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The He2* metastable molecule
Helium dimers, have been subjects of investigations since the first decades of the 20th century [Cur13]. However, the work has been mainly concentrated on the molecules that consist of the most common 4He isotope [Phe55, BG71] (natural abundance in EarthŠs atmosphere is 1 million higher than that of 3He). The main source of available amount of lighter isotope, tritium decay being present in dismantled nuclear weapons, so investigations performed on molecular 3He begun after 1950 [DR50], resulting unfortunately in poor amount of references and spectroscopic data.
Binding between two He atoms
Helium molecule is an example of a dimer that consists of atoms belonging to the noble gases group. In this case completely filled electronic shells of the neutral atoms exclude formation of both covalent and ionic bonds but, still, existence of stable diatomic molecule is possible.
The reason for that is presence of van der Waals interactions (vdW), resulting from the weak, long-distance, attractive electrostatic coupling between atomic permanent or transiently induced dipolar momenta. In the case of helium, ground state atom has an electronic cloud that has spherical symmetry. However, fluctuations give rise to finite time-dependent dipols that provide same attractive contribution t the atom-atom interaction potential. As for other elements, the binding f the two He atoms depends on the existence of a net attractive potential as a result of the short-range strong repulsive interaction and the weak long-range vfW interaction.
Composition of these two results in molecular potential curve describing the dependency of interaction on the interatiomic distance. It can be purely repulsive or an attractive well, related to the reference energy being taken as the dissociation energy, can be present at the distance re. Since 1928 the existence of the singlet bound state X1Σ+g has been theoretically considered.
The early calculations, in fair agreement with the more recent [WHD+10] (2010) modeling of the two ground state helium atom system interaction potential, indicated the shallow potential well of the De = 11 K depth at the equilibrium interatomic distance re=5.6 a0 (Bohr radius).
In 1982 it has been reported [US82] that this potential well supports one and only one weakly (binding energy of 0.8 mK was provided) bounded state. This weakly bound state was observed experimentally [LMK+93] after electron impact ionization of a supersonic expansion of helium with translational temperature near 1 mK, as the binding energy had been estimated between 0.8 and 1.6 mK. This is in agreement with the recent (2009) theoretically determined value 1.3 mK [Gel09]. Such a low energy excludes presence of the ground state molecules at room dissociation energy repulsive interaction, blue – induced dipole attraction – van der Waals force. The addition of these interaction results in the characteristic shape of the potential (black curve) with well of De depth (in respect to the dissociation energy limit that is put = 0) at the re equilibrium bond (interatomic) distance temperature. In contrast to the X1Σ+g state, a deep attractive potential exists for the interaction between a metastable helium atom and a ground state one. The metastable state atom can exist in either the singlet or the triplet state, so that A1Σ+u and a3Σ+u can be formed respectively (see the potential curves on Fig. 2.2). As it is clear from this figure also atoms in other excited states give with the ground state atoms a binding potentials, for instance the b3Πg molecular state formed by the 23P and the ground state atoms. However, forbidden radiative transition to the ground atomic state from the triplet metastable atomic state, privileges this population over the atoms in the singlet metastable state and other highly excited states when the number densities are compared in the plasma conditions of this work. Thus, the higher temperature regime (≫ 1mK) the a3Σ+u is often considered as the lowest helium molecular state in fact being the first excited molecular state above the mentioned X1Σ+g . The potential curve of the a state (marked with solid red line on the 2.2) with potential well (at 1Å) depth is ∼ 15700 cm−1 [Yar89], is result of interaction between one ground state He atom (11S 0) and helium atom in the metastable state (23S 1).
Formation rates for He2∗
The association of these two atoms however requires presence of one additional ground state atom so that three body collision occurs. He(23S ) + 2He(11S ) → He2(a3Σ+u ) + He(11S ) (2.1)
This requirements is the result of presence of the repulsive barrier (484±48 cm−1) at intermediate nuclear separations (2.75±0.03Å) [KTZ+90,Yar89], as a result of the competition between the He+2 (1σ2g 1σu) core attraction and He(11S)+He(23S) exchange repulsion.
The role of the third body can be explained on the basis of the simple dynamical model in which three-body collision is pictured as two binary collisions in rapid succession as presented by Köymen et al. [KTZ+90]. The temperature dependency of the rate constant δm(T) for the formation of He2 molecule has been measured for 4He:
δm(T) = T[8.7 exp(−750/T) + 0.41 exp(−200/T)]Ö10−36cm6s−1 (2.2)
The temperature dependent rate constant δm(T) is a coefficient in differential equation for temporal change of metastable helium atoms density nm due to diffusion process (Dat – diffusion coefficient normalized to unit density) and associative ternary collisions with the ground state helium atoms (ground state atoms density N ∝ P, P-gas pressure): ∂nm ∂t = Dat∇2nm − δmP2nm (2.3)
The product of δmP2 is the frequency for destruction of 23S atoms in three-body collision [KTZ+90,ZSB+93] – see also subsection 4.5.2.2 for reaction eq.(2.1)
The values of δm(T) for 3He isotope are measured to be ≈ 33% higher than given by eq.(2.2) due to increased thermal velocity of the lighter isotope. [KTZ+90] Temperature dependent rate δP(T) for molecule formation from 23P state in reaction: He(23P) + 2He(11S ) → He2(b3Πg) + He(11S ), (2.4) is given by: δP(T) = (2.5 + 267T−1)Ö10−32cm6s−1, (2.5) because no repulsive potential barrier for the interaction between helium atoms in 11S and 23P states is existing in that case [ZSB+93].
