^{1}, Wenxi Liang

^{1}and Ahmed H. Zewail

^{1,a)}

### Abstract

Recent studies in ultrafast electron crystallography (UEC) using a reflection diffraction geometry have enabled the investigation of a wide range of phenomena on the femtosecond and picosecond time scales. In all these studies, the analysis of the diffraction patterns and their temporal change after excitation was performed within the kinematical scattering theory. In this contribution, we address the question, to what extent dynamical scattering effects have to be included in order to obtain quantitative information about structuraldynamics. We discuss different scattering regimes and provide diffraction maps that describe all essential features of scatterings and observables. The effects are quantified by dynamical scattering simulations and examined by direct comparison to the results of ultrafast electron diffraction experiments on an *in situ* prepared Ni(100) surface, for which structuraldynamics can be well described by a two-temperature model. We also report calculations for graphitesurfaces. The theoretical framework provided here allows for further UEC studies of surfaces especially at larger penetration depths and for those of heavy-atom materials.

This work was supported by the National Science Foundation (NSF) and the Air Force Office of Scientific Research (USAFOSR) in the Center for Physical Biology at Caltech supported by the Gordon and Betty Moore Foundation. One of the authors (S.S.) gratefully acknowledges a scholarship from the Alexander von Humboldt-Foundation.

I. INTRODUCTION

II. EXPERIMENTAL METHODS

III. THEORETICAL METHODS

A. Dynamical diffraction theory

B. The two-temperature model and structuraldynamics of Ni(100)

IV. RESULTS AND DISCUSSION

A. Qualitative features in dynamical scattering from static crystal surfaces

B. Temporal changes of the scattering intensity

C. Connection between the diffraction map and the diffraction pattern

D. Dynamical scattering in nickel and graphite

1. Dynamics of a clean Ni(100) surface

2. Dynamics of single crystalline graphite

V. CONCLUSION

### Key Topics

- Surface dynamics
- 42.0
- Electron scattering
- 32.0
- Surface scattering
- 28.0
- Graphite
- 20.0
- Multiple scattering
- 20.0

## Figures

Schematics of the diffraction geometry. An electron beam with wave vector impinges on the sample with an incidence angle θ. Due to the scattering potential within the sample a set of diffracted electron beam is generated in transmission () and reflection (). Above and below the sample the scattering potential *V* is zero and the wave function is a superposition of non-interacting plane waves. Inside the sample the wave function can be described by using a 2D-Bloch wave approach.

Schematics of the diffraction geometry. An electron beam with wave vector impinges on the sample with an incidence angle θ. Due to the scattering potential within the sample a set of diffracted electron beam is generated in transmission () and reflection (). Above and below the sample the scattering potential *V* is zero and the wave function is a superposition of non-interacting plane waves. Inside the sample the wave function can be described by using a 2D-Bloch wave approach.

Ni(100) diffraction map for the specular diffraction spot. (a) The calculated (ϕ, θ)-dependent reflectivity of the specular diffraction spot from a Ni(100) surface near the [110] azimuth is shown on the top right. The qualitative features of the map can be rationalized by considering the Bragg angles (blue lines) and surface wave resonance conditions (red lines) depicted on the top left (see text and Appendix A for details). (b) The ϕ-dependence of the calculated reflectivity (shaded gray area) is compared to the experimentally determined intensity (blue line) of the specular spot for a fixed incidence angle θ = 5.3°, corresponding to the (0010) diffraction condition. ϕ-values that allow SWR according to Eq. (8) are indicated by vertical black lines. (c) Schematical comparison of the involved beams in a direct Bragg scattering and a surface wave resonance regime.

Ni(100) diffraction map for the specular diffraction spot. (a) The calculated (ϕ, θ)-dependent reflectivity of the specular diffraction spot from a Ni(100) surface near the [110] azimuth is shown on the top right. The qualitative features of the map can be rationalized by considering the Bragg angles (blue lines) and surface wave resonance conditions (red lines) depicted on the top left (see text and Appendix A for details). (b) The ϕ-dependence of the calculated reflectivity (shaded gray area) is compared to the experimentally determined intensity (blue line) of the specular spot for a fixed incidence angle θ = 5.3°, corresponding to the (0010) diffraction condition. ϕ-values that allow SWR according to Eq. (8) are indicated by vertical black lines. (c) Schematical comparison of the involved beams in a direct Bragg scattering and a surface wave resonance regime.

