Neutron Crystallography and Laue Diffraction

$Page 1: Neutron Crystallography and Laue Diffraction$
Tim Grüne

Thursday, January 13th, 2011

Neutron Crystallography and Laue DiffractionTim Grüne

Shelx Group Seminars

http://shelx.uni-ac.gwdg.de

Neutron Crystallography and Laue DiffractionShelx Group Seminars 1/34

http://shelx.uni-ac.gwdg.de

Tim Grüne

The de–Broglie Wavelength

The wave-particle duality associates a wave with every quantum mechanical particle of mass m, the de–Brogliewavelength

λ =h

mv

√1−

v2

c2

For neutrons, a wavelength of λ = 1− 2Å corresponds to a velocity of v = 4− 2kms∗. Neutrons in this energy

range are called thermal neutrons for the energy corresponds to about 290K.

∗h = 4.135 · 10−15eV s,mn = 939.5 · 106eV/c2

Neutron Crystallography and Laue Diffraction 2/34

Tim Grüne

Neutron Crystallography

“Wave” is a very general concept and the fact the a particle is associated with a wavelength does not imply thevalidity of the concepts we are used to from X-ray crystallography.

However, starting from the Schrödinger equation(−

h̄2

2mn∆ + V (r)

)Ψ(r) = EΨ(r)

and a few simplifying assumptions, one can actually derive the same equations for the interaction betweenparticle waves and matter that Ewald derived for electronmagnetic radiation.

For the not too difficult derivation, see e.g. V.F. Sears, “Neutron Optics”, Oxford University Press, 1989, Ch. 2.


Tim Grüne

Nuclear Scattering of Neutrons

In the Born approximation, the potential responsible for the scattering of a neutron by a nucleus is described bythe Fermi pseudopotential

V (r) =2πh̄2

mnbδ(r) (1)

where

b = bc +2bi√

I(I + 1)s · I

is called the bound scattering length. The second term takes the interaction between the neutron spin s and thenucleus’ spin I into account. According to the “Intl. Tables C”, b is independent of the scattering angle θ ∗.

∗When absorption of neutrons can occur, b is complex (e.g. 3He, 6Li) in which case the Fermi pseudopotential is an invalid approxi-mation [1].


Tim Grüne

Scattering Amplitude

The X-ray scattering amplitude f(θ) decreases with the scattering angle θ.

The neutron scattering amplitude by the nucleus calculates as

f(θ) = −4π2mn

h̄2

⟨k′|V̂ (r)|k

⟩= −

1

(2π)3

∫drei(k−k

′)·rbδ(r)

= −b

I.e., the scattering amplitude f is independent of the scattering angle θ


Tim Grüne

The Advantage of Neutron Scattering

Bond scattering lengths of some atom types:

bc [fm] bi [fm] bc [fm] bi [fm]1H -3.74 25.27 2H 6.67 4.0412C 6.65 0 (I = 0) 14N 9.37 2.016O 5.80 0 (I = 0) 31P 5.13 0.2

The scattering amplitude section of photons is proportional to the number of electrons per nucleus, which is whywe see hydrogens only at very high resolution, i.e. from very ordered crystals.

The scattering amplitude of neutrons varies from nucleus to nucleus, but the order of magnitude for 1H and 2H

is comparable to that of C, N, O. . . , i.e. hydrogens are visible even at moderate resolution.

The main application of neutron scattering (in biology) is therefore the study of interaction between ligand andreceptor.


Tim Grüne

Neutron Production

There are two main processes to produce neutrons:

Fission the “common” chain reaction, used e.g. in nuclear power stations

n+ 235U −→ 236U −→ 92Kr + 141Ba+ 3n

Thermal neutrons are required to maintain the chain reaction. The product neutrons have energies about1MeV and need to be slowed down to meV by means of moderators, often D2O. The temperature of themoderator influences the peak of the energy distribution of the neutrons.

Spallation High-energy protons (GeV) hit a metallic target (U, W, Pb, Hg). The excited nuclei “boil off” par-ticles, including ≈ 20 high-energy neutrons. Thermodynamically this process is one order of magnitudemore efficient than fission, even though energetically there is not much gain due to the energy required toaccelerate the protons.

There are no “home sources” for neutrons. . .


Tim Grüne

Neutron Sources

This is a (non-comprehensive) list of neutron sources around the world:

• ILL Grenoble, France: reactor• FRM II Munich-Garching, Germany: reactor• ISIS Rutherford Appleton Lab, UK: spallation• SNS Oakridge, USA: spallation• J-PARC Japan: spallation• planned: ESS (European spallation source) in Lund, Sweden to become the

strongest spallation source

The ILL is the strongest fission source and now at the thermodynamic limits of heat transfer (1.5MW/l).

