P. JAUHO and P. PIRILA: Coherent and Incoherent Cross-Sections 757

phys. stat. sol. 42, 757 (1970)

Subject classification: 6; 22

Department of Technical Physics, Technical University of Helsinki, Otanierni (Helsinki)

The Separation of Coherent and Incoherent Cross-Sections in Inelastic Neutron Scattering Using Polarized Nuclear Targets

BY P. JAUHO and P. PIRILA~)

The coherent and incoherent neutron scattering cross-sections have different dependences on the polarization of neutrons and target. This fact can be used to separate the coherent and incoherent part of the inelastic cross-section and to enhance the otherwise very weak coherent scattering in the cases of H1 and V51. Some details of this method are studied and the results are applied to the case of ZrH,. It seems evident that the dispersion curves of phonons in ZrH, can be measured by this method.

Die koharenten und inkoharenten Neutronen-Streuquerschnitte besitzen unterschied- liche Abhangigkeiten von der Polarisation der Neutronen und des Target. Das kann benutzt werden, urn den koharenten und inkoharenten Anteil des inelastischen Wirkungsquer- schnitts zu trennen und die sonst sehr schwache kohkente Streuung fur H1 und V51 anzu- heben. Einige Details dieser Methode werden untersucht und die Ergebnisse werden auf ZrH, angewendet. Es scheint, daB die Dispersionskurven der Phononen in ZrH, mit dieser Methode gemessen werden konnen.

1. Introduction Previous work done with polarized neutrons and targets has usually dealt

with the dependence of the absorption cross-section on the polarization of neutrons and nuclei; the main field of study has been the determination of the spins of resonances. The polarized nuclei have usually been in a ferromagnetic material, which makes the polarization relatively easy, but is not very suitable for scattering measurements (see Section 2).

Recent developments in cryogenics (especially the construction of dilution cryostats) have now made it possible t o produce polarized nuclear targets by the “brute-force” method and to preserve the polarization for many hours. This affords several new possibilities for slow neutron scattering measurements. It is now possible to determine the spin dependence of scattering from many nuclei. The most easily studied are those nuclei in metals which have a relatively high magnetic moment. Measurements with polarized nuclei also make the separation of coherent and incoherent scattering possible, as the two depend on polarization in different ways. The scattering from some nuclei (especially H1 and VS1) is almost purely incoherent, so that it is not possible to observe the peaks due to the coherent scattering in inelastic cross-section. This situation can be removed using polarized nuclei, and the dispersion curves can thus be measured.2)

l) Present address : Research Institute for Theoretical Physics, University of Helsinki. ,) Another method that can be used for similar measurements has been recently presented

by Moon, Riste, and Koehler [l]. 49 physica 42/4

758 P. JAUHO and P. PIRILA

In this paper we study some aspects of the use of polarized targets in neutron scattering measurements. Section 2 treats the methods used in polarization of nuclei and neutrons. In Section 3 we present the formulas for coherent and incoherent inelastic scattering and discuss the methods for the separation of the coherent and incoherent components. Finally, as an example, we calculate the dependence of the scattered intensity on the degrees of polarizations for ZrH,.

2. Polarization of Nuclear Targets and Neutrons Nuclear polarization methods are usually divided into two groups ; dynamic

and equilibrium. The highest polarizations have been achieved with dynamic methods, but in this way only protons in certain specific substances (e.g., La,Mg,(NO,),, - 24 H,O) or very small metal pieces can be polarized. This severely limits their use in producing neutron targets.

Equilibrium polarization is possible a t 0.1 to 2 OK if the target consists of a ferro- or paramagnetic substance in which the internal magnetic field is a t least of order 1 MG. Thus i t is possible to polarize almost any nucleus as a dilute solution in a ferromagnetic substance, but such solutions are not practical as neutron targets. Further the scattering from paramagnetic ions complicates the analysis of results.

The “brute-force” method requires a very low temperature (about 0.01 OK or less) and a strong magnetic field (about 50 kG) if one wants to obtain polari- zations of over 10%.

The equilibrium value of polarization fl = (I ,) / I is I -t coth (T 2 I + 1 x) - 1 coth-z 1 ,

2 fi =

where

,u is the magnetic moment of nucleus, I its spin, and T the temperature. When I = 1/2, formula (1) is equivalent to

(3) 1 2 fi = tanh-x .

