Fine Structure of the Metastable a 3Σu State of the Helium Molecule

3 + Fine Structure of the metastable a Σu state of the helium molecule

Rui Su and Charles Markus 12/4/2014

Abstract The original article written by Lichten, McCusker, and Vierima reported the 3 + measurement of the fine structure of the v=0 vibrational state of the a Σu metastable state of He2 by a molecular beams-magnetic resonance (MBMR) machine from the afterglow of a pulsed helium plasma discharge. By fitting the measured zero field intervals in the N=1 and N=3 rotational levels, the author deduced the spin-spin interaction constant λ and spin-rotation interaction constant γ of these corresponding rotational levels. In this paper we follow the steps to get the same results, skipping the process of doing the real experiment.

Introduction It had been more than half a century since the discovery of the helium band spectrum by Curtis and Goldstein1 when the article was written. The existence of stably bound, excited states 3 + of He2 had been confirmed along with the existence of a metastable a Σu state, which raised the possibility of measuring the fine structure in helium molecule. However a definite claim of such a measurement was absent until the authors of this article made a breakthrough by means of a molecular beam-magnetic resonance (MBMR) technique. In this paper we first briefly talk about the structure of helium molecule and the experimental setup, then follow the steps of making theoretical estimates and reproducing the interaction constants from experimental measurement.

The Structure of helium molecule 1 + The ground state of helium molecule He2, X Σg , consists of four electrons filling two lowest atomic orbitals 1sA and 1sB, forming a closed shell. Under linear combination of atomic orbitals (LCAO) approximation, four electrons occupy the two lowest molecular orbitals

(MO) σg1s and σu1s, which are 1s +1s 1s -1s s 1s = A B ; s 1s = A B (1) g 2 u 2 where g and u means ‘gerade’ or ‘ungerade’, representing the parity, the symmetry properties of the wavefunction under inversion about the enter. 3 + The excited a Σu has the electron configuration as 2 (s g1s) (s u1s)(s g 2s), where the s g 2sMO was constructed from two 2s atomic orbitals in a similar way Figure 1. Excited energy levels of the 2 He2 molecule to construct s g1s. The two unpaired spins of the s u1s and s g 2s form a triplet state with spin angular 1 + 3 + momentum S=1. Considering S=0 for the X Σg ground state, selection rule DS = 0makes a Σu a metastable state. Literature reports for an isolated molecule the life time is at least 100 msec3,4. 3 + Figure 1 shows some of the excited energy levels of the He2 molecule including the a Σu state 3 + (A cascade mechanism which populates the metastable a Σu state is also shown).

The experimental method In order to understand the measuring process in the article it is better to briefly explain how the experiment was set up, which is shown in figure 2. Helium gas flows through the 3mm diameter hollow cathode made by drilling in a 1cm-thick pure aluminum. The pressure in the discharge tube is typically 10 torr. Region I, II, III, as designated in figure 1, are pumped to the pressure of 0.13 torr, 5×10-5 torr, and 8×10-6 torr. A 1750 V high voltage is used for discharge, with pulse length less than 1 μsec, repetition rate 3-10 kHz.

Figure 2. Schematic of the MBMR instrument used for this work2

In the afterglow of the discharge, the molecules are first regulated by a slit 1cm away from the nozzle, and then pass through the A magnet and is refocused by the B magnet onto the stop wire located in front of the detector. When a radio-frequency magnetic field of proper frequency and intensity is applied to magnet C, the molecules undergo a transition to a state with a different high field magnetic moment, which will not be refocused by B, and thus will miss the stop wire and pass into the detector.

Predicting the Fine Structure Spacing The goal of this experiment was to determine 3 the fine structure splitting of metastable 푎 Σ He2 by directly observing transitions between the energy states which arises from spin-spin and spin-rotation interaction. For diatomic molecules, there are a number of ways these parameters can couple with each other. To help understand this, Hund determined five extreme cases which allowed for approximation of the coupled terms, referred to as 5 3 cases a,b,c,d, and e. For 푎 Σ He2, case b is the most appropriate. This case applies to situations where there is no orbital angular momentum (Σ states) with nonzero total spin. If the state has orbital angular momentum, case b may still apply if spin-spin coupling is low. The interaction in this case arises Figure 3. Vector diagram of Hund’s from the nuclear rotation (R) and the total spin (S) of coupling case b5 the electrons. Electrons are coupled to the internuclear axis through the internal magnetic field produced by orbital angular momentum, which in this case is 0. In this case, the total angular momentum J is the sum of S and the total rotational angular momentum (N) which is defined as the total angular momentum excluding electron spin, and for this coupling scheme, the basis functions are of the form |N, S, J, mJ⟩. However, if the fine structure constants are unknown it can be difficult to find transitions between different |J, mJ⟩ states. The authors solved this issue by using the coupling of adjacent states due to a perturbation to determine the fine structure spacing. When a strong magnetic field H is applied to a molecule or an atom, previously degenerate mJ states will split into different levels, which is called the Zeeman effect. The perturbation occurs by the magnetic field interacting with the molecule’s magnetic dipole moment μ⃗ , with H(1) = -μ⃗ ∙ H.6 This dipole moment can be related to a state’s total angular momentum by the following equation.

