Chemistry in Action: Making Molecular Movies with Ultrafast Electron Diffraction and Data Science [1 ed.] 9783030548506, 9783030548513

Table of contents :
Supervisor's Foreword
Preface
Acknowledgements
Parts of This Thesis Have Been Published in the Following Journal Articles
Contents
List of Abbreviations
1 Introduction
1.1 The Scientific Question
1.2 Overview of Thesis
1.3 Author's Contributions
2 Methods: Experimental Techniques and Data Science
2.1 Transient Absorption Spectroscopy
2.1.1 Overview
2.1.2 Experimental Setup
2.1.3 Data Analysis
2.2 Ultrafast Electron Diffraction
2.2.1 Overview of Crystallography
2.2.2 Physics of Electron Scattering
2.2.3 Kinematical Scattering Versus Dynamical Scattering
2.2.4 Electrons versus X-rays and Neutrons
2.2.5 Experimental Setup
2.2.6 Data Analysis: Part 1
2.2.7 Data Analysis: Part 2
2.2.8 Data Analysis: Part 3
3 Ultrafast Structural Dynamics of (EDO-TTF)2X
3.1 Overview of (EDO-TTF)2X Molecular Systems
3.2 Mapping Molecular Motions in (EDO-TTF)2PF6
3.2.1 Methods
3.2.2 Experimental Results and Discussions
3.2.3 Summary and Conclusions
3.3 Counterion Effect in (EDO-TTF)2SbF6
3.3.1 Methods
3.3.2 Experimental Results and Discussions
3.3.3 Summary and Conclusions
4 Photocyclization Dynamics of Diarylethene
4.1 Overview of Diarylethenes
4.2 Methods
4.3 Experimental Results and Discussion
4.4 Summary and Conclusions
5 Photoinduced Spin Crossover in Iron(II) Systems
5.1 Overview of Spin Crossover
5.1.1 Theoretical Considerations
5.1.2 Light-Induced Excited Spin-State Trapping
5.1.3 Photoinduced Spin Crossover in Optical Experiments
5.1.4 Photoinduced Spin Crossover in X-ray Experiments
5.1.5 `Alternative Facts' of Photoinduced Spin Crossover
5.2 Spectral Signatures of Spin Crossover in [FeII(bpy)3](PF6)2
5.2.1 Methods
5.2.2 Experimental Results
5.2.3 Discussions
5.2.4 Summary and Conclusions
5.3 Structural Dynamics of Spin Crossover in [FeII(PM-AzA)2(NCS)2]
5.3.1 Methods
5.3.2 Experimental Results
5.3.3 Discussions
5.3.4 Summary and Conclusions
6 Future Work
6.1 Rotation UED and Charge Flipping
6.2 Dimensionality Reduction of UED Image Stack
6.3 Frequency Analysis of [FeII(bpy)3]2+ TA Data
6.4 Structural Dynamics of Spin Crossover in [FeII(bpy)3](PF6)2
6.5 Application of Deep Learning to UED
6.6 Other Molecular Systems of Interest
7 Conclusion
A Timeline of Time-Resolved Crystallography
B Time and Length Scales of UED
C Crystallographic Information for Referenced Systems
D Calculation for the Crystallographic Interplanar Distance
E Derivation of the Mott-Bethe Formula
F Electron Elastic Scattering Factor of Select Elements
G The Extended Peierls-Holstein-Hubbard Model
H Crystal Field Splitting Energy
I Atomic and Molecular Terms Symbols
J Theory of Spin Crossover
K Sample Excitation Condition in SCO Literature
L SVD Analysis of [FeII(bpy)3](PF6)2 TA Data
References

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Lai Chung Liu

Chemistry in Action: Making Molecular Movies with Ultrafast Electron Diffraction and Data Science

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field.

