Biological Macromolecular Dynamics, (1996) p.1 Adenine press, Ed. S. cusack et al.
Iterative Calculation of the Vibrational Density of States from Incoherent Neutron Scattering Data with the Account of Double Scattering 1 M. Settles and W. Doster*
Technische Universität München Physik-Department E13, D-85748 Garching, Germany
Abstract Transmission coefficients in the vicinity of 0.9 for biological samples are considered to be a reasonable compromise between optimizing the count rate and minimizing multiple scattering. However, even with such samples between 10 and 20 % of the incoming neutrons are multiply scattered. This fact should be accounted for in the data analysis. Our numerical calculations for disk-shaped samples show, consistent with experimental results on myoglobin, that multiple scattering gives rise to an approximately angle independent background. This affects in particular experimental quantities which have to be evaluated at small scattering vectors Q, such as the density of states. We propose a new iterative scheme which allows to derive the vibrational density of states taking into account explicitly second scattering. The method converges rapidly which is illustrated by experiments on hydrated myoglobin. We recover the otherwise obscured librational bands of hydration water and of the methyl groups.
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presented at the Workshop on Quasielastic and Inelastic Neutron Scattering in Biology, Grenoble, Oct.96 To whom correspondence should be addressed, fax: +49 89 2891 2473, email:
[email protected] *
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1. Introduction
The effect of multiple scattering in neutron spectroscopy was noticed for the first time in 1948 by Wollan and Shull (1). Initial theoretical calculations go back to Vineyard (2). A very systematic analytical treatment of this subject was published by Sears in 1975 (3), which is often referenced but rarely applied (4). In biological applications of neutron spectroscopy multiple scattering corrections have recieved little attention. Cusack and Doster (5) determined a low frequency density of states of myoglobin accounting for second scattering (3). They found that second scattering events of the kind elastic-inelastic or vice versa introduce a Q-independent inelastic background. The background becomes relevant at low Q since incoherent inelastic spectra increase with Q2. Also, the evaluation of mean square displacements and elastic incoherent structure factors at ω = 0 require an extrapolation to Q = 0. We have performed calculations similar to those of Sears using an infinite, flat sample disk. The spatial integrals are calculated analytically, starting from the point of incidence of the neutron.The Sears procedure starts from the point of exit and works backwords. The angular integrations are performed numerically. Fig.1 shows the relative scattering fraction of single, double and higher order scattering versus the transmission of the sample. The effect is not small: At t= 90%, the fraction of singly scattered neutrons is only 0.8. The figure also implies that for practical purposes it should be allowed to ignore terms higher than second order. In the following we intend to calculate the one phonon density of states using an iterative scheme which takes into account explicitly the second scattering.
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2. The multiphonon expansion
The intermediate scattering function of atom i, in a harmonic system consisting of N atoms can be written in a very concise way as follows (6):
( )
( ( ) ( )) ( (
( )
( )))
i i i 2 i i I Q , t = exp I Q, t − I Q ,0 ≡ exp Q ⋅ γ Qˆ , t − γ Qˆ ,0 (1) (1) with the one phonon correlation function for atom i, Qˆ ξ µ
i2
2 2 h Q 2 i ˆ ∑ Q γ Q, t = 2 m µ hω µ i
( )
The sum
∑µ
[1]
γ i (ˆQˆ , t ) :
n (ω ) ⋅ e − jωµ t + n (ω ) + 1 ⋅ e jωµ t µ µ
[2]
has to be extended over the 3N -6 normal modes of the system.
The vibrational amplitude of atom i participating in mode µ is given by the eigen vector ξi µ . The term, exp( − I
i (Qˆ ,0 )) ,denotes the usual Debye-Waller-factor, which is different in (1)
general for each atom and also depends on the direction of the scattering vector
Qˆ = Q / Q.