In Fig. 1 we present the polarization of protons as a function of BIT.

However, it is not only sufficient that the equilibrium value of polarization is large enough, for in many cases the spin lattice relaxation times of nuclei are very long (e.g., several years). This is the case in insulators under conditions, where f, calculated from (1) is not very small, as there are almost no lattice vibrations a t these temperatures and the possible paramagnetic ions are com-

Fig. 1. The polarization of protons in a magnetic field as a B / l ( W b / d deg) - function of BIT

The Separation of Coherent and Incoherent Cross-Sections 759

pletely polarized. Due to Fermi-Dirac statistics of conduction electrons, the spin lattice relaxation time in metallic conductors is not so strongly dependent on temperature and is usually shorter than one hour even a t the lowest tempera- tures. The dependence on magnetic field is contrary to the case of insulators with paramagnetic ions either weak, or such that the relaxation time becomes shorter with increasing field [2].

Besides metals many metal hydrides (e.g., ZrH,, Pd,H) are also metallic conductors, so that it should be possible to polarize their protons by the brute- force method.

The conditions needed for brute-force polarization are now obtainable with a dilution cryostat and a superconducting magnet. The warming due to ab- sorbed neutrons is not negligible, but with a suitable design of cryostat its effect on the obtainable temperature is relatively small for most target materials.

Highly polarized (polarization degree about 97 yo) monochromatic neutron beams can be produced through scattering from a magnetized Feo.gzCoo.o~ single crystal. Calculations show that the neutrons can be conducted into target posi- tion without destroying the polarization. The direction of the polarization can be reversed with a spin flipper between the polarizer and the target.

The equipment can be constructed so that transmission experiments, dif- raction experiments, and inelastic scattering studies are possible.

3. Scattering of Polarized Neutrons from Polarized Nuclei

The scattering of polarized neutrons from polarized nuclei has been considered by Schermer and Blume [3]. They have assumed that the scattering cross-section can be derived from a spin-dependent scattering length

6 , = bfii + bsi s - I , ,

where i refers to the i-th nucleus, s is the spin vector of neutron, and Ii that of the nucleus. Because there can be more than one isotope from each element, we must take isotope averages which we denote by (). Because the Larmor precession of neutrons in the strong magnetic field needed to polarize nuclei is very fast, it is possible to use only parallel or antiparallel polarizations for neu- trons with respect t o nuclear polarization. We denote the polarization degree of neutrons by f , and that of the i-th nucleus by f x i . The inelasticity effects in scattering due to the motions of atoms are not connected with the nuclear spins, so that the results of Schermer and Blume can immediately be generalized to inelastic scattering. As in the non-polarized case the cross-section can be divided into a coherent and incoherent part as follows :

(4)

49.

760 P. JAUHO and P. PIRILA

where It,) and It) are the initial and final spatial states of the target, (. . .)., means an average over initial states it,) weighted with their probability. E, is the energy of the target in the state It). k, and k are the initial and final wave vectors of the neutron, and E, and E the corresponding energies; x = = k - k,. Ri is the position of the i-th nucleus.

The separation of the coherent and incoherent scattering can be most easily performed using two different values of neutron polarization and subtracting the observed, properly weighted cross-sections from each other. This way either incoherent or coherent scattering can be almost totally eliminated. The polari- zation of the target is more difficult to change so that it is advantageous to keep it constant. Suitable values of neutron polarization are 0 and fl (more exactly about & 0.95). When one separates incoherent and coherent scattering by this method, the most important parameter j s the ratio of coherent and incohe- rent scattering. The larger the change of this ratio is, the easier the separation can be performed.

In crystals containing hydrogen the scattering from protons usually domi- nates so that we are interested of the ratio B E H I B H . (BH H and B, are calculated

BH +BHH

-7 t

8 I

a5 7 4 -

Fig. 2. Cross-section factors for hydrogen, Bsz and nH + BnH, as a function of proton polarization

for neutron polarizations in = 0, 0.5, and 1

Fig. 3. .The differences e and e’ (formulas (9) and (10)) as a function of the proton polarization

The Separation of Coherent and Incoherent Cross-Sections 7 61

from formulas (7) and (8) respectively (Fig. 2).) In Fig. 3 we represent t,he differences

and

as function of the polarization of protons fa. It is interesting to note that for f H = t 0 . 2 the coherent scattering totally disappears.

