Magnon scattering of slow neutrons on a pyrrhotite single crystal

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Magnon scattering of slow neutrons on a pyrrhotite single crystal

Adam Wanic

To cite this version:

Adam Wanic. Magnon scattering of slow neutrons on a pyrrhotite single crystal. Journal de Physique, 1964, 25 (5), pp.627-634. �10.1051/jphys:01964002505062700�. �jpa-00205842�

MAGNON SCATTERING OF SLOW NEUTRONS ON A PYRRHOTITE SINGLE CRYSTAL

By ^ADAM WANIC,

Institute of Nuclear Physics, Cracow, ^Poland.

Institute of Nuclear Sciences " Boris Kidric ", Vinca, Yugoslavia (1).

Résumé. ²⁰¹⁴ La diffusion magnétique inélastique de neutrons de longueur ^{d’onde 03BB =} 1,376 Å

a été étudiée dans un monocristal naturel de pyrrhotine. ^{La surface} de diffusion liée au point réciproque ^{03C4 =} (001) ^{a été examinée à l’aide du} spectromètre à neutrons de Cracovie installé dans la Pile TVRS de Vinca.

L’analyse ^de l’énergie ^{dans le} faisceau diffusé et la méthode de diffraction ont été utilisées

conjointement.

Les résultats sont interprétés dans le cadre de la théorie des ondes de spins. On obtient une rela- tion de dispersion ^{d’ondes de} spins pour la branche acoustique ^{et on} prouve ^son anisotropie.

L’existence de la branche optique ^{a été} également ^{mise en évidence. Les résultats sont} comparés

avec le calcul théorique ^{utilisant un modèle} simplifié.

Abstract. ²⁰¹⁴ The magnetic ^inelastic scattering ^{of a} monoenergetic ^{beam of} neutrons, 03BB0 ^{= 1.376} Å, ^{on a natural} single crystal ^of pyrrhotite, Fe1201403B4 S, ^was investigated. ^The scattering

surface connected with 03C4 ⁼ (001) ^{was examined} by ^{means of the Cracow neutron} spectrometer

installed at the TVRS reactor in Vinca. The energy analysis together with the diffraction method were applied ^{and are} described. The results are interpreted ^{in the frame of} spin ^wave theory. ^A spin ^wave (magnon) dispersion relation for the acoustical branch was obtained and its

anisotropy discovered. Evidence for the optical ^{branch was} also found. A model is proposed

which agrees with the observed phenomena ^when relevant calculations are performed.

PHYSIQUR 25, 1964,

Introduction. ^- Among the very few possibi-

lities of obtaining information about the behaviour of the single group of excitations, ^{in a} system ^of spins strongly coupled by exchange forces, ^neutron

methods occupy the dominant place. ^In principle they ^{are able to} give ^{the full and} complete spec- trum of excitations i. e. the dispersion ^{curves for}

all branches of spin ^{waves. At} present ^{no other}

method can accomplish this task. The spin ^wave

resonance method [30] ^cannot compete ^{in this} respect ^{since at} present ^it is limited to acoustical branch and small q-values. However, ^neutron

methods have one serious drawback : they ^are

very expensive ^{and not} easily applicable. ^{As a}

rule they ^demand samples ^{in the form of} large single crystals ^of good quality ^{(small mosaic} spread) ^{and neutron beams of} hight intensity ^{i. e.}

high ^{flux reactors.}

For this reason the list of experiments ^{done so}

far is short and comprises only ^{a few} magnetics : Fe,0, [6,25], Fe,0, [26], ^Fe [19], Co0.92Fe0.08 [29], [23]. ^{In the author’s} opi-

nion progress will depend ^{on the} supply ^of parti-

cular samples. However, ^{before a} variety ^of large

artificial crystals ^becomes accessible, experimentors

have to focus their interest on natural ones.

