Small-angle critical magnetic scattering of neutrons in iron

J. Phys.

Chem. Solids

Pergamon

Press 1967. Vol. 28, pp. 1947-1955.

SMALL-ANGLE

CRITICAL

MAGNETIC

OF NEUTRONS D. BALLY,

B. GRABCEV,

A. M. LUNGU,

21 November

SCATTERING

IN IRON M. POPOVICI

Institute of Atomic Physics, Bucharest, (Received

Printed in Great Britain.

1966; in revisedform

and

M. TOTIA

Rumania 21 March

1967)

Abstract-Measurements of the temperature variation of small-angle critical scattering of 1.25 A neutrons by iron were performed. Taking into account the inelasticity of the scattering, the parameters which define the instantaneous spin correlation function of the Omstein-Zemicke type were determined. This form of the function turned out to be valid at distances larger than 15-20 A. For smaller distances, it has been confirmed that the instantaneous correlation function has the form of a simple exponential. From the data for T- Tc > 10” it has been deduced that the temperature variation of the static magnetic susceptibility is of the form (T- Tc) -aa l 0.04). In the immediate proximity of the critical temperature Tc from this dependence no significant deviation was noticed.

INTRODUCTION

THE PHENOMENON of critical magnetic

scattering of neutrons has been studied extensively, since it was first observed,(1*2) with particular emphasis on iron.c3- Q, In experiments of this type, the most complete information which can be obtained is the determination of the time dependent spin correlation function.(lO) Detailed investigations have been performed at small scattering angles with cold neutrons,(5-8) in order to measure the inelasticity of the scattering and, thus, to determine the temporal dependence of the correlation function. It has turned out, however, that the deduction of the spatial dependence of the correlation function by cold neutron scattering is complicated by difficulties related to the influence of the inelasticity on the measured angular distributions. For independent determinations of the spatial behaviour of the spin correlation, the scattering of small wavelength neutrons (A N 1 A) has been measured and analysed in the static approximation.(3*4~Q) The results of the experiments performed with cold neutrons indicated, however, that the inelastic scattering intensity, primarily at temperatures near the critical point, was essentially greater than was predicted by the calculations of VAN HOVE and DE GENNES.(~~) For this reason, it appears as 1947

if the use of the static approximation for the explanation of the angular distributions of scattered small-wavelength neutrons is not justified. The aim of the present paper is to study the temperature dependence of the angular distributions of small-angle critical scattering of nuetrons in iron in order to: (1) determine the instantaneous spin correlation function, taking into account the recent results regarding the scattering inelasticity;(8) (2) follow the behaviour of the magnetic susceptibility at temperatures close to the critical temperature.

EXPBRIMENTAL

PROCEDURE

AND

RESULTS

The small-angle magnetic critical scattering measurements were performed with a 1 a25 A monochromatic neutron beam, obtained by Bragg reflexion from a lead single crystal. For our reactor beams and experimental conditions, the effect of the critical scattering is maximum at a neutron wavelength of about 14 A. The choice of a 1.25 _k wavelength for our experiment represents a compromise between the requirement of obtaining a high scattered intensity and that of decreasing the effect of inelasticity and beam contamination by parasite reflexions. The latter effect was estimated as negligible.

1948

D.

BALLY,

B.

GRABCEV,

A.

M.