More information about this series at http://www.springer.com/series/8790

Lai Chung Liu

Chemistry in Action: Making Molecular Movies with Ultrafast Electron Diffraction and Data Science Doctoral Thesis accepted by University of Toronto, Canada

Lai Chung Liu University of Toronto Toronto, ON, Canada

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-54850-6 ISBN 978-3-030-54851-3 (eBook) https://doi.org/10.1007/978-3-030-54851-3 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns: the ones we don’t know we don’t know. DONALD H. RUMSFELD United States Secretary of Defense (2001–2006) Department of Defense News Briefing, February 12, 2002

Supervisor’s Foreword

I am very pleased to give some insight into the importance of the PhD work of Dr. Lai Chung (Nelson) Liu as part of its selection to be published as a Springer book. To put his work in context, I think it is fair to say that one of the dream experiments in science has been to directly observe atomic motions during the primary events governing chemistry and biology. Given the extremely high and simultaneous requirement of sub-Å and 100 femtosecond (fs, 1 fs = 10−15 s) spacetime resolution, this prospect seemed to be destined to remain as the purest form of a thought experiment. The major technical obstacle was really source brightness. As with any movie camera, as one goes to shorter and shorter exposure times, one needs a brighter and brighter light source to resolve the image of interest in motion. We solved the space charge problem associated with electron sources (effects of Coulomb repulsion) to generate sources operating at the absolute limit of source brightness before the onset of space charge aberrations in either spatial or temporal resolution. These sources are sufficiently bright to literally light up atomic motions. We increased our electron source brightness with respect to spatial coherence to deliberately study how chemistry scales with complexity. How can so many possible degrees of freedom be directed to a given product state? As discussed in Nelson’s PhD thesis, we systematically studied systems involved in ring-closing reactions with conserved stereochemistry (bond formation), to bond breaking events, to electron transfer reactions, and spin transitions of increasing complexity. The surprise was that, for all these systems, effectively independent of complexity, the structural transition reduced to 5}. Next, the wavelength-dependent shift in time-zero9 is considered. This varying arrival time of the probe pulse is caused by group velocity dispersion (GVD) that occurs when the broadband probe light propagates through transmissive optical elements of the experimental setup. To quantify this effect, TA measurements are made on samples10 that are not resonant within the probe wavelength region (see 8 WLG

is the conversion of laser light into light with a very broad spectral bandwidth (spanning hundreds of nanometers) via strongly nonlinear interactions in a medium [6, 389]. 9 ‘Time-zero’ is the time delay at which there is temporal overlap of the pump and probe pulses. 10 For the experiments described in Sect. 5.2, the non-resonant samples are just blank sample media: deionized water in a quartz flow cell for the aqueous case, a sapphire window for the crystal case.

2.1 Transient Absorption Spectroscopy

15

Fig. 2.3 Illustration of the steps for artifact removal in TA data. Step 1: (a) restrict TA data to wavelength regions where SNR(λ) > 5 using (b) the SNR estimated from Eq. (2.5) for all probed wavelengths. Step 2: (c) correct the GVD distortion of the TA data by reversing (d) the wavelength dependence of the time-zero and IRF width derived from a Hermite fitting of the CPM signal as in Eq. (2.6). Step 3: (e) obtain the final artifact-free TA data by (f) integrating and then subtracting all pre-CPM signals via Eq. (2.7)

Ref. [158] for more details). The correct time-zero t0 (λ) is estimated by fitting the CPM signal ACPM (t, λ) with a model function that is a linear combination of the first four Hermite polynomials Hn for each wavelength: C(t, λ, {an }, t0 , τ ) =

4  n=0

  an (λ)Hn (x)

(2.6) x=(t−t0 (λ))/τ (λ)

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2 Methods: Experimental Techniques and Data Science

Then, each time series of the sample data is translated temporally such that all the time-zeros are aligned: A(t, λ) → A(t − t0 , λ). For time-zero shifts which are not integer multiples of the time delay step size, super-resolution alignment is achieved through spline interpolation. Incidentally, the parameter τ (λ) corresponds to the temporal width of the instrument response function (IRF)11 of the experimental setup [362]. Another TA artifact is the negative signals caused by scattered pump light and fluorescence from the sample. Since they occur independently of the probe pulse and their amplitude is mostly determined by the geometry of the experimental setup, they are present and nearly constant at all time delays. Thus, they are removed by averaging the pre-CPM differential absorbance and subtracting it from the rest of the TA values: A(t, λ) → A(t, λ) −

