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Experimental values of spin dependent nuclear scattering lengths of slow neutrons
H. Glättli, G.L. Bacchella, M. Fourmond, A. Malinovski, P. Mériel, M. Pinot, P. Roubeau, A. Abragam
To cite this version:
H. Glättli, G.L. Bacchella, M. Fourmond, A. Malinovski, P. Mériel, et al.. Experimental values of spin dependent nuclear scattering lengths of slow neutrons. Journal de Physique, 1979, 40 (7), pp.629-634.
�10.1051/jphys:01979004007062900�. �jpa-00209147�
LE JOURNAL DE
PHYSIQUE
Experimental values of spin dependent nuclear scattering lengths of slow neutrons
H. Glättli, G. L. Bacchella, M. Fourmond, A. Malinovski, P. Mériel,
M. Pinot, P. Roubeau and A. Abragam
DPh-G/PSRM, CEN Saclay, B.P. n° 2, 91190 Gif-sur-Yvette, France (Reçu le 3 janvier 1979, accepté le 14 mars 1979)
Résumé. 2014 Après avoir rappelé le principe de la mesure par la méthode du pseudomagnétisme, des longueurs de
diffusion neutroniques dépendant du spin des noyaux on donne un tableau qui rassemble les résultats obtenus par cette méthode. Il contient des valeurs déjà publiées. Mais certaines ont cependant subi des corrections à la suite de nouvelles mesures plus précises. Le tableau contient en outre des résultats nouveaux.
Abstract. 2014 The principle of the method of pseudomagnetism for measuring spin dépendent scattering lengths of slow neutrons on nuclei is recalled. A table is given which reviews the results obtained with this method. It contains previously published values, some of them corrected following careful remeasurement, as well as other hitherto unpublished values.
Classification
Physics Abstracts
25.40
1. Introduction. - Slow neutron scattering has
become a widely used tool of research in physics, chemistry and biology. In spite of this, one of the two parameters which characterize the interaction between the neutron and the nucleus, i.e. the spin dependent scattering length, is often unknown. A few years ago,
we started systematic measurements of spin dependent scattering lengths [ 1 ]. Many of them have already been published [2, 3] but, in the meantime, some new values
have been obtained. It is the purpose of this paper to
gather all the results in a single table. Preceding the table, the necessary definitions will be recalled and the method of measurement briefly outlined. More
details can be found in refs. [4] and [5].
2. Définitions. - The scattering of slow neutrons by fixed nuclei can be described by a bound scattering length
S and 1 are the spin operators for the neutron and the nucleus respectively. The so called spin dependent part, bN, is related to the scattering lengths b t in the
two possible spin channels J = I ± 12 by :
It has been explained in [4] how, inside a polarized target, this spin-dependent part gives rise to a pseudo-
magnetic field hT * which modifies the Larmor pre- cession frequency of the neutron by an amount [6]
where l’n = - 2 ’Tt X 2 917 (s . gauss)-1 is the. gyro-
magnetic ratio of the neutron.
The pseudomagnetic field can be written :
where the sum is over the different types of nuclei in the sample, with number densities Ni, polarizations Pi, and so called pseudomagnetic moments JlT defined
as :
Ils is the Bohr magneton,
the classical radius of the electron and
the value of the neutron magnetic moment expressed
in nuclear magnetons junm.
Inside the sample, the change in the precession frequency of the neutron due to the nuclear pola-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004007062900
630
rization P leads to an extra precession angle (for one
nuclear species).
where 1 is the sample length and v the neutron velocity.
3. Expérimental method. - The precession angle
is measured by the standard beam method of Ramsey [7]. A monochromatic neutron beam of polarization Pno passes inside a constant magnetic field Ho succes- sively through a first rf-coil, through the sample and through a second identical coil. Both coils are driven at the neutron Larmor frequency, i.e.
The rf currents in both coils are so adjusted that the rotating fields Hi they produce satisfy the relation :
where i is the time of flight of the neutron through
either coil. Upon leaving the first coil, the neutron spins are at an angle of 03C0/2 with respect to Ho. They
then precess around Ho over a distance L before entering the second coil. The total precession angle
is a = - y. Ho t where t = L/v is the transit time between the coils. During the time t, the rf field of the second coil, driven at a constant phase difference dao with respect to that of the first coil, has acquired
a phase shift CXr( = Wn t + Dao = a + dao. Its phase
difference with the precession of the neutrons is thus
constant and equal to Aao, independent of the neutron velocity. The effect of the second rf field is to rotate the component of the neutron spin polarization that is orthogonal to it by a second angle n/2, aligning it along Ho with a longitudinal polarization which,
in the absence of nuclear polarization in the sample,
is given by :
The additional phase shift Dap due to the nuclear polarization results in a changed longitudinal neutron polarization along Ho :
A measurement of the longitudinal neutron polari-
zation along Ho is made with a FeCo single crystal
in the (200) reflection condition, followed by a neutron
counter. This gives a direct measurement of 1 L1cxp 1 and, by virtue of (6), of I 1 Jl* 1, if the nuclear polarï-
zation is known.