μ⃗ = −μBgJJ (2)

g (J(J+1)+S(S+1)−N(N+1) g (J(J+1)+N(N+1)−S(S+1) = −μ ( s + s )J (3)2 B 2J(J+1) 2J(J+1)

If we consider the direction of the H field to be the z axis, the component of 휇 along the magnetic field is −μBgJMJ. Therefore, the first order perturbation acts on a particular state as follows.

(1) H |N, mN⟩|S, mS⟩= −μBgJ(m푁 + mS) H |N, mN⟩|S, mS⟩(4)

It should be noted that the perturbation is determined from the uncoupled state, since gJ is dependent on N and S. However, the unperturbed Hamiltonian H(0) is determined in the coupled basis of |N, S, J, MJ⟩. Since wave functions of the same J value have the same energy without the present of a magnetic field, perturbation theory for degenerate states is required. To calculate the energy splitting of the fine structure, a matrix of the following elements must be diagonalized where ψi, ψj are the coupled wave functions.

Wij = ⟨ψi|H|ψj⟩ (5)

To do this, the coupled states should be written in terms of the uncoupled states using coefficients from the Clebsch-Gordan. If for example, N=S=1, then the 1x1 table could be used. As can be noted from the Clebsch-Gordon table, if two coupled wave functions do not share the same mJ, then they will not share an uncoupled wave function and the term will go to zero. Therefore the only states sharing the same mJ will remain. Values along the diagonal will be of the following form. ⟨N, S, J, mJ|H|N, S, J, mJ⟩ = E0(N, S, J) + mJgjμB H (6)

Where E0 is the energy of the unperturbed state. Licthen et al used the following general expression to calculate matrix values for states which shared an mJ.

1 (퐽−푁+푆)(퐽+푁−푆)(퐽+푁+푆+1)(푁+푆+1−퐽) ⟨N, S, J, m |H|N, S, J − 1, m ⟩ = [ (퐽2 − 푚2)]2 (푔 − 푔 )휇 H (7) J J 4퐽2(2퐽+1)(2퐽−1) 퐽 푠 푁 퐵

We were able to numerically calculate this splitting for N=1 with the assumption that gs>>gN and g (J(J+1)+S(S+1)−N(N+1) therefore 푔 ≅ s . With this, we constructed the following matrix. To 퐽 2J(J+1) simplify the format, we set 휇퐵푔푠 = 1, since they are simply constants.

The eigenvalues of this function are functions of the magnetic field H and are plotted in figure 4. This demonstrates that in a weak magnetic field the energy levels split and change linearly as E0(N, S, J) + mJgjμB H. However, as the magnetic field increases the coupling terms begin to dominate and the energies greatly deviate. This deviation will be dependent on the energy spacing between fine structure levels as well as the strength of the magnetic field. The authors used this dependence on the spacing between fine structure levels to predict the frequency of the fine structure transitions. This dependence is shown in figure 6. The authors also diagonalized the N=3 perturbed Hamiltonian.

Figure 5. Two energy levels of J=2

as a function of spacing E0(J=2)-

E0(J=1) at constant H

Figure 4: The energies for N=1 as a function of magnetic field from the diagonalized Hamiltonian Figure 6: The transition energies between Zeeman levels in J=2 as a function of magnetic field

To determine the fine structure energy spacing from transitions between Zeeman levels, the following approximation was used to estimate the second order correction to the fine structure energy levels.

2 2 ⟨푁,푆,퐽,푀퐽|퐻|푁,푆,퐽−1,푚퐽⟩ ⟨푁,푆,퐽,푀퐽|퐻|푁,푆,퐽+1,푚퐽⟩ 2 퐸(푁, 푆, 퐽, 푀푗) = 퐸0(푁, 푆, 퐽) + 푚퐽푔퐽휇퐵 H + + (8) 퐸0(푁,푆,퐽)−퐸0(푁,푆,퐽−1) 퐸0(푁,푆,퐽)−퐸0(푁,푆,퐽+1)

The energy spacing between fine structure levels can be determined by measuring transitions between Zeeman levels sharing the same J value. The frequency of the transitions can then be used with the previous equations to calculate E0(N, S, J) − E0(N, S, J − 1). Using the previous assumption that gs>>gN, the g factor for the transitions is approximately gs/2, and therefore the transitions will be approximately half the frequency of atomic Figure 7: Spectrum of transitions |2, −1⟩ → |2, −2⟩ helium. This allowed the authors to find and |1,1⟩ → |1,0⟩ in N=1.2 three transitions for both N=1 and N=3. Figure 7 shows an example scan over two of these transitions. Without the coupling terms, these transitions would have had the same frequency. However, due to the higher order terms they diverged, which can be explained by looking at figure 6. Initially, the transition energies are nearly the same, but at higher magnetic field they quickly become different. Looking at equation 8, not only does the divergence give information on the relative energies it also allows for determining the absolute order of states from highest to lowest. The sign of the contributions of J-1 and J+1 depend on which energy level is higher. Using the measurements of six different transitions, the authors estimated the fine structure energy levels to an uncertainty of 30 MHz.

