Theoretical Cross Sections of the Inelastic Fine Structure Transition M

Article pubs.acs.org/JPCA

Theoretical Cross Sections of the Inelastic Fine Structure Transition M(2P1/2) + Ng ↔ M(2P3/2) + Ng for M = K, Rb, and Cs and Ng = He, Ne, and Ar Charlton D. Lewis and David E. Weeks* Department of Engineering Physics, Air Force Institute of Technology, Wright-Patterson AFB, 45433-7765 Ohio, United States

ABSTRACT: Scattering matrix elements of the inelastic fine structure transition M(2P1/2) + Ng ↔ M(2P3/2) + Ng are computed using the channel packet method (CPM) for alkali-metal atoms M = K, Rb, and Cs, as they collide with noble-gas atoms Ng = He, Ne, and Ar. The calculations are performed within the block Born−Oppenheimer approximation where excited state VA2Π1/2(R), VA2Π3/2(R), and VB2Σ1/2(R) adiabatic potential energy surfaces are used together with a Hund’s case (c) basis to construct a 6 × 6 diabatic representation of the electronic Hamiltonian. Matrix elements of the angular kinetic energy of the nuclei incorporate Coriolis coupling and, together with the diabatic representation of the electronic Hamiltonian, yield a 6 × 6 effective potential energy matrix. This matrix is diagonal in the asymptotic limit of large internuclear separation with eigenvalues that correlate to the 2Pj alkali atomic energy levels. Scattering matrix elements are computed using the CPM by preparing reactant and product wave packets on the effective potential energy surfaces that correspond to the excited 2Pj alkali states of interest. The reactant wave packet is then propagated forward in time using the split operator method together with a unitary transformation between the adiabatic and diabatic representations. The Fourier transformation of the correlation function between the evolving reactant wave packet and stationary product wave packet yields state-to-state scattering matrix elements as a function of energy for a particular choice of total angular momentum J. Calculations are performed for energies that range from 0.0 to 0.01 hartree and values of J that start with a minimum of J = 0.5 for all M + Ng pairs up to a maximum that ranges from J = 450.5 for KAr to J = 100.5 for CsAr. A sum over J together with an average over energy is used to compute thermally averaged cross sections for a temperature range of T = 0−400 K.

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a population inversion between the 2P1/2 level and the ground S1/2 level. In this paper, we use the time dependent channel packet method18−24 (CPM) to compute scattering matrix elements for the inelastic collision M(2P1/2) + Ng ↔ M(2P3/2) + Ng, where M = K, Rb, or Cs and Ng = He, Ne, or Ar. The scattering matrix elements are calculated within the block Born−Oppenheimer approximation where a basis corresponding to Hund’s case (c)25 is used to obtain a 6 × 6 diabatic representation of the body-fixed M + Ng Hamiltonian. This diabatic representation of the electronic Hamiltonian is block-diagonal with two

INTRODUCTION

2

The nonadiabatic dynamics of various atomic species in open shell electronic states, as they collide with noble gas atoms, has enjoyed a long history of experimental1−6 and theoretical7−11 interest. Progress in the development of diode pumped alkali lasers12−17 (DPAL) has generated renewed interest in inelastic collisions that specifically involve alkali-metal atoms in excited 2 P electronic states. In simple DPAL systems, a low concentration of K or Rb atoms is vaporized into a buffer gas, typically He. The alkali atoms are then optically pumped from the ground 2S1/2 level to the excited 2P3/2 level. Nonadiabatic collisions between the excited alkali atoms and atoms of the buffer gas cause the alkali atoms to make a fine structure transition to the 2P1/2 level at a sufficiently fast rate to establish This article not subject to U.S. Copyright. Published XXXX by the American Chemical Society

Received: December 22, 2016 Revised: April 2, 2017 Published: April 3, 2017 A

DOI: 10.1021/acs.jpca.6b12801 J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

Article

Here, R̂ is the internuclear separation, P̂ R is the radial momentum conjugate to R̂ , L̂ is the orbital angular momentum of the nuclei, and μ is the reduced mass of the M + Ng pair. Ĥ 0MNg is the sum of the electronic kinetic energy and the total Coulomb potential energy, and Ĥ so = ̂ is the spin−orbit Hamiltonian where l ̂ and ŝ are a(R) l·ŝ the electronic orbital and spin angular momentum, respectively, and a(R) is the radially dependent spin−orbit splitting parameter. The total angular momentum J ̂ = L̂ + j,̂ is the sum of the orbital angular momentum of the nuclei L̂ and the total electronic angular momentum j ̂ = l ̂ + ŝ. Expressing the orbital angular momentum of the nuclei as the difference between the total angular momentum and the total electronic angular momentum, L̂ = J ̂ − j,̂ yields