Averaging the individual Gaussian Q2-dependences over all atoms and orientations of the powder sample may lead to deviations from the pure Q2 law (9). The multi phonon expansion in the time domain follows from equ.[1]:
I
i (n)
(Q , t ) =
I i (Q , t ) n! (1) 1
n
[3]
The n-phonon spectrum in frequency space is given by the n-fold convolution of the one-phonon spectrum with itself: i
S (Q , ω ) = e
−Q2 I
(1)
(Qˆ ,0 )
2n ∞ Q i i ⋅ (δ (ω ) + ∑ (γ (Qˆ , ω ) ⊗ ... ⊗ γ (Qˆ , ω ))) ( 1 ) ( 1) n=1 n!
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[4]
Expanding this equation in powers of Q2 leads to the so-called Placzek-expansion. The Q2term is a linear function of the squared eigenvectors,
i 2
Qˆ ξ µ
, which makes it easy to
calculate the average in equ.[6]. The higher order terms contain products of sums of the squared eigen-vectors. The scattering of neutrons by organic matter specifically selects the dynamics of the protons due to their dominant (incoherent) cross-section. Therefore the Q2 term gives exactly the one phonon spectrum, averaged over all protons and directions, which yields the proton weighted the density of states:
1 h2 n(ω ) S(1) (ω ) = g prot (ω ) 6 m prot hω
[5]
and
ρ 2 g prot (ω ) = ∑ µ ξ µi
prot
⋅ δ (ω − ω µ ),
[6]
which is normalized to 3. To derive the density of states in general one has to evaluate the limit limQ →0 S(1) (Q, hω ) / Q 2 (7). In practice this method is hampered with difficulties: First of
all, the neutron kinematics forces Q away from zero with increasing energy exchange which introduces the need to extrapolate. Second, the accurate determination of this limit is corrupted by multiple scattering. To account for these effects one usually assumes a constant background and introduces higher order terms in Q2 (5,8). An iterative method to determine the density of states from a constant angle spectrum was discussed in ref.(3,4). In this work we use the Placzek-expansion to fit experimental data. The multiple scattering background is calculated quantitatively with iterative improvement.
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3. Second order scattering We consider the case of an infinite plane slab sample of thickness d. For second scattering one obtains as a function of the two scattering angles and the energy exchange normalized to the total number of the scattered neutrons the following expression (4):
s2 (θ1 ,θ 2,ω1 ,ω2 ) = 2α 0 ⋅ e (α 0 − α 2 )
Σ( E 0 ,θ1 ,ω1 ) Σ( E1 ,θ 2 ,ω2 ) ⋅ ⋅ Σ0 Σ1
α1 ( α α − α1 2 1) sinh(α 1 − α 0 ) sinh(α 2 − α 0 ) α ⋅ e 1 ⋅ − α 2 − α1 α1 − α 0 α2 − α0
[7]
Σi denotes the total cross section of the i-times scattered neutron, which depends through the neutron kinematics on the energy Ei before the scattering event. ∑ ( Ei ,θ i +1 , hωi +1 ) represents the dynamical structure factor which determines the scattering angles and energy exchanges of the scattered neutrons depending on the dynamics of the sample.. The other terms describe the attenuation of the neutrons in the sample due to further scattering and the geometrical factors of the plate. The quantity α i =( Σ ) / (2 cos(θ i )), gives the effective d diameter of the sample for a given flight direction, divided by two. θi denotes the angle of the i-times scattered neutron relative to the sample normal.
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The final goal is to calculate the second order scattering in a particular direction Ω and the ∞
energy exchange hω. To this end one has to integrate over
∫ d (hω1,2 ) and all
− E0
directions ∫ dΩ1,2 under the conditions, hω1 + hω2 = hω and Ω1 + Ω2 = Ω. The last equation is symbolic in the sense that scattering angles should not simply be added, but have to be transformed between the coordiate systems of the once and twice scattered neutrons using rotation matrices . Since the energy integrals can be considered as convolutions, it turns out that the combination with the elastic scattering term δ ( hωi ) , and also elastic-elastic and elastic-inelastic , can be evaluated analytically. Only the integration over all directions has to be performed numerically. Fig.2 shows, in comparison with experimental data, the components elastic-inelastic at fixed energy exchange, versus Q2 calculated without any adjustable parameters. To compute the elastic and inelastic scattering, the sample was assumed to be homogeneous and isotropic, and the density of states of D2Ohydrated myoglobin at 180 K was used as input: S (θ1, hω = 0) = e
S (θ 2 , hω ) = e
− Q2 ( E0 ,θ ,hω = 0)⋅γ (0)
− Q2 ( E0 ,θ 2 ,hω )⋅γ (0)
∞ dt Q2 ( E0,θ 2,hω )γ (t ) ⋅ ∫ e jωt e [8] −∞ 2πh
The main difference between elastic-inelastic and inelastic-elastic consists in the modified kinematics of the neutron after an energy gain: Because of the larger values of Q 2 ( E0 + hω , θ 2 ,0) , the probability to be scattered elastically is reduced. Fig.2 shows that
double scattering leads to a nearly Q-independent ( but energy dependent) background which comes close to the extrapolated result A(ω) of the Plazcek- expansion.