Spallation sources are thermodynamically about 10 times more efficient and therefore the favoured neutronsources.


Tim Grüne

Neutron Optics

Just like in the context of synchrotron/ X-ray radiation, the term “optics” refers to guiding and focussing neutrons,and to wavelength selection.

Neutrons are electrically neutral and therefore their trajectory cannot easily be controlled (unlike electrons orprotons).

Neutron radiation is also not as directed as e.g. synchrotron radiation, which is one reason for the relatively highinefficiency of neutron sources.

Neutrons are handled similarly to X-rays by mirrors and reflections at crystals (monochromators).


Tim Grüne

Neutron Optics

There are two main devices used to control neutrons:

Monochromator Wavelength Selection, Focussing. Multilayer crystals are used just like for X-rays, e.g. LADI-III at the ILL has a Ni/Ti multilayer band-pass filter. The bandwidth is achieved through a controlled mosaicityof the crystals.

θ1λ1

θ2

λ2

"white" neutrons

Two slightly inclined mosaic blocks select to slightlydifferent wavelengths from the white beam.λi = 2d sin θi

Collimator Direction and Divergence of beam.

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a2a1

L

n−absorbing

material

The final neutron beam has a maximum divergence ofαmax = (a1 + a2)/L

(Intl. Tables. Cryst. C, Ch. 4.4)


Tim Grüne

Neutrons and (Quasi–)Laue Diffraction

• Neutrons are rare• Monochromatic data collection would take very long (weeks to months)

Solution: use energy range of neutrons• Whole spectrum (so-called “white beam”): Laue diffraction• Bandwidth between λmin and λmax: quasi-Laue, or “pink beam” diffraction.

Even with quasi-Laue diffraction, a neutron data set collection takes several days.

Feasibility studies:

• Kalb(Gilboa) et al., “Neutron Laue diffraction experiments on a large unit cell: concanavalin Acomplexed with methyl-α-D-glucopyranoside”, JAC (2001), 34, p. 454• Coates et al., “A Neutron Laue Diffraction Study of Endothiapepsin: Implications for the Aspartic

Proteinase Mechanism”, Biochem (2001), 40, p. 13149


Tim Grüne

Ewald Sphere — Monochromatic Case

Reciprocal Lattice with origin (000).


Tim Grüne


*d = 1/dReciprocal Lattice with origin (000).

Lattice defects limit resolution to d∗ = 1/d


Tim Grüne


*d = 1/d

S

|S0| = 1/ λ



The Ewald sphere with radius 1/λ touchesthe lattice origin. Lattice points on the surfaceof the sphere fulfill the Laue conditions.

a · S = h

b · S = k

c · S = l


Tim Grüne


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��*d = 1/d

S

|S0| = 1/ λ



The Ewald sphere with radius 1/λ touchesthe lattice origin. Lattice points on the surfaceof the sphere fulfill the Laue conditions.

a · S = h

b · S = k

c · S = l

By rotation about one axis one can record allreflections within the shaded “torus”.


Tim Grüne

Ewald Sphere — quasi–Laue Case

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|So| = 1/λmin

|S′o| = 1/λmax

d∗ = 1/d

Without rotation, all reflections inside theshaded region, limited by 1/λmin, 1/λmax,and d∗, fulfill the Laue conditions.

These reflections are fully recorded.


Tim Grüne

Ewald Sphere — quasi–Laue Case

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Incident beam direction

Without rotation, all reflections inside theshaded region, limited by 1/λmin, 1/λmax,and d∗, fulfill the Laue conditions.

These reflections are fully recorded.

The centroid of each reflection corresponds toexactly one wavelength, because there is onlyone circle through the lattice point and the lat-tice origin, and with its centre on the incidentbeam direction.


Tim Grüne

Number of Images

In the monochromatic case, one collects adjacent frames

1. Data completeness, i.e. full coverage of reciprocal space2. Collect full reflections which are usually spread across frames.

In the case of (quasi-)Laue, the second point is automatically taken care of by the wavelength range.

For a complete coverage of reciprocal space, still images must be recorded in steps of

∆φmin = sin−1(λmax/2d)− sin−1(λmin/2d)

This results in data with significant overlap. Just for overall completeness, a frame every

∆φmax = π∆λ/d

is actually sufficient (see [4]).

For a 2 Å dataset with ∆λ = 1Å, this makes four images every π/2 = 90◦, but high-resolution data areincomplete in this case, so a compromise between ∆φmin and ∆φmax has to be found.


