2021 AIAA Excitation Line Optimization for Krypton Tagging Velocimetry and Planar Laser-Induced Fluorescence in 200-220 nm Range

Excitation Line Optimization for Krypton Tagging Velocimetry and Planar Laser-Induced Fluorescence inof two-photon cross-sections for five Krypton lines, an excitation spectrum is constr

Excitation Line Optimization for Krypton Tagging Velocimetry and Planar Laser-Induced Fluorescence in

of two-photon cross-sections for five Krypton lines, an excitation spectrum is constructedand compared against excitation spectrum data, with encouraging results From this workand the successful comparison to experiment from our lab and those in the literature, weconclude that the optimal line is 212.556 nm for Kr-PLIF and single-laser KTV For KTVwhere the read step in performed with a continuous wave (CW) laser diode, the 216.667 nmwrite-laser excitation is optimal

Ze = Effective Nuclear Charge

α = Fine Structure Constant, α = e2/(4πo¯hc)

ao = Bohr Radius, (cm), ao= 100¯h/(αmec)

dD = Debye Length, (m)

Ry = Rydberg Constant, (J), Ry = ¯h2/(2mea2) = (1/2)meα2c2

kb = Boltzmann Constant, (J/(atom·K))

r = Radius, (Bohr Radii, ao)

θ = Azimuth Angle, (rad)

φ = Polar Angle, (rad)ˆ

 = Polarization Unit Vector of Laser Electric Field

q = Polarization Componentˆ

 · ~r = Dipole Operator, (Bohr Radius)

D = Matrix Representation of Dipole Operator, (Bohr Radius)

G = Matrix Representation of Green’s Function Operator, (s/rad)

Mf g(2) = Two-Photon Transition Matrix Element from states |gi to |f i, (a2o· s)

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Wf,g = Two-Photon Excitation Rate, (1/s)

Wpi = One-Photon Ionization Rate, (1/s)

σ(2)o = Two-Photon Cross-section, (cm4)

σ(2) = Two-Photon Rate-coefficient, (cm4·s)

σpi = One-Photon Ionization Cross-section, (cm2)

I = Laser Intensity (J/(cm2·s))

|ki = Intermediate State k

|gi = Ground State

|f i = Final Two-Photon Excited State

Ek = Energy of level k, (eV)

ωL = Angular Frequency of Laser Excitation, (rad·s−1)

ωk = Angular Frequency for Intermediate Energy State k, (rad·s−1), ωk = Ek/¯h

ωg = Angular Frequency for Ground Energy State g, (rad·s−1), ωg= Eg/¯h

ωij = Observed Angular Frequency for Transition from states i to j, (rad·s−1), ωij= (Ei− Ej)/¯hg(2ωL) = Lineshape Function, (s)

F = Laser Photon Flux, (photon/s), F = I/(¯hωL)

ˆi = Vector Representation of State |ii

λij = Transition wavelength from state i to state j, (nm)

Aij = Einstein coefficient (transition probability) for transition from state i to j, (s−1)

n = Principal Quantum Number

n∗ = Effective Principal Quantum Number

l = Angular Momentum Quantum Number

l∗ = Effective Angular Momentum Quantum Number

m = Azimuth Angular Momentum Quantum Number

L = Total Orbital Angular Momentum Quantum Number for an Atom

S = Total Electron Spin Quantum Number for an Atom

J = Coupled Angular Momentum Quantum Number, J =

~

J =

~

L + ~S

Krypton fluorescence experiments have attracted great interest over the last decade because of their promise

in making fundamental contributions in subsonic and supersonic combustion in addition to supersonic andhypersonic aerodynamics.1 Two such experiments are krypton planar laser-induced fluorescence (Kr-PLIF)and krypton tagging velocimetry (KTV) Kr-PLIF and KTV are performed by the addition of a small molefraction of Kr to a high-speed/reacting flow This strategy has enabled the non-intrusive measurement of

important quantities such as density, temperature, mixing-fraction, and velocity that were not previouslypossible in difficult-to-measure gas flows.

