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## Background
In an attoclock measurement, the maximum of the photoelectron momentum
distribution is displaced from the direction expected from the laser vector
potential at ionization. That displacement is often discussed as a timing
observable, but the outgoing electron also scatters from the residual binding
potential.
## Research problem
For a short circularly polarized pulse, determine whether the intensity
dependence of the offset angle can be accounted for by a post-release
central-potential scattering model. Establish how far a conclusion obtained
for an unscreened Coulomb target remains valid as the interaction is screened,
and state what the resulting evidence does and does not establish about a
universal interpretation of the attoclock offset.
Choose the derivation, numerical evidence, comparisons and presentation needed
to make that assessment scientifically convincing. The number and arrangement
of figures are for you to decide.
## Physical inputs
Work in atomic units. Use a circular driver of angular frequency
$\omega=0.057$ and the intensity convention
$$
I=2E_0^2,\qquad A_0=\frac{E_0}{\omega},
$$
where $E_0$ and $A_0$ are the peak electric-field and vector-potential
amplitudes. Study a one-electron target with ionization potential
$I_p=0.5$ under both
$$
V_C(r)=-\frac{1}{r}
$$
and the screened family
$$
V_{\lambda}(r)=-\frac{Z_{\lambda}}{r}e^{-r/\lambda}.
$$
For every finite $\lambda$, determine $Z_{\lambda}$ so that the field-free
ground-state binding energy remains $I_p=0.5$. Include the Coulomb limit, the
strong-screening case $\lambda=1$, and enough intermediate screening lengths
of your choice to test whether an inference from either endpoint transfers
across the family.
## Method and scope boundary
Build the reduced model from the standard weak-field Keldysh exit scale, the
stated vector-potential momentum scale, and the outgoing half of a
central-potential scattering trajectory. Derive and declare the resulting
angle convention. Standard Rutherford and screened central-potential
scattering theory may be derived or cited. State the approximation level of
the screened calculation you use.
Choose an intensity domain that resolves the regimes needed for your claim
while remaining consistent with the physical assumptions you invoke. Numerical
grids, root solvers, quadrature, sampling, tolerances, plotting ranges and
reliability tests are your choices. They are not grading targets in themselves;
their consequences for the physical evidence are.
High-cost TDSE/iSURF propagation and CTMC regeneration are outside the required
task. They may be discussed as external tests, but neither is required and
unsupported numerical data from those methods must not be presented as your
own result.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package. Include the task-specific derivation, code, numerical data
underlying material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise environment
and run information. No fixed file tree, filename scheme, figure count or
complete exploration history is required.
Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary and temporary-storage allowance are binding.
No separate aggregate CPU-hour limit applies.
Optional extensions must be clearly separated and do not earn extra credit
merely by using more computation.
## Reference and integrity policy
Complete this task without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with this task, on those offline materials or
software documentation. Record every such source actually consulted. For formulas,
methods, constants, or conventions used from general knowledge rather than a
consulted source, state enough of the method basis, definition, value, units,
and convention to make the calculation auditable; do not invent a citation.
You are not required to identify or prove a publication date or to retrieve
literature while completing this task.
The hidden target article and its identifying metadata, figures or captions,
supplementary material, author code or data, target-specific reproduction
code, solution or scoring material, online solutions, and worked
reproductions of this case must not be used. All submitted scientific figures
and numerical data must be generated by your own implementation of the stated
problem.
See [submission requirements](../../submission_requirements.md) for the common submission entry points.
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