Tim Grüne

Problems with (Quasi-) Laue diffractions

Compared to monochromatic data collection, Laue diffraction suffers from some disadvantages:

1. Spatial Overlaps2. Harmonic Overlaps3. (Wavelength-) Scaling

This is especially true for X-ray Laue diffraction and to a much lesser extent for neutron Laue diffraction.


Tim Grüne

Spatial Overlaps

Already for monochromatic data overlapping reflections can be a serious problem, especially in the case of largecell dimensions and a high degree of mosaicity - both rather common for macromolecules.

(Quasi-)Laue diffraction makes this a lot worse.

The situation can be improved by a larger sample-to-detector distance. Overlaps can be estimated by profilefitting, because the overlap is usually not perfect.

0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1

Inte

nsity

Detector

With an initial guess of position, standard deviation, and amplitude, the two spots can be “deconvoluted” byfitting one normal distribution each. Of course for “distorted” reflections this is not ideal.Neutron Crystallography and Laue Diffraction 20/34

Tim Grüne

Spatial Overlaps

Already for monochromatic data overlapping reflections can be a serious problem, especially in the case of largecell dimensions and a high degree of mosaicity - both rather common for macromolecules.

(Quasi-)Laue diffraction makes this a lot worse.

The situation can be improved by a larger sample-to-detector distance. Overlaps can be estimated by profilefitting, because the overlap is usually not perfect.

0

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1

Inte

nsity

Detector

µ1= 0.50σ1= 0.05A1= 3.0

µ2= 0.60σ2= 0.04A2= 4.0

With an initial guess of position, standard deviation, and amplitude, the two spots can be “deconvoluted” byfitting one normal distribution each. Of course for “distorted” reflections this is not ideal.Neutron Crystallography and Laue Diffraction 21/34

Tim Grüne

Harmonic Overlaps

Harmonic overlaps result from an ambiguity in Bragg’s Law (without multiple reflections)

λ = 2d sin θ

If it holds for a pair (λ, d), it also holds for (λ/2, d/2), (λ/3, d/3), . . . with the same diffraction angle, e.g. theseharmonic reflection overlap perfectly.

However, the number of single reflections is greater than 70 % even for (white) Laue and increases to 95 % ifλmax/λmin ≤ 2 (Cruickshank et al., Acta Cryst (1987), A43).


Tim Grüne

Harmonics in the Ewald Sphere

(−7,7)

(−6,6)

(−5,5)

(−4,4)

(−1,1)

The reflections 4(−1,1), 5(−1,1), 6(−1,1), and7(−1,1), are all in reflection position. Since theiroutgoing wave vectors S are all parallel, their scatter-ing angle θ is identical and they all map to the samespot on the detector.


Tim Grüne

Dealing with Harmonics

The number of harmonic reflections can be reduced by an appropriate choice of the band-width. Not only forthis reason neutron diffraction data are often recorded with λmin = 3 Å and λmax = 4 Å.

Advantages:

• Reducing number harmonic overlaps• Long wavelength increases the spatial separation of Bragg

peaks• Small bandwidth of neutrons reduces the amount of diffusely

scattered neutrons (background noise).

The resulting resolution limit of 1.5 Å is sufficient for most macromolecular studies.

Neutron diffraction can deal with harmonics even better by use of Time-of-Flight Spectrometers.


Tim Grüne

Neutron Laue and Time-of-Flight Instruments

Photons have a constant velocity and photon detectors used in crystallography are not energy sensitive. Henceharmonic reflections are real overlaps that one can only deconvolute theoretically.

Because the energy/wavelength of neutrons corresponds to their velocity (E = 12mnv2), the wavelength of

each reflection can be determined from the time it takes from the sample to the detector.

Such instruments are called time-of-flight spectrometers, e.g. available at the ILL in France and the ISIS in theUK and under construction at the new spallation source at Oakridge.

Time-of-flight instruments also facilitate the indexing of reflections because the wavelength λ for each reflec-tion is known and hence conventional indexing methods as used in monochromatic crystallography can beemployed.


Tim Grüne

Scaling Laue-Data: Wavelength Normalisation

An (hkl)-file does not contain the wavelength the data were collected at, and all reflections must be normalisedas though collected at one wavelength. The intensity of a reflection is wavelength dependent:

• The intensity I = I(λ) of the beam is not constant.• The detector sensitivity is wavelength dependent.

The spectra of different insertion devices at ID09, ESRF Greno-ble show strong variation with energy, making wavelength nor-malisation difficult.Scaling is performed using symmetry equivalents collected atdifferent wavelengths. The program LAUENORM [5] divides thebandwidth into bins, and applies a polynomial fit to the scale fac-tors with order of at least 4 (which seems optimistic consideringthe picture).

Currently I do not know whether for neutrons it might be appropriate to assume e.g. a Maxwell-Boltzmanndistribution of the velocities which could be fitted more easily.