Initial Kr-PLIF work was performed at 214.7 nm,2 6which now includes thermometry.7 9 Additionally, the204.196 nm line has also been used for Kr-PLIF.10 – 12Experimental Kr-PLIF excitation line comparisons havebeen performed by,13 , 14 with the observation that the 212.556 nm was superior High-speed Kr-PLIF wasperformed at 212.556 nm by Grib et al.15 Original KTV work relied on write-line excitation at 214.769 nm togenerate the metastable Kr state.16 – 25 In more recent KTV work26 – 28 and in this paper, we observe higherSNR for single-laser, unfiltered KTV with a 212.556 nm write-line excitation; additionally, we observe thattwo-photon excitation at 216.667 nm is optimal for KTV where the read step uses a laser diode

In this paper, we calculate the two-photon cross-sections of Kr to (1) remove any ambiguity in the superiority

of the 212.556 nm line for Kr-PLIF and single-laser KTV; (2) provide fundamental physical insights toverify the Richardson et al.13 excitation spectrum; (3) provide reliable cross-sections for modeling other Krexcitation schemes; and (4) prepare a framework for calculating multiphoton excitation spectra for othernoble gas atoms Herein, we detail our calculation method and compare the results of those calculations toexperimental results with success Additionally, we present time- and pressure- resolved experimental data

of excitation performed with a near IR laser diode, for which the 216.667 nm line KTV is optimal

The current state of KTV rests on (2 + 1) resonant enhanced multiphoton ionization (REMPI) to partiallyionize Kr gas and observe a long-lasting afterglow produced by electron-ion recombination and its resultingradiative cascade.29 REMPI is a compound process consisting of two-photon excitation followed by a one-photon ionization It is magnitudes more efficient than direct three-photon ionization.30 In Table1, thereare multiple excitation lines for the two-photon excitation of Kr in the 190-220 nm range that are accessiblewith commercially available optics and laser systems Krypton atoms can be excited to any of these levelsduring the write step to form the tagged tracer This paper considers and compares the last three lines:212.556 nm, 214.769 nm, and 216.667 nm

Table 1: Accessible Kr levels with two-photon excitation Racah nl[K]J notation

λL (nm) Energy Level (-) E (cm−1)192.749 6p[1/2]0 103761.6336193.494 6p[3/2]2 103362.6124193.947 6p[5/2]2 103121.1419202.316 5p0[1/2]0 98855.0698204.196 5p0[3/2]2 97945.1664212.556 5p[1/2]0 94092.8626214.769 5p[3/2]2 93123.3409216.667 5p[5/2]2 92307.3786Following the transitions in the energy level diagrams in Fig.1 along with the relevant transition data inTable2, the three KTV schemes are performed as follows

1 λL= 216.667 nm

Write Step: Excite krypton atoms with a pulsed tunable laser to form two tagged tracers,metastable Kr and Kr+, through (2+1) photoionization Two-photon excitation of 4p6(1S0) →5p[5/2]2 (216.67 nm, transition A) and subsequent one-photon ionization31 to Kr+ (216.67 nm,transition B) occur This is followed by decay to metastable 5p[5/2]2 → 5s[3/2]o (transitionD) and resonance states 5p[5/2]2 → 5s[3/2]o (transition C), and other transitions, J, K and Lresulting from the recombination process,32 , 33I Using a camera oriented normal to the flow, theposition of the write line is recorded by gated imaging of the laser-induced-fluorescence (LIF)from transitions (C, D, J, K, L)

Read Step: After a prescribed delay, record the displacement of the tagged metastable kryptonand Kr+ With an additional tunable laser, excite 5p[3/2]1level by the 5s[3/2]o→ 5p[3/2]1 tran-sition (769.454 nm, E), which is followed by decay to metastable 5p[3/2]1→ 5s[3/2]o(829.81 nm,G) and resonance 5p[3/2]1 → 5s[3/2]o (769.454 nm, F) states The position of the read line ismarked by gated imaging of the LIF from transitions F and G and the residual fluorescence fromtransitions J, K and L that result from the recombination process, I.

