One and Two-Photon Transitions in Ammonia Experimental and Simulated Line Lists in the Search for Drifting Constants

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One and Two-Photon Transitions in Ammonia

Experimental and Simulated Line Lists in the Search for Drifting

Constants

Merit van der Lee

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Abstract

Molecular vibrations are sensitive to changes in the reduced mass which makes them sensitive to changes in the proton-to-electron mass ratio (µ). Possible variations of µ could thus be de-tected with accurate measurements of near infrared (NIR) transitions in NH3 using a molecular

fountain set-up. To this end a list of experimental and simulated one-photon transitions in the 6230 to 6711 cm−1range is presented here. We also give a list of potential two-photon transition that could be used in this experiment. These two-photon transitions are from the same ground level as the one-photon transitions, the asymmetric J=1 K=1 level, and could be made with the same NIR laser.

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List of Figures

2.1 Symmetry operations in C2v and D3h symmetry groups[12]. . . 6

2.2 The symmetry labels in this figure are for the C3v pointgroup[10] . . . 8

2.3 The potential function for the inversion motion ν2 of ammonia [11] . . . 9

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List of Tables

2.1 Character table for C3v point group and C3v(M ) MS group. . . 7

2.2 Character table for D3h point group and D3h(M ) MS group. . . 8

2.3 Term values, a-s splittings and symmetries for the fundamental vibration of am-monia in the ground state. . . 8

2.4 Product table for the D3h point group (and MS group). . . 9

2.5 Symmetries of vibrational dipole transitions in NH3. . . 10

2.6 spectroscopic transition rules [16] . . . 10

3.1 Term values, a-s splittings and symmetries for the combination bands of ammonia in the ground state. . . 13

3.2 List of experimental lines for NH3 from different sources . . . 14

3.3 15 lines that possibly give the best two-photon transitions. Data from Yurchenko et al.[29] . . . 15

6.1 First (top) and second (bottom) transitions in possible two-photon transitions. Data from Yurchenko et al. [29] . . . 20

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Chapter 1

Introduction

The proton-to-electron mass ratio (µ) and the fine-structure constant (α) are considered to be fundamental constants of nature. The values of these constants make life as we know it possible. Even the slightest variation could result in enormous changes in our universe. It is assumed that these constants do not change over time or space, however, the Standard Model does not put any constraints on the value or constancy of these constants [1]. One of the biggest mysteries in modern day science is whether or not the apparent fine-tuning of these constants is coincidental or if it can be explained by a theory beyond the Standard Model [2]. There are several theories that predict spatial and temporal variations in α and µ [3-5].

Since different spectroscopic transitions are affected differently by a change in α , the shift in the frequency of these lines can be used to identify a possible change. On the other hand, vibration, rotation and tunnelling of a molecule is influenced by the reduced mass of a molecule. A change in the proton-to-electron mass ratio will change the reduced mass and thus shift the wavenumber (ν) of these transitions in the infrared (IR) and near infrared (NIR) spectrum. The change in wavenumber of well-known transitions in the NIR can thus be used to look for changes in µ. Also, frequency (and thus wavenumber)is the most accurately measured quantity in physics and thus is one of the best ways to measure any change in µ.

To enable a sensitive detection of a varying µ on short (eg. a few years) timescales, it is imperative to use spectroscopic transitions with a high sensitivity coefficient Kµ. Because of

this, ammonia would be a good candidate for these studies since it has vibrational levels that are split into doublets due to the inversion motion of the molecule. This inversion splitting is very sensitive to changes in and as a result has a relatively high sensitivity coefficient [6].

Ammonia is abundantly present in our universe and to this date more than 40000 lines have been recorded [7] in the HIgh resolution TRANsmission molecular adsorption database (HITRAN) [8]. In this work we will identify lines that can be used in molecular fountain experiments [9] in the search for a drifting µ. These will be lines in the spectral range from 6230 to 6711 cm−1 that originate from the anti-symmetric J=1, K=1 level.

Apart from experimentally measured lines from literature we will also look into lines resulting from quantum mechanical simulations. An extensive list of lines [29] will be investigated for possible one- and two-photon transitions in the before mentioned wavenumber range.

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Chapter 2

Theory

2.1 Symmetry

Molecules in their equilibrium geometry possess a certain symmetry. This symmetry is similar to the symmetry of a macroscopic body. The symmetry of a specific molecule can be described in terms of rotational axes and reflection planes. Reflections through these planes and rotation around these axes keep the molecule unchanged, hence the word symmetry operation. Since all these operations occur through one point, the molecule is said to belong to a certain point group. Each point group consist of a specific set of symmetry operations (the rotations and reflections) [11]. With the use of these point groups the vibrational motions, and thus energy levels, of a molecule can be given a symmetry label which is of great use in molecular spectroscopy. This comes from the fact that spectroscopic transitions can only occur between levels with a specific symmetry. Which symmetry this is depends on the symmetry of the transition dipole-moment, which in turn depends on the point group of the molecule. The ammonia molecule belongs to point group C3v (Fig.2.1a) which consists of 3 rotations of 120 degrees around the

symmetry-axis (C3) and 3 reflections through planes perpendicular to the symmetry axis (σv). However,

it can be more appropriate to use the point group D3h (Fig. 2.1b) for the ammonia molecule.

This group also contains the rotations and reflections from the C3v group, but since it is planar

it also has 3 C2rotation axes and it has a horizontal (with respect to the principal C3 symmetry

axis) reflection plane σh.