3 × 3 blocks that correspond to positive and negative projections of the electronic angular momentum j onto the internuclear axis. The 3 × 3 blocks are further split into a 1 × 1 and a 2 × 2 block corresponding to projections of j onto the internuclear axis with values of ±3/2 and ±1/2 respectively. The 1 × 1 block together with eigenvalues of the 2 × 2 block are matrix elements of the adiabatic representation of the electronic Hamiltonian and provide analytic expressions for the adiabatic potential energy surfaces in terms of the diagonal and off-diagonal matrix elements of the diabatic representation of the electronic Hamiltonian. When inverted these expressions yield the diabatic matrix elements of the electronic Hamiltonian in terms of the M + Ng adiabatic potential energy surfaces. For these calculations, adiabatic VA2Π1/2(R), VA2Π3/2(R), and VB2Σ1/2(R) potential energy surfaces computed at the spin−orbit multiconfiguration interaction (SOCI) singles and doubles Def2TZVPP level26 are used to determine matrix elements of the diabatic representation of the electronic Hamiltonian as a function of the internuclear separation R. The angular kinetic energy of the nuclei is computed using angular momentum calculus to obtain diagonal centrifugal matrix elements and off-diagonal Coriolis coupling terms, and is combined with the electronic Hamiltonian to yield a 6 × 6 effective potential energy matrix. At sufficiently large internuclear separations, R ≈ 100 bohr, the Coriolis coupling terms go to zero, and the effective potential energy matrix is diagonal with eigenvalues that correspond to six effective potential energy surfaces that weakly depend on R. In the asymptotic limit of R → ∞ these six surfaces correlate to the excited 2P3/2 and 2P1/2 alkali energy levels. To perform the scattering matrix calculation, reactant Möller states are propagated forward in time using the split-operator method together with a unitary transformation between diabatic and adiabatic representations.27 Nonadiabatic coupling terms in the electronic Hamiltonian and Coriolis coupling associated with the rotational kinetic energy of the nuclei will cause the evolving reactant Möller state to couple onto all six effective potential energy surfaces. The Fourier transformation of the time correlation function between the evolving reactant Möller state and the stationary product Möller states is divided by the momentum representation of the initial wave packets expressed as a function of energy to yield state-to-state scattering matrix elements. Calculations are performed for energies that range from 0.0 to 0.01 hartree and values of the total angular momentum J that start with a minimum of J = 0.5 for all M + Ng pairs up to a maximum that ranges from J = 450.5 for KAr to J = 100.5 for CsAr. A sum over J together with an average over energy is used to compute thermal average cross sections for a temperature range of T = 0−400 K.

2 2 2 L̂ = J ̂ − 2J ̂·j ̂ + j ̂ 2 + − 2 = J ̂ + j ̂ − 2Jẑ jẑ − j−̂ J ̂ − j+̂ J ̂ ̅ ̅

where Jẑ ̅ and jẑ ̅ are the component of the total angular momentum and the component of total electronic angular momentum along the internuclear axis, respectively. The raising and lowering operators of the total electronic angular momentum j±̂ operate in the usual way while the raising and lowering operators of the total angular momentum J±̂ are reversed as the result of the body-fixed angular momentum calculus.28 The first three terms of eq 2 are identified as the centrifugal potential, and the last two terms are identified as Coriolis coupling. Hund’s case (c) provides a convenient basis, labeled j ⟩, within which to represent the Hamiltonian. |ψc⟩ = R , J , ω Ω In a DPAL system, the nonadiabatic dynamics of M + Ng occurs primarily within the 2Pj alkali manifold and, using the block Born−Oppenheimer approximation, the total electronic angular momentum j may be restricted to values of 3/2 and 1/2, with a projection ω restricted to values of ±3/2 and ±1/2. This yields for each value of the total angular momentum J a set of six basis vectors labeled J 3/2 ⟩, J 3/2 ⟩, J 1/2 ⟩, Ω 3/2 Ω 1/2 Ω 1/2 J 3/2 ⟩, J 3/2 ⟩, and J 1/2 ⟩, where R has Ω −3/2 Ω −1/2 Ω −1/2 been omitted for notational clarity. This basis is used to represent eq 1. The representation of Ĥ e in the Hund’s case (c) basis is obtained by first considering the representation of Ĥ 0MNg in an uncoupled basis lλ σs ⟩, where λ and σ are projections of the electronic orbital and spin angular momentum onto 0 the internuclear axis. The lλ σs ⟩ are eigenstates of Ĥ MNg where

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THEORY Body-fixed coordinates are used to compute the M + Ng scattering matrix elements where the body-fixed z-axis is collinear ̅ with the internuclear axis and the body-fixed Hamiltonian that governs the M + Ng dynamics is given by

l s 0 ĤMNg λ σ

2 2 PR̂ L̂ 0 + + (ĤMNg + Ĥ so) 2 2μ ̂ 2μR

= Wλ , σ , l , s(R )

l s λ σ

(3)

and the Wλ,σ,l,s(R) are adiabatic potential energy surfaces labeled Π for λ = ±1 and Σ for λ = 0. The lλ σs ⟩ representation of Ĥ 0MNg in eq 3 is then transformed to the Hund’s case (c) basis ̂ using Clebsch−Gordan coefficients Clsj λσω = ⟨lλsσ|lsjω⟩ and He 7−9 becomes