Table I. Predicted and experimentally determined transition frequencies between fine structure levels N Transition Predicted Experimental Frequency/MHz Frequency/MHz 1 J=0 to J=1 2240 2199.968 J=1 to J=2 890 873.668 3 J=2 to J=3 1350 1323.911 J=3 to J=4 984 964.992

These estimations allowed for directly measuring transitions between the fine structure levels. Two transitions between each allowable fine structure transition were used to determine the unperturbed energies. These transitions were still Zeeman split, and the magnetic field was kept low to reduce the higher order terms in the Hamiltonian. These values could then be used with confidence to determine fine structure coefficients.

Deducing Constants from experimental data For a certain vibrational level of a 3 åstate, the energy levels caused by rotation, spin- spin interaction and spin-rotation can be expressed by a sum of the following terms according to Kramer’s theory7, 2 H = BN(N +1)+ l[3S2 - S2 ]+g (S×N) (9) 3 z where B is the rotational constant, S is the total spin angular momentum, N is the rotational angular momentum, λ is the spin-spin interaction constant, and γ is the spin-rotation constant. 2 The fine structure part of H, l[3S2 - S2 ]+g (S×N), can be expressed in terms of N,8 3 z 2l(N +1) F (N) = F(J = N +1) = - +g (N +1) (10) 1 2N + 3

F2 (N) = F(J = N) = 0 (11) 2lN F (N) = F(J = N -1) = - -g N (12) 3 2N -1 which gives the interval between three split states for a given rotational angular momentum N of 3Σ states. In addition, another set of term values given by Schlapp were also used in the article,8

2 2 2 F1(N) = F(J = N +1) = (2N + 3)B - l - (2N + 3) B + l - 2lB +g (N +1) (13)

F2 (N) = F(J = N) = 0 (14) 2 2 2 F3(N) = F(J = N -1) = -(2N -1)B - l + (2N -1) B + l - 2lB -g N (15)

Next, we plug the measured intervals from table I into these two sets of formulas to get λ and γ, and the results are shown in table II. In fact, two things need be to be addressed before getting the results of table II. The first is the relative order of every three J levels, which is not indicated by the measured intervals shown in table I but will influence the sign and value of the λ and γ we will get. This is solved in the previous part by estimating the fine-structure intervals along with the relative orders from the detection of Zeeman transition. The second issue is that from the intervals of two different rotational levels we can get a slightly different set of λ and γ for each rotational level. They are expressed in the form of

l(N) = l0 + l1N(N +1) (16)

g (N) = g 0 +g 1N(N +1) (17)

as shown in table II. Apparently, λ0 and γ0 are the fine-structure constants when no rotation is present (N=0 in the table), and λ1 and γ1 are related to the rotation when N is present. In fact, λ1 and γ1 are called centrifugal stretching coefficients.

Table II: Calculated constants for the rotational fine structure2 Rotational level Centrifugal N=1 N=3 N=0 stretching Constants(MHz) (extrapolated) coefficient Spin-spin λ From Kramers’ -1098.668 -1096.958 -1099.010 +0.1710 formula From Schlapp’s -1098.773 -1096.803 -1099.167 +0.1970 formula Spin-rotation γ From Kramers’ -2.633 -2.520 -2.653 +0.0113 formula From Schlapp’s -2.421 -2.414 -2.422 +0.0007 formula

According to previous calculation, the measured spin-spin interaction constant λ has a very small magnitude, ~1100 MHz, compared with that of other molecules, for example the ground state of O2 has a 2 3 + value of ~59550 MHz. This can be explained by the high symmetry of the a Σu molecular structure, mainly due to the probability distribution of the unpaired s u1s and s g 2s electrons. An Interested reader can find the detailed explanation in the original article.

Conclusion 3 + The article we chose measured the fine structure of the metastable a Σu state of the helium molecule in two different rotational levels (N=1 and N=3), as a fruit of theoretical analysis such as Hamiltonian matrix diagonalization and the application of perturbation theory. We followed the steps and examined the influence of several factors on the fine structure of rotational level N=1. Spin-spin interaction and spin-rotation constants are deduced as a result of the measurement and the ultra small spin-spin interaction constant can be explained by sketching the highly symmetric electron distribution of the electronic state of this molecule.

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