Instrumental Resolution

Transcription

1 Instrumental Resolution MLZ Triple-Axis Workshop T. Weber Technische Universität München, Physikdepartment E21 April 3 4, 2017

2 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

3 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

4 Introduction to resolution calculation Scattering triangle: k i k f = Q and h²/(2m n ) ki 2 h²/(2m n ) kf 2 = E Variances in k i and k f add up to a variance in Q and E. k f Q 2θ k i

5 General formalism Overview General formalism presented here is based on Violini [1] and Eckold [2] papers. Assume Gaussian transmissions Starting with known variances of components Need result in a (δq,δe) coordinate system: ( R (δq,δe) = R 0 exp 1 ( ) t ( δq δq R 2 δe δe With δq = Q Q 0 and δe = E E 0. ) r ( δq δe ) ) c.

6 General formalism Important quantities R (δq,δe) = R 0 exp ( 1 2 ( δq δe ) t ( δq R δe ) ( δq r δe ) ) c R is called the resolution matrix. r is a vector shifting the mean positions Q 0 and E 0. R 0 is often called the resolution volume. c is a factor diminishing the scattering intensity.

7 General formalism Covariance R is the inverse covariance matrix: R = C 1. Reminder: The covariance of two random variables X and Y is defined as: cov(x,y ) = (X X )(Y Y ). It is a measure of the correlation between X and Y.

8 General formalism Instrument parameters Start with known variances σi 2 of instrument parameters p i : C instr = diag ( σ1 2, σ2 2,...,σN 2 ) Assuming no correlation: C instr is diagonal. The parameters p i are defined in terms of k i and k f.

9 General formalism Jacobian Transform the instrument parameters to a system given by Q and E: p 1,p 2,...,p N Q x,q y,q z,e Covariance matrix C in new Q x,q y,q z,e system with x along k i and y perpendicular to k i : C = T C instr T t. With the Jacobian: T = Q x p 1 Q y p 1 Q z p 1 E p 1 Q x p 2... Q y p 2... Q z Q x p N Q y p N Q z p N p 2... E p 2... E p N

10 General formalism Rotation We could stick with the absolute Q x,q y,q z,e system (in fact, Violini [1] does). A more convenient system is the local Q,Q,Q z,e system with x along Q : k f 2θ Q (k i,q) Q k i, θ k i,

11 General formalism Rotation The coordinate systems are rotated by the angle between k i and Q: Q x,q y,q z,e Q,Q,Q z,e. The final covariance matrix is: With the rotation matrix: S = C = S C S t = S T C instr T t S t. cos (k i,q) sin (k i,q) 0 0 sin (k i,q) cos (k i,q)

12 General formalism Contour line We set the general expression equal to 0.5 to get the HWHM contour of the Gaussian (c = 0): ( R (δq,δe) exp 1 ( ) t ( ) ) δq δq R 2 δe δe ( ( )) δq exp r 1 δe 2 ( δq δe ) t ( δq R δe ) ( δq + 2r δe ) = 2ln2

13 General formalism Quadric This contour outlines a four-dimensional quadric, namely an ellipsoid: ( ) t ( δq δq R δe δe }{{} orientation ) ( δq +2r δe } {{ } translation ) = 2ln2. The lengths and the orientation of the ellipsoid axes are obtained via the principal axis theorem, i.e. calculating the eigenvectors and -values of R.

14 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

15 Minimal example Definitions Assume the variances of k i and k f are known and are the only parameters, the Jacobian is: T = Q x k i,x Q y k i,x Q z k i,x E k i,x Q x k i,y Q y k i,y Q z k i,y E k i,y Q x k i,z Q y k i,z Q z k i,z E k i,z Q x k f,x Q y k f,x Q z k f,x E k f,x Q x k f,y Q y k f,y Q z k f,y E k f,y Q x k f,z Q y k f,z Q z k f,z E k f,z.