Ĥ = HR̂ + HL̂ + Ĥe =

(2)

(1) B



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where Ĥ so is diagonal in the Hund’s case (c) basis with eigenvalues a and −a. Note that eq 4 is a 3 × 3 block diagonal element 2 of a full 6 × 6 representation of Ĥ e corresponding to ±mj blocks. Diagonalization of the matrix in eq 4 yields the electronic eigenvalues, Σ1/2, Π3/2, and Π1/2, in the adiabatic representation as a function of Σ, Π, and a. Expressions for the eigenvalues are then inverted to obtain Σ, Π, and a as a function of the Σ1/2, Π3/2, and Π1/2 surfaces in eq 5. Σ=

Figure 1. Π3/2, Σ1/2, and Π1/2 adiabatic potential energy surfaces26 for KHe. The zero of energy (dashed line) is set at the average between the 4-fold degenerate atomic 2P3/2 and the 2-fold degenerate atomic 2 P1/2 alkali states.

2Π1/2 − Π3/2 + 2Σ1/2

3 1 [(Π1/2)2 + 2Π1/2Π3/2 − 2(Π3/2)2 − 4Π1/2Σ1/2 + 2Π3/2Σ1/2 + (Σ1/2)2 ]1/2 3 1 Π = {Π1/2 + 4Π3/2 + Σ1/2 6 − [(Π1/2)2 + 2Π1/2Π3/2 − 2(Π3/2)2 − 4Π1/2Σ1/2 + 2Π3/2Σ1/2 + (Σ1/2)2 ]1/2 } +

1 {−Π1/2 + 2Π3/2 − Σ1/2 3 + [(Π1/2)2 + 2Π1/2Π3/2 − 2(Π3/2)2 − 4Π1/2Σ1/2 + 2Π3/2Σ1/2 + (Σ1/2)2 ]1/2 }

a=

(5)

M + Ng adiabatic potential energy surfaces, Σ1/2 = VB (R), Π3/2 = VA2Π1/2(R), and Π1/2 = VA2Π1/2(R), computed at the SOCI Def2-TZVPP level,26 are used together with eq 5 to yield the diabatic matrix elements in eq 4. These adiabatic surfaces are presented as supplementary material in ref 26. The adiabatic surfaces for KHe are shown in Figure 1, and Σ, Π, and a are plotted in Figure 2. The off-diagonal elements in eq 4 are also plotted in Figure 2, and identify the approximate range of R for which radial coupling is important. Note that the spin−orbit parameter a(R) is not in general constant for all nine M + Ng pairs and deviates from the asymptotic atomic value, a(∞), for R ≲ 15 bohr. Figure 3 shows the difference between a(R) and a(∞). The radial dependence of the spin− orbit parameter stems from the perturbation to the alkali electronic wave function caused by the noble gas atom at smaller values of R. The representation of Ĥ L, using eq 2, is given by 2

+ Σ1/2

J′

j′

Ω′ ω′

HL̂

J

j

Ω ω

=

J′

j′

Ω′ ω′

+ − 2 J ̂ + j ̂ − 2Jẑ jẑ − j−̂ J ̂ − j+̂ J ̂ J 2

̅ ̅

2μR̂

2

2

Figure 2. Σ, Π, a, and 3 (Σ − Π) surfaces for KHe as determined by eq 5 together with the Π3/2, Σ1/2, and Π1/2 surfaces shown in Figure 1.

The representation of Ĥ R in the Hund’s case (c) basis takes the following form:

j

Ω ω

(6)

The first three terms in eq 6 yield diagonal matrix elements and correspond to a centrifugal potential. The remaining two terms yield Coriolis coupling matrix elements, corresponding to angular derivative coupling, where J±

J

j

Ω ω

J

⟩ = [(J ± Ω)(J ∓ Ω + 1)]1/2

j

Ω∓1 ω

J

j

Ω ω

⟩ = [(j ∓ ω)(j ± ω + 1)]1/2

J

Λj ′ ω ′ , jω = 2Fj ′ ω ′ , jω

⟩

∂ + Gj ′ ω ′ , jω ∂R

The radial derivative coupling terms are expressed in integral form:

(7)

Fj ′ ω ′ , jω =

∫ d r ⃗ Φ*j′,ω′( r ⃗; R) ∂∂R Φj,ω( r ⃗; R)

Gj ′ ω ′ , jω =

∫ d r ⃗ Φ*j′,ω′( r ⃗; R) ∂∂R2 Φj,ω( r ⃗; R)

j

⟩ Ω ω±1

(9)

where

and j±

2

j′ PR̂ J j Ω′ ω′ 2μ Ω ω ⎤ 1 ⎡ ∂2 = −∑ ⎢δj ′ ω ′ , jω 2 + Λj ′ ω ′ , jω⎥ ⎦ ∂R j , ω 2μ ⎣ J′