This work tries to prove that the possibilities

(1) ^{This work was} performed ^{at the} Institute of Nuclear Sciences " Boris Kidric ^" at Vinca (Yugoslavia) and spon- sored by ^the Polish Government Commission for the Use of Nuclear Energy ^{and the} Federal Nuclear Energy ^Com-

mission of Yugoslavia,

are still not exhausted in the problem ^{itself nor in}

the methodology. ^{It was carried out in Vinca} using ^{neutrons f rom the} Yugoslavian heavy ^water

moderated TVRS reactor and the Polish " Cracow Neutron Spectrometer ". This project ^was spon- sored by both Polish and Yugoslavian ^Nuclear Energy Commissions according ^{to the mutual} agreement ^on cooperation ^{in science.}

Some general information. ^- The principles ^of

the phenomenon ^{of the} magnetic ^{inelastic scat-} tering ^of neutrons, strictly valid much below the critical point ^of spin alignment, ^{are best of all}

elucidated in the works of Elliott and Lowde

[10, 18]. ^Earlier treatments of the subject by Avakyants [2] ^{and Moorhouse} [22] ^{are restricted}

to the limit _{of very} long ^{neutron wave} lengths.

Further theoretical works in the field were carried out by ^Maleev [20, 21], ^Sa6nz [27, 28] ^and Izyumov [13]. Maleev and Sa6nz developed ^{the formulas}

which take explicitly ^{into account} polarization changes ^{of the neutron beam} inelastically ^scattered

and comprising ^a larger ^{class of} magnetic ^struc-

tures [Sa6nz].

Izyumov introduces a more general approach, using ^the temperature dependent ^{Green functions} method, ^which extend the theory ^to higher tempe-

ratures. The result is, that in the whole tempe-

rature range, where the spin alignment exists, ^the magnetic ^inelastic scattering ^{of neutrons consists}

in the creation and [or] annihilation of spin ^waves

i. e. has the character of magnon scattering. ^Thus

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002505062700

628

the picture given by ^{Elliot and Lowde not} only

retains its validity ^{but its use} (with slight ^modifi- cation) ^{in the} region ^of higher temperatures ^is

allowed. This was assumed earlier on the basis of Dyson’s [9] considerations on spin ^{wave interac-}

tions and experimental ^{facts obtained} by ^{the neu-}

tron method. Finally ^{the works of Kociriski} [14]

and Low [17], who considered some special ^cases

of magnon scattering, ^should be mentioned.

The common conclusion from all the theoretical considerations is’that the double differential cross

section of the neutron scattering ^is directly ^{related to the} crystal ^structure (geometry)

of the spin system ^{and to the} dispersion ^{relations of}

magnons ^at the temperature ^of investigation. If, leaving ^aside the variance of structure, ^{atomic and} other factors, which influence the intensity ^of

magnon scattering, ^we describe the phenomenon

in the reciprocal space (neutron ^momentum space)

the end points ^{of the wave} vectors of scattered neutrons do not fill out the space but ^form certain surfaces (see fig. 3) ^the shapes and dimensions of which follow from the conservation laws of energy :

and momentum :

. where : e: === :i 1,

kol ^{k : neutron wave vectors} correspondingly ^before

and after scattering.

AE : neutron energy gain.

m : neutron mass.

magnon energy.

to(q) dispersion relation for the given magnon branch.

q ^{: wave vector} of the created or annihilated magnon.

T : reciprocal ^{lattice vector of} magnetic lattice,

I’t’hkd ⁼ 1/dhkl ^{where d =} interplanar ^spa- cing ^with Miller indices (h ^k l).

The experiment ^{is aimed} mainly ⁱⁿ the locali- zation of the so-called scattering ^{surfaces or}

their fragments using ^{different neutron} techniques.

When the critical point ^{is not too} low and the energy ^{of neutrons} used is not too high, ^{the scat-} tering surfaces connected with, ^at least, ^the

accoustic branch are closed, surrounding ^{the end} point ^{of the T-vector. In such a case it is} possible

to gain ^some information from the angular ^distri-

butions of the scattered neutrons without _any energy analysis. ^The feasibility ^{of such measu-}

rements was predicted by Elliott and Lowde [10],

and the first experiments ^{were carried out} by

Lowde [18] (on iron) ^{and Riste} [25] (on magnetite) by ^means of the white neutron beam technique.

The situation is improved when instead of the

white impinging ^{beam one works with the mono-}

chromatic one. During ^{the work on} pyrrhotite

the diffraction technique ^{with a} monochromatic beam was applied together ^{with the} energy analysis

method. It appeared ^{that for} the acoustic branch and small q-values the resolution of the diffraction

technique ^can surpass that of energy analysis.