(a) Ammonia molecule with C3v

(b) Boron trifluoride with D3h symmetry

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CHAPTER 2. THEORY

The use of D3h as a point group for ammonia is justified by the fact that the inversion

motion (right molecule in Fig.2.2a) can be thought of as a vibration trough the plane of the “flat” molecule [11] in which the flat molecule is the equilibrium configuration. For each point group there exist a character table. In such a table, the rows are labelled by the irreducible group representations, the columns by classes of symmetry operations of that specific group and the entries are the traces of the irreducible matrixes. The character tables for C3v and D3h[11]

are given in Table 2.1 and Table 2.2 respectively. These point groups are very useful for rigid molecules, however, when the molecules are not rigid or tunnelling occurs we need a different way to label their symmetry. For this reason the CNPI groups were developed. CNPI stands for Complete Nuclear Permutation Inversion. The CNPI group of a molecule consists of all permutations of identical nuclei, the inversion E* and the product of E* with all permutations [26]. A subgroup of this CNPI group is the MS (Molecular Symmetry) group, which, for a rigid non-linear molecule, is the same as the point group. It thus has the same name as the point group but with an (M) added to it. The MS group is created by removing all unfeasible operations from the CNPI group. These unfeasible operations are for example all operations that move the molecule from one minima to the next (see figure 2.3). If the energy barrier between these minima is sufficiently high tunnelling between these minima is highly unlikely and thus can all these operations be discarded. From this it is also clear that it is wrong to use the C3v(M ) group for ammonia since this group does not include all operations in both

minima. All these operations need to be included since the barrier between the two minima is low enough to allow for tunnelling even in the lowest energy levels. This tunnelling occurs with a frequency of 24 GHz (for J=1 K=1)[27]. The D3h(M ) group does include all

C3v E C3(z) σv C3v(M ) E (123) (23)* Nr of elements 1 2 3 A1 1 1 1 Tz , αzz , αxx + αyy A2 1 1 -1 Jˆz , Γ* E 2 -1 0 (Tx, Ty), ( ˆJx , ˆJy), (αxz , αyz), (αxx αyy , αxy)

Table 2.1: Character table for C3v point group and C3v(M ) MS group.

2.2 Normal modes

The normal mode of a molecule is a vibrational motion in which all (or some) of the nuclei vibrate together with the same frequency. The number of normal modes of a non-linear molecule is equal to the number of vibrational degrees of freedom of the molecule. This is 3N-6, where N stands for the number of nuclei in the molecule. For ammonia this means that there are 3*4-6 = 6 normal vibrations, which are depicted in Figure 2.2.

From these 6 normal modes in Figure 2.2, the ones in Figure 2.2(b) and 2.2(c) are degenerate (ν3 and ν4), which means that they have the same frequency but different vibrations. These

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CHAPTER 2. THEORY D3h E C3 C20 σh S3 σv D3h(M ) E (123) (23) E* (123)* (23)* Nr of elements 1 2 3 1 2 3 A’1 1 1 1 1 1 1 αzz , αxx + αyy A’2 1 1 -1 1 1 -1 Jˆz E’ 2 -1 0 2 -1 0 (Tx, Ty), (αxx αyy , αxy) A”1 1 1 1 -1 -1 -1 Γ* A”2 1 1 -1 -1 -1 1 Tz E” 2 -1 0 -2 1 0 ( ˆJx , ˆJy), (αxz , αyz)

Table 2.2: Character table for D3h point group and D3h(M ) MS group.

(a) ν1(left) and ν2(right)

(b) the two degenerate ν3 normal modes (c) the two degenerate ν4normal modes

Figure 2.2: The symmetry labels in this figure are for the C3v pointgroup[10]

(0, 1, 0, 0). ..

cm−1 Symmetry Splitting

(J=0, K=0) (D3h) (cm−1) Motion Ref

ν1 s 3336.11 A’1 0.99 Symmetric stretch (fig. 2.1(a) left) [18]

a 3337.10 A’1

ν2 s 932.4338 A”2 35.688 Symmetric bend, umbrella motion (inversion) (fig. 2.1(a) right) [18]

a 968.12 A”2

ν3 s 3443.63 E’ 0.37 Anti-symmetric stretch (fig. 2.1(b)) [18]

a 3444.00 E’

ν4 s 1626.30 E’ 1.00 Anti-symmetric bend (fig. 2.1(c)) [18]

a 1627.30 E’

Table 2.3: Term values, a-s splittings and symmetries for the fundamental vibration of ammonia in the ground state.

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CHAPTER 2. THEORY

Figure 2.3: The potential function for the inversion motion ν2 of ammonia [11]

2.3 Overtones and combination bands

Overtones and combination bands are the result of excitations of the normal modes by more than one quanta. This can be by excitation of one normal mode by more than one quanta (overtone) or by excitation of two (or more) different normal modes by one or more quanta (combination bands). The resulting symmetry of these overtones and combination bands can be calculated using formula 2.1 and Table 2.4. Table 2.4 gives the products of multiplication of the different symmetries of the D3h (M) MS group. An example of this calculation will be

given in the results section.

Γ(ν1, ν2, ν3, ν4) = [Γ1]ν1 ⊗ [Γ2]ν2⊗ [Γ3]ν3 ⊗ [Γ4]ν4⊗ (2.1)

A’1 A’2 E’ A”1 A”2 E”

A’1 A’1 A’2 E’ A”1 A”2 E”

A’2 A’2 A’1 E’ A”2 A”1 E”

E’ E’ E’ A’1+A’2+E’ E” E” A”1+A”2+E”

A”1 A”1 A”2 E” A’1 A’2 E’

A”2 A”2 A”1 E” A’2 A’1 E’

E” E” E” A”1+A”2+E” E’ E’ A’1+A’2+E’

Table 2.4: Product table for the D3h point group (and MS group).