2

(8) C

(10)



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approximation is now given. Ĥ R is a 6 × 6 diagonal matrix with diagonal elements:

where Φj,ω(r;⃗ R) are electronic states and r ⃗ are electronic coordinates. For these calculations F and G are assumed to be neglij ⟩.7,29−31 gible in the Hund’s case (c) basis J , ω Ω Noting that the projection of L̂ onto the internuclear axis is zero and Ω = ω, the representation of the alkali metal−noble gas Hamiltonian, eq 1, in the block Born−Oppenheimer

ℏ2 d2 HR̂ = − 2μ dR2

Ĥ L is a 6 × 6 matrix (eq 12), and Ĥ e is a block diagonal 6 × 6 matrix with two 3 × 3 blocks given by eq 4.

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̂ Ψ+⃗ (R , Δt ) = e−(i/2ℏ)Veff

S-MATRIX ELEMENTS Scattering matrix elements for the M(2P1/2) + Ng ↔ M(2P3/2) + Ng collision are computed using the channel packet method18−24 (CPM):

ℏ2(|k′||k|)1/2 (2πμ)η−*( ±k′)η+( ±k)

+∞

∫−∞

× -[DAT e−(i/2ℏ)eff

e iEt C γ ′ γ(t ) dt

(14)

ator method

̂ ̂ ̂ Û ≈ e λVeff /2e λT e λVeff /2 + 6(λ 3)

ℏ

(R )Δt

Ψ+⃗ (R , 0)]}

(16)

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is the correlation function between evolving reactant and product Möller states |ψγ±(t)⟩.18−23 The label γ specifies the internal quantum state, j and ω, of the asymptotic M + Ng configuration where the Ng atom is assumed to be in the ground state, the radial momentum is PR = ℏk, and the η±(k) are the expansion coefficients of the reactant (+) and product (−) Möller states,32 typically chosen to be Gaussian functions in the momentum representation. To compute the correlation function in eq 14, the reactant it γ |ψ+⟩ is propagated under Û = exp − Ĥ using the split oper-

(

A

where Ψ⃗ +(R,0) is a six-component wave vector at t = 0. The transformation DAT is the diabatic to adiabatic (DAT) transformation34 of eff (R ) such that †DAT eff (R )DAT = effA (R ), A where eff (R ) is diagonal and the - and - −1 are fast Fourier Transforms (FFT) between the momentum and coordinate representations. The time step Δt is chosen such that terms of 6(λ 3) and higher in eq 15 are negligible and the reactant Möller state is propagated until the correlation function in eq 14 converges to zero. Absorbing boundary conditions24 are used to absorb the reactant Möller state as it reaches the end of the computational grid.

where

33

(R )Δt 2

(13)

⎛ it ⎞ C γ ′ γ(t ) = ⟨ψ−γ ′|exp⎜ − Ĥ ⎟|ψ+γ ⟩ ⎝ ℏ ⎠

A

× †DAT - −1{e−iℏ(k /2μ)

S±γ ′kγ′± k = ⟨k′, γ ′|S|̂ k , γ ⟩ =

(11)

COMPUTATION The S-matrix calculations are carried out in two separate phases. In the first phase (MS) Möller states are calculated, and in the second phase (CF) the correlation function is computed. This is done to take advantage of the fact that the potential in the asymptotic region does not change rapidly; thus a big time step can be used when computing Möller states. Initial reactant and product wave packets used to compute the Möller states are chosen to be Gaussian functions. At t = 0 these initial wave packets are centered at R0 = 100 bohr with nuclear kinetic energy ranging from 0 to 0.01 hartree. The choice of R0 = 100 guarantees that the Coriolis coupling matrix elements in eq 12 are negligible over the entire range of R for which the initial wave packets are nonzero. As a result, the Coriolis and radial coupling matrix elements are not included when computing Möller states. The initial wave packets are analytically propagated backward/forward in time from t = 0 until the centrifugal potential is negligible over the entire range for which these intermediate wave packets are nonzero.

)

(15)

i where λ = − ℏ Δt . The kinetic energy, T̂ , is a 6 × 6 diagonal matrix with diagonal elements given by eq 11 and the effective potential energy, V̂ eff, is given by the sum of eqs 4 and 12. Wave packets are then propagated using

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Figure 4. Sum of Π3/2(R) = Π(R) + a(R) from eq 4 and the cen3/2 ⟩ matrix element in trifugal potential given by the diagonal J 3/2 3/2 eq 12 is shown for J = 0.5, J = 50.5, J = 100.5, and J = 150.5. At smaller values of J < 50.5 the surfaces are determined to a large degree by Π3/2. As J becomes large, the surfaces are dominated by the centrifugal potential and reach their asymptotic value at larger values of R.

3/2 , + reactant Möller states are shown for Figure 5. KHe R J 3/2 3/2 J = 0.5, J = 50.5, J = 100.5, and J = 150.5. As the intermediate Gaussian wave packet is propagated from the asymptotic region back to t = 0, the centrifugal potential causes the evolving wave packet to become asymmetric.