The sample. ^{- The} monocrystallic sample ^of pyr- rhotite weighing ^{about 50} g ^{was cut out} from a

large lump ^{of mineral} originating ^{from the Morro}

Welho deposit. ^{It formed} approximately ^a cylin-

der 6 cm high ^{and L . 5 cm in} diameter ^{with the} geometrical ^axis being nearly ^{normal to the} [00.1]

direction. The formula ascribed to natural pyrrho-

tites is Fei-aS with 8 enclosed within the limits

(0 .13 ^- 0.17). ^{The formula in} the ionic notation appears ^as follows : Fe’+ ^where 0 represents ^a vacancy ^{or a} hole in the iron sublattice.

The question whether the holes are ordered or not

depends ^{on the} magnitude ^of &. For 8 0.09 the substance is antif erromagnetic ^and above 0 . 09

ferrimagnetic [11, 24, 31]. However its critical

point ^{does not seem to} depend ^{much on S and}

amounts to about 320 OC.

Analysis (2) gave 8 ⁼ 0.12, ^{and the measu-}

rement of saturation magnetization ^{at room} tempe-

rature revealed the non-vanishing ^{value of 13}

gauss CM3g-1. ^These results show that the struc- ture of the sample ^{should not diff er} markedly

from the generally accepted ^one corresponding ^to

the formula Fe7S8 [3, 11, 12] (see fig. 1).

FIG. 1. ^- a) The structure ascribed to pyrrhotite, assuming formula Fe,S, [3], [11], [12], [~3].

b~ Simplified ^{model used for} calculation of the dis-

persion ^{relation. The} sulphur ^{atoms are not shown.}

(2) The author is indebted to Dr. Wolski, ^{head of the} Magnetochemistry Laboratory of the Po~nan University,

for the analysis.

Measurements. ^- The measurements were per- formed at room temperature ^{in two} ways, namely by :

a) The diffraction technique ^{and b) the} spectro- metric technique.

In the two cases the same wave length ^{of incident}

neutrons ?, ⁼ 1.376 A was used. The arran-

gement ^{of the} apparatus ^{at the} place ^{of measu-}

rements is presented ⁱⁿ figure ^{2. For the instru-}

FIG. 2. ^- Simplified ^{draft of the} experimental arrange- ment. The situation of the spectrometer corresponds

to the energy analysing technique. ^When using ^the

diffraction technique ^the spectrometer ^{was shifted} linearly

to the position in which the axis of T-I was right ^{in the}

beam.

1) ^Borated paraffm ^{bricks ;} 2) ^The sample crystal ; 3) ^{Cadmium boxes filled with} B4C ; 4) Analysing ^Zn crystal ; 5) ^The spectrum ^of acoustical magnon neutron

scattering ^{as seen} by ideal resolution : 1. Elastic inco- herent peak ; ^{2. Inelastic} from the front of scattering surface ; ^{3. Inelastic} froin the back of the scattering

surface.

mental details and setting up ^see [16, 8]. ^{An Al} single crystal ^{was chosen as} monochromator since its mosaic spread ^{matched best the} onminal diver- gence [13] ^{of the} Soller-type ^{beam hole collimator}

with an aperture ^{of 40 X 50 mm. The} crystal

had Fankuchen cut and reflection from (111) planes ^was used. The flux of the monochromatic beam was about 1 X 108 neutrons per minute.

The divergence ^{of the beam can be estimated from}

figure ^4. The term " primary beam " will be used frequently ⁱⁿ place ^{of "} monochromatic beam ". Although ^{one looses in} intensity ^{as com-} pared ^{with the} original ^{white beam} technique [18, 25], however, ^{the new} technique ^is superior

from the methodological point of view and provides

certain valuable possibilities (investigation ^{of ani-} sotropy).

a) ^The pyrrhotite sample ^{was attached to a} goniometer ^{head and} placed ^{on the axis of table I.}

The connection between the sample ^and gonio-

meter formed a 1.5 mm thick and 20 mm long

aluminium rod. Thegoniometer ^head was screen-

ed from the primary ^beam by ^a cadmium sheet.