In contrast to measurements of cross-sections, the pseudomagnetic precession is able to give the sign of
the scattering amplitude. Still, unless some care is exercised, it is possible to obtain the wrong sign,
as had happened to ourselves for a couple of nuclei.
This is why we outline in some detail the procedure
for an unambiguous determination of the sign of g*.
For a given nuclear polarization, eao is adjusted
so that the total phase shift (Aao + Aap) is close to (2 p + 1) n/2. The applied magnetic field Ho is then
increased by a small amount AHO leading to an extra phase shift :
between 0 and n/2, say n/4. The longitudinal neutron polarization is then :
Similarly, a small positive change AM in the nuclear
magnetization results in a change AH* in the pseudo- magneticfield,anextraphaseshiftL1cxH* = - Yn Jazz and a change DPH* N ( - 1 )p sin (L1CXH*) in the longi-
tudinal neutron polarization. L1P H* will have the same sign as L1PH if AH*/AM is positive, that is if Jl*/JlN (where MN = yN iiI is the true magnetic moment of
the target nuclei) is positive.
The nuclear polarization at thermal equilibrium
is given, in the high temperature approximation, by :
and (6) can be rewritten as :
with
For our experimental set up,
sl is the slope of the straight line which results if one
plots Aa against (1/T). It is directly proportional to u*.
The high temperature approximation is not valid
for two of the cases studied by us, i.e. 59Co and 165Ho,
where the hyperfine fields lead to large nuclear pola-
rizations in our temperature range.
If there are several isotopes present in the sample,
our method gives the sum of the different contribu- tions :
To achieve high enough nuclear polarizations,
very low temperatures are often required. Lattice temperatures of 30 mK are achieved with a dilution
refrigerator [8]. In metals, the thermal equilibrium
between the nuclear spins and the lattice can in general be obtained rapidly even at these low tempe-
ratures thanks to the conduction electrons. The well known Korringa relation for the spin-lattice relaxation
time Ti gives :
In most cases, the constant is known and of the order of 1 s. K. If it is unknown, the evolution of the neutron
precession angle after a sudden temperature change gives a measurement of Tl (e.g. Bi) or, if too short,
an upper limit equal to the thermal constants of the refrigerator (e.g. As).
In insulators, the spin-lattice relaxation times Tl
become hopelessly long at low temperatures (apart
from a few exceptions). Only a dynamic method can give high enough nuclear polarizations. Such a method, the solid effect [9], has been used for five nuclei 6Li and ’Li in LiF [10], 19F in CaF2, 2’Al in A1203 and 13C in BaC03.
4. Explanation of table I. - The first column gives
the isotope used. ,
The negative sign in front of the spin in the second
column indicates a nuclear magnetic moment of sign opposite to the spin (yN 0). This is important since sign (p*) = sign (yN). sign (s). The third and the fourth
columns give the compound used and the number of nuclei of interest per cm3 in the sample. e means ele-
ment. Natural isotopic abundances have been used except where noted enr(enriched). Where possible,
the room temperature density (and sample length)
has been corrected for thermal contraction. This is indicated by an asterisk. The fifth column gives sl, the product of the slope per unit length, and 1, the sample length used and it is the parameter actually
measured. It gives an indication of the sensitivity of
the method. The accuracy of a typical measurement is limited to A(sl) - 0.2 K.deg., resulting mainly
from magnetic field drifts ; d.p. means dynamic polarization.
The resulting pseudomagnetic moment and its corresponding spin dependent scattering length b+ - b_ are given in the sixth and seventh columns.
The eighth column gives the concentration of
hydrogen impurities, when an analysis was both
necessary and possible. The number in parenthesis
are the limits of solubilities for H [11]. Since the
accuracy of our measurements is limited to
à(si) - 0.2 K.deg., the minimum hydrogen content capable of affecting our results is NAminH - 1019 cm - 3 (for 0( 1 /T ) N 20 K -1 and 1 = 2.5 cm).