Note that this region where the centrifugal potential becomes negligible changes with J as shown in Figure 4. Once the intermediate wave packets are in the asymptotic region, they are computationally propagated forward/backward in time back to t = 0 using eq 16 under the diagonal matrix elements of Ĥ R and Ĥ L to yield Möller states for all J values of interest. KHe reactant Möller states are plotted in Figure 5 for several different values of J. The reactant Möller state for J = 0.5 does not differ much from the original Gaussian; however, as the nuclear angular momentum increases the Möller state becomes asymmetric as they are computationally propagated through the centrifugal potential. Reactant and product Möller states are computed for each of the six diabatic surfaces up through Jmax for each M + Ng pair.

Figure 3. Spin−orbit parameters, a(R), minus the spin−orbit parameter in the asymptotic limit, a(∞), where aCs(∞) = 1.68 × 10−3, aRb(∞) = 7.21 × 10−4, and aCs(∞) = 1.75 × 10−4 bohr. E



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Figure 7c−e shows state-to-state transmission and reflection coefficients for KHe at J = 50.5 where the Coriolis coupling in eq 12 and radial coupling in eq 4 are similar in magnitude. In this case all six Hund’s case (c) states are coupled and the reactant Möller state exits on all six surfaces. Since the S-matrix is unitary, all state-to-state reflection and transmission coefficients must sum up to unity for all energies. Figure 7f shows this to be the case and is an important check of the calculation. This check was verified for all values of J in each of the nine M + Ng calculations. State-to-state cross sections are computed by summing S-matrix elements over the total angular momentum J

The reactant and product Möller states are then used in the CF phase to compute the correlation function given by eq 14. For these computations the reactant Möller state is selected to 1/2 , start on the surface corresponding to the state J −1/2 −1/2 ⎛ ⎞ 0 ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ 0 Ψ+⃗ (R , 0) = ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ J 1/2 ⎜ ⎟ , +⟩⎟ ⎜⟨R − − 1/2 1/2 ⎝ ⎠ (17)

σj ′ , ω ′← j , ω(E) =

π

∞

∑

kj2′ , ω ′ J = 0.5

(2J + 1)|SjJ′ , ω ′← j , ω(E)|2 (18)

In practice it is not necessary to sum to infinity, rather only up to Jmax, where the centrifugal potential prevents the system from further coupling for a given energy. For example, the σ3/2,3/2←1/2,1/2(E) cross section of KHe is plotted in Figure 8 as a function energy for six different values of Jmax. For a particular energy the cross section converges to a maximum value for some value of Jmax. In general, the value of Jmax used in eq 18 required for a converged cross section increases with energy. Converged cross section at lower energies will include fewer S-matrix elements. As a result, resonance features exhibited by single S-matrix elements survive the sum over J. For example, the sharp peaks in Figures 8 and 9 between E = 0.00 and E = 0.005 × 10−2 hartree are single S-matrix resonances that survive a sum over J from 0.5 to Jmax ≊ 25.5. Cross sections for which the sum over J is truncated prematurely, for example the Jmax = 50.5 result at E > 0.1 × 10−2 hartree, oscillate because of common features of the S-matrix elements that survive the truncated sum. Values of Jmax for each of the nine M + Ng pairs are selected empirically by observing the convergence of the state-to-state cross sections. The value of Jmax is driven by two competing factors. As the reduced mass increases, the centrifugal potential will become smaller for the same value of J allowing the evolving reactant Möller state to more completely explore the radial coupling region. This suggests that Jmax for heavier systems will be larger than Jmax for lighter systems. This trend is counterbalanced by the effectiveness of the radial coupling to transfer probability from the Π1/2 to the Σ1/2 surface. As a result, the Cs and Rb calculations are more resilient to residual exploration of the coupling region by the evolving wave packet and smaller values of Jmax can be used to perform the sum in eq 18. The collisional cross section for the 2P3/2 ← 2P1/2 transition is required in order to compare to experiment. Taking

The six-component diabatic wave vector in eq 17 is transformed to an adiabatic wave vector using †DAT and propagated forward in time using eq 16 under the full 6 × 6 Hamiltonian for use in the CF phase. The time step, ΔtCF, must be smaller than the MS phase because the potential varies significantly as R → 0. Values of the time step and total propagation time for the MS and CF phase, and the spin−orbit splitting for each M + Ng pair, are tabulated in Table 1. The size of the computational grid for the CF computations was 16 384 grid points where Rmin = 3.0 bohr and Rmax = 1641.4 bohr with a grid step size of ΔR = 0.1 bohr. Energy content of the Möller states is between 0.0 and 0.01 hartree. The evolving reactant Möller state is plotted in Figure 6 for KHe at J = 0.5 and t = 0.024, 0.073, and 0.145 ps. As the reactant Möller state evolves in time it is radially coupled to 0.5 3/2 the − 1/2 −1/2 ⟩ state as seen in Figure 6a. For J = 0.5 the Coriolis coupling is negligible and no wave packet amplitude is 3/2 0.5 1/2 ⟩, 0.5 3/2 ⟩, and 0.5 coupled to the 1/2 −3/2 −3/2 ⟩ 1/2 1/2 1/2 states. As a result, the correlation function given by eq 14 is 0.5 1/2 0.5 1/2 nonzero for the − 1/2 −1/2 ⟩ → −1/2 −1/2 ⟩ (reflection) 0.5 1/2 0.5 3/2 and − 1/2 −1/2 ⟩ → −1/2 −1/2 ⟩ (transmission) transitions and zero for all other transitions. The correlation function 0.5 1/2 0.5 3/2 for − 1/2 −1/2 ⟩ → −1/2 −1/2 ⟩ is shown in Figure 7a, and the absolute value squared of the corresponding S-matrix element is shown in Figure 7b.