16 Minimal example Jacobian ( Using Q = k i k f and E = h2 2m n k 2 i kf 2 ) : T = h 2 h m n k 2 h i,x m n k 2 h i,y m n k 2 h i,z m n k 2 h f,x m n k 2 f,y m n k f,z.

17 Minimal example Covariance The covariance matrix then reads: C = T diag ( σ 2 k i,x,σ2 k i,y,σ2 k i,z,σ2 k f,x,σ2 k f,y,σ2 k f,z) T t. C = C xx 0 0 C xe 0 C yy 0 C ye 0 0 C zz C ze C xe C ye C ze C EE C qq = σ 2 k i,q + σ 2 k f,q,, with C EE = h4 ( k 2 mn 2 i,x σk 2 i,x k2 f,x σ k 2 f,x +...), C qe = h2 ( ki,q σk 2 m i,q + k f,q σ 2 ) k f,q. n

18 Minimal example Correlation The covariance matrix is not diagonal, there are q,e correlation terms: C qe = h 2 m n }{{} 4.1meVÅ 2 ( ki,q σk 2 i,q + k f,q σk 2 ) f,q. In the standard Q,Q,Q z coordinate system (we re still in Q x,q y,q z,e ) this leads to the important focusing formula for transverse scans: δe [mev] [ [ δq Å 1] 4 k fix Å 1].

19 Minimal example Correlation

20 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

21 Monte-Carlo method In Monte-Carlo simulations (e.g. McStas [3]) we can directly obtain k i and k f for each neutron and immediately get the covariance matrix: C = N j=1 p j (Q j Q ) (Q j Q ) N j=1 p j where j numbers the N Monte-Carlo events. The Q j are the four-vectors: Q j = ( k k f,j ] h 2 2m n [ki,j 2 k2 f,j ).

22 Monte-Carlo method

23 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

24 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

25 Time-of-flight spectrometers Detectors 2θ Q ls D Sample θ l 2 S Chopper 2 Chopper 1 l 1 2 l 1 S Source

26 Time-of-flight spectrometers Energy expression TOF resolution: Violini method [1]: E = h2 ( k 2 2m i k 2 ) m n f = n 2 = m ( ) n 2 l1 2 2 = m n 2 t i ( l1 2 t i ( v 2 i vf 2 ) 2 l S D t f }{{} =t S D 2 ) 2 l S D t 2 D t }{{} 2 S. =l 2 S /v i

27 Time-of-flight spectrometers Momentum expression TOF resolution: Violini method [1]: Q = k i k f = m n h (v i v f ) = m n h l 1 2 t i l S D t f cos2ϑ cos2φ sin2ϑ cos2φ sin2φ. 2ϑ is the in-plane and 2φ the out-of-plane scattering angle.

28 Time-of-flight spectrometers

29 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

30 Triple-Axis spectrometers Q 2θ S Sample θ S η S α 1 α 2 Detector α 3 2θ A η A Ana. 2θ M η M Mono. α 0 Source

31 Triple-Axis spectrometers Algorithms Comparison of TAS resolution algorithms: Method Description Cooper-Nathans [4, 5] Popovici [6] only considers collimators, no geometry collimators + instrument geometry Eckold-Sobolev [2] collimators + geometry + + non-centered beams The Eckold-Sobolev algorithm is the only TAS algorithm considering the linear part of the quadric. It thus allows to calculate the resolution for off-centre neutron paths, non-centered samples and for monochromator and analyser focusing.