1 T



t  t

A(t, λ)dt

(2.7)

where t ∈ [t  , t  ] and T = t  − t  refer to the time delay interval before time-zero during which the CPM signal is not yet present. Now that the TA data is artifact-free, a technique known as global analysis (GA) is applied to reduce A(t, λ) into a sum of exponentially decaying spectral components that could be assigned to specific electronic states and photophysical processes [388, 458, 533, 538]. The first step of GA is to recognize that A(t, λ) is just a M × N data matrix A, where M and N are the number of wavelength and time points respectively, ⎤ A(t1 , λ1 ) · · · A(tN , λ1 ) ⎥ ⎢ .. .. .. A(t, λ) → A = ⎣ ⎦ . . . A(t1 , λM ) · · · A(tN , λM ) ⎡

(2.8)

and it can be factored using singular value decomposition (SVD) into separate spectral and temporal features: A = USVT

(2.9)

where U = U (λ) and V = V (t) are orthogonal matrices of size M × M and N × N respectively; S is a M × N diagonal matrix of rank P . This factorization can be rewritten to give an expansion of A as a weighted ordered sum of ‘singular components’ Xi :

11 The

IRF here is assumed to be Gaussian, given that it is a convolution of the nearly Gaussian temporal profile of the pump and probe pulses.

2.1 Transient Absorption Spectroscopy

17

Fig. 2.4 A low-rank approximation of a data matrix using SVD, as per Eq. (2.10), is demonstrated using the TA data shown in Fig. 2.3f. The contribution of each singular component Xi is weighted by the associated singular value si , which is shown here on a log-log plot in (a) after normalization. Two linear trendlines can be fitted piecewise to this curve; they are overlain to highlight the sudden change in slope as the normalized singular values decay to zero. Summing the singular components before and after this flexion yields the A(t, λ) values in Panels (b) and (c) respectively. By inspection, it is clear that the index at which the flexion occurs marks the point beyond which the remaining Xi only contribute irrelevant features and incoherent noise

A =

P 

si Xi

i=1

(2.10)

Xi = ui ⊗ vTi where ui = ui (λ) and vi = vi (t) are the column vectors of U and V; they are known as the left and right ‘singular vectors’ and, specifically in TA, the ‘basis spectra’ and ‘kinetic traces’ of the overall dataset, respectively; si is defined as the diagonal entries of S and known as the ‘singular values’ of the data matrix. In general, the set of singular values is sorted in descending order and an approximation of the data matrix can be obtained by truncating this summation at some P   P , thus keeping only those singular components that contribute significantly to the overall signal. This concept is particularly useful in deconvolving incoherent noise from signals and it is illustrated in Fig. 2.4. After SVD and truncation, the TA dataset is now reduced to a few basic spectra ui (λ) and kinetic traces vi (t). These are not readily interpretable in terms of photophysical processes since they are only orthogonal bases that jointly span the data matrix. To proceed further, a global lifetime analysis (GLA) fit is done by assuming an exponential model for the population dynamics of optically active species. Each kinetic trace is assumed to be a weighted sum of Q exponential decay

18

2 Methods: Experimental Techniques and Data Science

functions switched on at time-zero by a Heaviside step function H (t) and convolved with a Gaussian IRF whose temporal width τIRF is obtained earlier by fitting the response of a non-resonant sample (see Fig. 2.3f); as before, the CPM signal is described by a linear combination of the first four Hermite polynomials Hn (t), Ci (t, {ain }) =

4 

ain (λ)Hn (t/τIRF )

n=0 1 2 2 /τIRF

IRF(t) = e− 2 t Ei (t, {bij }, {kj }) =

  bij H (t)e−kj t ∗ IRF(t)

Q  j =1

fi (t, {ain }, {bij }, {kj }) = Ci (t, {ain }) + Ei (t, {bij }, {kj }) where fi , Ci and Ei are the model functions for the kinetic trace v i (t), and its CPM and exponential contributions.12 To track the changes in A that are associated with each exponential process, a decay-associated spectrum (DAS) Dj (λ) for each decay constant kj is calculated as a linear combination of the basis spectra ui (λ) weighted by the fitted amplitude bij and singular values si : 

Dj (λ) =

P 

(2.11)

bij si ui (λ)

i=1

This calculation is demonstrated using the TA data in Fig. 2.4b and the results are plotted in Fig. 2.5. Finally, the global fit G(t, λ) (as shown in Fig. 2.5d) are calculated by recombining each Dj (λ) with its corresponding exponential function with decay constant kj : G(t, λ) =

Q 

 Dj (λ)

  H (t)e−kj t ∗ IRF(t)

(2.12)

j =1

The residuals of the fit, R(t, λ) = A(t, λ) − G(t, λ), contain signals not captured by the exponential model. As seen in Fig. 2.5e, these features include the CPM near time-zero, lattice heating effects at late times, and various oscillatory dynamics in between. For the latter, relevant frequencies can be extracted by calculating the power spectrum of the residuals at each probe wavelength using Welch’s method for spectral density estimation [554] or a continuous wavelet transform (see Sect. 6.3).