Table 1. M + Ng Computational Parameters Used for MS and CF Computation along with the Spin−Orbit Splitting, ΔE, for Each Alkali Atoma

a

M + Ng

ΔtMS

t∓max

KHe KNe KAr RbHe RbNe RbAr CsHe CsNe CsAr

0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19

145.08 217.62 314.34 120.90 241.80 435.24 48.36 241.80 435.24

ΔtCF 4.83 9.68 1.94 4.83 1.94 1.94 4.83 1.94 1.94

× × × × × × × × ×

TCF

10−4 10−4 10−3 10−4 10−3 10−3 10−4 10−3 10−3

36.27 120.90 241.80 48.36 145.08 241.80 72.54 145.08 241.80

Eγint

ΔE

−1.75 × 10−4, 8.76 × 10−5

2.26 × 10−4

−7.22 × 10−4, 3.61 × 10−4

1.08 × 10−3

−1.68 × 10−3, 8.41 × 10−4

2.52 × 10−3

All energy values are in atomic units and time is in picoseconds (ps). F



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Figure 9 shows the 2P3/2 ← 2P1/2 cross section for the case of both radial and Coriolis coupling (RCC) and for radial coupling only (RC). For KHe the RCC cross section is approximately twice as large as the RC cross section.

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RESULTS AND DISCUSSION Thermally averaged cross sections, Q(T), are computed by multiplying the cross sections σ(E) in eq 19 with the Maxwell− Boltzmann distribution, and integrating over E at each temperature T: Q (T ) = (k bT )−2

(20)

CONCLUSION Thermally averaged cross sections are computed for the M + Ng fine structure transition where M = K, Rb, and Cs and Ng = He, Ne, and Ar. The cross section calculations employ scattering matrix elements computed using the channel packet method for values of J ranging from 0.5 for all pairs to 250.5 for M = K and Rb and 100.5 for M = Cs. This range of J values is required to accommodate collisional energies that range from 0.0 to 0.01 hartree. A sum over scattering matrix elements together with an average over the Boltzmann speed distribution provides thermally averaged cross sections for temperatures ranging from T = 0 to T = 400 K. These cross section calculations are in general agreement with experimental observations for all nine M + Ng pairs over nearly 6 orders of magnitude. As expected, the size of the cross

1/2

(1/2)[σ3/2, −3/2 ← 1/2, ω(E)

ω =−1/2

+ σ3/2, −1/2 ← 1/2, ω(E) + σ3/2,3/2 ← 1/2, ω(E) + σ3/2,1/2 ← 1/2, ω(E)]

Ee−E / kbT σ(E) dE

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the fine structure population density to be equally distributed between ω = ±1/2, the asymptotic cross section is given by

∑

∞

Here kb is the Boltzmann constant and Eint is the internal energy of the alkali atom. These cross sections are computed for all nine M + Ng pairs for temperatures that range from T = 0 K to T = 400 K with the exception of RbAr (0−350 K). The predicted results of all nine M + Ng pairs are compared with experimental results in Figures 10−12 and at specific temperatures in Table 2. The Q(T) for all nine pairs in Figures 10−12 can be organized by the spin−orbit splitting of the alkali atom from smallest to largest. The thermally averaged cross sections for K + Ng, which have the smallest ΔE, are on the order of 101 bohr2. By comparison the thermally averaged cross sections of Rb + Ng, with a larger spin−orbit splitting, range from 10−1 to 10−3 bohr2. Then finally Cs + Ng, which has the largest spin− orbit splitting, exhibits thermally averaged cross sections that lie in the range 10−4−10−6 bohr2. The Q(T) in Figures 10−12 may be further organized by the noble gas atom, where for K and Rb the largest thermally averaged cross section corresponds to He, followed by Ne, and then Ar. We attribute this pattern to the reduction in Coriolis coupling that occurs as the mass of the noble gas increases. This pattern is broken for Cs, which has the largest spin−orbit splitting. For all M + Ng pairs except CsNe and CsAr RCC has the effect of significantly increasing the cross section over RC. In some cases RCC more than triples the prediction of RC at a given temperature. It is notable that there is a significant discrepancy between some of the theoretical cross sections and experimental measurements. These discrepancies suggest that the theoretical results in general underestimate the M + Ng cross sections. The largest source of error in these calculations are likely due to the shoulder height overestimation of the M + Ng Σ1/2 surfaces.26