Changes of the rocking curves upon heating and lattice expansion for different diffraction conditions. The rocking curves for a Ni(100) surface at equilibrium conditions (*T* = 300 K, equilibrium lattice constants, blue lines) are compared to the rocking curves for a lattice which is 10% expanded perpendicular to the surface (green lines), and a lattice at *T* = 600 K with the room temperature lattice constants (red lines). The azimuthal direction of the incident electron beam is either in the [110] zone axis (a), or 7° and 9.45° off the zone axis (b,c). As a reference, also the rocking curves in the kinematical scattering approximation are given in (d). The expected peak shift upon lattice expansion and intensity drop upon heating, according to the kinematic approximation, are not reproduced in (a) and at high incidence angles in (b) due to the prevalence of multiple scattering events in these diffraction conditions. However, if the azimuthal angle is off the zone axis and for intermediate values of the incidence angle a kinematic behavior can be qualitatively recovered as can be seen in (b).

Changes of the rocking curves upon heating and lattice expansion for different diffraction conditions. The rocking curves for a Ni(100) surface at equilibrium conditions (*T* = 300 K, equilibrium lattice constants, blue lines) are compared to the rocking curves for a lattice which is 10% expanded perpendicular to the surface (green lines), and a lattice at *T* = 600 K with the room temperature lattice constants (red lines). The azimuthal direction of the incident electron beam is either in the [110] zone axis (a), or 7° and 9.45° off the zone axis (b,c). As a reference, also the rocking curves in the kinematical scattering approximation are given in (d). The expected peak shift upon lattice expansion and intensity drop upon heating, according to the kinematic approximation, are not reproduced in (a) and at high incidence angles in (b) due to the prevalence of multiple scattering events in these diffraction conditions. However, if the azimuthal angle is off the zone axis and for intermediate values of the incidence angle a kinematic behavior can be qualitatively recovered as can be seen in (b).

Photo-induced intensity change of the (0010) Bragg spot for different azimuthal angles. (a) The simulated ϕ-dependence of the (0010) intensity for two different temperatures (*T* = 300 K (blue line); *T* = 600 K (red line)) is compared to the experimentally measured intensity before and 15 ps after laser excitation (estimated *T* ≈ 550 K, see below). Both the shift of the SWR maxima towards the zone axis and the smaller intensity drop compared to kinematical scattering (green line) is reproduced in the experiment. (b) Change of the rocking curve at a SWR (ϕ = 9.4°) upon a change of the lattice constant *c*. (c) The intensity drop of the (00 10) Bragg spot upon heating is compared for different azimuthal angles ϕ. The intensity drop in a SWR condition approximately follows a Debye-Waller-like behavior (Eq. (9)) with an effective *q* _{eff} which strongly depends on the azimuthal angle. The intensity increase at ϕ = 3.05° with raising temperature can be explained by multiple, competing scattering channels.

Photo-induced intensity change of the (0010) Bragg spot for different azimuthal angles. (a) The simulated ϕ-dependence of the (0010) intensity for two different temperatures (*T* = 300 K (blue line); *T* = 600 K (red line)) is compared to the experimentally measured intensity before and 15 ps after laser excitation (estimated *T* ≈ 550 K, see below). Both the shift of the SWR maxima towards the zone axis and the smaller intensity drop compared to kinematical scattering (green line) is reproduced in the experiment. (b) Change of the rocking curve at a SWR (ϕ = 9.4°) upon a change of the lattice constant *c*. (c) The intensity drop of the (00 10) Bragg spot upon heating is compared for different azimuthal angles ϕ. The intensity drop in a SWR condition approximately follows a Debye-Waller-like behavior (Eq. (9)) with an effective *q* _{eff} which strongly depends on the azimuthal angle. The intensity increase at ϕ = 3.05° with raising temperature can be explained by multiple, competing scattering channels.