The proper orientation ^{of the} crystal ^{within the}

beam was assured by optimization ^{of the} (0 0.1) Bragg ^reflexion intensity. ^{Then could start the} scanning ^{of the} intensity distribution within the

scattering ^cone connected with t ⁼ (0 0.1 ) (see fig. 3), ^{which was of} purely magnetic origin. ^It

was performed for different A6 on both sides of

FIG. 3. ^- The horizontal cross-section along the centre line of the magnon scattering ^cone. The draft is not in scale for any particular ^{case ; the} dimensions of the

scattering surface have been exaggerated for sake of

claritv. ^The shape of the surface does not have to be

spherical ^but depends ^{on the} form of the magnon dis-

persion ^relation. 1) Sphere ^of reflection ; 2) Scattering surface ; 3) Detecting system ; 4) ^{The result of the} angular scanning by ^ideal resolution _{in every} respect ; 5) ^The same, but with the real resolution, ^i.e. by

collimation in the horizontal plane only.

630

the reflexion sphere. ^The position corresponding

to A6 ⁼ 0 was found by taking ^the peak ^{value of}

a rocking ^{curve for the} (0 0.1 ) ^reflexion.

For scanning ^{in the} horizontal plane ^a high ^slit

detector was used. This consisted of a tray ^of

three BF ^{neutron counters} (NTI - ⁶²⁾ placed

one above the other in ^a B 4C ^filled casette, ^inside

a shielding ^{of borated} paraffin ^{wax, on the} spectro-

meter arm, and provided ^{with an entrance colli-}

mator of 20’ nominal divergence ^{and 20 X 1~.0 mm}

window (aperture). The aim of this innovation

was to gain ⁱⁿ intensity ^{without loss} of resolution and to satisfy the condition of integration along

the vertical plane. ^{For each preset value of ð6}

the spectrometer ^{arm rotated in} predetermined steps ^{and for} every angular position C ^{the number}

of counts N in the preset ^{time was} automatically

recorded. Then the values of 1V versus (D (for given DO) ^were plotted (see fig. 4). For every

curve so obtained its centre Oc was plotted ~ fig. 5)

as a f unction of A6.

FIG. 4. ^- Some exemplary peaks ^{of magnon} scattering

and the elastic peak profile (for ^{A0 =} 0), ^all being ^nor-

malized to the same height.

The agreement ^with the theoretical dependence

of Oc on AO is good, proving ^the correctness of the method. Some ot the diffuse peaks ^had superim- posed ^elastic Bragg reflexions from certain tiny misaligned crystallites. They were narrow and

unsymmetrical ^with respect ^{to Cc. Their} origin

was confirmed by energy analysis ^{of some of them.}

FIG. 5. ^- dependence. ^{The solid line} represents

the formula

+ experimental points.

In the monochromatic beam technique they ^are

easier to identify ^{and to} subtract from the effect.

For each of the diffuse peaks ^{the total} experimental

width rexp ^{was found} by extending ^the slopes (see fcg. 4) ^to the abcissa axis.

The background ^of mainly incoherent elastic

scattering, ^was previously subtracted, ^{under the} assumption ^{of its} constancy ^{over the} region ^{of the} peak ^{and its} nearestneighbourhood. ^{This was in}

fact observed with the exception of small 1> (large

when the detector system approached ^too

close to the primary ^beam.

From the ₂ 0) ^{was subs-}

tracted and the difference, ^{after a} small correction for mosaic spread ^{of the} sample, ^was plotted ^versus

Do (see fcg. 6).

FIG. 6. ^- The widths Fcorr. ^{of the diffuse} peaks ^{vs misset-} ting angle ^04.

+ and 0 : results obtained from horizontal scanning, roi, ^{= 0.74°. A : results} obtained from vertical scan-

ning, r 01. ^{= 1.02o.} The solid lines do not represent

any theoretical dependence, ^{but are to} help ⁱⁿ focusing

the reader’s eye ^on the three distinct groups of points.

631 The same procedure ^was repeated along ^the

direction normal to the previous ^one by using ^the

"

magnonoskop " device described elsewhere [16]

i. e. by scanning ^{of the scattered beam} along ^the

vertical plane ^{with a} climbing ^{counter. The} primary ^{beam was} additionally narrowed in a

vertical direction by putting ^a collimator with horizontal slits of 30’ nominal divergence ^between

the monochromator and the sample. ^{The colli-}

mator in front of the detector (this ^{t°me a} single counter) ^was replaced by ^{another one with 20’}

divergence and horizontal slits also. As a result the instrumental width of the system ⁼ 0)

was about 30 % ^{higher than in the case of hori-}

zontal scanning. ^The vertical widths I‘1 ^{of the}

magnon scattering peaks ^are presented ⁱⁿ figure ⁶ together ^{with the} points ^{for the} horizontal widths

rll.

b) ^The spectrometric technique [5] ^was applied

in a conventional _{way. The} crystal sample ^was placed ^{on the} table II and the scattered beam

analyzed by ^{a Zn} crystal analyser placed ^{on table I}

with its reflecting surface, ^{12 cm} high ^{and 7 cm} wide, parallel ^{to the} (0 ^{0 1)} plane.