0.5 1/2 Figure 6. Initial reactant Möller state, Ψ+(R ) = R − 1/2 − 1/2 , +⟩, 0.5 1/2 is chosen to start on the surface corresponding to the − 1/2 − 1/2 ⟩ state as shown in (b). The evolving reactant Möller state is also shown in (b) at t = 0.024 ps and t = 0.073 ps. As the evolving reactant Möller state evolves it is radially coupled to the surface corresponding to 0.5 3/2 the − 1/2 − 1/2 ⟩ state and is shown in (a) at t = 0.073 ps and t = 0.145 ps. Also shown in (a) is the stationary product Möller state 0.5 3/2 Ψ−(R ) = R − 1/2 − 1/2 , −⟩. The Fourier transform of the correlations between the evolving reactant Mö ller state and stationary product Möller state is shown in Figure 7a. The magnitude of the evolving reactant Möller state in (a) is magnified by a factor of 110 relative to |Ψ−(R,0)|2 and in (b) by a factor of 22 relative to |Ψ+(R,0)|2.

σ 2P3/2 ← 2P1/2(E) =

∫0

(19) G



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Figure 7. (a) KHe correlation function as determined by the evolving reactant Möller state and stationary product Möller state. (b) Scattering matrix 0.5 1/2 0.5 3/2 elements for the transition from the − 1/2 − 1/2 ⟩ state to − 1/2 − 1/2 ⟩ state are computed from the Fourier transform of the correlation function in 50.5 1/2 ⟩ to 50.5 1/2 ⟩ (reflection), and to 50.5 1/2 ⟩, 50.5 3/2 ⟩, and (a). (c−e) J = 50.5 state-to-state scattering matrix elements from − 1/2 − 1/2 − 1/2 − 1/2 1/2 1/2 ± 1/2 ± 1/2 50.5 3/2 ± 3/2 ± 3/2 ⟩ (transmission). A sum over all S-matrix elements over the energy range of interest shown in (f) and is unity as expected. H



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section is driven primarily by the spin−orbit splitting of the alkali atom. These calculations also indicate that the Coriolis coupling is an important factor in the cross section calculation, increasing the cross section in some cases by a factor of 2 or 3. This increase in cross section caused by the Coriolis coupling is associated with a complicated state-to-state flow of probability as a M + Ng collision evolves in time. For wave packets starting on the Π1/2 surface, radial coupling promotes wave packet amplitude to the Σ1/2 surface. Coriolis coupling then moves wave packet amplitude from the Σ1/2 surface to the Π3/2 surface. Since there is no coupling between Π3/2 and Π1/2, the Π3/2 surface acts as a reservoir of probability and for the 2 P3/2 ← 2P1/2 scattering cross section. Finally, we note that the time required to compute state-tostate S-matrix elements for energies that range from E = 0.00 to E = 0.01 hartree is about 2−3 h per J on a desktop machine. The total propagation times, TCF, used for each J are listed in Table 1 and are required to resolve the resonances with widths ΔE ≃ 10−5 hartree as seen at low energies in Figures 8 and 9.

Figure 8. KHe cross sections σ3/2,1/2←1/2,1/2(E) for Jmax = 10.5, 50.5, 100.5, 150.5, 190.5, and 250.5 in the radial plus Coriolis coupling case. As Jmax increases, the cross sections converge at higher energies. At this scale the Jmax = 190.5 and 200.5 state-to-state cross sections are indistinguishable.

Figure 10. Thermally averaged cross sections for the 2P3/2 ← 2P1/2 transition, (a) KHe, (b) KNe, and (c) KAr, are compared with experimental observation, I (ref 4), II (ref 5), III (ref 3), and IV (ref 2).

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Figure 9. KHe cross sections for the 2P3/2 ← 2P1/2 transition for both RCC and RC cases. The RCC result is approximately 2 times larger than the RC result.

AUTHOR INFORMATION

Corresponding Author

*E-mail: david.weeks@afit.edu. I



Article

Figure 12. Thermally averaged cross sections for the 2P3/2 ← 2P1/2 transition, (a) CsHe, (b) CsNe, and (c) CsAr, are compared with experimental observation, IV (ref 2). The dashed and dotted red line is from ref 1, with the uncertainty limits in dotted black.

Figure 11. Thermally averaged cross sections for the 2P3/2 ← 2P1/2 transition, (a) RbHe, (b) RbNe, and (c) RbAr, are compared with experimental observation, IV (ref 2) and V (ref 6). The dashed and dotted red line is from ref 1, with the uncertainty limits in dotted black.

