On the evaluation of neutron scattering elastic scan data

ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 603 (2009) 439–445

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

On the evaluation of neutron scattering elastic scan data Reiner Zorn  ¨ lich, IFF, D-52425 Ju ¨ lich, Germany Forschungszentrum Ju

a r t i c l e in fo

abstract

Article history: Received 9 January 2009 Received in revised form 20 February 2009 Accepted 26 February 2009 Available online 13 March 2009

Neutron scattering elastic scans are a common technique to obtain an overview of the microscopic dynamics over a wide range of a control parameter, in most cases the temperature. The data evaluation usually aims at calculating the mean-square displacement hr 2 i of the scatterers. Unless there is a clear separation of fast and slow dynamics, this mean-square displacement has to be treated as a timedependent quantity hr 2 iðtÞ. In this paper a definition of the timescale t res of an elastic scan from the energy resolution of the instrument is proposed. A common assumption in the evaluation of elastic scans is that the Q-dependence of the elastic scattering decays like a Debye–Waller factor. Here, it is shown that this can only be expected under certain conditions. In general there will be a non-linearity in plots ln Iel ðQ Þ vs. Q 2 because hr 2 iðtÞ is not constant close to t res , and Iel ðQ ¼ 0Þa1 due to multiple scattering. Finally, a procedure is suggested to calculate hr 2 i taking these facts into account. & 2009 Elsevier B.V. All rights reserved.

Keywords: Inelastic neutron scattering Elastic scan Mean-square displacement

1. Introduction Inelastic neutron scattering is an important tool in the investigation of microscopic dynamics of condensed matter. The scattering itself, quantified by the scattering function SðQ ; EÞ in terms of the momentum transfer _Q and the energy transfer E ¼ _o, is the Fourier transform in space and time of the van-Hove correlation function Gðr; tÞ. Therefore the scattering contains comprehensive information about the microscopic dynamics. In practical experiments it is often, due to time and intensity limitations, not possible to get the full spectral information, SðQ ; EÞ. Instead, one often picks a certain energy transfer E, in most cases E ¼ 0 for elastic scattering, and records the scattered intensity depending on a physical control parameter, e.g. the temperature T. This technique is called ‘‘fixed-window scan’’ or for E ¼ 0 ‘‘elastic scan’’. The importance of elastic scans on backscattering instruments becomes clear from a statistics of experiments. In 2007, 50% of all experiments on the high-flux backscattering spectrometer IN16 at Institut Laue-Langevin Grenoble involved elastic scans and 25% performed elastic scans only [1]. On instruments with lower flux the fractions are even higher, 86%/57% for IN10 and 86%/79% for IN13. The more it is surprising that most of the studies only use the most simple form of evaluation, namely a calculation of the meansquare displacement of the scatterers by a semilogarithmic plot of the elastic intensity vs. the square of the scattering vector Q. As outlined below this procedure can lead to significant errors in the

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calculation of the microscopic quantity underlying an elastic scan, the mean-square displacement. These errors are not only due to multiple-scattering effects which were discussed in an earlier paper [2]. Instead, some are intrinsic to the method of elastic scans. Firstly, it is often overlooked that the mean-square displacement is a time-dependent quantity, hr 2 iðtÞ. The question, what the time t is for an elastic scan on a given backscattering instrument and for a given sample is not trivial. Secondly, the proportionality ln Iel / Q 2 is only valid at rather low Q and including the whole detector set of a backscattering instrument may lead to wrong results. In addition to demonstrating the significance of these effects, this paper will present a suggestion of evaluating elastic scans by assuming an underlying model of fractal diffusion.

2. Standard evaluation of elastic scans The standard evaluation of elastic scans starts from the expression of the Debye–Waller factor which reduces the elastic scattering towards higher scattering angles: DWF ¼ expðhu2 iQ 2 =3Þ.

(1)

Here, Q is the scattering vector depending on the angle 2y and incident wavelength l by the usual expression for elastic scattering: Q¼

4p

l

sin y.

(2)

Its maximal value is given by the wavelength as Q max ¼ 4p=l. hu2 i is the mean-square offset of the atomic positions from their equilibrium positions (e.g. lattice positions in a crystal). To

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R. Zorn / Nuclear Instruments and Methods in Physics Research A 603 (2009) 439–445

generalise this expression to fluids where such equilibrium positions are not defined, one introduces the mean-square displacement between positions of the same atom at different times separated by a timespan t, hr 2 iðtÞ. For a solid (crystal or glass) for sufficiently long times hr 2 iðtÞ ¼ 2hu2 i will hold. For a fluid (liquid or gas) in contrast, hr 2 iðtÞ will increase without bounds for long times. From this fact, immediately, the question arises: What is the time t at which hr 2 iðtÞ is measured by an elastic scan? Ignoring this question, there is a simple way to calculate hr 2 i. From Eq. (1) with hu2 i replaced by hr 2 i=2 one expects the logarithm of the elastic intensity to be linear in Q 2 with the slope yielding hr 2 i ¼ 6

d ln Iel ðQ Þ 2

dðQ Þ

.