2 λL= 214.769 nm

Write Step: Excite krypton atoms with a pulsed tunable laser to form two tagged tracers,metastable Kr and Kr+, through (2+1) photoionization Two-photon excitation of 4p6(1S0) →5p[5/2]2(214.769 nm, transition A†) and subsequent one-photon ionization31to Kr+(214.769 nm,transition B†) occur This is followed by decay to metastable 5p[3/2]2→ 5s[3/2]o (transition N)and resonance states 5p[3/2]2→ 5s[3/2]o

1(transition O), and other transitions, J, K and L resultingfrom the recombination process,32,33I The position of the write line is marked by gated imaging

of the LIF from these transitions (N, O, J, K, L), recorded with a camera positioned normal tothe flow

Read Step: After a prescribed delay, record the displacement of the tagged metastable kryptonand Kr+ With an additional tunable laser, excite 5p[3/2]1level by the 5s[3/2]o→ 5p[3/2]1 tran-sition (769.454 nm, E), which is followed by decay to metastable 5p[3/2]1→ 5s[3/2]o(829.81 nm,G) and resonance 5p[3/2]1 → 5s[3/2]o (769.4547 nm, F) states The position of the read line ismarked by gated imaging of the LIF from transitions F and G and the residual fluorescence fromtransitions J, K and L that result from the recombination process, I

3 λL= 212.556 nm

Write Step: Excite krypton atoms with a pulsed tunable laser to form two tagged tracers,metastable Kr and Kr+, through (2+1) photoionization Two-photon excitation of 4p6(1S0) →5p[1/2]0(212.556 nm, transition A∗) and subsequent one-photon ionization31to Kr+(212.556 nm,transition B∗) occur This is followed by decay to the resonance state 5p[1/2]0→ 5s[3/2]o

1 sition M) and other transitions, J, K and L resulting from the recombination process,32,33 I Themetastable state is formed through transition J The position of the write line is marked by gatedimaging of the LIF from these transitions (M, J, K, L), recorded with a camera positioned normal

(tran-to the flow

Read Step: After a prescribed delay, record the displacement of the tagged metastable kryptonand Kr+ With an additional tunable laser, excite 5p[3/2]1level by the 5s[3/2]o→ 5p[3/2]1transi-tion (769.454 nm, E), which is followed by decay to metastable 5p[3/2]1→ 5s[3/2]o(829.81 nm, G)and resonance 5p[3/2]1→ 5s[3/2]o(769.454 nm, F) states The position of the read line is marked

by gated imaging of the LIF from transitions F and G and the fluorescence from transitions J, Kand L that result from the recombination process, I

The extent of ionization in all three schemes is proportional to the intensity of the laser beam, which

is limited by the available laser power and the experimental setup (ex window transmission and laserbeam splitting) Lower laser power reduces (and can effectively eliminate) ionization and its subsequentradiative cascade, which may or may not be good for tracing At low power, fluorescence from transitions

J, K and L become insignificant At the write step, this is not an issue in the three schemes because thefluorescence from transitions C, D, N, O and M dominates that of transitions J, K and L At the read step,the schemes behave differently The schemes that use λL = 214.769 and 216.67 nm create metastable Krthrough transitions D and N, which do not rely on ionization The fluorescence from the re-excitation of themetastable state, transitions F and G is often sufficient on its own without the need for the fluorescence fromtransitions J, K and L Therefore, these two schemes can be used even without ionization However, the λL=212.556 nm scheme is completely reliant on recombination processes and their resulting radiative cascade tocreate fluorescence at the read step Metastable Kr in this scheme is produced though recombination, I, andsubsequently, transition J Hence, if there is no ionization, I and J do not occur Then at the read step, there

is no metastable Kr to re-excite (transitions E, F and G do not occur), and there would be no fluorescencefrom transitions J, K and L Therefore, this scheme requires the Kr atoms to be ionized to form Kr+ andmetastable Kr as the tracers Consequently, the power requirement for this scheme is higher than that of

Figure 1: Energy diagrams (not to scale) with Racah nl[K]J notation for the three excitation schemes.Left: 212.556 nm Center: 214.769 nm Right: 216.667 nm Transition details in Table 2 States 5pand 5s represent the numerous 5p and 5s states (tabulated in Mustafa et al.27) that are created by therecombination process, I Transitions J, K and L represent the numerous transitions in the 5p-5s band 14.0

eV marks ionization limit of Kr

Table 2: Relevant NIST Atomic Spectra Database Lines Data, labels match Fig 1 Racah nl[K]J tion Transition I is not listed because it is not an atomic-level transition It represents the recombinationprocess Entries in the J/K/L row represent ranges and order of magnitude estimates since J, K and L inFig.1 represent numerous transitions in the 5p-5s band k and i denote the upper and lower energy levelsrespectively

nota-Transition λair (nm) Nature Lower Level Upper Level Aij (1/s) Ej (cm−1) Ei (cm−1)

the other two

A simplified version of KTV that utilizes only a write laser26 , 27can also be implemented by omitting theread laser and its re-excitation of the metastable state (transition E) Therefore, in all three schemes, thefluorescence imaged at the read step is generated solely from transitions J, K and L As mentioned earlier,transitions J, K and L result from the radiative cascade of a cold Kr plasma While the use of only one laseroffers significant reductions in cost and experimental complexity, the use of a single laser necessitates highlaser power, sufficient to ionize krypton atoms