12 For

     2 1 2 2 1 2 2 . reference, H (t − t0 )e−kt ∗ e− 2 t /τ = 12 e 2 k τ −k(t−t0 ) 1 + erf t−t√0 −kτ 2τ

2.2 Ultrafast Electron Diffraction

19

Fig. 2.5 Truncating the SVD sum of the TA data matrix (from Fig. 2.4c) at the point of flexion leaves (a) eight principal basis spectra and (b) eight kinetic traces in the TA data matrix. Two exponential components (τ1 = ∞, τ2 = 72 fs) are needed for the GLA fit of the kinetic traces (Eq. (2.11)) and their respective DAS (Eq. (2.11)) are plotted in Panel (c). The resulting global fit and residuals are shown in Panels (d) and (e)

2.2 Ultrafast Electron Diffraction Since its discovery in 1923–1927 [179], electron diffraction became an indispensable weapon in the arsenal of those seeking to investigate the structure of materials on the atomic level. The advent of time-resolved laser spectroscopy and the introduction of the pump–probe methodology involving ultrashort pulses to this technique further extended its capabilities, which now includes the study of ultrafast atomic motions and structural dynamics during physico-chemical processes, hence conferring it the titular specialization of ultrafast electron diffraction (UED).

20

2 Methods: Experimental Techniques and Data Science

Given the roots of UED, some basic tenets of crystallography and electron scattering physics is necessary to fully understand UED as an experimental technique and interpret its measurements. As such, a brief introduction to these two topics— based on Refs. [13, 16, 145, 552] and [111, 180, 289, 452, 521, 587]—is given as follows.

2.2.1 Overview of Crystallography A common feature of physical objects is that their constituent atoms are not distributed randomly but are bound together as molecules and arranged in a regular periodic array. Materials structured as such are known as crystals13 and they are the subject of study for crystallographers. A fundamental concept to describe crystal structures is the ‘Bravais lattice’,14 which is an infinite array of points that maps back onto itself under discrete translations in three-dimensional space. The position of every point in a Bravais lattice can be stated as R = n1 a 1 + n2 a 2 + n3 a 3 , where nj are integers and a j are the three non-coplanar primitive vectors that generate the lattice. The region of space spanned by the vectors a j is known as the ‘unit cell’ of the lattice; it is the spatial volume that is repeated throughout the lattice and preserved under its translational symmetry. Although there is no unique way to choose the primitive vectors and the unit cell of a lattice, the conventional approach is to pick the most orthogonal set of lattice-generating vectors and the parallelepiped in the first octant of the resulting coordinate system. As seen in Fig. 2.6a, the unit cell and its primitive vectors are specified by a set of six lattice parameters:15 the length of the three vectors (a1 , a2 , a3 ) and the internal angles (α23 , α31 , α12 ), where cos αij = |aˆ i · aˆ j |. Herein, the components of a j can be expressed as a 1 = a1 [1

0

0]

a 2 = a2 [cos α12  a 3 = a3 cos α31

sin α12

0]

cos α23 − cos α31 cos α12 sin α12

V a1 a2 a3 sin α12



(2.13)

where V is the volume of the unit cell, 13 Since

the discovery of ‘quasicrystals’ in 1984 by Israeli material scientist Daniel Shechtman (1941–present) [480], crystals are now more accurately described as “any solids having an essentially discrete diffraction diagram” wherein 3D periodicity may be absent [240]. 14 Moritz L. Frankenheim (1801–1869) first discovered in 1826 that they are no more than 15 distinct types of lattices in three dimensions. Auguste Bravais (1811–1863) later correctly revised this number to 14 [16]. 15 In conventional crystallographic notation, the six lattice parameters are denoted by a, b, c, α, β, γ respectively. Here, this is done using the index notation for convenience.