(3)

Because in practice the plots are never strictly linear for reasons to be discussed later, it is common to use the low-Q limit for a standard definition: 2

hr i0 ¼ 6 lim

Q !0

d ln Iel ðQ Þ dðQ 2 Þ

because limQ !0 Iel ðQ Þ ¼ 1. Insertion of Iel ðQ Þ from (8) gives  Z 1 dSðQ ; tÞ ^ dt. RðtÞ ¼  dðQ 2 Þ Q 2 ¼0 0

(11)

The (incoherent) intermediate scattering function can be expanded in a series in Q 2 containing the even moments of the displacement [4], SðQ ; tÞ ¼ 1 

hr 2 iðtÞ 2 Q þ OðQ 4 Þ 6

(12)

from which the derivative can be calculated in the limit Q ! 0, yielding lim

Q!0

d ln Iel ðQ Þ 2

dðQ Þ

Z ¼ 0

1

2 ^ hr iðtÞ dt. RðtÞ 6

Insertion into (4) shows that Z 1 2 ^ hr 2 i0 ¼ iðtÞ dt RðtÞhr

(13)

(14)

0

.

(4)

In the following, the expected hr 2 i0 will be calculated from the scattering function SðQ ; oÞ and compared to the actual physical quantity hr 2 iðtÞ. For simplicity, the scattering will be assumed completely incoherent. The experimentally measured scattering function on a neutron spectrometer is the convolution with the instrumental resolution: Z 1 ~ ; oÞ ¼ Rðo0 ÞSðQ ; o  o0 Þ do0 . (5) SðQ 1

(Assuming identical energy selection for all detectors, the energy resolution of the instrument is defined independent of Q.) The elastic intensity is proportional to the value at o ¼ 0, i.e. Z 1 ~ ; o ¼ 0Þ ¼ RðoÞSðQ ; oÞ do. (6) SðQ

thus the commonly determined hr 2 i0 is an average of the physical, time-dependent hr 2 iðtÞ weighted with the Fourier transform of the resolution function. Therefore, hr 2 i0 is not the mean-square displacement at a defined time. Nevertheless, a justification for assigning a timescale is given in the next section.

3. Time scale associated to an elastic scan As the probably most simple example, the exact Q dependence of an elastic scan given by Eq. (8) will be calculated for diffusion characterised by hr 2 iðtÞ ¼ 6Dt

(15)

SðQ ; tÞ ¼ expðDQ 2 tÞ.

(16)

1

As a first step of data evaluation usually a normalisation to a low temperature measurement is done.1 In classical approximation SðQ ; oÞ ¼ dðoÞ holds at T ¼ 0 yielding R1 ~ ; o ¼ 0Þ RðoÞSðQ ; oÞ do SðQ Iel ðQ Þ ¼ . (7) ¼ 1 Rðo ¼ 0Þ S~ T¼0 ðQ ; o ¼ 0Þ Using the convolution theorem of Fourier transforms, Eq. (5) reduces to a product of the intermediate scattering function SðQ ; tÞ and the inverse Fourier transform of the resolution function, RðtÞ [3]. The values at zero frequencies needed for (7) can be calculated as integrals over time of the inverse-Fourier transformed quantities. This leads to the following expression: R1 Z 1 RðtÞSðQ ; tÞ dt ^ ¼ Iel ðQ Þ ¼ 0 R 1 RðtÞSðQ ; tÞ dt (8) 0 0 RðtÞ dt ^ is the alternatively normalised time-domain resolution where RðtÞ function: RðtÞ RðtÞ ^ ¼ . RðtÞ ¼ R1 p Rð o ¼ 0Þ RðtÞ dt 0

(9)

With expression (8) it is now possible to calculate the limit needed in Eq. (4): lim

Q !0

d ln Iel ðQ Þ dðQ 2 Þ

¼ lim

1

Q!0 Iel ðQ Þ

dIel ðQ Þ dðQ 2 Þ

¼ lim

Q !0

dIel ðQ Þ dðQ 2 Þ

.

(10)

1 This step is actually not necessary if the scattering is completely incoherent because later on the logarithm will be taken. But if a coherent contribution exists it removes its most prominent effect, the modulation of the Q-dependence with SðQ Þ.