2.4 Two-photon transitions

The first experimental demonstration of two-photon absorption was given by Kaiser and Garrett in 1961 [30]. However, it was already described theoretically by Goeppert-Mayer in 1931 [31] but could only be tested after the invention of the laser. In two-photon absorption processes two photons are absorbed simultaneously from either the same or two different lasers. The absorbed

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CHAPTER 2. THEORY

photons can thus have the same or different energies. Two-photon absorption is a second-order non-linear process and depends on I2_{. For this reason high powered pulsed laser are usually}

needed. Here, two-photon transitions will be between two levels with a real intermediate state, since the power of the laser that will be used (CW NIR laser) is not high enough to use virtual levels. The real intermediate state will thus enhance the two-photon transition to make it sufficiently strong.

2.5 Transition rules and probabilities

From the character tables we can deduce the symmetries of the possible dipole transitions. The symmetries (irreducible representation Γ) for vibrations of the ammonia molecule in the C3v and D3h point group are given in table 2.5.

The selection rules for an harmonic oscillator are : ∆ν = ±1. Based on symmetry, transitions only occur when R ΓiΓtrΓf includes the totally symmetric A1 or A1 in the case of ammonia.

This condition is met when ΓiΓf = Γtr.

C3v D3h Motion

* A2 A”1 Electrical dipole Γinv) A1 A”2 Inversion

Tz A1 A”2 Translation z, dipole

Tx E E’ Translation x, dipole

Ty E E’ Translation y, dipole

Table 2.5: Symmetries of vibrational dipole transitions in NH3.

Type Parallel (non-degenerate) Perpendicular (degenerate)

Species A E

0, ±3, ±6, ±1, ±2, ±4,

∆J and ∆K For K=0, ∆K=0;∆J=±1 For all K, ∆K=±1; ∆J=0, ±1 For K6=0, ∆K=0; ∆J=0, ±1

inversion a↔s a↔a; s↔s

∆(l − K) 0 0, ±3 (perturbation-allowed) Nuclear spin weights S for (J, K)a/s S=2 for K=3, 6, 9,

S=1 for K6=3, 6, 9,

For K=0; S=0 for even J with s; odd J with a For K=0; S=2 for odd J with s; even J with a

Table 2.6: spectroscopic transition rules [16]

The ground level in this work is J=1, K=1, asymmetric (-) which has symmetry E” in the ground state [19]. Allowed transitions will thus be to levels with E’ symmetry (in the absence of an electric field) [11]. Since angular momentum has to be conserved and photons have angular momentum of ± 1, one-photon processes are only allowed for ∆l = ±1. When a transition is made with two photons the allowed transition will thus be for ∆l = 0, ±2 in order to conserve angular momentum.

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CHAPTER 2. THEORY

2.5.1 Two-photon transitions

From Linskens et al. [36] and [38] we have that the two-photon rate Γ2ph, if the detuning from an

intermediate level is large compared to the linewidth of the laser and the one-photon transition, Rabi-oscillations will occur between the ground and exited state at a frequency given by:

Ω2ph=

Ω01Ω12

2δij

(2.2) With Ω01and Ω12the Rabi frequencies for the transition from level 0-1 and 1-2, respectively,

and δ the detuning from the intermediate level.

We also know that [33] Ωij is proportional to pAji and we can use the following equation

to get a feel for the two-photon rate.

Γ2ph ∝

µ01µ12

2δij

(2.3) Here, µij is the transition dipole moment between level i and j in s−1and δij in the detuning

between levels i and j in Hz. µij can be calculated from Aji using [35]:

µij = s Aji 3ohc3 2ω3 ji (2.4) This gives µij in C.m

rad1_.5 , which can be converted into Debye by dividing by 3.33 ∗ 10 −30_.

In experiments performed in Paris [39], the P(4)E two-photon transition of the 2ν3 mode in

SF6 was probed with a detuning of about 76 MHz from the intermediate state. In a molecular

beam travelling at a speed of 400 m/s, an intensity of about 15 mW in a laser beam with a waist of 3-4 mm is required to make a π

2 pulse. For comparison, on the single photon P(4)E transition an intensity of ∼ 10 µW is sufficient to drive a π

2 pulse, using similar molecular beams and laser geometries. The transition used in the SF6 experiment has µ01 ≈ µ12 ≈ 0.4 Debye

and a detuning of 76 MHz, i.e., µ01µ12

2δ ≈ 5 ∗ 10

−4_Debye2_{/M Hz. In ammonia it should not}

be difficult to have a 100 times higher intensity, furthermore as the molecules have a velocity 1m/s, the interaction time is 400 times larger. Combining these factors, we will be interested in transitions that have µ01µ12

2δ on the order of 1 ∗ 10

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CHAPTER 2. THEORY

2.6 Molecular Fountain

Figure 2.4: Schematic depic-tion of a molecular fountain [37]

The resolution of a measurement is limited by the time the molecule spends in the measuring device. The longer the surement the better the resolution will be. However, most mea-surement techniques that keep atoms or molecules ”trapped“ in the devise require some kind of trapping potential, which will perturb the energy levels and thus leads to less accurate mea-surements. A way around this is the atomic or molecular foun-tain. Here, molecules are slowed down to a velocity of 1 m/s (in our case [9])before they fly upward for a short distance (ap-proximately 30 cm) and then fall down under gravity. During this flight they pass a microwave cavity twice, once on the way up and once on the way down. The two interactions in this cavity lead to Ramsey fringes [34] with a frequency width of ∆ν = 1/(2∆t). In this microwave cavity the molecules are excited with a laser, the excited state population progresses along the Bloch sphere and the molecules are excited again when they pass the cavity the second time. The accuracy of such a transition is determined

by ∆ν/ν. At v = 3 m/s and a distance of 30 cm ∆t will be 0.5 s , which gives a ∆ν of 1 Hz. At a typical NIR wavenumber of 6600 cm−1 (198 THz) this gives a accuracy of 5.05 ∗ 10−15.