2.2 Ultrafast Electron Diffraction

21

Fig. 2.6 Crystal structures as described by a Bravais lattice and an associated basis. The unit cell is orange parallelepiped; atoms inside/outside are filled/empty circles; the basis atoms are outlined in blue. (a) The six parameters (a1 , a2 , a3 , α23 , α31 , α12 ) that specify the shape of an unit cell are defined. The crystal structures of (b) diamond and (c) hexagonal graphite are depicted. The first is a face-centered cubic lattice with a two-atom basis; the second is a hexagonal lattice with a four-atom basis

V = |a 1 · (a 2 × a 3 )|  = a1 a2 a3 1 + 2 cos α23 cos α31 cos α12 − cos2 α23 − cos2 α31 − cos2 α12 (2.14) The structure of a physical crystal can be described by a Bravais lattice wherein the repeating arrangement of atoms is enclosed within the volume of a particular unit cell and this subset of atoms defines a ‘basis’ relative to each lattice point. Under this scheme, the position of any atom j in the crystal is given by r j = R n1 n2 n3 + i , 3  where R n1 n2 n3 = nj a j is the position of the lattice point to which the origin of

r

j =1

r

the unit cell is defined and i is the position of the basis atom i that is equivalent to atom j under symmetry. The facets of the crystal are related to families of parallel equidistant planes which intersect the Bravais lattice at (at least) three non-collinear lattice points. a Within the unit cell, such a lattice plane has intercepts hjj on the three lattice vectors and can be specified by the notation (h1 h2 h3 ), where hj are coprime integers (see Fig. 2.7a) and they are known as the ‘Miller indices’16 of the plane. A set of vectors 3  can be constructed17 as bi = Vπ ij k a j × a k , where ij k is the Levi-Civita j,k=1

symbol. Under this definition, the three vectors bi have the property of bi · a j = 2πδij , where δij is the Kronecker delta, with the corollary |bi | = 2π/|a i |. Since 16 William

H. Miller (1801–1880) introduced this notation for crystal planes in 1839. The letters h, k, l are used therein most often; however, the labels h1 , h2 , h3 are used instead here for convenient indexing. An overline indicates a negative number, h = −h. 17 A computationally more useful definition of the reciprocal lattice vectors is [b b b ] = 1 2 3 2π [a 1 a 2 a 3 ]−1 , where the vectors are in column representation.

22

2 Methods: Experimental Techniques and Data Science

Fig. 2.7 (a) Family of lattice planes specified by the Miller indices h1 , h2 , h3 = 2 is depicted, along with the reciprocal lattice vector g that corresponds to the normal of the planes. The path difference that leads to diffraction in crystals is shown according to the formulation of (b) Max v. Laue and (c) William L. Bragg

their magnitude is the reciprocal of that of the crystal lattice and they can generate 3  their own Bravais lattice where each vector g = hi bi , the vectors bi are known i=1

as ‘reciprocal lattice vectors’; the reciprocal vector g corresponds to the normal of a plane (h1 h2 h3 ) in the ‘direct lattice’ (as opposed to the reciprocal lattice) and its magnitude is proportional to the reciprocal of the interplanar distance dh1 h2 h3 , |g| = 2π/dh1 h2 h3 . This geometric relationship between planes in the crystal lattice and points in the reciprocal lattice provides a convenient basis to discuss the interference of waves reflected by the bulk crystal. When a wave penetrates a material, new waves are produced through scattering and they can interact with the original wave. In the case of a crystal, the periodic features that underlie the crystal structure generate many scattered waves with discrete phase differences. Under certain conditions, there is constructive interference between these waves and the incident one and diffraction occurs. There are two formulations of these diffraction conditions: one conceived by Max v. Laue and the other by William L. Bragg, both18 in 1912. As seen in Fig. 2.7b, the first approach considers the crystal to be lines of scatterers, equally spaced by δR = a j ; assuming elastic scattering, the phase difference between the transmitted wave and the scattered one is φ = (k sc − k inc ) · δR and thus, the Laue condition for diffraction is q · a j = 2πhj

(2.15)

where h¯ q = h(k ¯ sc − k inc ) is the momentum transferred during the scattering event and hj are integers. In the Bragg formulation (Fig. 2.7c), the crystal is composed of parallel reflective planes evenly separated by the interplanar distance d; the incident wave interferes with the reflected waves only if the path difference is an integer

18 Max

v. Laue (1879–1960) was awarded the 1914 Nobel Prize in Physics for the discovery of X-ray diffraction. William L. Bragg (1890–1971), along with his father William H. Bragg (1862– 1942), won the same award in 1915 for their work in X-ray crystallography [400].

