c e d u r e , w i t h d e r i v a t i v e s o f t h e f u n c t i o n a l

dependence of observables on variables calculated

480 ENERGY DEPENDENCE OF NEUTRON SCATTERING LENGTH OF Eu

numerically using a step length of the variable pro- vided as input. Variables could be held fixed at their input values, or at a value determined at a certain stage of the calculation, which could be reiterated.

After many trials, a final iteration was carried out with all Rex and Sex parameters for '5'Eu held fixed at zero. The only parameters to be varied were the level energy E~, and reduced width y~( , , )of the bound level. The final fit to the data gave a X 2 value of 38 for 22 data points. The values of the bound level parameters were determined to be:

EA = --0-034 (2) eV

"y~(,,) = 0"067 (2) eV.

This fit gives o ,~=0.4473pm 2 at thermal [cf 0.4565 (100) pm 2, measured], acoh=6"76 fm at 0.0719eV [cf 6.69 (3) fm] and acoh=5"94fm at 0.145 eV [cf. 5.97 (2)fm]. The fit to the total cross- section data is shown in Fig. 1.

1.6

1.4

'~ 1.2

(D 1.0

"r-

0.8 g 0.6

~ 0.4

0.2

1 I I I 0.1 0 .2 0 .3 0 .4

N e u t r o n e n e r g y

.o 015 o16 0.7 .0

Fig. 4. The real part of the coherent scattering length of natEu. The curves have the same significance as in Fig. 2:

-0.2

-0.4

c:~ -06 - c

~ -0.8 -

c -I .0 -

o -1.2 " 0

-I .4 -

-1 6 1 1 1 1 I I .0 0.1 0.2 0.3 0 4 0 5 0.6 0 7

N e u t r o n e n e r g y

Fig. 5. The imaginary part o f the coherent scat ter ing length of "a'Eu. The curve has the same significance as ~n Fig. 2.

Table 2. The calculated coherent scattering length of "~' Eu

E Re (a~oh) lm (a¢o~) (eV) (10 -12 cm) (10 -12 cm)

0"0141 0.740 - 0 . 1 5 2 0.0281 0.728 -0 -119 0"0421 0.712 - 0 . 0 9 6 0"0561 0-694 -0-081 0.0701 0.678 -0 -071 0.0841 0-661 -0"065 0.0981 0-646 -0 .061 0" 1121 0.630 -0"058 0"1261 0.615 - 0 . 0 5 7 0" 1401 0"599 - 0 . 0 5 8 0" 1541 0"582 - 0 . 0 5 9 0" 1681 0.567 - 0 . 0 6 2 0.1821 0.549 - 0 - 0 6 6 0-1961 0.530 -0"071 0-2101 0.509 - 0 . 0 7 9 0.2241 0-487 -0"089 0.2381 0-462 - 0 . 1 0 2 0.2521 0"435 -0 -121 0.2661 0"406 - 0 . 1 4 8 0.2801 0.379 - 0 . 1 8 6 0.2941 0.360 - 0 . 2 3 7 0.3081 0.365 -0"296 0.3221 0.394 -0 -335 0-3361 0.416 -0"338 0-3500 0.401 -0 .331 0- t640 0-356 -0 -343 0.3780 0.295 -0"388 0.3920 0.229 -0 .471 0-4060 0.171 - 0 . 6 0 7 0.4200 0.150 - 0 . 8 1 0 0.4340 0"226 - 1.08 0-4480 0"482 - 1 "35 0.4620 0.903 - 1.44 0.4760 1-29 - 1.27 0-4900 1-49 - 0 . 9 8 7 0-5040 1"54 - 0 . 7 2 6 0.5180 1.51 -0"533 0.5320 1 "45 - 0 . 4 0 0 0.5460 1 "39 - 0 . 3 0 8 0" 5600 1.34 - 0 . 2 4 3 0.5740 1.29 - 0 . 1 9 6 0-5880 1.25 - 0 . 1 6 2 0-6020 1"21 - 0 . 1 3 6 0.6160 1.18 -0" 116 0-6300 1" 15 - 0 . 1 0 0 0.6440 1.13 - 0 - 0 8 7 0.6580 1-11 - 0 . 0 7 7 0.6720 1 "09 - 0 . 0 6 8 0.6860 1 "07 - 0 . 0 6 2 0.7000 1.05 - 0 - 0 5 6

These parameters are found to fit the data on '5 'Eu reasonably well.