LUNGU,

incident The and scattered beams, of 30x 30 mm2 cross section, were defined by identical Soiler collimators having cadmium slits of 1” vertical and 30’ horizontal angular divergences. The samples were of polycrystalline Armco iron (99.98 per cent). They were slab shaped, 30 x 50 mm2 cross section and 2.5 mm thick. At such a thickness the critical multiple scattering may be neglected.(8y1a) It can be estimated that the multiple Bragg scattering is of no importance when relative intensity measurements are made at very small angles, The samples were heated in a vacuum aluminium oven by direct radiation from a long molybdenum winding, surrounded by two cylindrical heat shields made of thin molybdenum foils. The temperature was controlled by a calibrated chromel-alumel thermocouple, mounted in a bridge set-up. Deviation of the compensation galvanometer spot from the zero setting supplied an error signal to an electronic contro11er,(13) which performed the sample temperature stabilization. The measurements of the scattered neutron intensities were performed at f?.xed scattering angles, 8, with stepwisely temperature variations. The results of measurements made below and above the Curie temperature, T,, for values of the scattering vector tc in the range 0.1435 A-l, are plotted in Fig. 1. The statistical errors of individual measurements were less than 1 per cent and 3 per cent respectively for the largest and smallest intensities. During the measurements in the neighbourhood of T, the temperature stability accuracy was + O*l”. One of the measurements of scattered intensity for a temperature very near the critical one is shown in Fig. 2. The absolute temperature scale may be inferred from the fact that the position of the maximum, showing the critical temperature (T, = 770”), corresponds to an apparent temperature of 7683”, as indicated by the thermocouple. In Fig. 3 the shape of the temperature dependence of the scattered intensity at different angles, as well as the marked discontinuity occuring at the u-y phase transition is shown in detail. The figure also demonstrates that the ferromagnetic short-range order is maintained up to the temperature of this transition (about 900”). The sharp intensity drop at temperatures just

M.

POPOVICI

and

M.

TOTIA

above T, (Fig. 3, curve a) is typical of the scattering at the smallest angles, for which the correlations at large distances are most important. This drop is determined by the decrease of the spin correlation range with temperature. At large measurement angles the intensity increases at

FIG. 1. Temperature dependence of the scattered neutron intensity for several values of the scattering vector.

temperatures above T, (Fig. 3, curve b). Such a result is not to be found in the measurements carried out in the same angular range with cold neutrons@ - 8, but with smaller corresponding values of K. This suggests that the intensity increase for T > T, is specific to the measurements carried out in the range of largest scattering vectors K. The result obtained may be assigned to the way in which the cross section o(8) characteristic to high K values depends on temperature, ANALYSIS OF DATA

Within the framework of VAN Hovn’s theory(lO) the differential cross section of small-angle neutron magnetic scattering for a polycrystalline sample of

SMALL-ANGLE

CRITICAL

MAGNETIC

SCATTERING

OF

NEUTRONS

IN

IRON

1949

FIG. 2. Scattered neutron intensity vs. temperature in the immediate proximity of the critical temperature.

cubic symmetry, consisting of N spins, in the static approximation, is given byc3) da -

=

m

s

8~ 3.N(yro)2f2wo-1*

sinKr

r2*y(r) -.dr.

dQ

r

(1)

KY

i-l “c

FIG. 3. Variation of the scattered neutron intensity with temperature for 0 = lo 25’ (empty circles) and 0 = 3” 12’ (full circles).

Here, y is the neutron magnetic momentum, Y,, is the electron classical radius, f is the magnetic scattering form-factor, os denotes the volume of the crystal cell, K is the scattering vector, and Y(Y) stands for the instantaneous correlation function of the pairs of spins separated by a distance r. For T > T,, from equation (1) we have,

where x is the static magnetic susceptibility of the spin system at a given temperature, x1 is the susceptibility of the system of independent spins at the same temperature and S the quantic spin number of the magnetic atoms. Equation (1) describes the scattering phenomenon around the origin of the reciprocal lattice only. It is valid as long as K is small enough and the variation of y(r) with Y is sufficiently weak to allow for the replacement of a summation over lattice points by the integral at the right side of the equation (1). The static approximation for the differential cross section is useful for the direct interpretation of the experimental data as long as the neutrons have enough speed so that the dynamics of the spin system has no influence on the scattering. Van Hove’s theory predicts that at the critical temperature the spin system should have an infinite relaxation time. This implies that the static

1950

D.

BALLY,

B. GRABCEV,

A.

M.