2.2 Ultrafast Electron Diffraction

23

Fig. 2.8 Construction of the Ewald sphere. (a) The Laue-Bragg condition is satisfied at scattering angles θ where the surface of the sphere coincides with reciprocal lattice points g; when projected onto the detector, these points indicate where diffraction spots may be observable. (b) Realistic incident beams with nonzero energy and momentum spreads, in addition to finite and imperfect crystals, loosen the condition for diffraction and lead to more diffraction spots than would otherwise be expected

multiple n of the wavelength λ and hence the Bragg condition for diffraction is 2d sin θ = nλ

(2.16)

The Laue and Bragg conditions are equivalent and this can be shown by noting that the Laue condition effectively constrains diffraction to only occur for k sc such that q is an integer multiple of a reciprocal lattice vector g of the crystal. A helpful way to interpret these conditions is through a geometric construct19 known as the ‘Ewald sphere’. In the space of wavevectors, the reciprocal lattice and a sphere— centered on k inc and of radius |k inc |—are drawn such that the latter is tangential to the lattice origin. The surface of this sphere represents all possible elastic scattering events such that |k sc | = |k inc |. As seen in Fig. 2.8, the locations on the surface of the Ewald sphere that overlap with any points of the reciprocal lattice specifies those scattering geometries wherein the transfer wavevector q and the scattering θ satisfy the Laue and Bragg conditions respectively. Propagating these beams onto the detector yields the spots of the diffraction pattern that can be expected by an observer in real space. In a perfect, infinite, and static crystal,20 few if any diffraction spots would be observable for a general monochromatic incident beam. It is unlikely that any integer solution (h1 h2 h3 ), other than the transmitted beam at (000), exactly satisfies the Laue-Bragg condition. In practice, incident beams are not composed of perfectly coherent waves; the resulting nonzero energy and momentum spread

19 Paul

P. Ewald (1888–1985) conceived of the sphere-in-reciprocal-lattice construct to geometrically visualize diffraction in 1912 [145]. 20 A ‘perfect’ crystal is one with unbroken discrete translational symmetry; ‘infinite’ refers to the absence of boundaries, and ‘static’, a lack of atomic motion.

24

2 Methods: Experimental Techniques and Data Science

Fig. 2.9 Samples of varying crystallinity and the resulting diffraction patterns: (a) a perfect single crystal, (b) a polycrystal, (c) a powder, and (d) an amorphous sample

manifest within the Ewald construct (Fig. 2.8b) as a thickening the surface of the sphere into a shell that can overlap with many more reciprocal lattice points. Furthermore, the finite dimensions and long-range disorder of the crystal effectively broaden the points into extended spikes21 that can more easily be intercepted by the Ewald sphere. Hence, experimental imperfections ensure that the requirements of diffraction are sufficiently relaxed for a multitude of spots to appear in a diffraction pattern. A diffraction pattern does not necessarily appear as the regular array of spots that the Ewald construct would suggest. The nature of the pattern can be an indication of the crystallinity of the target. As shown in Fig. 2.9, a perfect single crystal gives a periodic arrangement of diffraction spots while fluids and amorphous22 materials produce highly diffuse rings centered around the transmitted beam. In between these two extremes in structural order, there are coarse- and fine-grained polycrystalline materials wherein the individual randomly-oriented crystallites contribute their own diffraction spots that sum to a set of spotty concentric circles, generally known as Debye-Scherrer23 rings.