Some cases need a particular explanation, referred
to by a number in the ninth column. These explana-
tions are given below :
1 :13C. - A compressed powder sample of
Ba 0 was used. 13C nuclear polarizations of
+ 10 % and - 15 % could be obtained by the solid
effect [9] through free radicals created by y-irradiation
of the sample at liquid nitrogen temperature. A phase
shift of the 135Ba and 13’Ba nuclei was undetectable.
The value obtained is in good agreement with a recent shell model calculation [26] which gave
Table 1.
632
It is, however, in disagreement with a previous unpublished experiment on a powder sample of U13C, in which we found b+ -b_ 0.02 x 10-12 cm.
A sintered sample was used, immersed in the dilute
phase of 3He in 4He. This results in a severe loss in neutron intensity due to absorption by 3He. The cooling, however, is very efficient, down to 40 mK.
This has been verified on a powder sample of VC,
where the 51V polarization served as a pseudo- magnetic thermometer. The spin-lattice relaxation
trime of 13C should be short. Ti T = 25 s . K has been found down to 4.2 K and this should be independent
of temperature [12]. It is therefore unlikely that the 13C nuclei were not polarized. But the negative phase
shift (u* 0) of 13C could be counterbalanced by hydrogen impurities (p* > 0). This would need a
concentration of 0.5 atomic per cent 1 H. An analysis
to confirm this has not been carried out.
2 : 19p. - A single crystal, doped with Tm2+ was
used. Both positive and negative polarizations of 10 %
were obtained. Since Jl*(19F) is only ten times larger
than the true nuclear magnetic moment, a small correction has been applied due to the nuclear magne- tism proper. In contrast to pseudomagnetism, this
correction depends on the sample shape, due to the long range of the magnetic interactions.
3 : 39K. - The measurement was performed on a
metallic sample of natural isotopic abundance. One
obtains, strictly speaking, a sum of the contributions of 39K and 41K, according to equation (9) :
Since a large majority of nuclei have 11* 1 PB, it is
quite safe to assume that the 41K contribution is small.
A separate measurement of either isotope would be
needed to confirm this. Such a measurement is planned
in the near future.
4 : 45Sc. - In the course of previous work [2],
the values u* = 1.95 Jls and b+ - b_ = 1.20 ± 0.03
were given for scandium. Due to a misinterpretation
of our experimental results, the plus sign is incorrect.
In the experimental set up used at that time, the
choice of the wrong sign led to some error in the
magnitude.
We are much indebted to R. E. Chrien who first
pointed out the sign error. His accurate measurements of the Sc cross-section in the 0.4-20 keV range were
definitely inconsistent with our proposed plus sign for b+ - b_ [13].
This was confirmed by a remeasurement of the nuclear spin contribution to the Bragg reflections of Sc at low temperatures by W. C. Koehler and R. M.
Moon, to whom we are also indebted for communi- cation of their results prior to publication [14]. They
found :
A correct reinterpretation of our previous data
showed that not only was the sign incorrect but also the magnitude. The reason was that in order to fill
as completely as possible the space in the dilution chamber of our 3He-4He cryostat on the neutron path,
the Sc sample was sandwiched between two pieces
of aluminium. The (known) AI contribution had to be subtracted from the observed neutron spin pre- cession if Sc and AI had pseudomagnetic moments
of the same sign, but had to be added if 1À* was negative.
New experiments were then recently carried out
with the sample previously used, either alone, or associated with a vanadium target of known thickness.
In the latter case, the experiment offered an unambi-
guous test of the sign of Ilt:, Il; being positive, and gave very simply the ratio usc* /uv*, whence 1À* , as Il; is
known. Both experiments confirmed the negative sign
of 1À* and the whole set of experiments leads to the
values :
This result is 16 % higher in magnitude than Koehler
and Moon’s value. As these authors pointed out,
contamination of the sample by hydrogen or para-
magnetic impurities may lead to erroneous results.
1) A spectrographic analysis of our sample (of
99.9 purity) was then carried out and revealed a hydrogen content of 0.3 % atom. This hydrogen
content would lead to a small correction
if its relaxation time Ti was short enough (see also
remark 9). Ti being unknown, we have included this correction in the error bars.
2) As concerns the magnetic impurities, magne- tization measurements were kindly performed at the
Service National des Champs Intenses in Grenoble, in fields up to 28 k0e and temperatures between 37 mK and 1.05 K. They showed that, at our field value
Ho = 25 kOe and in the temperature range of our
measurements, the magnetization of the sample
varies by 0.017 uem/g, which would lead to a correction of the b + - b _ value of the order of 0.006 x 10- 12 cm, which is negligible.