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Fig.3 shows the energy dependence of the second scattering for the particular scattering angle of 90o in comparison with the to Q=0 extrapolated experimental spectrum. This result confirms the the notion of an approximately Q-independent multiple scattering spectrum (5).
4. The iterated density of states According to equ.[5] the density of states gprot(ω) follows from S(1)(ω) = B(ω), the coefficient of the Q2 term in the Plazcek expansion. In the zero order iteration one determines the offset A(ω) at Q = 0 directly from the data (symbols in fig.2) according to: S (Q, hω ) = A(hω ) + B(hω ) ⋅ Q 2 + C (hω ) ⋅ Q 4 [9]
Above hω = 40 meV the last term is omitted, since the scatter of the data does not allow to determine the fourth order term accurately. With the resulting B(ω) and thus g 0prot (ω ) , the zero order density of states, we calculate the double scattering background Adsc.(ω) according to equ.[7].. With A(ω)=Adsc fixed we now determine B(ω) and C(ω) again by fitting the data to: equ.9. This procedure is repeated until convergence is achieved, which occurs typically after 3 to 7 cycles. Fig.4 shows the intermediate steps of the iteration working on data obtained with a hydrated myoglobin sample for t = 92 %. Fig.5 summarizes the final gprot(ω) of dry and hydrated myoglobin showing the libration bands of H2O and D2O at 70 and 50 meV and the methyl libration band of the protein at 30 meV.
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Figures
Fig.1: The calculated relative scattering fraction of single, double and multiple scattered neutrons versus sample transmission. At t = 0.9, 80 % is due to 1x- scattering, 14% of the neutrons are scatterered twice and 3% is due to higher order scattering events..
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Fig.2: The dynamical structure factor at fixed energy exchange ( 3, 10 and 30 meV) versus Q2 illustrating the use of equ.[9], full symbols: experimental data (D2O-hydrated myoglobin, T = 180 K), dashed line: the extrapolated A(ω), open circles and open triangles: the elastic-inelastic and inelastic-elastic scattering event respectively and their sum Adsc(full line).
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Fig.3: The energy dependence of Adsc (solid line) at the scattering angle of 90o calculated without adjustable parameters in comparison with extrapolated (Q=0) experimental data.
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Fig.4: Intermediate steps of the iterated density of states, the method has converged after seven iterations.
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Fig.5: Iterated density of states of hydrated and dry myoglobin showing the librational bands of H2O, internal H2O, D2O and the methyl groups of the protein._
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References (1) E.O. Wollan and C. G. Shull, Phys.Rev., 73:830, (1948) (2) G.H. Vineyard, Phys. Rev. 91:239, (1953) (3) F.V. Sears, Adv. in Phys., 24:1, (1975) (4) M. Bee, in Quasielastic Neutron Scattering. Adam Hilger, Bristol and Philadelphia,,p.107(1988) (5) S. Cusack and W. Doster Biophys.J. 58:243, (1990) (6) W. Marshall and S.W. Lovesey in Theory of thermal neutron scattering. Clarendon Press, Oxford (1971) (7) P.A. Egelstaff in Dynamics of Disordered Materials, 37:1 ( 1989) Springer Proc in Phys. (8) J. Wuttke, W. Petry, G. Coddens and F. Fujara, Phys.Rev. E, 52,4:4026 (1995) (9) M.Settles and W.Doster, Faraday Discussion 103 (1996) 269.
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