Tim Grüne

Real Sample Images

Courtesy M. Blakeley, ILL Grenoble


Tim Grüne

(Simulated) Sample Image

Laue diffraction pattern simulated withLAUEVIEW from a rhombohedral cella = 325.5 Å, α = 61.7◦

λ = 0.4− 1.4 Ådmin = 3 Å (from [4]).The ellipses are so-called zones, theisolated reflections at the intersection ofzones nodal points.


Tim Grüne

Zones

A “zone” is the collection of reflections originating from crystal planes with one common direction

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Crystal Oθ

θ

S

T

R

ψ

• Let (ST ) one direction in a crystal plane, with reflection R.• Let the plane rotate about the direction (ST ). Each time

it crosses a lattice point, it becomes a crystal plane andcreates a reflection.• All reflections lie on a circle centred at T , the semi-rays

(SR) lie on a cone.• The detector intersects the cone, which results in an ellipse

(Ψ < 45◦), a parabola (Ψ = 45◦), or a hyperbola (Ψ >45◦).Note that θ is the diffraction angle between (SR) and the crystal plane,changing as the plane rotates about (ST ), whereas Ψ is the anglebetween the line (ST ) and (SO) being constant for all reflections onthe circle, i.e. θ 6= Ψ.

Zones can be used for indexing Laue-data (Ravelli et al., “Towards Automatic Indexing of the Laue DiffractionPattern”, J. Appl. Cryst. (1996), 29, pp. 270–278) and for orienting crystal w.r.t. symmetry elements.


Tim Grüne

Laue/Neutron Software: Processing

Laue data (X-ray and Neutron) are mostly processed using the Daresbury Laboratory Laue Software Suite,http://www.srs.ac.uk/px/jwc laue/laue top.html, by John W. Campbell, now maintained by Prof. Quan Hao (Uni-versity of Hong Kong) (private communication M. Blakeley, ILL).

The last update of this page was in 2001

There is also a commercial product, “precognition” from renzresearch.com (Z. Ren), with its latest release from2005.

Neither of these programs are specialised for neutrons or adjusted for time-of-flight spectrometers.


http://www.srs.ac.uk/px/jwc_laue/laue_top.html

http://renzresearch.com

Tim Grüne

Laue/Neutron Software: Refinement

Refinement of neutron data is possible with shelxl (Coates et al., “A Neutron Laue Diffraction Study of Endoth-iapepsin: Implications for the Aspartic Proteinase Mechanism”, Biochem (2001), 40, p. 13149), but nowadayspeople mostly use phenix.refine (P. Afonine et al., Acta Cryst. D (2010), p. 1153, private communication F.Meilleur (Oakridge) and M. Blakeley (ILL Grenoble)).

phenix.refine uses the combined X-ray and neutron target function

T = wnxc (wxrayTxray + wneutronTneutron) + wcTgeom

The above reference does not mention how Tgeom in phenix.refine treats e.g. the different lengths forhydrogen atoms in X-ray and neutron data.


Tim Grüne

1H vs. 2H

The negative bond scattering length of hydrogens results in negative density which cancels with the positivebond scattering lengths of other atoms (N,C,O, . . .). To improve the visibility, hydrogen 1H can be replacedwith deuterium 2H. This is achieved by soaking the crystal in D2O for several months (see e.g. the ’feasibilitystudies’ above).

Calculated neutron maps at 1.5 Å

Hydrogenated Deuterated

Afonine et al., Joint X-ray and neutron refinement with phenix.refine, Acta Cryst. D66 (2010)


Tim Grüne

Summary

Neutron diffraction shows hydrogen positions even at medium and low resolution.

Because neutrons are difficult to produce, quasi-Laue diffraction is applied.

Time-of-flight spectrometers record the energy of neutrons facilitating indexing and deconvolution of harmonicoverlaps.

Current software, especially for data integration, does not reflect the hardware capabilities.


Tim Grüne

References

1. V. F. Sears, Neutron Optics, Oxford University Press 1989

2. J. L. Amorós, M. J. Buerger, M. C. de Amorós, The Laue Method, Academic Press 1975

3. G. Eckolf, H. Schober, S. E. Nagler (editors), Studying Kinetics with Neutrons, Springer 2010

4. Z. Ren et al., Laue crystallography: coming of age, J. Synchr. Rad. (1999), 6, pp. 891–917

5. J. W. Campbell, Daresbury Laboratory Laue Software Suite, http://www.srs.ac.uk/px/jwc laue/laue top.html


http://www.srs.ac.uk/px/jwc_laue/laue_top.html

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Neutron Crystallography and Laue Diffraction