2.2.2 Physics of Electron Scattering The Laue-Bragg condition determines the geometry in which waves scattered by a crystal can constructively interfere and give rise to diffraction. To calculate the amplitude of these waves, consider a prototypical scattering experiment: a free

21 These

shapes are known in the literature as reciprocal lattice rods, or ‘relrods’ for short. amorphous material is one that lacks the long-range order of a crystal. 23 Paul Scherrer (1890–1969), under the guidance of Peter Debye (1884–1966), and Albert W. Hull (1880–1966) developed in 1915–1917 the method for analyzing X-ray crystal structures using powder samples instead of single-crystal ones [145, 229]. 22 An

2.2 Ultrafast Electron Diffraction

25

Fig. 2.10 Schematic of the geometry of a prototypical three-dimensional scattering experiment

electron impinging on a single atom in three dimensions. As depicted in Fig. 2.10, the incident matter wave ψinc (r) is taken to be a plane wave traveling along the z axis and the scattered wavefunction is a linear combination of the original plane wave and an outgoing spherical wave that is produced by the interaction: ψ(r) ∼ ψinc (r) + ψout (r) = Aeikz + f (k, r)

eikr r

(2.17)

where f (r, k) is the scattering amplitude of the interaction. To calculate the scattering amplitude, let us start with the time-independent Schrödinger24 equation, 

 h¯ 2 2 − ∇ + U (r) ψ(r) = Eψ(r) 2me

(2.18)

where me is the mass of the electron, U (r) is the electrostatic potential energy of the atom. This can be re-written as 

 2me ∇ 2 + k 2 ψ(r) = 2 U (r)ψ(r) h¯ 2me Lψ(r) = 2 U (r)ψ(r) h¯

(2.19)

√ where k = 2me E/h¯ is the magnitude of the wavevector of the incident electron and L is the linear differential operator ∇ 2 + k 2 . A formal solution of this equation can be expressed as

24 Erwin

Schrödinger (1887–1961) won the 1933 Nobel Prize in Physics along with Paul A.M. Dirac for formulating the wave equation that now bears his name [400].

26

2 Methods: Experimental Techniques and Data Science

ψ(r) = ψ0 (r) +

2me



h¯ 2

G(r, r  )U (r  )ψ(r  )d3 r 

(2.20)

where ψ0 (r) is a solution of the homogeneous equation Lψ(r) = 0 and G(r, r  ) is the Green’s function25 of L, i.e. a solution of the equation Lψ(r) = δ 3 (r − r  ). Solving these two coupled differential equations yields ψ0 (r) = Aeikz G(r, r  ) = −



eik|r−r | 4π|r − r  |

(2.21)

for some constant A. Since the atomic potential energy U (r) is highly localized to microscopic distances, the range of the r  integral is limited to the region of r   r and some approximations can be made by neglecting terms of order (r  /r)2 and higher as    2   r r r  |r − r | = r 1 − rˆ · + (2.22) ≈ r 1 − rˆ · r r r and similarly, 1 1 ≈  |r − r | r

(2.23)

Furthermore, the energy E of the incident electrons in a UED experiment is about 105 eV, whereas the atomic potential energy U is on the order of 101 –102 eV; Given that E  |U |, the Born approximation can be applied to the expression  for the scattered wavefunction by replacing ψ(r  ) with ψ0 (r  ) ∝ eikz . Therefore, by substituting in Eq. (2.21) and applying these approximations, Eq. (2.20) becomes    ik|r−r  | e ψ(r) = Aeikz + 2 − U (r  )ψ(r  )d3 r  4π|r − r  | h¯   ⎛  ⎞  ikr 1−ˆr · rr   2me ⎝− e ⎠ U (r  ) eikz d3 r  ≈ Aeikz + 2 4πr h¯ 2me

(2.24)



me eikr   U (r  )ei(kz −k rˆ ·r ) d3 r  2πh¯ 2 r  ikr   me  i(k inc −k sc )·r  3  e = Aeikz + − )e d r U (r r 2πh¯ 2 = Aeikz −

25 George

Green (1793–1841) made important contributions to mathematical physics in an essay on electricity and magnetism in 1828; these include the relationship between properties inside a volume and those on its surface (Green’s theorem) and a potential function that can be used to impose boundary conditions (Green’s function) [67, 185].