Fig.2shows the time resolved fluorescence signal from schemes that utilize λL= 212.556 and 214.769 nm.This data is from the single-laser version of an excitation scheme with no read laser, and was taken in a 99%

N2/1% Kr gas mixture at 5 torr The yellow region in the graph indicates the camera gate at the write step,which is typically a 5 ns exposure The two green regions are indicative of the camera gate at the read step

with a 500 (left) and 1000 (right) ns delay, and a 50 ns exposure The results show that the signal-to-noiseratio, SNR, of the 212.556 nm scheme is higher relative to the 214.769 nm scheme when no laser diode isused.

Figure 2: Time-resolved Kr Fluorescence Signal in a P = 5 torr, 99% N2/1% Kr gas mixture using 212.556 nmand 214.769 nm two-photon excitation wavelengths with no read laser The yellow region is representative

of the camera gate at the write step The two green regions are representative of the camera gate at theread step with a delay of 500 and 1000 ns respectively

By definition, the fluorescence signal, Q, from an atomic transition is calculated per Eckbreth34as,

where h is Planck’s constant, fe is the frequency of emitted light, Nu is the population of the upper level,

A is the overall Einstein coefficient, Ω is the collection solid angle, and V is the emitting volume As Eq 1

In Eq 5, Ze = 1 is the charge of the Kr ion, Ry is the Rydberg constant, and Ef is the energy of thefinal state The one-photon photoionization cross-section σpi is approximately the same for the different Krexcitation lines because of the closely clustered energies of the eight states Therefore, the two-photon cross-section σ(2)o is the most significant in determining the excitation spectrum for the Kr lines Researchers, such

as Saito et al.30 and Khambatta et al.,35respectively developed detailed analytical and numerical populationmodels, featuring Eq.2 In this work, the solution to Eq.2is not explored beyond Eq.3

RangeMethods for calculating two-photon cross-sections include first-order perturbation theory, the Green’s func-tion method, R-matrix theory, and time-dependent density-functional theory (TDDFT) First-order pertur-bation theory for multiphoton excitation and ionization is described by Lambropoulos36 who provides athorough review of multiphoton processes and calculations, and demonstrates the matrix mechanics nature

of the problem Khambatta et al.35,37 uses the first-order perturbation theory of Lambropoulos36 and theoscillator formulas from Hillborn38 to calculate two- and three-photon cross-sections for argon and krypton

He presents both a single-path and multi-path calculation However, that calculation is limited by the ability of tabulated Einstein coefficients Additionally in that work, the dipole-matrix element is asymmetric,thus unable to capture the mathematical symmetry of the two-photon transition matrix element A similarsingle-path calculation for the excitation of Kr to the 6p level was made by Bokor et al.39 The calculations

avail-in Bokor et al.39 and Khambatta et al.35 , 37 serve as important benchmarks for the two-photon cross-sectioncalculation and (2+1) photoionization modeling Mustafa et al.27 used the single-path approximation toestimate the two-photon cross-section for the 212.556 nm excitation line for krypton An additional moti-vation for the current work was to assess the validity of the results of Mustafa et al.27 and explore if otherexcitation lines might result in higher fluorescence

A two-photon cross-section calculation was conducted using multi-path, first-order accurate perturbationtheory The matrix mechanics formulation of Lambropoulos,36 who provides a thorough review of multi-photon processes and calculations, is used because it obtains all excitation pathways for a finite basis ofstates A Hartree-Fock radial wave function of the krypton ground state (4p6 1S0) a was assumed,40 andoscillator-strength (OS) formulas were used upon the availability of NIST transition probabilities and data.41