LUNGU,

approximation is valid at temperatures sufficiently close to T,. The experiments of small-angle cold neutron scattering@ - 8, demonstrated, according to KOCINSKI’S(~*) theoretical prediction, that the inelasticity is considerable even at temperatures very close to T,. The results reported by KAWASAKI show, however, the need of recalculating the temperature dependence of the inelasticity. Therefore the use of static approximation in analysing the results obtained at any temperatures introduces errors, which are essential for cold neutrons. To avoid them without nevertheless, giving up the advantages of the data interpretation within the static approximation, it is necessary to correct the data for inelasticity by a factor Fit defined as the proportionality factor between the actual differential cross section and that of (1). In addition, one must take into account the considerable distortions which the finite angular resolution of the device introduces in the angular distributions measured at small angles. Then, it is necessary to correct the experimental data for the geometry of the measurement device by a geometrical factor F,. For explicit calculation of the Fi and FE factors it is necessary to choose a calculable expression for the spin correlation function. The experiments performed thus far 3-9 proved that the critical scattering from iron at the smallest angles is well described by the Van Hove theory if an instantaneous spin correlation function of the Ornstein-Zernicke type is included. In Van Hove’s notation its form is exp(; y(r) = [r+s( S + 1)/4?rr,2]-----

k-,r)

(3)

Here the parameters K1-l and y1 represent the range and strength, respectively of the spin correlations. For such a correlation function, equation (1) gives the differential cross section, within the static approximation, in the following form: do -= dfl

2N

Using the form of equation (3) for the instantaneous correlation function, NIELSEN’~“) computed numerically the inelasticity correction

M. POPOVICI

and M.

TOTIA

factor Fi (R-in Nielsen’s notation) for several values of the dimensionless parameter t.~ which describes the inelasticity. This parameter, which is inversely proportional to the relaxation time of the magnetization fluctuations, is a measure of the full width at half maximum Ah of the spectral distribution of neutrons scattered at very small angle 0, resulting from a perfectly monochromatic incident beam: Ah/h 1~ ,LL~~. Inelasticity measurements for cold neutron scattering’s) indicated a probable value of 11 for the parameter p, independent of the temperature. When correcting the data for inelasticity we used this value for all temperatures and used the corresponding factor tabulated in Ref. 16. The influence of the Fi factor may be attenuated by decreasing the neutron wavelength. This decrease, however, increases the relative magnitude of the geometrical corrections. This is because measurements must be made at smaller angles where the angular resolution is bad as a result, primarily, of the vertical divergences of the collimators. (The divergences are difficult to reduce for intensity reasons.) In calculating the geometrical factor Fg employed in the present paper, the authors neglected the variation with 8 of the factor F, within the angular interval defined by resolution and have used the Ornstein-Zernicke function for the instantaneous correlations. The factor Fg was computed numerically for several situations of practical interest, under the assumption that the angular transmission functions of the collimators have Gaussian shapes with half-widths equal to the geometrical horizontal aH and vertical ay divergences. The details of the computation are described in Ref. 17. An interesting conclusion to be drawn out from these computations is that, when aX/av = $, at temperatures near the critical, the geometrical factor is close to unity even at very small angles, due to the compensation of the distortion effects related to the horizontal and vertical divergences. As an illustration, in Fig. 4 several results of computations are shown. Taking into account that both correction factors depend not only on the measurement angle, but also on the correlation range Klvl, which will be inferred from experimental data, it was necessary to use a method of successive approximations in determining the parameter Kl and these factors.

SMALL-ANGLE

CRITICAL

MAGNETIC

Uncorrected experimental data were used to obtain the zero order approximation. The corrected -inverse values of the measured intensities should present, according to relation (4), a linear dependence on K2. This dependence I

I

I

1

I=

SCATTERING

OF

NEUTRONS

IN

IRON

1951

The inelasticity influence upon data interpretation turns out to be, for X = 1.25 A, considerably smaller than it is for cold neutrons. Therefore the uncertainty in the value of the parameter p does not affect essentially the determination of spincorrelation range. The results of Fig. 5 show a somewhat different than that variation of K12 with temperature observed by other authors.(5-8) The K12 values deduced in the static approximation (without the

as61

Scattering angle B in degrees

FIG. 4. Some values of the computed geometrical factor Fg.