Ewald construction for the specular spot. (a) Due to energy conservation, the incident and outgoing wave vectors *k* and *k* ^{′} ^{′} lie on a sphere in reciprocal space (green circle). The periodicity of the surface structure leads to a quantized momentum transfer parallel to the surface, with allowed valued according to reciprocal lattice rods *B* _{ i } (gray lines). The wave vectors of observed diffracted beams lie at the intersection of the Ewald sphere with reciprocal lattice rods. For an idealized system with a well-defined incidence wave vector *k* and perfect periodicity parallel to the surface, the wave vector *k* ^{′} ^{′} of the diffracted specular beam is sharply defined with an outgoing angle θ^{′} ^{′} equal to the incidence angle θ (independent of structural dynamics). (b) If the quantization condition of the parallel momentum transfer is partly relaxed by Δ*B* due to surface defects, the specular beam shows a relative spread of Δ*k* ^{′} ^{′}/*k* = θ^{′} ^{′} = Δ*B*/θ*k* in the outgoing wave vectors. The intensity within this allowed range of outgoing angles depends on the modulation along the reciprocal lattice rod. (c) Similarly, an incident beam with a range Δ*k* of wave vectors (convergent beam) gives a distribution of outgoing wave vectors with Δ*k* ^{′} ^{′} = Δ*k*.

Ewald construction for the specular spot. (a) Due to energy conservation, the incident and outgoing wave vectors *k* and *k* ^{′} ^{′} lie on a sphere in reciprocal space (green circle). The periodicity of the surface structure leads to a quantized momentum transfer parallel to the surface, with allowed valued according to reciprocal lattice rods *B* _{ i } (gray lines). The wave vectors of observed diffracted beams lie at the intersection of the Ewald sphere with reciprocal lattice rods. For an idealized system with a well-defined incidence wave vector *k* and perfect periodicity parallel to the surface, the wave vector *k* ^{′} ^{′} of the diffracted specular beam is sharply defined with an outgoing angle θ^{′} ^{′} equal to the incidence angle θ (independent of structural dynamics). (b) If the quantization condition of the parallel momentum transfer is partly relaxed by Δ*B* due to surface defects, the specular beam shows a relative spread of Δ*k* ^{′} ^{′}/*k* = θ^{′} ^{′} = Δ*B*/θ*k* in the outgoing wave vectors. The intensity within this allowed range of outgoing angles depends on the modulation along the reciprocal lattice rod. (c) Similarly, an incident beam with a range Δ*k* of wave vectors (convergent beam) gives a distribution of outgoing wave vectors with Δ*k* ^{′} ^{′} = Δ*k*.

Connection between the diffraction map and the diffraction pattern. In the left two panels the schematic shift of the Bragg condition before (blue) and after excitation (red) is shown. The shaded area signifies the effective divergence angle of the electron beam. The effective divergence angle is larger in the top panel compared to the bottom panel. The center of the distribution can be adjusted by tilting the sample whereas the width is, in the present setup, dominated by the sample quality (see Fig. 5(b) and discussion in the text). The right panels show the resulting change of the Bragg spot in the diffraction pattern. In the case of a large effective divergence (top panels) the shift in the Bragg spot qualitatively matches the shift in the diffraction map. For a small effective beam divergence, the Bragg spot is almost not moving after excitation (corresponding to Fig. 5(a)), but the Bragg spot intensity decreases significantly since the Bragg condition changes due to an expanding lattice.

Connection between the diffraction map and the diffraction pattern. In the left two panels the schematic shift of the Bragg condition before (blue) and after excitation (red) is shown. The shaded area signifies the effective divergence angle of the electron beam. The effective divergence angle is larger in the top panel compared to the bottom panel. The center of the distribution can be adjusted by tilting the sample whereas the width is, in the present setup, dominated by the sample quality (see Fig. 5(b) and discussion in the text). The right panels show the resulting change of the Bragg spot in the diffraction pattern. In the case of a large effective divergence (top panels) the shift in the Bragg spot qualitatively matches the shift in the diffraction map. For a small effective beam divergence, the Bragg spot is almost not moving after excitation (corresponding to Fig. 5(a)), but the Bragg spot intensity decreases significantly since the Bragg condition changes due to an expanding lattice.

