o ak r idge n ational l aboratory u. s. d epartment of e nergy 1 handling single crystals in...

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OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY

Handling Single Crystals in Inelastic Neutron Scattering Experiments

Mark Lumsden

Center for Neutron Scattering

ORNL

To be published, Journal of Applied Crystallography With Lee Robertson and Mona Yethiraj

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Outline:

History and current state Better way – what it enables in experiments

– how to use it in practice Basic concept Application to triple-axis Application to time-of-flight spectrometers

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How single crystals were handled in the past (and present) Triple-axis: usually have to work with a set of

orthogonal vectors. If symmetry is less than orthorhombic, this is very difficult. Measurements are restricted to a plane.

TOF spectrometers: very little capability for handling single crystals during an experiment. Most things need to be calculated in a manual manner. Some limited abilities in data analysis/visualization.

Must be a better way!!

UB matrix

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What it enables in experiments:

Ability to handle any crystal symmetry

(becoming increasingly important)

For triple-axis, allows movements (in h,k,l,E space) out of scattering plane

For TOF spectrometers, allows full mapping of measured intensity into h,k,l,E space Simplifies data analysis/visualization Requires less intervention from user Provides considerable ease-of-use

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How do you do an experiment on MAPS• Complicated sample alignment using GENIE.

• From this alignment, you can manually calculate what is needed to align some direction with incident beam

• Manually insert a bunch of information into mslice to allow you to look at data (extremely difficult with lower than orthorhombic sym.)

Possible experiment with UB matrix• Measure 2 or more reflections and tell the computer what their indices are

– UB matrix calculation completed!!• Tell the computer where (in reciprocal space) you’d like to go• (e.g. put c-axis along ki with a-axis vertical )• Convert full measured data set into h,k,l,E• Run ‘mslice’ with very little data input – only what you want to look at (e.g.

I want a slice in the (hk0) plane with a specific E and l range). NOTE: this would now work for ANY crystal symmetry!!

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UB Matrix Myths

• Only works for 4-circle diffractometer

• Only works for diffraction

NO – can easily be adapted to any goniometer

NO – we have extended the formalism to handle inelastic case

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UB Matrix – basic conceptWilliam R. Busing and Henry A. Levy, Acta. Cryst. 22, 457 (1967)

Based on rotation matrices 2 important matrices involved:

B transforms lattice constants into orthonormal space (crystal coord. system)

U transforms from crystal coord. system into instrument coord. system

What does this mean?? Coordinate mapping from angle space into reciprocal space!!

Basic Equation: QL=NUBQ What do you need to do to implement it:

Scheme to calculate U from measured reflections (well tested schemes already exist – need Q=UBQ)

Convert from angles to h,k,l,E (simple Q=(UB)-1Q) Covert from h,k,l,E to angles

- problem is usually overdetermined – need some appropriate constraints

B=b1 b2 cos β3 b3 cos β2

0 b2 sin β3 −b3 sin β2 cosα 1

0 0 1/ a3

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Triple-axis implementation• Busing and Levy’s formalism dealt with 4-circle diffractometer• Needed to derive expressions for different goniometer• Needed to generalize implementation to handle inelastic case

N=1 0 00 cos ν −sin ν0 sin ν cosν

M=cos μ 0 sin μ0 1 0−sin μ 0 cos μ

=cosω −sin ω 0sin ω cosω 00 0 1

Θ=cosθ −sin θ 0sin θ cosθ 00 0 1

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Triple-axis implementation (cont.)

QL=ki−kf

¿ 0k i0 −−k f sin φk f cos φ

0

¿ k f sin φk i−k f cos φ0

QL=Θ MNUBQ

Qθ= ΘQL= MNUBQ=q00 Qθ= MNQν

Q v= N M Qθ

uν= N M 100 =cos μcosω−cos ν sinωsin ν sin μcosωsin ν sinωcos ν sin μcosω

All of Busing and Levy’s calculations use this vector!

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Triple-axis implementation (cont.)