The definition of δP(T) rate coefficient is analogous as for δm(T) but the differential equation similar to the eq. 2.3 concerns the change of the density of atoms in 23P state (nP instead of nm) and the diffusion process is described by the coefficient DP (instead of Dat)
The ratio of rates for formation of metastable molecules given by (2.2) and (2.5) at room temperature ( 300K) is δP(300K)/δm(300K) = 3.4·10−32 277·10−34 ≈ 122, increased by two orders of magnitude.
In this case the helium molecule in higher excited state b3Πg is formed that,however, de-excites to metastable state He2(a3Σ+u ) [EADL88,RBBB88].
Hence, enhanced formation of metastable molecule is expected in presence of 1083 nm laser light, that resonantly promotes atoms from 23S1 state to He(23P) state during MEOP. (inducing transition He(23S)→He(23P) ) and possibility of nuclear orientation of polarized helium atoms lose in collisions with molecular species gave an indication and motivation for investigations of relation between observed light-induced relaxation of atomic nuclear polarization and molecular formation dynamics.
Theoretical description of molecular dimer
Theoretical models describing pair of atoms associated in diatomic molecules has been already described in many literature sources thus only those issues that are needed for further understanding of the dissertation are presented.
Exact description of diatomic molecule in quantum mechanics comes down to solution of a many-body problem including interaction of two nuclei and N=4 electrons. The Schrödinger equation (omitting the spin interactions between nuclei and electrons) has a following form: are Laplace operators corresponding to kinetic energy of electrons with mass m and each of nucleus having mass M1 and M2 forming the molecule. The total potential energy of a dimer V(⃗re, ⃗rn), which graphical representations for He2 are given on 2.2, and the total wave function of electrons and nuclei Ψ(⃗re, ⃗rn) depend on electrons ⃗re and nuclei ⃗rn spatial coordinates.
Exact solving of the equation (2.6) is not possible as its analytical solution does not exist.
However, using certain assumptions, it can be simplified to the form that can be solved. Details of such procedures are well described in the literature, i.e. [Her50] thus only main steps will be shown leading to the final solutions. Born and Oppenheimer [BO27] assumed that the variation of internuclear distance (nuclei motion), which due to their mass is relatively slow in comparison with motion of light electrons, has a negligible influence on the latter. This allows separation of Ψ(⃗re, ⃗rn) into a product of electronic Ψe(⃗re; ⃗rn) and nuclear Ψn(⃗rn) wave functions:
Ψ(⃗re, ⃗rn) = Ψe(⃗re; ⃗rn)Ψn(⃗rn). (2.7)
With this assumption the eq. (2.6) splits into two independent equations describing electrons movement in electrostatic field of motionless nuclei (2.8a) and nuclei motion (vibration and rotation) in effective potential Vn(⃗rn) + Ve(⃗rn) (2.8b) Ψe(⃗re;⃗rn) = EelΨe(⃗re;⃗rn) , (2.8a)
Eel and E are the electrons energy in the nuclei field and total energy of molecule respectively, while Ve and Vn are electronic and Coulomb internuclear interaction potentials. It has to be pointed out that in Ψe(⃗re; ⃗rn) the ⃗rn is not a variable but a parameter which is the consequence of variation of Ve with internuclear distance.
Solutions of eq.(2.8b) are of the special meaning in terms of interpretation of complex molecular absorption and emission spectra that exhibit the structure related to the relative movement of nuclei. However, symmetries of electronic wave functions Ψe, solutions of eq.(2.8a), decide about characteristics of the whole molecular state. Transformation of coordinate system in eq.(2.8b) into the center of the mass (CM) allows separation of relative nuclei motion from motion of the molecule as a whole. In the CM system RCM and R coordinates appears which are describing position of the mass center and the intermolecular distance, as well as reduced mass μ and total mass of nuclei M, so that (2.8b) takes form: ⃗RCM, ⃗R
The wave function ˜Ψn can be decomposed into the part describing the mass center motion exp(i⃗k⃗RCM) and relative nuclear χ(R) motion: n = exp(i⃗k⃗RCM)χ(R), (2.10) with ⃗k – wave vector describing the momentum of the molecule. Eigenvalues of eq.(2.9) are given by the sum of kinetic energy of the molecule and its internal energy Eint: 2M + Eint (2.11)
Considering only this part of (2.10) which depends on internuclear distance it is worth changing the Cartesian into spherical coordinate system as the potential V(⃗R) depends only on relative nuclei positions. In this way wavefunction is factorized: χ(⃗R) = f (|⃗R|)F(θ, φ) and transformation of Laplace operator in (2.9) leads to f (|⃗R|)F(θ, φ) = Eint f (|⃗R|)F(θ, φ), (2.12) where ˆL2 is a square of orbital angular momentum operator. Choosing the form of the angular part of the eigenfunction: F(θ, φ) = FL,M so that it is the eigenfunction of ˆL2, fulfilling following eigenequations 2FL,M = ~L(L + 1)FL,M (2.13a) quantization rules for orbital angular momentum quantum number L and its projection M on quantization z axis appears:
L = 0, 1, 2, …, (2.14a)
−L ≤ M ≤ L, M ∈ Z. (2.14b)
which does not contain any angular coordinate and describes only the radial part of the wave function χ(⃗R). Its dependency on the value of quantum number L and M that has 2L+1 possible values leads to (2L + 1)-fold degeneracy.