Time-dependent rocking curve diffraction spot movement. (a) The experimental rocking curve of single crystalline graphite [near (0014) Bragg angle, SWR] before (blue curve) and after excitation (red curve) is depicted. The two peaks in the rocking curve correspond to two single crystalline grains with a slight relative inclination of ∼0.25°. The rocking curve after excitation shows an overall decreased intensity due to an increased atomic mean-square displacement and a uniform shift of 0.015° in the peak maxima indicating a lattice expansion of 0.36%. (b) The right panel shows the corresponding position of the (0014) Bragg spot (quantified by the scattering angle θ^{′} ^{′} + θ) before (blue crosses) and after excitation (red crosses). As expected, the scattering angle of the Bragg spot decreases after excitation lattice expansion. However, contrary to the uniform shift of the rocking curve in (a), the two grains show a different spot movement. This can be explained by the different peak widths in the rocking curve of the grains together with the imaging mechanism shown in Fig. 6.

Time-dependent rocking curve diffraction spot movement. (a) The experimental rocking curve of single crystalline graphite [near (0014) Bragg angle, SWR] before (blue curve) and after excitation (red curve) is depicted. The two peaks in the rocking curve correspond to two single crystalline grains with a slight relative inclination of ∼0.25°. The rocking curve after excitation shows an overall decreased intensity due to an increased atomic mean-square displacement and a uniform shift of 0.015° in the peak maxima indicating a lattice expansion of 0.36%. (b) The right panel shows the corresponding position of the (0014) Bragg spot (quantified by the scattering angle θ^{′} ^{′} + θ) before (blue crosses) and after excitation (red crosses). As expected, the scattering angle of the Bragg spot decreases after excitation lattice expansion. However, contrary to the uniform shift of the rocking curve in (a), the two grains show a different spot movement. This can be explained by the different peak widths in the rocking curve of the grains together with the imaging mechanism shown in Fig. 6.

Wave function of the diffracted electron for nickel (a) and graphite (b) depending on the depth *z* within the sample. (a) The wave function coefficients |*c* _{ i }|^{2} (Eq. (2)) for *B* _{ i } = (00) (green and blue lines) are calculated for an incidence angle θ = 5.4° ((0010) Bragg spot) and azimuthal angles of ϕ = 3.5° (blue line), corresponding to a resonance, and ϕ = 12° (green line). The red line shows the depth-dependence of the surface beam for ϕ = 3.5°. It can be seen that within the SWR the wave function is localized within 1 nm with a substantial transfer of population from the incident *c* _{(00)} beam to the tangential beam. Outside of a SWR (green line) the wave function coefficient *c* _{(00)} decays slower within the solid and the coefficient gains no significant amplitude (not shown). The inset shows the average scattering potential depending on the depth *z* inside the sample. (b) In the same way, *c* _{(00)} was calculated for graphite using θ = 3.995° ((0014) Bragg spot) and azimuthal angles of ϕ = 1.275° (relative to [110] zone axis, SWR) and ϕ = 7° (no SWR). The qualitative picture is similar to (a) although, due to the smaller scattering potential (see inset), the length scale over which the wave function decays is stretched.

Wave function of the diffracted electron for nickel (a) and graphite (b) depending on the depth *z* within the sample. (a) The wave function coefficients |*c* _{ i }|^{2} (Eq. (2)) for *B* _{ i } = (00) (green and blue lines) are calculated for an incidence angle θ = 5.4° ((0010) Bragg spot) and azimuthal angles of ϕ = 3.5° (blue line), corresponding to a resonance, and ϕ = 12° (green line). The red line shows the depth-dependence of the surface beam for ϕ = 3.5°. It can be seen that within the SWR the wave function is localized within 1 nm with a substantial transfer of population from the incident *c* _{(00)} beam to the tangential beam. Outside of a SWR (green line) the wave function coefficient *c* _{(00)} decays slower within the solid and the coefficient gains no significant amplitude (not shown). The inset shows the average scattering potential depending on the depth *z* inside the sample. (b) In the same way, *c* _{(00)} was calculated for graphite using θ = 3.995° ((0014) Bragg spot) and azimuthal angles of ϕ = 1.275° (relative to [110] zone axis, SWR) and ϕ = 7° (no SWR). The qualitative picture is similar to (a) although, due to the smaller scattering potential (see inset), the length scale over which the wave function decays is stretched.

