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

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Abstract

This chapter focuses on the methods—experimental and analytical—used to attain the results presented in Chaps. 3–7. Here, two experimental techniques will be described in detail: transient absorption (TA) spectroscopy and ultrafast electron diffraction (UED). I will outline the theoretical framework of the probe-sample interaction that is at the heart of each technique. I will also give an overview of the instrumentation that makes up the lab setups and detail the data-driven procedures used to characterize the experimental conditions, correct measurement artifacts, improve signal-to-noise ratios, and finally extract the relevant information from models for further interpretation.

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Notes

  1. 1.

    Absorbance A is often used interchangeably with the term ‘optical density’ (OD); intensity I refers to the irradiance of the laser [239].

  2. 2.

    Developed by James Franck (1882–1964) and Edward Condon (1902–1974) in 1926, this principle states that an electronic transition is most likely to occur without changes in the nuclear positions and the transition probability is proportional to the square of the overlap integral between the two vibrational wavefunctions involved [107, 163].

  3. 3.

    Originating from American spectroscopist Michael Kasha (1920–2013) [280]. This ‘rule’ is a consequence of the relative density of states involved in these different types of vibrational-electronic (vibronic) transitions.

  4. 4.

    For a more detailed description of CPM, see Ref. [254].

  5. 5.

    Scottish physicist John Kerr (1824–1907) discovered in 1875 that the index of refraction of material can be modified by an electric field to the second order: . Kerr-lens modelocking involves an optical pulse so intense that it self-focuses under its own induced Δn and is self-selected when coupled with an aperture that blocks any other mode that fails to self-focus [45].

  6. 6.

    German physicist Friedrich Pockels (1865–1913) discovered in 1893 that the index of refraction of material can be modified by an electric field to the first order: . This effect can be used to quickly and precisely control the polarization of a light beam [45].

  7. 7.

    SHG is a nonlinear optical process whereby two photons with the same frequency interact in a medium to generate a single photon with double the frequency [45].

  8. 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. 9.

    ‘Time-zero’ is the time delay at which there is temporal overlap of the pump and probe pulses.

  10. 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.

  11. 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.

  12. 12.

    For reference, \( \left ( H(t - t_0) \text{e}^{-k t}\right ) \ast \text{e}^{- \frac {1}{2} t^2/\tau ^2} = \frac {1}{2} \text{e}^{\frac {1}{2} k^2 \tau ^2 - k (t - t_0)} \left ( 1 + \text{erf}\left ( \frac {t - t_0 - k \tau ^2}{\sqrt {2} \tau } \right ) \right )\).

  13. 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. 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. 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.

  16. 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 h 1, h 2, h 3 are used instead here for convenient indexing. An overline indicates a negative number, \(\overline {h} = -h\).

  17. 17.

    A computationally more useful definition of the reciprocal lattice vectors is , where the vectors are in column representation.

  18. 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].

  19. 19.

    Paul P. Ewald (1888–1985) conceived of the sphere-in-reciprocal-lattice construct to geometrically visualize diffraction in 1912 [145].

  20. 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.

  21. 21.

    These shapes are known in the literature as reciprocal lattice rods, or ‘relrods’ for short.

  22. 22.

    An amorphous material is one that lacks the long-range order of a crystal.

  23. 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].

  24. 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].

  25. 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].

  26. 26.

    The atomic form factor can be made complex through the addition of a imaginary component when one wants to phenomenologically include inelastic or diffuse scattering in this derivation.

  27. 27.

    Under the ergodic hypothesis, given a stationary random process, the time average of an observable is the same as its ensemble average [360].

  28. 28.

    Named after Ludwig E. Boltzmann (1844–1906) for his use of this statistical method to derive the thermodynamics of a system from the properties of its microscopic constituents [360].

  29. 29.

    In the case of strong lattice vibrations and correlated motions, the following expression leads to thermal diffuse scattering wherein scattered intensity is observed as lines between diffraction spots, along .

  30. 30.

    Ivar Waller (1898–1991) followed up on discussion by Peter Debye (see footnote 23) on the effect of lattice vibrations on X-ray crystallography and gave it a complete mathematical formulation in 1923 [335].

  31. 31.

    This parameter is often referred among protein crystallographers by the ‘B-factor’: .

  32. 32.

    Neutrons have a non-zero magnetic moment and can also be scattered by unpaired electrons [498].

  33. 33.

    For X-rays, the scattering potential depends on the electron number density ; in the case of neutrons, it is Fermi pseudopotential , where m n is the neutron mass and b is the bound coherent neutron scattering length [446].

  34. 34.

    This criterion specifies the minimum resolvable separation between two point sources, first proposed by John William Strutt, the 3rd Baron Rayleigh (1842–1919) [400].

  35. 35.

    Louis de Broglie (1892–1987), won the 1929 Nobel Prize in Physics for introducing the idea of matter waves in 1924 [400].

  36. 36.

    Nevill F. Mott (1905–1996) and Hans A. Bethe (1906–2005) first discovered this relation between the electron and X-ray scattering factors in 1930 [336, 337]. For reference, the derivation is summarized in Appendix E.

  37. 37.

    Assuming molar absorption coefficient 𝜖 abs ∼ 105 M−1 cm−1 and absorber molar concentration c abs ∼ 2.0 M.

  38. 38.

    As reviewed in Refs. [21, 75, 94, 139, 140, 201, 457, 573], there are tabletop femtosecond X-ray sources which are based on laser-driven plasma generation; however, their brightnesses are 104–109 less than the XFEL ones [140], requiring much higher peak pump-laser intensities to achieve good SNR [167, 575, 576].