We note that a Kr gas mixture with naturally-occurring isotope mole fractions was considered because theNIST line spectra database presents spectroscopic data for a naturally-occurring mixture of Kr,41 and thelaser pulse width is at least two orders of magnitude greater than the isotopic shifts of Kr Additionally,quantum-defect theory (QDT) was used to calculate electric dipole matrix elements hi| ˆ · ~r |ji when NISTtransition probabilities were unlisted This last inclusion of QDT is key to the success of our approach as itenabled the inclusion of additional excitation pathways not included in previous works; and it determinedthe sign of all pathway contributions to the two-photon matrix element

When QDT is used to evaluate the purely radial matrix elements hri, scaled hydrogen radial wave functionsare constructed to represents excited Kr states This is because a Hartree-Fock calculation showed thatexcited krypton states exhibited hydrogenic behavior and could be approximated well by quantum-defectradial wave functions that are calibrated by NIST line data With the aid of QDT, a truncated spectralexpansion of a Green’s function was effectively constructed from a basis of intermediate Kr states (5s, 6s,7s, 4d, 5d, and 6d states) that approximately satisfy the nonrelativistic Schr¨odinger equation Within the

framework of matrix mechanics, this expansion ultimately allowed the evaluation of the two-photon-transitionmatrix element.

The two-photon cross-section σ(2)o is independent of laser intensity, time, and Kr concentration It is aconstant, and it is a solution to the time-independent, non-relativistic Schr¨odinger equationb At the risingedge of the laser pulse, σo(2) ∝ σ(2) ∝ Q ∝ SNR.34 The two-photon cross-section σ(2)o is related to thetwo-photon excitation rate-coefficient σ(2) via the lineshape function g(2ωL) as

two-is hri = n2/Ze in Bohr radii.43 Per Park,44 the hri is proportional to the Debeye length dD:

nmax=

s

ZedD10ao

tempera-by ΦD = 1/r exp(−rao/dD) For the (2+1) resonance-enhanced multiphoton excitation (REMPI) of Kr atlaser wavelength λL= 212.556 nm, room temperature T = 298 K, and pressure P = 1 torr, the electron tem-perature is Te= 27626 K and number densities are calculated as Ne/V = Ni/V = 1.62 × 1021electrons/m3

The electron temperature was obtained from 2(3¯hωL−|Eion|)/3kb, and number densities were obtained viathe analytical population model of Saito et al.30 Assuming Ze= 1 for the Kr ion, the result is nmax= 7.42.Therefore, N accommodates all states with a principal quantum number equal to or less than 7: n ≤ 7 This

is convenient because NIST transition probability data is limited for states with n ≤ 8.41

An approximate Green’s function, expressed as a truncated spectral expansion, is nested in the center ofthe expression for Mf g(2):

Since Green’s functions are symmetric about variable exchange (~r ↔ ~r0), G(~r, ~r0) = G(~r0, ~r), so Mf g(2)=

Mgf(2) This mathematical property is a fundamental deviation from the oscillator-strength approach inKhambatta et al.,35 which is one-sided and asymmetric Therefore, the use of oscillator formulas, whilevalid, causes the loss of symmetry in the transition-matrix element This symmetry loss is problematic indescribing higher-order multiphoton excitation (three-photon and higher)

Mf g(2) is a double tensor contraction of an infinite matrix space M = DGD More importantly, due to theinvariance of multiphoton-excitation with respect to reference frame and basis |ki (See Appendix A for aproof.), M = DGD is a symmetric, rank-2 tensor

The evaluation of Mf g(2) requires the evaluation of two reduced matrix elements of the form

where ˆei is a unit vector that identifies the state of the system For example, the vector representations ofstates |gi, |1i, |2i, and |N i are

.0

.0

.0

.1

Eq.19substantiates to a rank 2 tensor contraction of the Green’s function matrix G The fthrow of matrix

D is post-multiplied by the matrix G, which is then post-multiplied by the gthcolumn of matrix D, resulting

in the scalar Mf g(2)