The fitting procedure was also carried out with the assumption that the bound level had spin s =2. In this case, with three parameters al lowed to vary, namely the level energy and reduced width of the bound level and A,.,=2, a fit with X 2= 37 was found. The bound level parameters were:

Ea = - 0 . 0 2 8 (7) eV, T2(,)=0.081 (13) eV

J. E. L Y N N 481

T a b l e 3. Single-level parameters for limited energy ranges of the cross section of ~5'Eu

Energy range (eV)

0-001-0.22 0-22-0.38 0.38-0-70 J = 2 J = 3 J = 3 J = 3

Ex (eV) - - -0 .0341 0.3159 0.46 ~/~ ~,,) ( e V ) - - 0 . 0 6 1 0 . 0 4 0 . 3 2 9

/-'~,~ ~) (eV) - - 0.090 0"08 0"087 Eo (eV) 0.11 0-11 0.25 0.46 A, 0.172 0.961 1.273 - 0 . 5 4 B~ 1.36 3.246 11.26 1.46 Cr 15-0 - 0 . 6 6 3"581 0.0 A~ -0 .01 0.19 0.29 0.0225 B, 0.196 0-961 1-92 0.0 C~ 6.25 3.77 73-1 0.0

and A ..... 2 = - 0 . 0 2 (5).

The thermal capture cross section, the coherent scat- tering lengths and the total cross-section fits were almost identical to the previous case.

Another possible variation in fitting the natural europium data is to fol low the inference of a single

bound level in 153Eu. This would imply:

B ..... 2 = 0 . 5 e V -1 and B~ .... 2 = - 0 . 0 3 eV-'

Attempted fits with these parameters fixed at about these values are poorer, resulting in particular in a lower scattering length at 0.145 eV. This seems to indicate that stronger more distant levels are dominat- ing A~.,=2 for this isotope.

5. R e c o m m e n d e d v a l u e s o f c o h e r e n t s c a t t e r i n g l e n g t h s

Using the fitted R-matrix parameters in (7), (6), (5), (16) and (17), I can now compute the coherent scatter- ing lengths at any energy within the fitted data range. The 153Eu coherent scattering length is constant, the real part having its experimental ly determined value; the complex value is

acoh.153 ---- 7"73 -- i0"08 fm.

The computed scattering length of ~5~Eu is shown in Figs. 2 and 3 and a tabulation of values at 14 meV intervals in Table 1 for the fit with the assumption

0.70

0 6 5 E ,,3

0 6 0

o ~ 055

r-

0 5 0 c .9.0 t:~ 045 c

¢~ 040

u~ 0 3 5 c o

030

~5 0 025

020

I I I I I I I I I

- - I

I [ l I I I I I I

0 002 004 006 008 010 012 014 0.16 018 0 2 0

Neutron energy

,.-,.

o

t -

co

-012 ~ . ~ -

-016

- 0 1 8

-0 20

-0 22

-C 24

-C 26

-0 28

-O 30

I I I ~ 18 I I I l 0 II "C3200 CC2 004 0C6 ~ 010 012014 016 8 0.20

Neutron e n e r g y

Fig. 6. Single-level fit to the calculated coherent scattering length of ~5~Eu over the energy range 0 to 0.22 eV. Open circles indicate selected values of the real part and full circles the imaginary part.

E o

o

t--

o) c (D

O O

0.3 i I i i I I i I

0.2 X 0.1

0

- 0 . I O O o o

-0.2

-0.3 " V O X

O -0.4 I I I I I I I I I

0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40

N e u t r o n e n e r g y

E (..) e~

O v

c

c

(D

,3

-0.1 w i l i w i w i

-0.2 -

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

°X -0.9

- I .0 ~ ~ , ~ J , ~ i L 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40

N e u t r o n e n e r g y

Fig. 7. Single-level fit to the calculated coherent scattering length of tS~Eu over the energy range 0.22 to 0.38 eV. Symbols as in Fig. 6.

482 ENERGY DEPENDENCE OF NEUTRON SCATTERING LENGTH OF Eu

that all three levels have the same spin s - - 3 . The alternative assumption that the bound level has s = 2 is shown as a broken curve in Figs. 2 and 4. The difference is small, and the broken curve is not used because it is known that the bound level has spin s = 3, but the comparison does give some indication of the degree of uncertainty associated with this kind of analysis.

A

O

t , ' -

( 1 ) D

" 2

t ' -

2.5

2.0

1.5

1.0

0.5

.0

-0.5

-1.0 0.35

I I

0.40 0.45

1 i i 1

0.50 0.55 0.60 0.65

Neutron energy

.70

E o -0.5

e ~

o , , , , - v

-1.0

~ -1.5

~ -2.0 .,..., g

~ -2.5 0

-3.0 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

Neutron energy

Fig. 8. Single-level fit to the calculated coherent scattering length of ~5~Eu over the energy range 0.38 to 0.70 eV. Symbols as in Fig. 6.

The computed scattering lengths of natEu are shown in Figs. 4 and 5 and in Table 2.

A single-level parametrization for these energy dependencies cannot be made over the full energy range illustrated in Figs. 4 and 5. I have therefore made parametrizations over three more limited ranges by refitting the calculated curves of Figs. 2 and 3 to the single-level form based on (8), (9), (10) and (11). The results of this procedure are given in Table 3 for ~5~Eu. The quality of these representations is shown in Figs. 6 to 8. They are probably adequate for many purposes, but are not perfect, especially in the mid- range 0.22 to 0.38 eV, which is strongly affected by the nearby level at 0.46 eV. Polynomial fits with a reasonable number of parameters are not adequate in general for these functions.

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