Throughout this paper we will assume that the resolution function is a Gaussian (which is usually the aim of construction for backscattering spectrometers):   1 o2 (17) RðoÞ ¼ pffiffiffiffiffiffi exp  2 2s 2ps   2 2 s t RðtÞ ¼ exp  (18) 2 rffiffiffiffi   2 s2 t 2 ^ . (19) s exp  RðtÞ ¼ p 2 The integral of expression (14) is simple to calculate and gives ! rffiffiffiffi rffiffiffiffi 2 1 2 1 . (20) s hr2 i0 ¼ 6 Ds ¼ hr 2 i

p

p

So it is just the value of the actual mean-square displacement hr 2 iðtÞ at a resolution-defined time rffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 4 p1 ln 2_ 1:24 meV ns t res ¼ s ¼ ¼ . (21) p DEFWHM DEFWHM (The last expression is converted from s to the more common ‘‘full width at half maximum’’ (FWHM) of the energy resolution.) Eq. (20) suggests that t res can be seen as the time scale of the elastic scan experiment. It is a general result that t res is inversely proportional to the width of the resolution but with a different prefactor if the resolution shape is not Gaussian. It is important to note that, strictly speaking, this result is particular for simple diffusion. To demonstrate this, one can change the assumption for the microscopic dynamics to

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fractal diffusion2 2

b

hr iðtÞ ¼ 6Dt .

1.8 (22)

If the distribution of displacements is still considered a Gaussian the intermediate scattering function is fully determined by hr 2 iðtÞ:

SðQ ; tÞ ¼ expððt=tðQ ÞÞb Þ

r 2 (tres)

Thus a mean-square displacement increasing as (22) yields a Kohlrausch function (stretched exponential) for the intermediate scattering function

1.4

0

(23)

1.6

1.2

r2

SðQ ; tÞ ¼ expðQ 2 hr 2 iðtÞ=6Þ.

441

(24)

with tðQ Þ ¼ Db Q 2=b . For b ¼ 12 this would correspond to the Rouse model of polymer dynamics [5], for b ¼ 23 to the Zimm model [5], for b ¼ 1 to simple diffusion [6], and for b ¼ 2 to the classical ideal gas [7]. Any value of b ¼ 0; . . . ; 1 can be found for glass-forming systems [8]. In that case a somewhat more complicated calculation leads to the result   6  2b=2 b 1 b þ s (25) hr 2 i0 ¼ pffiffiffiffi DG 2 2 p

1.0

0.8

0

1 β

2

Fig. 1. Ratio of the apparent mean-square displacement determined from an elastic scan using (4) (hr2 i0 ) and the true mean-square displacement at t res given by Eq. (21) for fractal diffusion with a time-exponent b.

0

Fig. 1 shows this ratio for b ¼ 0; . . . ; 2. It can be seen that the error of interpreting hr 2 i0 as hr 2 iðt res Þ is less than 8% for the more common values of b ¼ 0; . . . ; 1. Comparing this with the errors due to other sources, the determination of the mean-square displacement from the Q ! 0 limit of the elastic intensity can be considered negligible.

-1 ln Iel(Q)

where GðxÞ is the gamma function. Comparison with (21) and (22) shows that   hr 2 i0 b 1 ðb=2Þ1=2 þ p G ¼ . (26) 2 2 hr 2 iðtres Þ

-2

4. Elastic intensity at non-vanishing Q Unfortunately, the situation that sufficiently low-Q detectors are provided to determine reliably the limit Q ! 0 is often not fulfilled. This can be straightforwardly demonstrated for the case of simple diffusion. There integral (8) can be calculated analytically (assuming a Gaussian resolution function): ! ! D2 Q 4 DQ 2 erfc pffiffiffi (27) Iel ðQ Þ ¼ exp 2s2 2s pffiffiffiffi R 1 where erfcðxÞ ¼ 1  erfðxÞ ¼ ð2= pÞ x expðt 2 Þ dt. Consistently with the result of the preceding section, the logarithm can be expanded into a series rffiffiffiffi 2D 2 Q þ OðQ 4 Þ. (28) ln Iel ðQ Þ ¼ 

-3 0

2

4

6

8

10

DQ2/σ Fig. 2. Q dependence of the elastic intensity for simple diffusion (continuous curve). Following the common procedure to determine the mean-square displacement, the logarithm ln Iel ðQ Þ is plotted vs. Q 2 (rescaled by D=s to a dimensionless quantity). The dotted line represents the low-Q expansion (28) and the dashed line the high-Q asymptote (29). The shaded region indicates the range where the low-Q series approximation is correct with an error o10%.