IV.A The Calculation of Dipole Matrix Elements Dij Using QDT

In this section, the dipole matrix elements Dij are calculated via the central-field approximation,43,46whichallows one to separate the effects of angular and radial components in the Schr¨odinger equation, expressed

in spherical coordinates This allows a state |ki to be expressed as a product of one-electron, radial wavefunctions Rnl(r) ·Q

pRp(rp) multiplied by a tensor spherical harmonic YJ MLS(θ, φ) Here, subscript p denotes

an unexcited krypton electron, and nl denotes the quantum numbers of the valence electron to be excited

by the laser This state is represented as |nLSJ M i, assuming LS spin-orbit coupling The radius of theexcited valence electron from the Kr nucleus is r The orientation of its angular momentum is described byazimuth angle θ and polar angle φ The set of all principal quantum numbers for the Kr atom is n, and theprincipal quantum number of the excited electron is n L is the total orbital angular momentum quantumnumber of the atom, and l is the single-electron angular momentum number of the excited electron S is thetotal electron spin quantum number of the atom For a true dipole moment transition, S remains constantbecause the dipole moment operator ˆ · ~r does not act on electron spin coordinates The dipole momentoperator is solely written in terms of scalar spherical harmonics:46

ˆ

 · ~r =

r4π

3 rX

q=(0,±1)

where the polarization component is q; q = 0 for linear polarization; q = 1 for right-handed circularpolarization; and q = −1 for left-handed polarization of the laser’s electric field.47 The orientation of thelaser electric field defines the orientation of the z-axis in the spherical coordinate system imposed on thenucleus of a Kr atom

To evaluate the reduced matrix elements Dij, a simplified expression must first be obtained By applyingthe Wigner-Eckart Theorem,47 Dij may be rewritten as

Using the following expression from Messiah (Eq C.89) for reduced matrix elements and irreducible tensoroperators of tensor rank k,

(25)

where hri = hi |r| ji is the purely radial matrix element The term δSiSj implies that the dipole momentoperator does not act on electron coordinates Next, using the Wigner-Eckart Theorem47 for the expectedvalue of a spherical tensor Yk of rank k,

hl1|Yk| l2i =

= (−1)l1

r(2l1+ 1)(2k + 1)(2l2+ 1)

Dij= δSiSjhriq(2Li+ 1)(2Lj+ 1)

× Li 1 Lg

!q(2Ji+ 1)(2Jj+ 1)

Table 3: Parity Table for term −1 j i Jj = 0, 1 correspond to 2-photon transitions, and Jj = 0, 1, 2correspond to 3-photon transitions The term −1−Jj −M i +1 does not contribute to the transition matrixelement summation because it is consistently the same value for each stage of a multiphoton transition forall possible pathways.

Si= Sj = 0 for all transitions because the Kr ground state has a total electron spin of zero, and the dipolemoment operator ˆ ·~r does not act on electron spin coordinates Liis the norm of the addition of two angularmomenta, Li= |~li+~lg|, which describes the angular momentum coupling between the excited electron and a4p valence electron of opposite electron spin Since the dipole moment operator does not operate on electroncoordinates, it turns out that Li = Ji for the dipole transitions we analyzed A cartoon summarizing howangular momentum changes during (2 + 1)-photoionization is shown in Fig 3, and an angular momentumtable is provided in Table4 to show how to calculate the coupled quantum L from the angular momenta oftwo electrons, each with an azimuth orbital quantum number m = 0

Figure 3: Angular momenta of a Kr atom during linearly polarized (2 + 1) multiphoton photoionization.This cartoon demonstrates LS spin-orbit coupling for each Kr state at each stage of excitation: ground state

|gi, intermediate state |ki, two-photon state |f i, and ionized state e− For dipole transitions, ∆S = 0 andconsequently, J = L

Table 4: Addition of the angular momentum of two electrons l1and l2: ~L = ~l1+~l2 m = 0 for both electrons

20

20)

Therefore, the simplified dipole matrix element is

phRi,p(rp)|Rj,p(rp)i = 1 due to the normalization ofthe radial wave functions

Excited states of noble gas atoms approximate one-electron atoms, and to first order, electric dipoles.Quantum-defect theory correctly assumes that the excited states of atoms exhibit scaled, hydrogen-likebehavior, as verified by our Hartree-Fock calculation shown in Fig.4 This observation was first made byRydberg48 and was later exploited by Bethe et al.,42Bebb et al.,49 and McGuire.50,51 While Hartree-Fockiterates for an explicit electron repulsion potential,40 , 46 quantum-defect theory directly incorporates theeffect of electron repulsion through the use of excited state energy as an input to scale the wave function.With the verified assumption of hydrogenic behavior for excited Kr states, quantum-defect radial wavefunctions can be used with confidence to describe the excited states of Kr

Properly normalized hydrogen radial wave functions52 are expressed as