3) Another source of error may come from the nature of the Sc sample and the relevant density.
It is well known that polycrystalline samples of Sc
are generally composed of two phases, one hexagonal
for which, from X-rays, d = 2.985 ± 0.001 g/cm3 [15],
the other FCC for which d = 3.20 ± 0.01 g/cm3 [16].
A neutron diffraction examination of our sample
showed that it was composed of 93 % hexagonal and
7 % FCC phases. Reflections appeared at the Bragg angles expected from X-ray data. Neither additional
reflections nor preferred orientations were found.
One would then expect a density of 3.00 g/cm3. It
is this density which has been used in our measurement.
5 : 51y. - The large and well known pseudo- magnetic moment of 51 Y was used to calibrate ln situ the carbon resistor (Speer 220 Q) which served as the thermometer in the dilution chamber. This calibration has to be repeated from time to time. In fact, a tempe- rature-resistance hysteresis has been observed to
develop with ageing on our first resistor.
6 : 5geO. - The value given for 59Co is a mean
value obtained from measurements on single crystals
of hexagonal and cubic cobalt. There are two sources
of error : the uncertainty of our precession measure-
ments and the knowledge of the polarizing field seen by the nucleus. A detailed discussion is given in ref. [3].
7 : 65eU. - The value for 65Cu has been obtained
by subtracting the contribution of 63Cu from the
slope measured in samples with natural isotopic abundance, according to eq. (9).
8 : 75 As. - The sample consisted of two blocks of massive metallic arsenic joined together by machin-
ed plane faces to give a total length of 2.3 cm. It hadla
stated purity of 99.999 9 % with no spectrographically
detectable hydrogen content.
9 : 89y, 91Zr, 139La.-Assumingthehydrogenhas
a short spin-lattice relaxation time, as e.g. in PdH [17],
its contribution can be subtracted by use of eq. (9)
which reads in this case :
where C = NH/N is the atomic fraction of hydrogen.
The assumption of fast relaxation can be checked by using two or more samples of different hydrogen
contents. This procedure worked quite well for ’9Y and 91Zr. Note that there is an error in sign in ref. [3]
for Zr. As for 139La, the value of 11* given in ref. [2]
is erroneous. An analysis of the sample, after publi- cation, revealed a hydrogen content of 250 ppm by weight. New experiments were undertaken with
samples degassed under vacuum, for which the
hydrogen content was only 50 and 18 ppm.
The analysis of the results showed that, contrarily
to what was observed with Y and Zr, one cannot
assume that the nuclear polarization of hydrogen was
that expected for free protons with fast relaxation times. The value given in the table takes into account this uncertainty in the hydrogen impurity correction.
10 : 107 Ag, 109 Ag. - It was possible to perform
three independent experiments on two isotopically
enriched samples and one natural sample. The 107
enriched sample contained 98.65 % of 107 Ag, the 109
enriched sample 99.26 % of 109 Ag (from mass spec- trometer analysis). The measurement on the natural
sample offered a supplementary linear relation between
the pseudomagnetic moments of the two isotopes.
Within the limits of accuracy, one finds a pretty good compatibility between the three measurements.
11 : 165Ho. - At present, a lot of measurements of the spin dependent scattering length of Ho have been performed :
Two determinations were done, in the beginning,
almost at the same time :
1) A neutron diffraction experiment on a Ho powder with unpolarized neutrons [18] gave
From new experiments and reinterpretation of the previous results, a revised value was published [19].
2) A neutron diffraction experiment with a single crystal of Ho and unpolarized neutrons [20] yielded.
3) Later on, in the course of experiments on
holmium ethylsulphate by neutron precession [19]
a much higher value was found :
4) More recently, from a polarized neutron study
on a single crystal of HoAl2 [21 J, a value :
was given.
5) An examination by the neutron precession
method of one of the HoAl2 samples used by
Boucherle [21], who kindly lent it to us, gave :
It is this value which is given in table 1.
12 : 209Bi. - As mentioned above, the time depen-
dence of the neutron precession enables us to measure long relaxation times. In the case of bismuth, Tl has been measured at five temperatures between 0.7 K and 90 mK. As expected, Tl is found to follow a Korringa law with :
The sample used was a rod of 99.999 purity.
5. Conclusion. - The method of neutron pre- cession has been developed into a convenient tool for measuring spin dependent scattering lengths of
slow neutrons on nuclei. This method is applicable
in a straightforward manner for nuclei which fulfil the following conditions :
- Only one isotope with non-zero spin in the sample.