ps

For large Q an asymptotic expression can be found: rffiffiffiffi 2 s 2 Q þ OðQ 6 Þ. Iel ðQ Þ ¼ pD

(29)

Fig. 2 shows the ln Iel vs. Q 2 plot for the elastic intensity of the simple diffusion together with the low Q expression and the high Q asymptote. The fact that the plot does not yield a straight line may lead to the wrong impression that the underlying process is non-Gaussian [9] (in the sense that the distribution of displacements is not Gaussian). Indeed, expansion (28) contains terms Q 4 and higher order. Nevertheless, Eqs. (15) and (16) for 2 A more complicated generalisation of the diffusion process based on a microscopic model is used in Ref. [3] followed by a similar calculation of the elastic intensity. Nevertheless, in order to demonstrate the general features visible in the elastic intensity this simple model is sufficient.

diffusion are related by Eq. (23), i.e. diffusion is a Gaussian process. So the curvature in plots like Fig. 2 is just due to the fact that Iel ðQ Þ is plotted and not SðQ ; tÞ. From Fig. 2 it can be seen that the range where the Q dependence of the elastic scattering can be used to determine hr 2 i the classical way using (4) is very limited. Iel ðQ Þ should not decay to less than 73% of its value at Q ¼ 0 in order to avoid an error of more than 10% in hr 2 i (shaded region in the plot). Conversely, this means that the highest hr 2 i which can be determined with an accuracy of 10% is about 1:9=Q 2max where Q max is the scattering vector of the highest-angle detector to be included in the evaluation. For common backscattering spectrometers using the 1 ˚ . If all Si111 reflection the highest accessible Q is about 2 A detectors should be included, this limits the reliable range of 2 ˚ . The situation can mean-square displacements to less than 0:5 A be significantly improved by only including low-Q detectors.

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0

0

-1

ln Iel(Q)

–ln Iel(Q)

-1 -2

-3 -2 -4

-3

-5 0

10

20 Q2

30 r2

40

0

1 (Q/Qmax)2

0

Fig. 3. Q dependence of the elastic intensity for fractal diffusion with b ¼ 1, 23, 12, and 13 (continuous curves, top to bottom). Following the common procedure to determine the mean-square displacement, the logarithm ln Iel ðQ Þ is plotted vs. Q 2 . The abscissa values are rescaled by hr 2 i0 from Eq. (25) so that all curves have the same initial slope 16. The dotted line represents the low-Q expansion (28).

Fig. 4. Multiple scattering influence on elastic scans. The calculation is based on simple diffusion (b ¼ 1), DQ max s ¼ 0:1, 1, 10 (top to bottom groups of curves), and 20% multiple scattering probability (s ¼ 0:2). The continuous curves result from applying the multiple scattering expression (30) before calculating the elastic intensity using (8). The dashed curves result from the approximation in Ref. [2] doing it in the reverse order. For comparison the dotted curves show the result without considering multiple scattering.

1

˚ required to get Assuming a minimum set of detectors up to 0:4 A sufficient data to determine the slope of ln Iel vs. Q 2 , the limit can 2 ˚ . be pushed to 12 A Interestingly, the situation ameliorates slightly for fractal diffusion with bo1 as Fig. 3 shows. E.g. for b ¼ 12 a decay down to 39% in Iel ðQ Þ can be used with an error limit of 10% in hr 2 i. This corresponds to a usable detector range which is roughly 1.7 times larger than in the case of simple diffusion. In the limit b ! 0 the curvature of the plots vanishes. This is expected because b ¼ 0 corresponds to a constant hr 2 iðtÞ. In reality this means that the motions leading to the displacement are much faster than the timescale of the instrument (tfast 5tres ) and if slow motions exist which increase it further they are much slower (t slow bt res ). In this case the Debye–Waller picture is exact, leading to Iel ðQ Þ ¼ expðhr 2 iQ 2 =6Þ. 5. Influence of multiple scattering A further issue complicating the interpretation of elastic scan data is the presence of multiple scattering. In the framework of the intermediate scattering function, multiple scattering can be easily taken into account assuming isotropy and equal probability for higher scattering orders. Under these conditions the intermediate scattering function including multiple scattering becomes ¯ 2  ; tÞ ¼ SðQ ; tÞ þ s  SðtÞ SðQ 1  s  S¯ ðtÞ