Calculate U:Measure N reflections and specify their h,k,l

Ways to calculate U:1. 2 non-collinear reflections measured and lattice

constants given2. 3 or more non-coplanar reflections given. This will

calculate U and fully refine the lattice constants.

h,k,l,E from given angles:1. From ki, kf, , calculate Q and

2. Measured s1 gives s13. together with , gives u

4. Q=(UB)-1Q gives h,k,l

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TOF Spectrometer Implementation

QL=k i−k f−k f cosψ sin { φ'

¿k f cosψ cos {φ

¿

¿ 0k i0 −¿ ' ¿−k f sin ψ ¿

k f cosψ sin { φ'

¿k i−k f cosψ cos {φ¿¿ ¿ ' ¿k f sin ψ ¿

tan θ=k i− k f cosψ cos { φ '

k f cosψ sin { φ '¿ tan χ=

k f sin ψ

Q∣∣≡Q ¿

Q∣∣¿Q∣∣=k i2−2k i k f cos ψ cos { φ ' k f

2 cos2ψ ¿ ¿ ¿

Qθ= ΘQL= MNUBQ=Q∣∣

0Q ¿

Qv= N M Qθ uν=

N M q Q∣∣

0Q¿

¿1q Q∣∣cos μcosωQ ¿sin μ

−Q∣∣cos ν sinωQ∣∣sin ν sin μcosωQ ¿cos μ sin ν

Q∣∣sin ν sinωQ∣∣cos ν sin μ cosωQ ¿cos μ cosν

Need to handle kf out of plane

uν=cos μ cosω−cos ν sin ωsin ν sin μ cosωsin ν sinωcos ν sin μ cosω

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TOF Spectrometer Implementation

• As we can now calculate u from measured angles, we can use identical procedures to calculate U.

• Q=(UB)-1Q provides a mapping from measurement to h,k,l,E space!! (Given ki,kf,,’,s1,,observation)

Q∣∣=k i2−2k ik f cosψ cos {φ'k f2 cos2ψ

¿Q ¿=k f sin ψ ¿ tan θ=k i−k f cosψ cos {φ'

k f cosψ sin { φ'¿ ¿ ¿ s1=θω ¿Q v= Q∣∣cos μcosωQ¿sin μ

−Q∣∣cosν sin ωQ∣∣sin ν sin μcosωQ ¿cos μ sin ν

Q∣∣sin ν sinωQ∣∣cos ν sin μcosωQ ¿cos μcos ν ¿¿

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TOF Angle Calculations (case 1)Place a plane horizontal

• Specify plane with either plane normal or a pair of vectors within the plane – either can be converted to a plane normal unit vector in the -coordinate system, u

(using Q=UBQ):

• To get plane normal vertical, we need:

QL=001 =Θ ΜΝuν⊥ ¿ { ΝΜ 001 = −sin μ

cos μ sin νcos μ cosν =uν⊥ ¿

1ν⊥¿ ; tan ν=u2ν⊥¿ /u3ν⊥¿

¿¿ sin μ=−u¿

¿

1ν⊥¿2ν⊥¿¿

u¿u¿ u3ν⊥¿

¿ν⊥¿≡¿u¿

¿

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TOF Angle Calculations (case 2)Case of no goniometer arcs, ==0

Q v= Q∣∣cos μcosωQ ¿sin μ

−Q∣∣cos ν sinωQ∣∣sin ν sin μcosωQ¿ cos μ sin ν

Q∣∣sin ν sinωQ∣∣cos ν sin μcosωQ¿ cos μcos ν μ=ν=0 Q v= Q∣∣cosω

−Q∣∣sin ω

Q¿

• Start by converting Q=(h,k,l) into -coordinate system

Q v=UBQ≡Q 1ν

Q 2ν

Q 3ν Q 3ν=Q¿=k f sin ψ q2=Q1ν

2 Q 2ν2 Q3ν

2 =Q∣∣2Q¿

2

cos ω=Q1ν

Q∣∣ and s in ω=

−Q 2ν

Q∣∣cos φ '=

ki2k f

2 cos2ψ−Q∣∣

2

2ki k f cosψ

tan θ=k i−k f cosψ cos { φ '

k f cosψ sin { φ '¿ and s1=θω ¿ All angles defined!!

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Conclusions

We have extended the UB matrix formalism of Busing and Levy to handle inelastic experiments on TOF spectrometers.

Implementation of this formalism at SNS should greatly simplify and enhance single crystal experiments on TOF spectrometers

o ak r idge n ational l aboratory u. s. d epartment of e nergy 1 handling single crystals in...

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