offers a convenient method for the determination of the parameters Kl and Ye, which define the instantaneous spin correlation function y(r). We have checked this dependence using the results obtained at the smallest scattering angles and ascertained, as did SPOONER and AVERBACH, that it truly holds in the range ~~ 5 0.025 AV2. Beyond, the experimental points deviate from the linear dependence predicted by (4), showing that the Ornstein-Zernicke correlation function is valid only for distances r > 15-20 A. The linear extrapolation to the axis K2 of the reciprocals of scattered intensity, corrected for geometry and inelasticity, has given us the values of the parameter K12, plotted in Fig. 5 as full circles. In order to give a picture on the differences introduced by taking into account the inelasticity in data interpretation, the values of K12 obtained in the static approximation are plotted too, in Fig. 5. 4

FIG. 5. The values of the parameter K12 obtained in the static approximation (empty circles) and taking into account the inelasticity (full circles).

inelasticity correction) are consistent with those obtained by SPOONER and A~ERBACH(~) in the same approximation, but with a somewhat different method-that of Fourier inversion of experimental data for a large angular range. In the range ~~ < 0.025 Aw2 our results, as well as the results of Averbach and Spooner, exhibit a linear dependence of the reciprocal intensity on K~, whereas the measurements carried out by Passe1 exhibit a deviation from linearity. Assuming that the individuality of the samples used in several works does not affect the observed values

1952

D.

BALLY,

B.

GRABCEV,

A.

M.

LUNGU,

of the correlation range, the difference between the results mentioned previously may be assigned, at least partially, to the greater contribution of the inelasticity factor in the measurements for small wavelengths and to the difficulties of introducing exactly these corrections, both for inelasticity and for experience geometry. However, it remains necessary to define more accurately to what extent the choice of h can affect the dependence of intensity reciprocal on tc2. In Fig. 6 rr2(T)/~r2(TC), the temperature dependence of the correlations strength rr, inferred from experimental data corrected from inelasticity, is plotted. The values obtained in such a manner are practically equal to those obtained in the static approximation. They are also in agreement with those obtained in paper (9). The cause of the faster variation with temperature of y12, relative to the temperature variation observed in the cold neutron experiments of ERICSON and JACROT~~)and PASTEL et uZ.(*) (see Fig. 6), is uncertain.

o l

o-

W

POPOVICI

and

M.

TOTIA

the form x N (T-T,)-O, the value dc 2: 1.33, in agreement with the experimental results of ARAJS and COLVIN(~O)as well as with the recent data of NOAKES, TORNBERG and ARROTT.(~~) The previous data of these authors indicated 0: = 1.37.(22) WOOD and RUSHBROOKE(~~)computed the value tc = 1.36 for the Heisenberg ferromagnetic model in the limit of infinite spin, while BAKER, GILBERT and RUSHBROOKE,(~~) determining more accurately a for a Heisenberg ferromagnet of spin +, obtained a equal to 1.43. Within the framework of the Ising model, Domb and Sykes obtained the value a = 1.25. In Fig. 7 the experimental values of the quantity TK12r12( T)/r12( T,), which is proportional to the reciprocal of the susceptibility (according to relations (2) and (4)), are plotted, in logarithmic scale, vs. log(T- T,).

o.o~flf~ 1

Spoonec Averbach Ourrtwlts

a

M.

SO

40

I

T-T,in f

FIG. 6. Variation of rla(T)/r12(Z’c)

with

temperature.

as

I

2

T-To in6C JO

so

FIG. 7. Plot of the logarithm of TK~2r~a(T)/r~“(~c) log(T-

According to equation (2), the extrapolation to zero scattering angle of the angular distributions measured for various temperatures allows for the determination of the temperature behaviour of the magnetic susceptibility and, hence, offers the possibility of checking the recent calculations on the nature of the susceptibility singularity at T = T,. The calculations of DOMB and SYKE@) and GAMMEL, MARSHALLand MORGAN for Heisenberg ferromagnets indicated, for a singularity of

20

VS.