The result is shown in Fig. 4. It can be seen that the striking difference is that limQ !0 Iel ðQ Þo1. This difference increases with increasing hr 2 i, i.e. for higher temperatures. In the experiment this may be mistaken as an error in normalisation or sample loss at high temperatures and thus be fitted by a ‘‘fudge’’ factor. In reality it is more likely a consequence of multiple scattering. Because neutrons scattered several times with Q 40 can end up in a low angle detector, the elastic intensity at apparent Q ¼ 0 will decrease with increasing temperature as that at finite Q does. In an earlier publication [2] the influence of multiple scattering on elastic scans was considered in a more approximative way. It was assumed that the scattering can be decomposed into an elastic and an inelastic component. This assumption may be justified if SðQ ; tÞ is flat close to t res . In that case the elastic intensity including multiple scattering can be calculated from that without multiple scattering by formula (30) with SðQ ; tÞ replaced by Iel ðQ Þ. In other words, multiple scattering was considered after calculating the integral (8) while the correct (but also more laborious) way is to do this before. For comparison Fig. 4 includes the result of this simpler calculation too. It can be seen that for low mean-square displacement there is not much difference in which way multiple scattering is considered but for high hr2 i and high Q a significant deviation may occur.

(30)

where s is the probability that a neutron which is scattered n times is scattered another time [10]. Note that due to the additional multiple scattering term this function is normalised to ð1  sÞ1 instead of one. S¯ ðQ ; tÞ is the intermediate scattering function averaged over the solid angle of scattering: Z Z 1 1 p=2 S¯ ðtÞ ¼ SðQ ðyÞ; tÞ sin 2y dy SðQ ðOÞ; tÞ dO ¼ 4p 2 0 Z Q max 1 ¼ SðQ ; tÞQ dQ . (31) 2Q max 0 The influence of multiple scattering in the elastic scan can be  ; tÞ in Eq. (8). taken into account by replacing SðQ ; tÞ by SðQ

6. Proposed evaluation scheme for elastic scans As the preceding sections show, there are several reasons why the conventional determination of the mean-square displacement by a plot ln Iel vs. Q 2 can be doubted. Therefore, it is proposed to do the evaluation by fitting a function Iel ðQ Þ calculated using a combination of fractal diffusion (24), multiple scattering correction (30), and evaluation of the elastic intensity by Eq. (8). The fit produces two values for each scan point: D and b. D can be converted to the conventionally derived hr 2 i0 by Eq. (25). Although there is the idea of fractal diffusion behind the fit procedure, this will in general not be the correct model for the physics. Therefore, b is rather an empirical parameter determining

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hr 2 iðtÞ ¼ hr 2 ifast þ 6Dt b .

(32)

All processes are assumed to be Gaussian so that the intermediate scattering function is: ! Q 2 hr 2 ifast expðDQ 2 t b Þ. SðQ ; tÞ ¼ exp  (33) 6 The fast process is assumed to be harmonic, leading to a linear dependence on temperature: hr 2 ifast ¼ A  T.

(34)

The temperature dependence of the time-scale of the slow process (the ‘‘a relaxation’’) is assumed to follow a Vogel–Fulcher law:   bB . (35) D ¼ D1 exp  T  T0 For the constants, typical values close to those of the glass2

˚ K1 , forming polymer polybutadiene were chosen: A ¼ 0:005 A 2 1 1=2 ˚ b ¼ 2, D1 ¼ 200 A ns , B ¼ 1000 K, T 0 ¼ 100 K. The instrument resolution was assumed to be Gaussian with tres ¼ 1 ns corresponding to an energy resolution of 1:24 meV (FWHM) according ˚ The to relation (21) and to operate at a wavelength of 6:27 A. multiple scattering probability was set to s ¼ 0:1 roughly representing the 10%-scatterer used in standard experiments. Fig. 5 shows the relative deviation of the hr 2 i0 calculated by the procedure proposed here (filled cricles). It can be seen that it vanishes for low T where only the constant part of hr 2 iðtÞ from Eq. (34) exists. For high temperatures the fit underestimates by less than 10%. The good performance of this fit may seem trivial, but the fit function is not identical to the test-SðQ ; tÞ. The latter includes the fast part in addition. This can be seen from the b values which are in the range of 0:40; . . . ; 0:45 in the temperature range of 200,y,250 K where this parameter can be fitted reliably. The somewhat smaller value shows that the fast process is included into the fit as a slight additional stretching of the decay of SðQ ; tÞ. The major reason for the remaining deviation of the fitted hr 2 i0 from the real is that the fitted b tends to the value 12 used to construct the data set and for ba1 (or 0) hr 2 i0 ahr 2 iðt res Þ (Fig. 1). Indeed, formula (26) could be used to correct this leading to the dashed curve in the plot showing a deviation of only 3%. It has to be conceded that this is not an option for real data with experimental errors because there b cannot be fixed with sufficient precision. For comparison, Fig. 5 also includes results from evaluations which avoid the calculation of the elastic intensity by integration. Clearly, the most simple linear fit of ln Iel vs. Q 2 drastically

r2 (tres) [%]