ORCID

Charlton D. Lewis: 0000-0003-2112-5921 J



Article

Table 2. M + Ng Thermally Averaged Cross Sections Compared with Experimenta M + Ng

temp (K)

theory

ref 2

ref 3

ref 4

ref 5

ref 6

KHe

333 342 368 380 333 342 368 380 333 342 340 373 340 373 340 311 311 311

137.50 140.29 148.74 151.85 20.27 21.19 23.89 25.17 16.05 16.54 0.39 0.45 0.0060 0.0071 0.0026 3.2 × 10−4 2.7 × 10−6 5.6 × 10−3

− − 212.48 − − − 51.07 − − − 0.27 − 0.0061 − 0.0036 2.0 × 10−4 6.8 × 10−5 5.7 × 10−5

293.91* − − − 77.92* − − − 189.27* − − − − − − − − −

− 93.92 − − − 22.85 − − − 57.14 − − − − − − − −

− − − 103.34 − − − 33.57 − − − − − − − − − −

− − − − − − − − − − − 0.36 − 0.017* − − − −

KNe

KAr RbHe RbNe RbAr CsHe CsNe CsAr

a Values with asterisks are converted from the reported 2P1/2 ← 2P3/2 experimental data using detailed balance. The results from ref 1 are omitted. Units are in bohr2.

(11) Delos, J. B. Theory of Electronic Transitions in Slow Atomic Collisions. Rev. Mod. Phys. 1981, 53, 287−358. (12) Zweiback, J.; Krupke, W. F. 28W Average Power HydrocarbonFree Rubidium Diode Pumped Alkali Laser. Opt. Express 2010, 18, 1444−1449. (13) Ehrenreich, T.; Zhdanov, B.; Takekoshi, T.; Phipps, S.; Knize, R. Diode Pumped Caesium Laser. Electron. Lett. 2005, 41, 415. (14) Sharma, A.; Bhaskar, N.; Lu, Y.; Happer, W. Continuous-Wave Mirrorless Lasing in Optically Pumped Atomic Cs and Rb Vapors. Appl. Phys. Lett. 1981, 39, 209−211. (15) Krupke, W. F.; Beach, R. J.; Kanz, V. K.; Payne, S. A. Resonance Transition 795-nm Rubidium Laser. Opt. Lett. 2003, 28, 2336−2338. (16) Krupke, W. F.; Beach, R. J.; Kanz, V. K.; Payne, S. A. Resonance Transition 795-nm Rubidium Laser. Opt. Lett. 2003, 28, 2336−2338. (17) Zhdanov, B. V.; Maes, C.; Ehrenreich, T.; havko, A.; Koval, N.; Meeker, T.; Worker, B.; Flusche, B.; Knize, R. J. Optically Pumped Potassium Laser. Opt. Commun. 2007, 270, 353. (18) Weeks, D. E.; Tannor, D. J. A Time-Dependent Formulation of the Scattering Matrix Using M?ller Operators. Chem. Phys. Lett. 1993, 207, 301−308. (19) Tannor, D. J.; Weeks, D. E. Wave Packet Correlation Function Formulation of Scattering Theory: The Quantum Analog of Classical S-matrix Theory. J. Chem. Phys. 1993, 98, 3884−3893. (20) Niday, T. A.; Weeks, D. E. Scattering Matrix Elements for the Fine Structure Transition B (2P1/2) + H2 (j = 0) B ↔ (2P3/2) + H2 (j = 0). Chem. Phys. Lett. 1999, 308, 106−114. (21) Weeks, D. E.; Tannor, D. J. A Time-Dependent Formulation of the Scattering Matrix for the Collinear Reaction H + H2 (n) ↔ H2 (n′) + H. Chem. Phys. Lett. 1994, 224, 451−458. (22) Calfas, R. S.; Weeks, D. E. A New Application of Absorbing Boundary Conditions for Computing Collinear Quantum Reactive Scattering Matrix Elements. Chem. Phys. Lett. 1996, 263, 292−296. (23) MacLachlan, M. J.; Weeks, D. E. A New Application of the Interaction Picture to Calculate Reactive Scattering Matrix Elements. J. Phys. Chem. A 1998, 102, 9489−9493. (24) Weeks, D. E.; Niday, T. A.; Yang, S. H. Inelastic Scattering Matrix Elements for the Nonadiabatic Collision B (2P1/2) + H2 (1Σ+g , j) ↔ B(2P3/2) + H2 (1Σ+g , j′). J. Chem. Phys. 2006, 125, 164301−164315. (25) Aquilanti, V.; Grossi, G. Angular Momentum Coupling Schemes in the Quantum Mechanical Treatment of P-state Atom Collisions. J. Chem. Phys. 1980, 73, 1165−1172.

David E. Weeks: 0000-0002-9074-1667 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank G. Perram for many useful discussions and also would like to thank the DoD Supercomputing Resource Center (DSRC), the Air Force Office of Scientific Research (AFOSR), and the High Energy Laser Joint Technology Office (HEL-JTO) for funding and support.

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REFERENCES

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L


Theoretical Cross Sections of the Inelastic Fine Structure Transition M

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