2.2 Ultrafast Electron Diffraction

27

where k inc = k zˆ and k sc = k rˆ are the wavevector of the incident and scattered electron respectively. By comparing the last equation with Eq. (2.17), an expression for the scattering amplitude is obtained as f (q) = −

me 2πh¯ 2



U (r)e−iq·r d3 r

(2.25)

where q = k sc − k inc is the wavevector associated with the momentum p = hq ¯ transferred during the scattering event. In effect, the scattering amplitude is just the Fourier transform of the electrostatic potential energy with respect to q. In electron diffraction, the scatterer is not a single atom but many atoms bound together in molecules and arranged in a crystal. To extend Eq. (2.25) to this case, two substitutions  are needed. The first is based on the independent-atom approximation, U (r) → m Um (r − r m ), where the electrostatic potential energy of the overall structure is simply a sum over the potential energy Um (r) of each atom m in its unperturbed state, displaced by its position r m relative to the origin: f (q) = −

me 2πh2

 

 Um (r − r m ) e−iq·r d3 r

¯ m     me −iq·(r−r m ) 3 − (r − r )e d r e−iq·r m = U m m 2 2π h ¯ m  = fm (q)e−iq·r m

(2.26)

m

where fm (q) is the scattering amplitude associated with atom m, also known as its ‘atomic form factor’. The second substitution is a consequence of the spatial symmetry inherent in the structure of a crystal. The position of every atom can be generated through discrete translation of the set of basis atoms that make up the unit cell of the crystal. Thus, for each atom m, r m = R n1 n2 n3 + i , where 3  R n1 n2 n3 = nj a j is the position of the associated lattice point defined by the

r

j =1

r

integer triplet (n1 , n2 , n3 ) and three lattice vectors a j , and i is the position of the basis atom i relative to the origin of the unit cell. Then, assuming a crystal in the shape of a parallelepiped with edges N1 a1 , N2 a2 , N3 a3 parallel to the lattice vectors, Eq. (2.26) becomes

28

2 Methods: Experimental Techniques and Data Science

f (q) =

N 3 −1  1 −1 N 2 −1 N   n1 =0 n2 =0 n3 =0

=

3  j =1

⎛

⎛

Nj −1

⎝



r)

fi (q)e−iq·(R n1 n2 n3 +

i

⎞

e−iq·nj a j ⎠

nj =0

⎞



r

fi (q)e−iq·

i

i

(2.27)

i

3  e−iNj q·a j − 1 ⎠ F (q) e−iq·a j − 1 j =1

=⎝

= S(q)F (q) where the scattering amplitude has been separated neatly into two terms: S(q) and F (q). The first function is known as the ‘shape factor’ and depends only on the geometrical shape of the crystal. The second one is called the ‘structure factor’ and it is the most relevant to structure determination since the positional information of all the basis atoms are contained within it. To get an intuitive grasp on how the observed quantity—the scattered intensity I (q) = |f (q)|2 —varies as a function of the many crystallographic parameters, let us now consider first the absolute square of the shape factor, 2 3  −iNj q·a j  e − 1    e−iq·a j − 1 

|S(q)|2 =

j =1

(2.28)

3  sin2 ( 12 Nj q · a j )

=

j =1

sin2 ( 12 q · a j )

For large N , this function is essentially zero everywhere except at q such that q · a j = hj π for some integer triplet (h1 , h2 , h3 ), where it sharply peaks with peak width of ∼ Nπaj . Recalling Eq. (2.15), this constraint is simply the Laue condition for diffraction, re-derived from first principles. Similarly, let us consider the absolute square of the structure factor.    |F (q)| =  fi (q)e−iq· 

2

r 

2

i

=



|fi (q)|2 +

i

i





r −r )

fi (q)fj∗ (q)e−iq·(

i

j

(2.29)

i,j =i

= Iat (q) + Ist (q) The first term Iat (q) is referred as the atomic contribution to the scattered intensity. It contains no structural information since it just a sum of the atomic form factor of

2.2 Ultrafast Electron Diffraction

29

the individual atoms. Assuming centrosymmetric atomic potentials, f (q) and thus Iat (q) are real, positive, and monotonically decreasing with q = |q|. On the other hand, the second term Ist (q) is the structural contribution to the scattered intensity. Since fi (q) is strictly real,26 it can be rewritten into a more intuitive expression 

Ist (q) =

i,j =i



=

r −r )

fi (q)fj∗ (q)e−iq·(

i

j

r −r ) + e−iq·(r −r )

 fi (q)fj (q) eiq·(

i

j

i

j

(2.30)

i,j