0

-10

40

-20 r2[Å2]

estimated -

r2 (tres) ) /

50

-30

30 20 10

( r2

how hr 2 iðtÞ is extrapolated from hr 2 iðtres Þ to the whole time range necessary to calculate integral (8). In addition the multiple scattering parameter, s, is treated as a fit parameter but this one has to be constant for the whole scan. It may be objected that the proposed evaluation scheme requires much more computation time, especially due to integral (8) which has to be carried out numerically. Nevertheless, the whole fit takes about 5 min for 250 scan steps and 10 detectors on a 2 GHz dual core Pentium processor so that this is not a strong argument. In order to test the validity of the evaluation scheme, firstly, a test scan with simulated data is constructed. To mimic the situation in glass-forming materials, where one often encounters a combination of a fast process which only depends on temperature by its intensity and a slow on whose time-scale is strongly temperature dependent, the following time dependent mean-square displacement was assumed:

443

0

-40

0 0

100

50

200

100

150

200

250

T [K] Fig. 5. Relative deviations of different methods to estimate hr 2 i from a simulated elastic scan (without experimental errors). Filled circles: fit by fractal diffusion as described in the text (empty symbols with an additional correction due to varying b). Filled diamonds: linear fit of ln Iel vs. Q 2 (without multiple scattering correction). Filled squares: fit with expression (36) and multiple scattering correction (empty symbols: multiple scattering parameter s determined from To100 K only). Filled triangles: fit with expression (37) and multiple scattering correction. The inset shows the actual values of hr 2 i resulting from the different procedures (labelled at the ends of the curves by the same symbols as in the main plot) together with the assumed hr 2 iðt res Þ (unlabelled, bold curve).

fails for high hr 2 i (diamond symbols). Even for the smaller values at lower temperature there is a systematic error of about 10% resulting from the fact that multiple scattering is not taken into account. A significant improvement is possible by adding a fourth-order term ln Iel ðQ Þ ¼ 

hr 2 i0 2 Q þ C  Q4 6

(36)

and calculating the multiple scattering contribution according to Ref. [2] (Fig. 5: filled squares). The main disadvantage of this form is that the Q 4 term alone does not reflect the high-Q elastic scattering correctly. This leads to an overestimation of s (0.122 in the given example) which in turn induces the deviation at low hr 2 i. As suggested in Ref. [2], it can be determined more reliably from low temperatures only. Here, using To100 K results in the exact value s ¼ 0:1000  0:0001. Using this value leads to an exact reproduction of hr 2 i for low temperatures but larger deviations for T4200 K (Fig. 5: empty squares). A more elegant way to reduce the deviation is to use the pseudo-Voigt function instead of the simple cumulant expansion (36) (Fig. 5: triangles):   hr 2 i0 2 1 þZ Q . (37) Iel ðQ Þ ¼ ð1  ZÞ exp  6 1 þ hr 2 i0 Q 2 =6 This function also has the low-Q limit (4) but a high-Q asymptote Q 2 as Eq. (29). Although at high hr 2 i this fit function by far does not perform as well as the proposed scheme, it may be the best choice if a closed form is required. An important criterion for the practicability of a data evaluation procedure is its performance with data containing statistical errors. Therefore, the construction of data was repeated with statistical errors added. The errors are Gaussian distributed

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100

1000 1

Iel(Q)

100

r2 [Å2]

10 1

r2

estimated

[Å2]

10

0

2

4

Q2 [Å2] 1

0.1

0.1

0.01 0

50

100

150

200

250

T [K] Fig. 6. Tests of estimations of hr 2 i with simulated data including statistical errors. Black dots (online: blue): fit by fractal diffusion as described in the text. Grey dots (online: green): fit fixing b ¼ 0 (multiple scattering parameter s determined from To100 K only). The expected result hr 2 iðt res Þ is plotted as solid curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the this article.)