Simulated heating dynamics of a Ni(100) surface (see text for details). (a) Temporal change of the electron (*T* _{ e }) and lattice temperature (*T* _{ l }) at the surface depending on the excitation fluence as predicted from a two-temperature model with temperature-dependent material constants. (b) The temporal change of the lattice constant for an excitation fluence of 7 mJ/cm^{2} at different depths as obtained by combining the two-temperature model with a one-dimensional spring model.

Simulated heating dynamics of a Ni(100) surface (see text for details). (a) Temporal change of the electron (*T* _{ e }) and lattice temperature (*T* _{ l }) at the surface depending on the excitation fluence as predicted from a two-temperature model with temperature-dependent material constants. (b) The temporal change of the lattice constant for an excitation fluence of 7 mJ/cm^{2} at different depths as obtained by combining the two-temperature model with a one-dimensional spring model.

Two-temperature model and Bragg peak intensity change. Comparison between the experimentally observed and theoretically predicted temporal drop of the (0010) Bragg spot intensity of a Ni(100) surface at an azimuthal angle of ϕ = 3.4°. (a) The theoretically predicted heating dynamics (dotted line) are obtained by using the temporal change of the structure as deduced from a TTM (Fig. 9) and calculate the scattered intensity within dynamical scattering theory. To account for the finite temporal resolution the theoretical predicted drop is convoluted with a Gaussian (4 ps standard deviation). It has to be noted that the experimental data are quantitatively reproduced for different fluence by only using the known material constants of nickel. On the contrary, a kinematical scattering model severely overestimates the intensity drop due to lattice heating, as shown in the inset. (b) In a similar manner the cooling dynamics of the surface at longer times is well described, following a typical behavior.

Two-temperature model and Bragg peak intensity change. Comparison between the experimentally observed and theoretically predicted temporal drop of the (0010) Bragg spot intensity of a Ni(100) surface at an azimuthal angle of ϕ = 3.4°. (a) The theoretically predicted heating dynamics (dotted line) are obtained by using the temporal change of the structure as deduced from a TTM (Fig. 9) and calculate the scattered intensity within dynamical scattering theory. To account for the finite temporal resolution the theoretical predicted drop is convoluted with a Gaussian (4 ps standard deviation). It has to be noted that the experimental data are quantitatively reproduced for different fluence by only using the known material constants of nickel. On the contrary, a kinematical scattering model severely overestimates the intensity drop due to lattice heating, as shown in the inset. (b) In a similar manner the cooling dynamics of the surface at longer times is well described, following a typical behavior.

Simulated dynamics of the relative Bragg angle change. Within kinematic scattering theory (dotted red line) the relative change Δθ/θ of the Bragg angle of a specular spot (due to lattice expansion) is independent of the order of the spot. Within dynamical scattering theory (solid lines) the relative shift of the Bragg spot depends on the azimuthal orientation ϕ and the incidence angle θ. A reduced shift is observed in a SWR (light blue and red line). In an off-resonance condition the shift predicted by dynamical scattering is close to the kinematical results (dark blue line).

Simulated dynamics of the relative Bragg angle change. Within kinematic scattering theory (dotted red line) the relative change Δθ/θ of the Bragg angle of a specular spot (due to lattice expansion) is independent of the order of the spot. Within dynamical scattering theory (solid lines) the relative shift of the Bragg spot depends on the azimuthal orientation ϕ and the incidence angle θ. A reduced shift is observed in a SWR (light blue and red line). In an off-resonance condition the shift predicted by dynamical scattering is close to the kinematical results (dark blue line).

Simulated heating dynamics of the in-plane and out-of-plane vibrational modes in graphite at an excitation fluence of 10 mJ/cm^{2}. The material constants were taken from Refs. 60–62.

Simulated heating dynamics of the in-plane and out-of-plane vibrational modes in graphite at an excitation fluence of 10 mJ/cm^{2}. The material constants were taken from Refs. 60–62.