Because proton polarizations f E 2 0.8 are not available with present methods, we can conclude that it is more efficient to use the reversing of neutron polari- zation than polarizations 0 and 1 (0 and -1 are still less favourable). The reversing of neutron polarization is easy to perform, and it is also possible to alternate the polarization with a short period by using a flipper chopper so that fluctuations of neutron flux can be eliminated [a].

4. Scattering from Polarized Zirconium Hydride As an example we examine scattering from polarized ZrH, single crystals.

Zirconium hydride is a suitable crystal for several reasons. It is a good conductor so that the relaxation time of protons should be short even a t the lowest tem- peratures. The total cross-section of Zr is much smaller than that of hydrogen so that the effects due to the polarization of protons are clearly seen. Because of the use of zirconium hydride in nuclear reactors its properties have been widely studied. All inelastic neutron scattering measurements have, however, dealt with the incoherent cross-section only, because the coherent cross-section is much smaller due to spin incoherence. However, zirconium hydride has some disadvantages, the most severy of which is that no ZrH, single crystals have, as far as we know, ever been produced. A polycrystalline sample with high degree of orientation should be sufficient for experiments outlined in this paper.

Slaggie [5] has presented a central-force lattice dynamical model which allows the calculation of all lattice dynamical parameters. This model has been used in the following calculations.

At the very low temperature necessary for polarization there are almost no phonons present so that only down-scattering is possible. The phonon spectrum of ZrH, consists of separate acoustic phonons with energies less than 35 meV and of optical phonons with energies between 110 and 160 meV. Because of the large mass difference the inelastic scattering from hydrogen proceeds mainly with the formation of optical phonons, whereas scattering from zirconium is usually connected with acoustical phonons. Due to this effect and the values of scattering cross-sections of Zr and H nuclei, the phonon expansion converges so rapidly that two-phonon scattering can be ignored exept in the case when both phonons are optical ones. In this case, however, the energy loss must be over 220 meV so that two-phonon scattering does not disturb measurements of one-phonon scattering. In the following we restrict ourselves to the calculation of one-phonon scattering.

The one-phonon coherent and incoherent cross-sections can easily be calculat- ed from the dynamical model of Slaggie. In this model the face-centered tetra- gonal unit cell of ZrH, (lattice constants a = b = 4.98 A, c = 4.45 .$) has been

762 P. JAUHO and P. PIRILA

approximated by a cubic lattice. For the lattice constant we use the value 4.80 A. Slaggie's model uses four force constants which correspond to the nearest-neighbour H-H, H-Zr, and Zr-Zr forces and to the second nearest- neighbour H-H force. From this model Slaggie has calculated phonon polari- zation vectors3) ek, (q) , eill(q), and eh2(q) for 3504 different values of q in 1/48 of the first Brillouin zone. Index s refers to be s-th phonon mode; s = 1, 2, and 3 are acoustical modes, and s = 4, . . ., 9 optical modes. Slaggie has further calculated the weighted phonon spectra ezr(&) and pa(&), which give the number of phonon modes per energy interval weighted by the squares of the corresponding polarization vectors for zirconium and hydrogen.

In a cubic lattice the Debye-Waller factors and incoherent one-phonon scat- tering cross-sections can be calculated from the formulas4)

00

0 Here

and

where r is the number of atoms in a primitive cell.

polarization vectors through formula The coherent one-phonon scattering cross-section can be calculated from the

where ei and e, are the positions of atoms in the unit cell. Integrating over energy we get

where q is equal to - x reduced to the first Brillouin zone, i.e.

q = 2 n t - x . (16) - -

3, The primitive cell of ZrH, contains one zirconium and two hydrogen atoms. Polari- zation vectors of hydrogen atoms are connected by e k l ( q ) = ek2(q)* . Slaggie has not calculated the complex vectors e:(q), but real nine component vectors connected to eT(q) by a linear transformation.