pffiffiffiffiffi with a spread proportional to Iel as it is expected from counting statistics. The standard deviation at highest intensity (Iel ¼ 1) was assumed to be 1% of the value. Fig. 6 shows that between 1 and 2 ˚ the proposed fractal diffusion fit gives excellent results. It 20 A 2 ˚ are rarely published has to be noted that values of hr 2 i45 A because standard evaluation procedures do not produce reliable 2 ˚ the procedure still produces results in this range. Above 20 A acceptable results but the statistical errors are increasing and the systematic deviation already visible in Fig. 5 becomes clear. It is somewhat disappointing that the procedure performs only mediocre in the low temperature range To150 K. In that range, the constant part (34) dominates hr 2 iðtÞ and therefore the fit should give b ¼ 0. Instead, random deviations due to the statistics may lead to a noticeable curvature in ln Iel vs. Q 2 which is followed by the fit setting b40. Because only values 42 can be ruled out a priori, any value b ¼ 0; . . . ; 2 may result. This in turn leads to the noticeable overestimation of hr 2 i0 on average. Clearly, if it is known that b ¼ 0 it is better to fix this value.3 The result is shown by the green dots in Fig. 6. One can see that in the low temperature range the fit follows the expected values closely and without systematic deviation. Of course, for high temperatures a systematic deviation occurs, because there the t dependence of hr2 iðtÞ becomes noticeable. In conclusion, for practical application one should consider to switch the fitting method at about hr 2 iQ 2max  5. Finally, a test with actual elastic scan data [11] is presented in Fig. 7 (the same data as used in Ref. [2]). The sample was a glassforming liquid, propyleneglycol (PG, 1,2-propanediol, chemical formula: C3 H8 O2 , CAS number: 57-55-6). PG is a glass-forming liquid with a calorimetric glass transition at T g ¼ 167 K. The

3 In this case the procedure reduces to the one suggested in Ref. [2] without non-Gaussianity.

0.01 0

100

200

300

T [K] Fig. 7. Test of elastic scan evaluation procedures with real data. Filled circles: fit by the procedure described in the text. Empty squares: fit as described in Ref. [2] (expression (36) with multiple scattering correction, multiple scattering parameter s determined from To258 K only: s ¼ 0:162). Empty triangles: linear fit of ln Iel vs. Q 2 (without multiple scattering correction). The dotted circles represent data hr 2 iðt res Þ calculated from a separate inelastic measurement. Inset: semilogarithmic plot of Iel vs. Q 2 at the scan point T ¼ 250:1 K. The curves denote the different fits (solid: procedure in text, grey: as in Ref. [2], dashed: linear fit of ln Iel vs. Q 2 ).

incoherent neutron cross-section of the molecule is 642 b—about 16 times higher than the coherent (39 b). The neutron scattering elastic scan data were obtained on the backscattering spectrometer BSS at Forschungszentrum Ju¨lich, Germany. In the standard configuration used here, BSS uses Si-111 monochromator and analysers implying a neutron wavelength of ˚ The instrument has 10 detectors distributed over an l ¼ 6:27 A. 1 ˚ . angular range of 92138 corresponding to Q ¼ 0:1621:87 A The inelastic resolution is 0:76 meV (FWHM) and the shape of the resolution function is approximately Lorentzian. The registration time per point was 14 min yielding about 15 000 counts per (high-Q) detector with the exception of the lowest temperature where a four times longer interval was chosen to improve statistics. The temperature was swept continuously at a ratio of 0.14 K/min. Concerning the quality of the fit, as the inset shows, the proposed method performs significantly better than the simple linear fit of ln Iel vs. Q 2 and also slightly better than the earlier procedure from Ref. [2] (expression (36) with multiple scattering correction, multiple scattering parameter s determined from To258 K only: s ¼ 0:162). Also it can be seen that the new procedure extends to larger hr 2 i values and produces less scatter in the low-hr 2 i range. A direct check of hr 2 i, as for the simulated data, is not possible because hr 2 iðTÞ of the system does not follow any known law. But the same sample was studied also by recording backscattering inelastic spectra at several temperatures on IN16 (ILL,  ; tÞ and Grenoble, France) [11]. These data were converted to SðQ corrected for multiple scattering. From the resulting SðQ ; tÞ, hr2 iðtÞ can be obtained in the range of 0:15; . . . ; 2 ns by fitting expression (23). The large dots in Fig. 7 and the values in Table 1 represent hr 2 iðt res Þ where the resolution time, because the resolution shape is Lorentzian here, is calculated by t res ¼ s1 ¼ 1:31 meV ns=DEFWHM .

ARTICLE IN PRESS R. Zorn / Nuclear Instruments and Methods in Physics Research A 603 (2009) 439–445

Table 1 hr2 i0 at T ¼ 250:1 K calculated by the three procedures also shown in Fig. 7 together with hr 2 iðt res Þ from inelastic neutron scattering via SðQ ; tÞ. T (K)

195 210 230 250 270

2

˚ Þ hr 2 iðt res Þ ðA from SðQ ; tÞ

1:00  0:07 1:22  0:05 1:73  0:10 3:69  0:09 11:6  0:6

2

˚ Þ hr 2 i0 ðA Fractal diffusion

MSC procedure Ref. [2]

Without MSC

1.25 1.33 2.33 5.32 11.6

1.06 1.16 1.83 3.61 7.37

0.55 0.78 1.07 1.90 3.70

It can be seen that only the procedure proposed here is able to reproduce the highest value of hr 2 i at 270 K. Nevertheless, at low temperatures (Tp250 K) this procedure performs worse than the one proposed in Ref. [2] resulting in deviations of up to 45% instead of 5–6%.