2.7 Simulations

In 2010 Yurchenko et al.[29] published a variationally computed line list of hot NH3 as an

extension to their 2009 T = 300 K list [28]. With this, an enormous amount of data on NH3

transitions became available. The list consist of over 109 transition between more than 106 energy levels in the 0 − 12000cm−1 range. The simulations were performed by first calculating a potential energy surface using coupled cluster CCSD(T). The resulting surface was then further refined by fitting it against experimental data using a least-squares approach and adjusting the equilibrium geometry of the molecule accordingly. The dipole moments were also calculated at the CCSD(T) level with a aug-cc-pVQZ basis set in the frozen core approximation.

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Chapter 3

Results and discussion

3.1 Symmetries

The symmetries of the combination bands ν1+ν4, ν1+ν3and ν3+2ν4were found by multiplying

the characters of the individual normal vibrations using the results from Table 4. For example: ν1+ ν4 = A01xE02 = A01xE0xE0 = A01xE0 = A01 + A02 + E0. The resultant symmetries for the

three combination bonds are given in Table 3.1. All three symmetries were calculated in the same manner by using the results from Table 3.2.

cm−1(J=0, K=0) Symmetry (D3h) splitting Motion Ref

ν1+ ν3 s 6608.817 E’ 0.930 Symmetric [25][18]

+ anti-symmetric stretch

a 6609.747 E’ [25] [18]

ν1+ 2ν4 s 6556.424 A’1 + A’2 + E’ 1.508 Symmetric stretch [18] [23]

(|l| = 2) + anti-symmetric bend

a 6557.932 A’1 + A’2 + E’ [18] [23] ν3+ 2ν4 s 6678.363 2A’1 + A’2 + 2E’ 0.790 Anti-symmetric stretch [24]

(l3= 1, l4= −2) + anti-symmetric bend

a 6679.154 2A’1 + A’2 + 2E’ [24]

Table 3.1: Term values, a-s splittings and symmetries for the combination bands of ammonia in the ground state.

3.2 Line list

In the 6230 to 6711 cm-1 spectroscopic range, a multitude of lines have been measured and identified [8,15,16] to date. These lines belong to at least three different bands; ν1+ ν4, ν1+ ν3

and ν3+ 2ν4, but possibly a lot more [17]. There are lines that originate from a range of J and K

values ranging from 0 till 10 or more and all of these lines have symmetric and anti-symmetric levels. For the molecular fountain experiments only lines originating from the anti-symmetric J=1, K=1 can be used so the line list is substantially reduced. Table 3.2 gives an overview of the lines that were found in the desired spectroscopic region with the correct J and K values. A few lines in Table 3.2 belong to the same spectroscopic transition but have different wavenumbers or have the same wavenumber but have been appointed to different transitions.

A Python script was made (see Appendix 6.0.1) to enable quick analysis of this extensive line list. Starting from the ground level K=1, J=1, a, 64 transitions where found within the wavenumber range of 6230 to 6711 cm−1. With this list all transitions from this first level to

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CHAPTER 3. RESULTS AND DISCUSSION

a second level within the same wavenumber range, so to levels with termvalues of 2 ∗ ν1, and

with ∆ν smaller then ±1cm−1 were selected. This gave a list of 134 lines (see Table 5.1 in the Appendix). The 15 lines with the highest Γ2ph can be found in Table 3.3. Not all the

information on the different levels is given in Table 3.3, however, it can be found in Table 6.1 of the Appendix.

A comparison between the experimental and simulated lines can be found in Table 6.2 of the Appendix. The lines were matched (as good as possible) on both assignment and wavenumber. Some experimental lines were not present in in the simulated list and δν was rather large, ranging from 0.16 to 3.63 cm−1. Assignment of the lines did also not always agree, which can be due to both wrong experimental assignment or wrong matching of the simulated line to the experimental line.