At this stage further solution of Schrödinger equation depends on choice of a function describing V(R) potential.
Approximations of V(R) potential
The eigenproblem (2.15) represents the motion in a given potential, consists of terms responsible for centrifugal force and binding potential between the nuclei, which has to have the minimum if the bound state is considered. Application of approximation of a small oscillations about re leads to approximation of V(R) by expansion around re up to the second (harmonic oscillator) and the third (anharmonic oscillator) order of magnitude terms. Despite the model of a simple harmonic oscillator has no physical meaning in terms of proper description of diatomic molecule potential, simple results that can be obtained in this way can build up ones intuition for further understanding of more accurate, however more complicated, models.
Table of contents :
1 Introduction
2 The He2* metastable molecule
2.0.1 Binding between two He atoms
2.0.2 Formation rates for He2∗
2.1 Theoretical description of molecular dimer
2.1.1 Approximations of V(R) potential
2.1.2 Electronic molecular states
2.1.3 Symmetries of the rotational levels and transition selection rules
2.2 Intensities of the rotational lines of a3Σ+u (0)- e3Πg (0) transition and molecular densities
3 Blue laser setup
3.1 Second Harmonic Generation in PP-KTP
3.1.1 Quasi-phase-matching
3.1.2 Conversion efficiency
3.1.3 Matching tolerances
3.2 465 nm laser setup – description
3.2.1 Laser setup components
3.2.1.1 Laser diodes
3.2.1.2 Nonlinear PP-KTP crystal
3.2.2 Setup for SFG – construction scheme
3.2.3 Laser output wavelength tuning
3.3 Laser performance – experimental results
3.3.1 Output power and efficiency
3.3.2 Tuning range and matching conditions
3.3.2.1 Laser wavelength calibration
3.3.2.2 Matching conditions
3.3.3 Matching tolerances in the experiment
3.4 Summary and conclusions
4 Absorption measurements on helium molecule – experimental results
4.1 Experimental setup
4.1.1 Cell and rf discharge
4.1.2 Laser frequency scans. Acquisition scheme
4.1.3 Frequency scans recordings procedure
4.1.4 Measurements of the density decay times
4.2 Data processing and data reduction
4.2.1 Processing of the recorded data files in numerical lock-in software
4.2.2 Absorption line profile position determination
4.2.3 Absorbance and demodulated signal relations
4.2.3.1 100% square modulation
4.2.3.2 Partial sine amplitude modulation
4.3 Absorption lines shape
4.4 Position of absorption lines
4.4.1 Isotopic mixture cell – recordings of multiple lines
4.5 Intensities of absorption lines
4.5.1 Rotational temperature determination from relative absorption rates
4.5.2 Dynamics of helium molecule
4.5.2.1 Molecular and atomic density decay curves
4.5.2.2 Atomic and molecular coupled rate equations
4.6 Summary and conclusions
5 Investigations on Metastability Exchange Optical Pumping dynamics with blue laser
5.1 Basics of MEOP
5.2 Achievements in MEOP in standard and non-standard conditions
5.3 MEOP angular momentum budget and laser induced relaxation
5.4 Description of MEOP experiment with blue light transmission measurement on molecular helium
5.4.1 Design of the MEOP experimental setup
5.4.1.1 General constraints
5.4.1.2 Nuclear polarization measurement method choice – pros and cons
5.4.1.3 Blue laser in MEOP setup
5.4.2 MEOP experimental setup – realization
5.4.3 Acquisition scheme and measurements protocols
5.4.3.1 Acquisition scheme and recorded signals
5.4.3.2 Measurements protocol
5.5 Experimental data reduction and processing
5.5.1 Polarization build-up and decay curve for Meq and Tdecay determination
5.5.2 Metastable atoms number density determination
5.6 Experimental results of MEOP with molecular density measurements
5.6.1 Discharge intensity
5.6.2 Results of MEOP – nuclear polarization values
5.6.3 Light induced relaxation and molecular absorption rates at Meq
5.6.4 Atomic and molecular metastable species density change with polarization
5.7 Absorption measurements in presence of OP pump beam at M = 0
5.8 Summary and the conclusions
6 Conclusion and outlook
A Reference data on molecular helium a-e transition spectral lines
B Principle of laser absorption technique
C Sellmeier equation and dispersion coefficients for PP-KTP
D Constants
Bibliography