Tc).

The points plotted as full circles were obtained from the analysis of the angular distributions and ensued from the results given in Figs. 5 and 6. It is worth pointing out the agreement with the data reported in paper,(*) although some discrepancies between the values of each of the parameters K12 and y12 obtained by us and that group (see Figs. 5 and 6) were noticed. The slope of the straight line in Fig. 7, which best approximates the points for T-T, > lo”,

SMALL-ANGLE

CRITICAL

MAGNETIC

SCATTERING

is equal to a = (1.30 f 0.04) perfectly agreeing with the value (1.305 0.04) reported by PASSEL et uZ.(*) and with that of 1.30 shown ixPa) as reported by Jacrot. The results interpreted within the static approximation give a value of (l-34 + OO4) for a. At temperatures very close to the critical one, the determination of the susceptibility from the extrapolation of angular distributions to the zero scattering angle leads to considerable relative errors due both to the statistical fluctuations of measurement8 and, especially, to the uncertainty in the knowledge of the angular dependence of the correction factors. Yet, for such temperatures the correlation strength r, is practically constant and it becomes possible to draw out information about the susceptibility directly from the temperature dependence of the intensity measured at a fixed scattering angle. Computations of the correction factors for our experimental device indicated that, at temperatures very close to T,, the product F,*FE is practically temperature independent. This has allowed us to extract, by measurement like the one shown in Fig. 2, the results plotted in Fig. 7 as empty circles. These results have been obtained under the assumption of K12(TC) = 0. The data on the parameter Kr2, extracted from the angular distributions, are not in contradiction with this assumption within the experimental errors. The errors in Fig. 7 describe only the statistical uncertainties and do not include the possible systematical errors due to the correction factors or to the nonzero value of K12(TC). The values drawn out from the temperature dependence of the scattering at fixed angles are in satisfactorily agreement with the straight line of slope 1.30, which extrapolates to small temperatures the results extracted from the angular distributions at T-T, 2 10”. Accordingly, if the susceptibility has indeed a singularity at T = T,, then our data exhibit for its temperature dependence the same form (T - T,) -f1*30*0’04)for all the studied temperature range 0.25 < T- T, < 50”. The high errors from the range T-T, c 1” do not exclude, however, the possibility of any deviations from this form in the immediate proximity of T,. AS we have already stated, the OmsteinZernicke type correlation function (3) describes correctly the phenomenon of magnetic critical

0F

NEUTRONS

m

IRON

1953

scattering only for small values of the scattering vector (in our experiment8 for K 5 0*16A-l). This is quite normal, considering that the expression (3) represents only the ~~ptotic~ form at large distance8 of the actual instantaneous spin correlation function. SPOONERand AVERBACH( using the data obtained from the Fourier inversion of the angular distributions, suggest that for distances smaller than 15 k the instantaneous correlation function, both at T > T, and at T c T,, could have the form: y(r) = S(S + l)A exp( -r/a)

(6)

Here the parameter A would be connected with the largest nearest-neighbour correlation, while 6 describes the correlation radius, with appro~ately the same meaning as Kr -I. If we admit the expression (6) for y(r), the differential cross section of the critical scattering, computed by means of the relation (l), becomes

dc = const. AG-1-f2-(rca+6-2)-2 dQ

(7)

This form of the cross section can explain the increase of intensity with temperature just above the critical point (Fig. 3, curve b) since, in this case, Ss2 & rc2 and du/dQ N AS-r/tc4. Then,

Table 1. Parameters of the instantaneous spin correlationfunction y(r) = S(S+ 1)-A exp( - r/S), extractedfiom the data plotted in Fig. 8 T- Tc f”C)

-30 -20 -10 0 +10

+20

t-30 +40

-4T>iA(Tc)

Correlation range (4

1.64 1.58 l-40 l-00 O-76 0.62 0.59 0~5.5

8.0 9.0 14.1 26.0 15‘4 10.6 9.0 7.7

according to the data of Table 1, the product A*S-l increases with temperature in the immediate proximity of the Curie point.