J’ K’ l’ J” K” l l wn obs line strength band source 0 0 1 1 6540.96698/6540.9688 [23] 5.888E-23 (296K) 1 + 24 [16] [18] [23] 0 0 1 1 6592.79 3.973E-22 (296K) / sat 1 + 3 [16] [17] [18] 0 0 1 1 6661.34697 2.303E-22 (296K) / sat 3 +24? [16] [16] 0 0 1 1 6664.90663 6.130E-22 (296K) [16] 0 2 1 1 6649.918 2.16E-07 1 + 3 [17] 1 0 1 1 6561.26444/6561.2661 [23] 7.300E-23 (296K) 1 + 24 [16] [18] [23] 1 0 1 1 6612.70367 /6612.695 [18] 7.098E-22 (296K)/ 2.18E-06 1 + 3 [16] [17] [18] 1 0 1 1 6668.9644 2.625E-02 (296K) /2.90E-07 3 +24? [16] [17] [16] 1 0 1 1 6681.0752 3.030E-22 (296K) 3 +24? [16] [17] [16] 2 0 1 1 6601.3919 1.45E-07 1 + 3 [17] 2 0 1 1 6602.078 1.070E-22 (296K) 1 + 24 [16] 2 0 1 1 6602.0855 1.35E-07 1 + 3 [17] 2 0 1 1 6652.46994 1.649E-22 (296K) / sat 1 + 3 [16] [17] 2 0 1 1 6688.3752 1 + 3 [17] 2 0 1 1 6709.4788 9.760E-02 (296K) 3 +24? [16] [17] [16] 2 0 1 1 6720.8651 3 +24? [16] [17] 2 1 1 1 -1 2 6699.00607/6699.0058 [24] 4.235E-22 (296K) 3 +24 [16] [24] 2 2 1 1 6575.71887/6575.7201 [23] 4.880E-22 (296K) 1 + 24 [16] [23] 2 2 1 1 1 6636.1907/6636.185 [25] 1.94E-06 1 + 3 [17] [25] 2 2 1 1 6699.00607 4.235E-22 (296K) 3 +24? [16] [16] 2 2 1 1 6699.0061 3 +24? [16] [17]

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CHAPTER 3. RESULTS AND DISCUSSION

nr A10 A21 ω01 ω12 µ01 µ12 δHz Γ2ph(From Eq. 2.3, in Hz)

1 8.95E-01 4.10E-02 1.26E+15 1.26E+15 3.10E-03 6.61E-04 3.54E+09 2.89E-10 2 3.18E-01 6.59E-02 1.24E+15 1.24E+15 1.90E-03 8.61E-04 4.42E+09 1.85E-10 3 3.18E-01 4.01E-02 1.24E+15 1.24E+15 1.90E-03 6.71E-04 5.14E+09 1.24E-10 4 2.32E+00 3.18E-04 1.25E+15 1.25E+15 5.04E-03 5.87E-05 1.35E+09 1.09E-10 5 8.95E-01 6.35E-02 1.26E+15 1.26E+15 3.10E-03 8.22E-04 1.67E+10 7.64E-11 6 2.32E+00 3.48E-03 1.25E+15 1.25E+15 5.04E-03 1.95E-04 8.46E+09 5.79E-11 7 9.47E-03 2.03E-03 1.23E+15 1.23E+15 3.31E-04 1.53E-04 4.52E+08 5.59E-11 8 2.22E-01 1.89E-03 1.26E+15 1.27E+15 1.53E-03 1.41E-04 2.08E+09 5.17E-11 9 7.53E-02 2.67E-02 1.26E+15 1.27E+15 8.93E-04 5.30E-04 5.32E+09 4.45E-11 10 1.13E+00 4.15E-02 1.25E+15 1.26E+15 3.50E-03 6.68E-04 2.64E+10 4.42E-11 11 6.19E-02 5.10E-02 1.24E+15 1.25E+15 8.30E-04 7.50E-04 7.21E+09 4.31E-11 12 4.53E-03 4.64E-02 1.23E+15 1.24E+15 2.27E-04 7.23E-04 2.02E+09 4.06E-11 13 1.13E+00 1.50E-02 1.25E+15 1.26E+15 3.50E-03 4.01E-04 1.88E+10 3.74E-11 14 7.74E-04 2.81E-03 1.24E+15 1.25E+15 9.26E-05 1.76E-04 2.53E+08 3.22E-11 15 5.72E-03 4.08E+00 1.24E+15 1.24E+15 2.53E-04 6.73E-03 2.68E+10 3.18E-11

Table 3.3: 15 lines that possibly give the best two-photon transitions. Data from Yurchenko et al.[29]

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Chapter 4

Conclusion

The goal of this work was to identify spectroscopic transitions of ammonia in the 6230 to 6711 cm−1 range. Since this range is in the near infrared this is where the overtones and combination bands appear. This makes this a very dense spectral region with a multitude of transition. For the molecular fountain experiments only lines originating from the anti-symmetric J=1, K=1 level are of interest. This substantially reduces the number of transitions occurring in this region and has resulted in a list of 21 spectral lines from literature. However, there are several lines in this list that belong to the same spectroscopic transition but have different wavenumbers or have the same wavenumber but have been appointed to a different transition.

A second part of this work was selecting possible two-photon transitions, originating from the same ground level, from a recently published list of over 109 simulated transitions. To this end a python script was written to search for these transitions, which resulted in a list of 134 possible transitions. Of these 134 only 4 transitions have a µ01µ12

2δij

that is in the order of 1 ∗ 10−10Debye

2

M Hz .

For some one-photon transitions there was a significant difference, in the order of several cm−1, between the experimental and simulated wavenumbers. This could have been due to wrong assignment of (probably) the experimental lines or due to wrong matching of the simu-lated lines to the experimental. Another possibility is that the experimental lines can contain interferences that have not been simulated, however, this has to be checked. It is not expected that these possible interferences will be present in the molecular fountain experiment.

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Chapter 5

Recommendations

The two-photon rate from Equation (2.3) still has Rad left in the units after conversion to Debye. It needs to be checked if this Rad can simply be dropped in this case or if a further conversion is necessary. Another important contribution to a correct value for the two-photon rate is the multiplicity of the levels. The calculations in this work have been done with a multiplicity of one which might not be a correct assumption for all levels. It is thus recommended to recalculate the rates for the lines of interest with the correct multiplicity.

If the experimental lines are going to be used as a reference they need to be checked more thoroughly for possible interferences.