32 Triple-Axis spectrometers Cooper-Nathans

33 Triple-Axis spectrometers Popovici

34 Triple-Axis spectrometers Eckold-Sobolev

35 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

36 Triple-Axis spectrometers Cooper-Nathans method Cooper-Nathans [5] resolution matrix in (δk i,δk f ) frame: ( ) ( ) RM RM,h R =, with R M = and R A similarly R M,h = R A R M,v ( ) ( 4tan 2 ϑ M εM tanϑ M ki 2 α( 0 2 (2η M ) 2 ) k( i 2 α0 2 2η) M 2 2ε M tanϑ M ki 2 α0 2 ki 2 α η 2 M α1 2 ηm 2 )

37 Triple-Axis spectrometers Cooper-Nathans method Transformation of resolution matrix from (δk i,δk f ) to (δq,δe) frame: cosϕ i sinϕ i 0 cosϕ f sinϕ f 0 sinϕ i cosϕ i 0 sinϕ f cosϕ f 0 = k i c 0 0 2k f c δq x δq y δq z δe δk ix δk iz δk ix δk iy δk iz δk fx δk fy δk fz using ϕ i = (k i,q), ϕ f = (k f,q), and c = h 2 /(2m n ).

38 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

39 Triple-Axis spectrometers Popovici method Popovici [6] resolution matrix in (δk i,δk f ) frame: [ ( ) ] } R = A{ D S + T T FT D T + G A T S: Covariance matrix of source, monochromator, sample, analyser, and detector sizes. F : Covariance matrix of crystal mosaics. G: Covariance matrix of collimators. D: Matrix of instrumental lengths. Transformation from (δk i,δk f ) to (δq,δe) frame as before in Cooper-Nathans.

40 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

41 Convolution The measured intensity is the convolution of the dynamical structure factor S with the instrumental resolution function R: I (Q,E) = d (δq)d (δe) S (Q + δq,e + δe) R (δq,δe). Convolution of a simple transverse-acoustic phonon branch: E (mev) S (q, ω) (a.u.) Convolution simulation, χ 2 /n df = q (rlu) E (mev)

42 Convolution Fitting Change model parameters and minimise χ 2. I (a.u.) E (mev) B (T) 0.2

43 Contents General formalism Minimal example Monte-Carlo method Time-of-flight spectrometers Violini resolution function Triple-axis spectrometers Cooper-Nathans resolution function Popovici resolution function Convolution Summary

44 Summary General formalism: Resolution = Covariance 1 For Monte-Carlo simulations the resolution can be directly obtained. Several different methods of increasing complexity available for triple-axis resolution calculation. Important instrument-independent rule of thumb: E [mev]/q [ Å 1] 4 k fix [Å 1]. Also valid for TOF!

45 Appendix References I [1] N. Violini, J. Voigt, S. Pasini, T. Brückel, A method to compute the covariance matrix of wavevector-energy transfer for neutron time-of-flight spectrometers, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 736 (2014) doi: /j.nima URL pii/s x [2] G. Eckold, O. Sobolev, Analytical approach to the 4D-resolution function of three axes neutron spectrometers with focussing monochromators and analysers, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 752 (2014)

46 Appendix References II doi: /j.nima URL pii/s [3] L. Udby, P. K. Willendrup, E. Knudsen, C. Niedermayer, U. Filges, N. B. Christensen, E. Farhi, B. O. Wells, K. Lefmann, Analysing neutron scattering data using McStas virtual experiments, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 634 (1, Supplement) (2011) S138 S143, proceedings of the International Workshop on Neutron Optics (NOP2010). doi: /j.nima URL pii/s

47 Appendix References III [4] M. J. Cooper, R. Nathans, The resolution function in neutron diffractometry. I. The resolution function of a neutron diffractometer and its application to phonon measurements, Acta Crystallographica 23 (3) (1967) doi: /s x URL [5] P. W. Mitchell, R. A. Cowley, S. A. Higgins, The resolution function of triple-axis neutron spectrometers in the limit of small scattering angles, Acta Crystallographica Section A 40 (2) (1984) doi: /s URL

48 Appendix References IV [6] M. Popovici, On the resolution of slow-neutron spectrometers. IV. The triple-axis spectrometer resolution function, spatial effects included, Acta Crystallographica Section A 31 (4) (1975) doi: /s URL