This document discusses wavefunction collapse in xenon ions. It explains that as xenon is ionized, the 4f wavefunctions move closer to the core due to an increased effective potential. This leads to overlap between the 4d and 4f wavefunctions, resulting in many unresolved transitions appearing as broad arrays in spectra called UTAs. The document then describes how the Cowan code was used to solve the Schrodinger equation and calculate Slater integrals and configuration interaction to model UTAs in xenon ions from Xe IX to Xe XVIII. It concludes that UTAs in xenon and other elements have applications in areas like nanolithography and microscopy.

1. Wavefunction
Collapse in Xenon
Ions JACK BROWNE
SUPERVISOR: DR REBEKAH
D’ARCY

2. Wavefunction Collapse
Aufbau Principle
Lanthanide Contraction
Exceptions to Aufbau Principle: e.g. Cr, Cu
Source: https://commons.wikimedia.org/wiki/File:Periodic_table_blocks_spdf_(32_column).svg
lanthanides
Xenon

3. Wavefunction Collapse
Source: https://commons.wikimedia.org/wiki/File:Periodic_table_blocks_spdf_(32_column).svg
Double Well Potential
Centrifugal Barrier
Keeps high angular momentum
wavefunctions out of core
4f and 4d wavefunctions separate
Through ionisation barrier falls due
to increased effective potential
4f wavefunction moves into core
region

4. 4d-4f Wavefunctions
Wavefunctions now occupy areas closer to
the centre.
They now overlap with other wavefunctions
High overlap => High chance of transition

5. 4d-4f Wavefunctions
Due to this, many transitions are seen at
around the same wavelength.
This results in a broad array of
transitions
In spectra these arrays of transitions are
unresolved.
This bunch of transitions is called a UTA
(Unresolved Transition Array).
Ion Stage <4d|r|4f>
Xe I 0.40%
Xe II 11%
Xe IX 87%
Xe X-XVIII 89%

6. UTA & Cowan
Need to solve the generalised Schrödinger Equation:
𝑖
𝑁
−
ℏ
2𝑚
𝛻𝑖
2
−
𝑍𝑒2
𝑟𝑖
+
𝑖>𝑗
𝑒2
𝑟𝑖𝑗
+
𝑖
𝜉𝑖(𝑟𝑖) 𝑙𝑖 𝑠𝑖 𝜓 = 𝐸𝜓
How?: Hartree - Fock Methods.
This was done for ten Xenon Ion stages(Xe IX – XVIII).
HF Methods: Self Consistent Field.
Iteratively solved until convergence.
Cowan Code implements these to solve the above.
Parts:
• RCN: solves for the wavefunctions for each
configuration.
• RCN 2: allows scaling factors to be applied and
calculates Coulomb C.I.
• RCG: sets up matrices and solves for eigenvalues.

7. Slater Integrals
Cowan Code allows scaling of the output via
Slater Integrals.
These describe Correlation Effects.
The incomplete basis set must be taken into
account.
Due to computational constraints a complete
basis is not possible.
These are important for calculating the average
energy.
Energy Equation:
𝐸 𝑖𝑗
=
𝑘=0
𝑓 𝑘
𝑖𝑗 𝐹 𝑘
𝑖𝑗 +
𝑘
𝑔 𝑘
𝑖𝑗 𝐺 𝑘
𝑖𝑗
+
𝑘
𝑟𝑑
𝑘
𝑅 𝑑
𝑘
+ 𝑟𝑒
𝑘
𝑅 𝑒
𝑘

8. Configuration
Interaction
Further Component of Correlation Effects.
Configuration is the way in which electrons have
filled the available energy levels.
These interact with each other such that when
run independently the results are not the same
This relates back to the incomplete basis set
Configurations are listed in the common
notation.
These files were generated taking into account
selection rules and the parity of the
configuration
With Configuration Interaction
Without Configuration Interaction

9. Configuration
Interaction
Further Component of Correlation Effects.
Configuration is the way in which electrons have
filled the available energy levels.
These interact with each other such that when
run independently the results are not the same
This relates back to the incomplete basis set
Configurations are listed in the common
notation.
These files were generated taking into account
selection rules and the parity of the
configuration

10. Conclusion
UTA in Xenon
Other elements: Cs - W
Including Lanthanides
Broadband EUV light
sources
Applications:
Nanolithography
Microscopy
Biomedical areas
Δn = 0 UTAs : Xe, Cs – W
Δn = 1 UTAs : Zr, Mo, etc.
Source: https://commons.wikimedia.org/wiki/File:32_column_PT.jpg