1954

D. BALLY,

B. GRABCEV,

A. M. LUNGU,

Considering the new form of the correlation function, the correction factors employed above in the experimental data interpretation are not valid any more. Admitting, however, that the geometry corrections are commonly not essential at sufficiently large scattering angles, and taking into

004

M. POPOVICI

1. PALEVSKYH.

and M. TOTIA

RBFBRENCES and ‘HIJGHBSD. J., PAYS. Rev.

92,

202 (1953).

2. SQUIREs G. L., Proc. phys. Sot., A 67, 248 (1954). 3. GER~CHH. A., SHULL C. G. and WILKINSON M. K., Phys. Rev. 103, 525 (1956).

QO8

810

x2 In A-’

FIG. 8. Reciprocal of the square root of the observed intensity against the square of the scattering vector to show the validity of the exponential form of the instantaneous correlation function. account the comparatively small differences associated with the inelasticity corrections of the data reported hitherto, we neglected both of these corrections. To check equation (7), we plotted the inverse of the square root of the scattered intensity vs. ~~ for various temperatures about the critical temperature. As illustrated in Fig. 8, for any of these temperatures, both smaller and greater than T,, one actually obtains linear dependence on K2, as predicted by the relation (7). This seems to confirm the assumption of Spooner and Averbach about the form of the instantaneous correlation function, at least in the range r N lo-15 A, to which the data reported in Fig. 8 refer. The values of the parameters A and 6, extracted from these data are summarized in Table 1. They correspond, in order of magnitude, to the data reported by Spooner and Averbach.

4. LOWKJE R. D., Rev. mod. Phys. 30, 69 (1958). 5. ERICSONM. and JACROTB., J. Phys. Chek. Solids 13,235 (1960). 6. JACR~TB.; KO&STANTINOVIC J., PARBTTEG. and CRIBIERD., Inelastic Scattering of Neutrons in Solids and Liquids, IAEA, Chalk River (1962). 7. PASSELL., BLINOWSKIK., BRIJNT. and NIELSEN P., J. appl. Phys. Suppl. 35, 933 (1964). 8. PASTELL., BLINOWSKIK., BRUN T. and NIELSEN P., Phys. Rev. 139, A 1866 (1965). 9. SPOONJIRS. and AV~RBACH B. L., Phys. Rev. 142, 291 (1966). 10. VANHOVEL., Phys. Rev. 95, 1374 (1954). 11. DE GENNE~P. G., Rapp. CEA No. 925 (1959). 12. ERICSON-GALULAM., Rapp. CEA No. 1189 (1959). 13. TARINA E. and TIMIS P., Preprint IFA (to be published). 14. K~CINSKIJ., Acta phys. pal. 24,273 (1963). 15. KAWASAKIK., Phys. Rev. 145,224 (1966). 16. NIELSENP., Risij Report No. 118 (1965). 17. POPOVICIM., Preprint IFA (to be published). 18. DOMB C. and SYI(ESM. F., Phys. Rev. 128, 1681

(1962).

SMALL-ANGLE

CRITICAL

MAGNETIC

19. GAMMELJ., MARSHALLW. and MORGAN L., PYOC.

R. Sot. A 275,257 (1963). 20. AMJS S. and COLVINR. V., J. appl. Phys. Suppl. 35,2424 (1964). 21. NOAKESJ. E., TORNBBRGN.

E. and ARROTT A.,

J. appl. Phys. 38,1264 (1966).

SCATTERING

OF NEUTRONS

IN

IRON

1955

22, NOAIQS J. E. and ARROTTA., J. appl. Phys. Suppl. 35,931 (1964). 23. WOOD P. J. and RUSHFJROOIQX G. S., Phys. Rev. Lett. 17, 307 (1966). 24. BAKERG. A., GILBERT H. E., EVE I. and RUSHBROOKE G. S., Phys. Let. 20,146 (1966).