Channeling radiation in oriented crystals arises from transitions between quantized transverse bound states in the MeV regime and is strongly affected by thermal diffuse scattering through population transfer and decoherence. A frozen-phonon multislice propagation framework is developed to track a reduced transverse Hilbert space spanned by selected bound-state manifolds using configuration-resolved projection amplitudes. Beyond reproducing transition energies, the method yields reduced manifold density matrices, thermal population kinetics, and depth-resolved coherence metrics. Applied to axial electron channeling in $\langle100\rangle$ diamond at 16.9 MeV, the results show approximately exponential population loss with strongly state-dependent feeding among low-lying manifolds. For an initial coherent superposition in the degenerate 2p manifold, the intra-manifold purity relaxes toward the maximally mixed limit, consistent with thermally induced random basis rotations. Under 1s initial excitation, population transferred into the 2p and 3d manifolds remains close to maximally mixed, while weak cross-manifold coherences persist. The framework enables quantitative analysis of thermal population dynamics, decoherence, and their links to spontaneous and coherently driven emission observables across a broad range of crystal structures.
Charged particle channeling in oriented single crystals offers a paradigmatic system for studying motion in strong, periodic potentials. For electrons and positrons from the few-MeV to GeV range, the transverse dynamics can be quasi-bound to atomic columns (axial channeling) or planes (planar channeling) [1][2][3]. Transitions between quantized transverse states generate channeling radiation in the X-ray to 𝛾-range, yet experimental spectral features are significantly affected by the coupling to the crystal lattice [4][5][6][7].
Classical trajectory descriptions are typically adequate at hundreds-MeV to GeV energies, where the density of bound transverse states becomes effectively quasi-continuous [8][9].
In the lower MeV regime, the transverse motion is strongly quantized, so quantum formulations are required to reproduce discrete transition lines. Standard theoretical approaches typically incorporate finite temperature effects via a static mean-field (Debye-Waller broadened) atomic potentials [6,[8][9][10][11][12][13]. While this correctly shifts eigenenergies, it does not capture dynamical thermal diffuse scattering (TDS) effects, such as population transfer between bound states and the associated decoherence. Numerical quantification of coherence dynamics in axial channeling has remained limited, yet coherence can influence radiation observables, such as polarization and angular distributions, and is central to assessing external-field driven channeling-emission schemes [14][15][16][17][18].
A complementary route, adopted here, is to simulate the high-energy coherent wave propagation directly through crystal potential. In the paraxial approximation, the resulting Schrödinger-like evolution along the beam direction admits a split-operator factorization.
The resulting multislice propagator naturally captures dynamical diffraction and channeling without additional coupling assumptions, and it generalizes to arbitrary crystal structures and lattice defects [19].
To include thermal effects beyond Debye-Waller smearing, the frozen-phonon (FP) method is employed [19][20][21]. FP reproduces TDS by sampling many static displaced lattices, propagating coherently for each configuration, and forming relevant observables via incoherent ensemble averages. In the channeling-radiation setting, this provides a microscopic description of TDS effects while retaining full wave mechanics.
In this article, a tracked transverse Hilbert space is defined in which selected bound-state manifolds are explicitly followed during FP multislice propagation. Depth-dependent manifold populations, reduced states within degenerate manifolds, and basis-invariant cross-manifold coherence measures are extracted from configuration-resolved projection amplitudes. The framework is applied to axial electron channeling in diamond in the 30-MeV-class regime, analyzing thermal population transfer and degenerate intra-manifold decoherence, as well as cross-manifold coherence for the radiative channels 2𝑝 → 1𝑠 and 3𝑑 → 2𝑝 under 1𝑠-dominated excitation.
A high-energy, spinless electron is treated within the standard relativistically corrected paraxial approximation. The wavefunction factors into a slowly varying transverse envelope 𝜓(𝒓, 𝑧) multiplied by a high-momentum planewave, 𝑘 0 , along the optic axis z.
Assuming constant total energy, z plays the role of the evolution parameter and the transverse envelope obeys Schrödinger-type equation:
(1)
where 𝛾 is relativistic factor and 𝜎 = 𝛾𝑚 0 /𝑘 0 ℏ 2 . The operator in brackets defines the transverse Hamiltonian 𝐻 ⊥ (𝑧). The formal solution is expressed via a z-ordered unitary propagator:
(2) 𝜓(𝒓, 𝑧) = 𝑈(𝑧, 𝑧 0 ) 𝜓(𝒓, 𝑧 0 ),
Numerically, the crystal is discretized into N slices of thickness Δz and a Suzuki-Trotter approximation is applied [22]. The slice potential is taken as the z-average over the slice:
Using the first-order (Lie-Trotter) split-operator, the per-slice propagator and wavefunction at thickness 𝑁Δ𝑧 are: )𝜓(𝒓, 0).
The split-step is implemented efficiently using FFTs: the kinetic factor is diagonal in transverse reciprocal space, while the potential factor is diagonal in real space. Crystal is constructed from the Doyle-Turner (DT) parametrization of atomic potentials, commonly used in channeling radiation simulations [23]. The isotropic Debye-Waller (DW) factor with the two-dimensional mean square amplitude of thermal vibrations, 𝑢 2𝑑 2 , is used to statically smear the DT atomic potential (numerical implementation described in Supplemental Materials).
Thermal diffuse scattering is treated here in the frozen-phonon approximation. The initial phonon state is a thermal distribution (statistical mixture), represented by 𝑝 𝑛 in diagonal basis {|𝜇 𝑛 ⟩}, and an electron in a pure state |𝜓(0)⟩, giving initial density matrix:
In the FP approximation, the thermal phonon mixture is represented, under the harmonic approximation, by an ensemble of static lattice configurations indexed
This content is AI-processed based on open access ArXiv data.