4, For the derivation of formulas (12) to (16) see, e.g., the work of Gurevich and Tarasov on low-energy neutron physics [61.

The Separation of Coherent and Incoherent Cross-Sections 763

The cross-section is non-zero only if the equation

is fulfilled. From equation (16) and general properties of o,(q) and e;(q) it follows that

%(!I) = ws(x) 3 (18)

e:(q) = e;(x)* . (19) In the case of ZrH, we use for the Bii's and Bt's the following numerical

values : B ~ H = (0.135 + 2.110 f& - 1.066 f, fa) b , BHZr = (-0.257 $- 1.017 f , f H ) b , Bzrzr= 0.49 b ,

B E = 2.110 (3 - f& - 2 f n fa) b , Bzr = (0.02 0.04) b .

These values have been calculated from the measured coherent and incoherent cross-sections neglecting the spin incoherence due to the Zrsl isotope. The small value of BZr shows that this spin incoherence does not appreciably effect the results. The relat(ive1y large error in the value of BZr does not disturb the cal- culation as BZr is anyway so small that the incoherent scattering from Zr can be totally ignored.

For ZrH, formulas (14) and (16) can be written in the forms

where a 4

a = - (x , + xy + x,) .

For the Debye-Waller factors we get the following numerical values: WH = 7.83 x lo-' x2 [A'] , wz, = 6.15 x 10-4 d [A'] .

The ratio of coherent and incoherent cross-sections for different polarizations can be most easily studied by defining

764 P. JATJHO and P. PIRILA

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766 P. JAUHO and P. PIRILA: Coherent and Incoherent Cross-Sections

These definitions are useful, because it follows from (23) that k

Rinc ( E ) = -f H ) (8 kLl

is independent of the degrees of polarization of protons and neutrons. In Fig. 4 we present Rcoll and Rinc(&) for some values of Eo and x . The polarization of neutrons is in all cases 0.95, and for the polarization of protons the values 0, *0.25, and h0 .5 are chosen. Because the dimension of Rcoh is eV-l, whereas that of Rinc(&) is e V 2 , Rcoh is presented by 2 meV broad bars. In real measure- ments the form of these bars would be determined by the resolution function of the spectrometer, their areas would, however, remain unchanged. The para- meters have been chosen so that in every figure there is one phonon, optical or acoustical, which is enhanced. The advantages of the use of polarized targets are clearly demonstrated by the figures which indicate an easy separation of the changing intensity of coherent phonon peaks from the incoherent background.

Although the resolution of real spectrometers would probably not be so good as 2 meV (in the acoustical region this resolution may be available), it seems evident that the separation of coherent and incoherent scattering could be easily done by comparing measurements with proton polarization 0.50 and neu- tron polarizations k0.95.

5. Conclusions The calculated coherent inelastic cross-sections for the scattering of polarized

neutrons from a polarized zirconium hydride target show that the separation of these cross-sections from experimental results should be possible providing that a value of a t least 0.3 can be achieved for the polarization of protons. A similar separation is possible also for some other metal hydrides and vanadium, which is a still more purely incoherent scatterer than hydrogen. In many cases, on the other hand, it is difficult to determine the incoherent cross-section due to strong coherent peaks and insufficient resolution of the spectrometer. In these cases, the polarization method can be used for elimination of the coherent scattering.

Acknou7ledgements

We would like to thank Dr. E. L. Slaggie from Gulf General Atomic, San Diego (California), for the calculation of phonon polarization vectors and spectra, which we have used in our calculations.

References [l] R. M. MOON, R. RISTE, and W. C. KOEHLER, Phys. Rev. 181, 920 (1969). [2] P. JAUHO and P. PIRILA, Phys. Rev. B1, 21 (1970). [3] R. I. SCHERMER and M. BLUME, Phys. Rev. 166, 554 (1968). [4] A. VIRJO and 0. STEINSVOLL, Kjeller Rep. (Norway) KR-125 (1968). [5 ] E. L. SLACGIE, J. Phys. Chem. Solids 29, 923 (1968); General Atomic Rep. GA-8132

[6] I. I. GUREVICH and L. V. TARASOV, Low-Energy Neutron Physics, North-Holland Publ. (1967).

Co., Amsterdam 1968.

(Received August 26, 1970)