Thus it could reflect all the neutrons travelling

within the cone of _magnon scattering. ^{Thus two} procedures ^{were feasible :}

1) analysis ^without any vertical collimation, 2) analysis ^{with a} vertical collimator placed

between the sample ^{and the} crystal analyser. ^In

the first case the intensity of the diffuse peak ^was integrated ^{in a} vertical direction because of the wide collimation in front of the detector. In the second case the narrow bundle within the scattered

cone could be chosen. The ratio of its angular

dimensions to the angular ^{width of the cone} speci-

fies the accuracy of q determination which, together

with the _energy resolution of the analyser, gives

FIG. 7. ^- Spectra of the neutrons scattered within the diffuse magnon peak analyzed along the centre line of the cone, for several missetting angles ^{~8 of ’t’ =} (00.1).

The vertical collimation : 1.5° for ~6 ⁼ 590 and 10 otherwise. The peaks ^{for 09 =} 20° and the elastic one

from vanadium, 1, ^are not in the scale written on the ordinate axis.

the accuracy of the ^method. This accuracy could

not be high ^{because of limited} intensity.

In order to test energy resolution a vanadium (3)

metal slab of 8 X 30 X 60 mm was placed ^{on the}

table II and the energy spectrum ^{of incoherent}

elastic scattering ^measured (see fcg. 7). It appear- ed that the energy resolution could be assumed to be independent ^{on the} value of the vertical collimation. The uncorrected spectra ^{of neutron}

magnon scattering ^are presented ⁱⁿ figure ^7.

They ^{do not show the existence of the minimum} intensity ^{at the centre of the} scattering ^{surface as} expected ^from theory ^{and found in the} case-of Fe3o4 [6] ^{when a} single acoustical branch is considered.

Analysis ^of the results. ^- Assuming ^{that the}

widths of the peaks ^obtained by ^the diffraction

technique ^are governed by ^the dispersion ^relation

of acoustic magnons, ^the energy hm versus wave

vector q dependence ^was calculated, taking

q = 1 2 _kc rcorr and

where, ^for given Eo and T, kc is ^a known function of AO.

Fexp ^was corrected for instrumental width and mosaic spread. ^{The former} (4) ^{was done} by ^sub-

tracting 1 ^Fo from the total widths Texp ^{of the} 2 ^{° ,}

experimental ^curves.

Thus the results were presented ^{in the form of} 1ïÜ)(q) (see fig. ^{8). The most} spectacular ^{is the} agreement of data obtained from missetting angles

of both signs ^{i. e.} corresponding ^to magnon creation and annihilation processes, in contrast to thedata

expressed ^{in the form of} dependence ⁱⁿ figure ^{6. This} agreement corroborates the correct-

ness of the methods here applied. ^{The f ormula} dependence ^{in the case of a linear} dispersion

relation cited by ^{Elliott and Lowde} [10] ^is simply

not accurate, ^this having been checked by ^calcu-

lations performed by ^{Krasnicki without} any appro- ximation. Thus, ^{even in the case of} strictly

linear dispersion ^{the widths r for the} same IAOI

but different signs ^{are not} equal.

(3) ^{The author} is indebted to Dr B. Jacrot from the

Saclay Centre for the vanadium sample.

(4) ^{It can} be shown that when a rectangular distribution is folded with a triangular ^{one the} total width of the resul-

ting ^curve rresult. ^{= r} + 2 ^{ro. In every case the width}

is defined as the angular distance between the points ⁱⁿ

which the slope lines intersect the abscissa axis (background level). ^The steepness ^{of the} slopes ^is taken in the middle of its span. For the ^case under investigation the assump- tion as to the shapes ^{of the curves} treated in this way had theoretical justification [10], ^{and was} quite ^{well fulfilled}

as can be seen in figure ^4.