7. Conclusions Neutron scattering elastic scans are a widely used method to assess the microscopic dynamics of a sample over a wide parameter (mostly temperature) range. For each scan point, the intensity scattered elastically within the resolution of the instrument is recorded in dependence of the scattering vector Q. Eq. (7) or (8) allow the calculation of the Q dependence of the elastic intensity if the scattering function of the sample and the resolution function of the instrument are known. From the result some general conclusions can be drawn: (1) There is a timescale associated with an elastic scan which is related by Eq. (21) to the width of the resolution function. Nevertheless, in general the initial slope of the plot ln Iel vs. Q 2 , hr 2 i0 =6, does not yield hr 2 iðtres Þ. It is rather resulting from average (14) which requires the knowledge of the function hr 2 iðtÞ. (2) Plots of ln Iel vs. Q 2 are not linear over a larger Q 2 range but in most cases bend upwards for high Q 2 . This is an intrinsic result from integral (8) and has nothing to do with the ‘‘nonGaussianity’’ (a fourth-order term in the Q series expansion of ln SðQ ; tÞ). The range which can be used for a linear fit of ln Iel vs. Q 2 is often rather small. (3) The reduction of the axis intercept in the ln Iel vs. Q 2 plot at Q 2 ¼ 0 is due to multiple scattering. It is not (necessarily) due to experimental shortcomings but cannot be avoided for finite-sized samples. Therefore it is not justified to treat it as a free fit parameter.

445

From these results, it is clear that the usual procedure to determine hr 2 i by a linear fit ln Iel vs. Q 2 is prone to systematic errors. Unfortunately, the results do not imply a definite solution how to calculate hr 2 i ‘‘correctly’’. This would require a knowledge of hr 2 iðtÞ (or even SðQ ; tÞ in the truly non-Gaussian case). As a practicable solution, a parsimonious description with two parameters is suggested: hr 2 iðtÞ ¼ 6Dt b . The test of this procedure shows that (for simulated as well as for real data) for large hr 2 i it is better than the standard procedure as well as one proposed earlier [2]. Using simulated data, where hr2 i is known from construction, the fit results follow the expected results with high precision even to highest values (hr 2 i  80=Q 2max ). For real data, there is a good agreement with the value obtained from inelastic spectra. At low temperatures (i.e. where hr 2 iðtÞ is small and nearly constant close to t res ) the procedure shows some drawbacks. There is a systematic error (too high estimates of hr 2 i) visible for simulated data with errors. The reason lies in the fit parameter b which is ill-determined in this range. Instead of assuming the appropriate value b ¼ 0 it fluctuates depending on the individual realisation of the statistical errors. For real data the same overestimation takes place with respect to the values from inelastic spectra. Therefore, for the application it may be better to switch between a ‘‘traditional’’ method (but including the multiplescattering effect as, e.g. in Ref. [2]) for low temperatures and the one proposed here for high temperatures.

Acknowledgement The programs to simulate elastic scan experiments and evaluate elastic scans were written using routines from the Computational Science and Engineering Department at CCLRC, Chilton, UK (‘‘Harwell Subroutine Library’’) [12]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

According to reports compiled at hhttp://club.ill.fri. R. Zorn, Nucl. Instr. and Meth. A 572 (2007) 874. G.R. Kneller, V. Calandrini, J. Chem. Phys. 126 (2007) 125107. A. Rahman, K.S. Singwi, A. Sjo¨lander, Phys. Rev. 126 (1962) 986. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. M. Be´e, Quasielastic Neutron Scattering, IOP Publishing, Bristol, 1988. S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, vol. 1, Clarendon Press, Oxford, 1984. J. Colmenero, A. Alegrı´a, A. Arbe, B. Frick, Phys. Rev. Lett. 69 (1992) 478. R. Zorn, Phys. Rev. B 55 (1997) 6249. For a derivation of this formula see the appendix of R. Zorn, B. Frick, L.J. Fetters, J. Chem. Phys. 116 (2002) 852. R. Zorn, M. Mayorova, D. Richter, B. Frick, Soft Matter 4 (2008) 522. hhttp://hsl.rl.ac.uk/archive/hslarchive.htmli.