Ultrafast electron diffraction (UED) is a method of directly imaging system dynamics at the atomic scale with picosecond time resolution. In this thesis I present theoretical analyses of the experimental processes, and construct models in order to better understand UED experiments and to guide future refinements. In particular, I derive a model of electron bunch propagation and a model of electron bunch diffraction, where both models take into account all bunch parameters. To analyse the propagation of electron bunches, I present a mean-field analytic Gaussian (AG) model. I derive a system of ordinary differential equations that are solved quickly and easily to give the bunch dynamics. The AG model is compared to N -body numerical simulations of initially Gaussian bunches, and I demonstrate excellent agreement between the two result sets. I also present a comparison of the AG model with numerical simulations of quasi-Gaussian and non-Gaussian distributions, extending the applicability of the AG model to the propagation of ``real-world'' bunches. During propagation, electron bunches can be shaped by electron-optic devices, which are necessary to attain high brightness, sub-100 fs bunches. I investigate two types of electron-optic devices: one is a magnetic lens used for collimating or focusing bunches, the other is a bunch compressor. I derive bunch parameter transformations for each of the electron-optic devices, and present numerical calculations using these transformations along with the AG model showing the effects of the devices on the evolution of the bunch parameters. To analyse electron bunch diffraction in UED experiments, I present a general scattering formalism. Using single-scattering and far-field approximations, I derive an expression for the diffracted signal that depends on the electron bunch properties just before scattering. Using this expression I identify the transverse and longitudinal coherence lengths and discuss the importance of these length scales in diffraction pattern formation. I also discuss the effects of different bunch parameters on the measured diffracted flux, and present sample numerical calculations for scattering by nanosize particles based on this model. This simulation demonstrates the cumulative effects of the bunch parameters, and shows the complex interplay of the bunch and target properties on the diffracted signal.

Ultrafast electron diffraction (UED) is a method of directly imaging system dynamics at the atomic scale with picosecond time resolution. In this thesis I present theoretical analyses of the experimental processes, and construct models in order to better understand UED experiments and to guide future refinements. In particular, I derive a model of electron bunch propagation and a model of electron bunch diffraction, where both models take into account all bunch parameters. To analyse the propagation of electron bunches, I present a mean-field analytic Gaussian (AG) model. I derive a system of ordinary differential equations that are solved quickly and easily to give the bunch dynamics. The AG model is compared to N -body numerical simulations of initially Gaussian bunches, and I demonstrate excellent agreement between the two result sets. I also present a comparison of the AG model with numerical simulations of quasi-Gaussian and non-Gaussian distributions, extending the applicability of the AG model to the propagation of ``real-world'' bunches. During propagation, electron bunches can be shaped by electron-optic devices, which are necessary to attain high brightness, sub-100 fs bunches. I investigate two types of electron-optic devices: one is a magnetic lens used for collimating or focusing bunches, the other is a bunch compressor. I derive bunch parameter transformations for each of the electron-optic devices, and present numerical calculations using these transformations along with the AG model showing the effects of the devices on the evolution of the bunch parameters. To analyse electron bunch diffraction in UED experiments, I present a general scattering formalism. Using single-scattering and far-field approximations, I derive an expression for the diffracted signal that depends on the electron bunch properties just before scattering. Using this expression I identify the transverse and longitudinal coherence lengths and discuss the importance of these length scales in diffraction pattern formation. I also discuss the effects of different bunch parameters on the measured diffracted flux, and present sample numerical calculations for scattering by nanosize particles based on this model. This simulation demonstrates the cumulative effects of the bunch parameters, and shows the complex interplay of the bunch and target properties on the diffracted signal.