  39. 39.

    THG is a nonlinear optical process that triples the frequency of an input laser beam. Here, it is achieved by first frequency-doubling the 800-nm input beam through SHG in a BBO crystal, temporally overlapping the 400-nm and remaining 800-nm light using a birefringent calcite crystal, and then combining them into 267-nm photons in another BBO crystal through sum frequency generation (SFG).

  40. 40.

    The photocathode in this setup is a 1-mm thick sapphire substrate with 20 nm of vapour-deposited gold, which has a work function of ca. 3.8–4.3 eV [3, 524]. considering the photon energy of the UV pulse, electrons would be emitted with excess kinetic energy that translates to significant momentum spread.

  41. 41.

    The pinhole blocks electrons that were generated with high transverse momentum; its diameter can be increased to exchange better transverse beam coherence for higher beam brightness, or vice versa.

  42. 42.

    The electrons are high energy but not relativistic; their Lorentz factor γ = 1 + U∕E 0 is less than 2.0 and their relativistic beta \(\beta = v/c = \sqrt {1 - 1/\gamma ^2}\) is only 0.54.

  43. 43.

    The local transverse coherence width \(L_T^{loc}\) is defined here as \(\hbar /\sigma _{p_x}^{\mathrm{loc}}\), where \(\sigma _{p_x}^{\mathrm{loc}}\) is the width of the electron momentum distribution in the transverse direction of the beam [365].

  44. 44.

    The camera parameter is the conversion factor for the position of diffraction features, between the camera frame (in pixels) and reciprocal space (in Å−1). Although determined empirically here, it can be calculated as 4πl∕(λL), where l is the camera pixel size (13.5 μm/px), λ is the electron de Broglie wavelength (0.038 Å), and L is the sample-camera distance (ca. 35 cm).

  45. 45.

    Pyotr L. Kapitsa (1894–1984) and Paul A.M. Dirac (1902–1984) developed the theory of electrons reflected by standing light waves in 1933 [278]. They were both awarded independently a Nobel Prize in Physics: Kapitsa in 1978 for his work in low-temperature physics (along with Arno A. Penzias and Robert W. Wilson for their discovery of the cosmic microwave background radiation) [336] and Dirac in 1933 (jointly with Erwin Schrödinger) for a fully relativistic quantum theory [400].

  46. 46.

    Considering the camera pixel size and sample-camera distance (see footnote 44), this translates to a deflection velocity of ca. 2.8 mrad/ps.

  47. 47.

    The images can be as large as the CCD sensor of the camera (2048 × 2048 pixels) and they are stored as an uncompressed TIFF (‘tagged image file format’) 6.0 file; the pixel values are unsigned 16-bit integers.

  48. 48.

    ‘Big data’ herein refers to datasets that are too large to fit entirely in the random-access memory of a typical personal computer (ca. 8 GB).

  49. 49.

    A program authored by Dr. Meng Gao actively monitors the pointing of the laser by imaging a beam spot using a high-speed camera (PointGrey Firefly); it then compensates for any significant deviation in spot position by controlling a pair of in-beam mirrors mounted on piezoelectric actuators [171].

  50. 50.

    First proposed in 1949 by Norbert Wiener (1894–1964), this image processing technique estimates the local mean and variance around each pixel and then applies more or less smoothing in response [326].

  51. 51.

    This computational technique was developed by John F. Canny (1958–present) in 1986; it detect edges by searching for connected pixels whose gradient magnitude is between two given thresholds [70].

  52. 52.

    Named after Georges Friedel (1865–1933), these pairs are the diffraction spots with Miller index (h, k, l) and \((\bar {h}, \bar {k}, \bar {l} )\) [16].

  53. 53.

    This is performed using the regionprops function in MATLAB R2017b.

  54. 54.

    For the molecule (EDO-TTF)2PF6, there are N = 35 non-hydrogen atoms in the asymmetric unit, hence 3N − 6 = 99 DOFs.

  55. 55.

    For reference, \( \left ( H(t - t_0) \text{e}^{-k t}\right ) \ast \text{e}^{- \frac {1}{2} t^2/\tau ^2} = \frac {1}{2} \text{e}^{\frac {1}{2} k^2 \tau ^2 - k (t - t_0)} \left ( 1 + \text{erf}\left ( \frac {t - t_0 - k \tau ^2}{\sqrt {2} \tau } \right ) \right )\).

  56. 56.

    Named after English mathematician Karl Pearson (1857–1936) [342].

  57. 57.

    On the order of the coherence length of the probe electrons.

  58. 58.

    This is true for all UED samples studied in this thesis, with the exception of [FeII(bpy)3](PF6)2 whose photoproduct state cannot be reached thermally.

  59. 59.

    Norwegian physicist Lars Vegard (1880–1963) observed a linear relation between the lattice parameters of ionic salt alloys and the concentration of each components using XRD [125, 543].

  60. 60.

    The letters u, v, w are used most often to refer to crystal orientations; however, numbered subscripts are used instead here for convenient indexing.

  61. 61.

    French mathematician Benjamin Olinde Rodrigues (1795–1851) discovered in 1840 this convenient formula to rotate a vector in three dimensional space given an axis and an angle of rotation [386].

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Liu, L.C. (2020). Methods: Experimental Techniques and Data Science. In: Chemistry in Action: Making Molecular Movies with Ultrafast Electron Diffraction and Data Science. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-54851-3_2

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