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BIBLIOGRAPHY

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Chapter 6

Appendix

nr Ground 1e Γ Termvalue n1 n2 n3 n4 l3 l4 τinv J K τrot Γvib

1 12151 21872 6 6698.624874 0 2 0 3 0 1 1 1 0 1 6 2 12151 21861 6 6578.573162 1 0 0 2 0 2 1 1 0 1 6 3 12151 21861 6 6578.573162 1 0 0 2 0 2 1 1 0 1 6 4 12151 51341 6 6652.275054 1 0 1 0 1 0 1 2 2 0 6 5 12151 21872 6 6698.624874 0 2 0 3 0 1 1 1 0 1 6 6 12151 51341 6 6652.275054 1 0 1 0 1 0 1 2 2 0 6 7 12151 7030 6 6678.466728 0 2 0 3 0 1 1 0 0 0 6 8 12151 51338 6 6619.298773 1 0 0 2 0 2 1 2 0 0 6 9 12151 51335 6 6570.618851 2 0 0 0 0 0 0 2 1 0 1 10 12151 7030 6 6678.466728 0 2 0 3 0 1 1 0 0 0 6 11 12151 21862 6 6604.265984 1 2 0 1 0 1 0 1 1 0 3 12 12151 51338 6 6619.298773 1 0 0 2 0 2 1 2 0 0 6 13 12151 51321 6 6371.94831 0 3 1 0 1 0 1 2 2 0 6 14 12151 51332 6 6499.693395 0 0 0 4 0 2 1 2 2 0 6 15 12151 51344 6 6668.898157 1 0 1 0 1 0 1 2 0 0 6

nr 2e Γ Termvalue n1 n2 n3 n4 l3 l4 τinv J K τrot Γvib

1 35932 3 13380.40432 1 4 0 4 0 2 0 2 0 0 3 2 2891 3 13140.03548 1 0 2 2 2 0 0 0 0 0 3 3 2892 3 13140.35426 1 4 0 4 0 2 0 0 0 0 3 4 13066 3 13287.54163 1 4 1 2 1 2 0 1 0 1 3 5 35933 3 13380.84252 0 0 3 2 -3 0 1 2 1 0 4 6 71734 3 13287.86901 1 0 2 2 2 2 1 3 1 0 6 7 13084 3 13339.08842 0 7 0 3 0 -3 1 1 1 0 4 8 71681 3 13221.87483 3 2 0 1 0 1 0 3 0 1 3 9 71612 3 13124.34175 2 0 2 0 2 0 0 3 0 1 3 10 13085 3 13339.34421 1 4 0 4 0 2 0 1 0 1 3 11 2898 3 13192.46154 3 2 0 1 0 1 0 0 0 0 3 12 35848 3 13222.09914 2 4 1 0 1 0 0 2 0 0 3 13 71358 3 12726.79443 1 1 0 5 0 5 1 3 1 0 6 14 12971 3 12981.81115 0 5 0 5 0 5 0 1 0 1 3 15 35896 3 13321.53277 1 2 0 5 0 5 1 2 1 0 6

Table 6.1: First (top) and second (bottom) transitions in possible two-photon transitions. Data from Yurchenko et al. [29]

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CHAPTER 6. APPENDIX nr νobs line strength band source 1 6540.96698/6540.9688 [23] 5.888E-23 (296K) 1 + 24 [16] [18] [23] 2 6592.79 3.973E-22 (296K) / sat 1 + 3 [16] [17] [18] 3 6661.34697 2.303E-22 (296K) / sat 3 +24? [16] [16] 4 6664.90663 6.130E-22 (296K) [16] 5 6649.918 2.16E-07 1 + 3 [17] 6 6561.26444/6561.2661 [23] 7.300E-23 (296K) 1 + 24 [16] [18] [23] 7 6612.70367 /6612.695 [18] 7.098E-22 (296K)/ 2.18E-06 1 + 3 [16] [17] [18] 8 6668.9644 2.625E-02 (296K) /2.90E-07 3 +24? [16] [17] [16] 9 6681.0752 3.030E-22 (296K) 3 +24? [16] [17] [16] 10 6601.3919 1.45E-07 1 + 3 [17] 11 6602.078 1.070E-22 (296K) 1 + 24 [16] 12 6602.0855 1.35E-07 1 + 3 [17] 13 6652.46994 1.649E-22 (296K) / sat 1 + 3 [16] [17] 14 6688.3752 1 + 3 [17] 15 6709.4788 9.760E-02 (296K) 3 +24? [16] [17] [16] 16 6720.8651 3 +24? [16] [17] 17 6699.00607/6699.0058 [24] 4.235E-22 (296K) 3 +24 [16] [24] 18 6575.71887/6575.7201 [23] 4.880E-22 (296K) 1 + 24 [16] [23] 19 6636.1907/6636.185 [25] 1.94E-06 1 + 3 [17] [25] 20 6699.00607 4.235E-22 (296K) 3 +24? [16] [16] 21 6699.0061 3 +24? [16] [17] nr 1e A01 J Γ T erm v alue n1 n2 n3 n4 l3 l4 τinv J K τr ot ν1 ν2 ν3 ν4 ν5 ν 6 Γv ib ν ( cm − 1) δ ( cm − 1 ) 1 7027 0.65027 0 6 6558.223253 1 0 0 2 0 2 1 0 0 0 0 1 0 0 2 1 6 6541.259878 -0.292 2 7028 3.6112 0 6 6609.243208 1 0 1 0 1 0 1 0 0 0 0 0 2 0 0 1 6 6592.279833 0.510 3 7030 1.1323 0 6 6678.466728 0 2 0 3 0 1 1 0 0 0 0 0 0 0 3 5 6 6661.503353 -0.156 4 7031 0.83856 0 6 6679.050034 0 2 0 3 0 1 1 0 0 0 0 0 0 0 3 5 6 6662.086659 2.820 5 6 21861 0.31753 1 6 6578.573162 1 0 0 2 0 2 1 1 0 1 0 1 0 0 2 1 6 6561.609787 -0.345 7 21865 1.7658 1 6 6629.138084 1 0 1 0 1 0 1 1 0 1 0 0 2 0 0 1 6 6612.174709 0.525 8 9 21872 0.89546 1 6 6698.624874 0 2 0 3 0 1 1 1 0 1 0 0 0 0 3 5 6 6681.661499 -0.586 10 11 51338 0.061934 2 6 6619.298773 1 0 0 2 0 2 1 2 0 0 0 1 0 0 2 1 6 6602.335398 -0.257 12 13 51344 0.29139 2 6 6668.898157 1 0 1 0 1 0 1 2 0 0 0 0 2 0 0 1 6 6651.934782 0.535 14 15 16 17 51348 0.0063957 2 6 6712.338981 0 0 1 2 1 2 0 2 1 0 0 1 0 0 2 0 2 6695.375606 3.630 18 51336 0.26659 2 6 6593.000211 1 0 0 2 0 2 1 2 2 0 0 1 0 0 2 1 6 6576.036836 -0.317 19 51341 2.3197 2 6 6652.275054 1 0 1 0 1 0 1 2 2 0 0 0 2 0 0 1 6 6635.311679 0.876 20 51349 0.73175 2 6 6718.264275 1 0 0 2 0 2 1 2 2 0 0 1 0 0 2 1 6 6701.3009 -2.295 21 T able 6.2: Comparison of exp erimen tal (top) and sim ulated [29](b ottom) lines