Influence of dynamical scattering on the observed photo-induced diffraction pattern changes in graphite. (a) Intensity of the (0014) Bragg spot at the -SWR for different temperatures *T*. The blue curve shows the intensity drop when the in-plane and out-of-plane vibrations are in thermal equilibrium with each other at a temperature *T*. The red (green) curve refer to non-equilibrium scenarios where the in-plane (out-of-plane) vibrations are at a temperature *T* and the out-of-plane (in-plane) vibrations are at a temperature of 300 K. For comparison, also the kinematically predicted intensity drop is shown (pink line) together with the calculated intensity drop for an azimuthal angle of ϕ = 7° (no SWR, light blue line). This shows that the reduced intensity drop upon heating in a SWR as discussed for nickel is also observed in graphite. Furthermore, due to the coupling of the incident electron beam to a tangential surface beam the intensity drop is not only dependent on the out-of-plane vibrational temperature but also on the in-plane temperature. (b) Simulated temporal intensity drop of the (0014) Bragg spot using the time-dependent temperatures shown in Fig. 12 showing the difference in the relative, temporal change of the Bragg area in the rocking curve (blue line), the relative change in the reflectivity for incidence angles of θ = 3.97° (green line) and θ = 4.02° (red line), and the intensity dynamics as predicted by kinematical scattering (pink line). The different behaviors can be explained by the interplay between lattice expansion and lattice heating and their effect on the rocking curve as displayed in Fig. 14.

Influence of dynamical scattering on the observed photo-induced diffraction pattern changes in graphite. (a) Intensity of the (0014) Bragg spot at the -SWR for different temperatures *T*. The blue curve shows the intensity drop when the in-plane and out-of-plane vibrations are in thermal equilibrium with each other at a temperature *T*. The red (green) curve refer to non-equilibrium scenarios where the in-plane (out-of-plane) vibrations are at a temperature *T* and the out-of-plane (in-plane) vibrations are at a temperature of 300 K. For comparison, also the kinematically predicted intensity drop is shown (pink line) together with the calculated intensity drop for an azimuthal angle of ϕ = 7° (no SWR, light blue line). This shows that the reduced intensity drop upon heating in a SWR as discussed for nickel is also observed in graphite. Furthermore, due to the coupling of the incident electron beam to a tangential surface beam the intensity drop is not only dependent on the out-of-plane vibrational temperature but also on the in-plane temperature. (b) Simulated temporal intensity drop of the (0014) Bragg spot using the time-dependent temperatures shown in Fig. 12 showing the difference in the relative, temporal change of the Bragg area in the rocking curve (blue line), the relative change in the reflectivity for incidence angles of θ = 3.97° (green line) and θ = 4.02° (red line), and the intensity dynamics as predicted by kinematical scattering (pink line). The different behaviors can be explained by the interplay between lattice expansion and lattice heating and their effect on the rocking curve as displayed in Fig. 14.

Influence of lattice expansion and lattice heating on time-dependent rocking curves for the (0014) Bragg spot in graphite. Rocking curves at delay times Δ*t* = −5 ps, 4 ps, and 50 ps using either only the predicted lattice expansion (left panel), only the lattice heating (middle panel) or both, lattice expansion and lattice heating.

Influence of lattice expansion and lattice heating on time-dependent rocking curves for the (0014) Bragg spot in graphite. Rocking curves at delay times Δ*t* = −5 ps, 4 ps, and 50 ps using either only the predicted lattice expansion (left panel), only the lattice heating (middle panel) or both, lattice expansion and lattice heating.

Geometrical construction of the surface wave resonance condition (see text for details). (b and c) Two-dimensional cuts of the Ewald construction depicted in (a).

Geometrical construction of the surface wave resonance condition (see text for details). (b and c) Two-dimensional cuts of the Ewald construction depicted in (a).

Sample characterization (a) Auger electron spectra of the Ni(100) sample at the S(KLL), Ni(LMM), C(KLL), and O(KLL) transition energies. (b) LEED pattern of the nickel without cleaning steps (showing a Ni(100)-c(2×2)-S adsorbate structure) and the diffraction pattern of the sample after several sputtering and annealing cycles.

Sample characterization (a) Auger electron spectra of the Ni(100) sample at the S(KLL), Ni(LMM), C(KLL), and O(KLL) transition energies. (b) LEED pattern of the nickel without cleaning steps (showing a Ni(100)-c(2×2)-S adsorbate structure) and the diffraction pattern of the sample after several sputtering and annealing cycles.

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