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CHAPTER 6. APPENDIX 6.0.1 Python script [language=P ython] mport numpy as np import os import tables import time def write_file(fi le, t, Edata, moederc ol, kin dcol, E dataNcol,Efilen, header, fmt, linedelimiter, coldelimiter, sortcol, Ngrandmother=N one, ng randchildren=1e6, EdataTermval uecol=None): #print "Writing export file ’"+fnout_.s plit(’\\’)[-1]+"’" #f=open(fno ut_,’w’) #if Ngrandmother is None: header_=heade r #else: header_=hea der+["dTermvalue(child-2*mother+grandmother)"] #file.write (coldelimiter.join(head er)+linedelimiter) maxdTermval =1 for Nmother in np.unique(t[:,moe dercol]): tmsk=t[:,moe dercol]==Nmother t_=t[tmsk] Edatamask_t_ =np.array(t_[:,kindcol] ,dtype=int)-1 if not (EdataNcol[Edatamas k_t_]==t_[:,kindcol]).a ll(): r aise Ex ception("This shortcut cannot be applied.") if Ngrandmother is None: irows=np.args ort(t_[:,sortcol])#[i[s ortcol] for i in rowc]) else: cTermvalues=E dataTermvaluecol[Edatam ask_t_]#[Edata[list(Eda taNcol).index(Nchild),4 ] for Nchild in t_[:,kindcol ]] gmTermvalue=E dataTermvaluecol[EdataN col==Ngrandmother][0]#E data[list(EdataNcol).in dex(Ngrandmother),4] mTermvalue=Ed ataTermvaluecol[EdataNc ol==Nmother][0]#Edata[l ist(EdataNcol).index(Nm other),4] dTermvalues=c Termvalues-2*mTermvalue +gmTermvalue#[i-2*mTerm value+gmTermvalue for i in c Termvalues] ngrandchildre n=(abs(dTermvalues)<max dTermval).sum() irows=np.args ort(abs(dTermvalues))[: ngrandchildren] #row=[Ngrandm other,Ngrandmother,0]+l ist(Edata[list(EdataNco l).index(int(Ngrandmoth er))][1:])+[fns[Efilen] ] #file.write(( coldelimiter.join([’{:’ +k+’}’ for k in fmt])).forma t(*(row))+linedelimiter ) #row=[Nmothe r,Nmother,0]+list(Edata [list(EdataNcol).index( int(Nmother))][1:])+[fn s[Efilen]] #file.write( (coldelimiter.join([’{: ’+k+’}’ for k in fmt])).form at(*(row))+linedelimite r) if Ngrandmother is None: for irow in irows: Nchild,Nmothe r,A,n=t_[irow] row=[Nchild,N mother,A]+list(Edata[lis t(EdataNcol).index(int( Nchild ))][1:])+[fns[int(n)]] #import pdb;pdb.set_ trace() file.write((c oldelimiter.join([’{:’+k +’}’ fo r k in fmt])).format (*(row))+linedelimiter) else: for irow in irows: Nchild,Nmothe r,A,n=t_[irow] Edatarow=np.a range(EdataNcol.size)[Ed ataNcol==Nchild][0] row=[Nchild,N mother,A]+[i for i in Edata[ Edatarow,1:]]+[fns[int( n)],dTermvalues[irow]] #import pdb;pdb.set_ trace() file.write((c oldelimiter.join([’{:’+k +’}’ fo r k in fmt+[".14g"]] )).format(*(row))+lined elimiter) #file.close () dir_=os.pat h.join("C:",os.sep,"Use rs","Merit.Van-Der-Lee" ,"Desktop","Ammonia") fns=["energ y_nh3_0-41.dat","a-0620 0.dat","a-06300.dat","a -06400.dat","a-06500.da t","a-06600.dat","a-067 00.dat"] fnsbin=[i.s plit(".")[0]+".hdf5" for i in fns] fns_=[os.pa th.join(dir_,i) for i in fn s] fnsbin_=[os .path.join(dir_,i) for i in fnsbin]

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CHAPTER 6. APPENDIX binfile=[] for n in range(len(fns)): if not os.path.exists(fns bin_[n]): print "preparing data file "+fns[n]+" for first use" data=np.load txt(fns_[n]) h5file = tables.openFile(f nsbin_[n],’w’) h5file.creat eArray(h5file.root, "data",dat a) h5file.flush () h5file.close () binfile+=[t ables.openFile(fnsbin_[ n],’r’)] #NB: energy file denoted as 1, transition files denoted as 2 Efilen=0#nu mber of the energy file Ncol=0#colu mn numb er of N in the energy file Termvalueco l=4#column number of Termvalue in the ene rgy fil e moedercol=1 ;kindcol=0;#colnrs in transition file Tfilenrs=ra nge(1,len(fns))#list of numbers of the transition files #each mask contains two elements: #-column number in the transition file where the filter is app lied (0 =Nhi=N"=kindcol, 1=Nlo=N’=moed ercol) #-energy file filter defined as a l ist of (col,va l) masks=[(moe dercol , [(5, 0), (6, 0), (7, 0), (8, 0), (11 ,1), (1 2,1), ( 13,1), ]), #(kindcol , [(0 ,12151) ,]), ] ngrandchild ren=10 header=["Nc hild","Nmother","A","J" ,"Tau","Nblock","Termva lue","n1","n2","n3","n4 ","l3", "l4","tau_i nv","J","K","tau_rot"," v1","v2","v3","v4","v5" , "v6","Tau_v ib","TransitionFilename ",] fmt=[".0f", ".0f",".10g",".0f",".0f ",".0f",".14g",".0f",". 0f",".0f",".0f",".0f"," .0f",".0f",".0f",".0f", ".0f",".0f",".0f",".0f" ,".0f",".0f",".0f",".0f ","s"] coldelimite r=" " linedelimit er="\n" fnout=[] for i,j in masks: fnout+=[header[i]]+[ ’_’.join([header[k+2],s tr(l)]) for k, l in j] fnout=’__’. join(fnout).strip("_") fnout+=".da t" fnout_=os.p ath.join(dir_,fnout) #if os.path.exists (fnout_): raise Exception("Results file consistent with these filters already exists. Run aborted." ) Edata=binfi le[Efilen].root.data EdataNcol=E data[:,Ncol] EdataTermva luecol=Edata[:,Termvalu ecol] print "Looking for children" d2=[] print "Starting search in",len(T filenrs),"transition files." for n2 in Tfilenrs: data2=binfi le[n2].root.data m2=np.ones( data2.shape[0],bool) for c2,m1s in masks: m1=True for col,val in m1s: m1&=(Edata[:, col]==val) d1vm=np.arra y(Edata[:,Ncol])[m1]

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CHAPTER 6. APPENDIX d2vu,d2vi=np .unique(np.array(data2[ :,c2])[m2],return_inver se=True) dvu=set(d2vu ).intersection(d1vm) m2u=np.array ([i in dvu for i in d2vu]) m2[m2]=m2u[d 2vi] d2+=[list(d ata2[i])+[n2] for i i n np.ar ange(m2.size)[m2]]#fill d2 with the transfile values (3) + transfile nr (1) print "Found",m2.s um(),"matches in",m2.size,"tra nsitions in fi le ’"+f ns[n2]+"’." t=np.array( d2)#contains child, mother, transfileval, transfil enr tum=np.uniq ue(t[:,moedercol]) print "Results summary: ",len(t )," mat ch(es),",tum.size," unique mother(s). " file2=open( fnout_,’w’) file2.write (coldelimiter.join(head er)+linedelimiter) write_file( file2, t, Eda ta, moe dercol, kindcol, EdataNcol,Efilen, header, fmt, linedelimi ter,coldelimiter, 2) file2.close () print "Looking for grandchildre n" for Nmother in tum: print "Mother N=",int(Nmo ther) fnout_=os.p ath.join(dir_,"Nmother" +str(int(Nmother))+".dat ") print "Writing export file ’"+fnout_.sp lit(’\\’)[-1]+"’" file=open(f nout_,’w’) file.write( coldelimiter.join(heade r)+linedelimiter) tmask_child ren=t[:,moedercol]==Nmo ther for tchild in t[tmask_children]: #tchild contains child, mother, transfileval, transfilenr Nchild=tchil d[kindcol] print "Child N=",int(Nchil d) d3=[] for n2 in Tfilenrs: data2=binfile [n2].root.data mgc=data2[:,m oedercol]==Nchild#grand child mask d3+=[list(dat a2[i])+[n2] for i in np.ara nge(mgc.size)[mgc]] tc=np.array( d3)#contains child, mother, transfileval, transfile nr print "Found ",np.unique(t c[:,kindcol]).size," grandchil d(ren)" write_file(f ile, tc , Edata , moede rcol, k indcol, EdataNcol,Efilen, header, fmt, linedelimite r,coldelimiter, 6, Nmother, ngrandchildren, EdataTermvaluecol) file.close( )

One and Two-Photon Transitions in Ammonia Experimental and Simulated Line Lists in the Search for Drifting Constants