MAGNETIC RESONANCE IN CHEMISTRYMagn. Reson. Chem.2000;38: S16–S19

Theoretical study on the characterization of the muoniumradical in sulfur using 33S hyperfine interactions

Brian Webster∗

Chemistry Department, University of Glasgow, Glasgow G12 8QQ, UK

Received 8 January 2000; accepted 20 January 2000

ABSTRACT: Configuration interaction molecular orbital calculations (CISD) are used to calculate the33S(I D 3/2)hyperfine interaction in the muonated sulfanyl radical SMu. After allowance for minor zero-point vibrational effects, itis found that an ALC spectrum for a33S enriched sulfur sample should show a resonance at an applied longitudinal fieldof around 1 T if an SMu radical forms. Alternately, for a muonium bridged S—S bond, a UHF calculation suggeststhat resonances should occur at applied fields of 0.32 and 0.50 T for33S nuclei at the bridging sites. Copyright2000 John Wiley & Sons, Ltd.

KEYWORDS: muonated sulfanyl radical; bridged muon sites in S8

INTRODUCTION

Positive muons,�C, are leptons, created naturally in theupper atmosphere on the decay of positive pions,�C.They are constituents of cosmic radiation. This processis replicated at muon factories where high-energy protonbeams impinge on a target, such as beryllium, to yielda pion beam. This beam is channelled into a supercon-ducting solenoid, of length about 8 m, where the muonsare born accompanied by a neutrino:�C ! �C C ��.As the neutrino is chiral, being left-handed, this prop-erty imprints on each new muon, which is spin polar-ized with a helicity of�1, a velocity of 0.27c, and amass of around 0.113 u. In the rest frame of the pion,muons are emitted in forward and backward directions.Usually the backward decay muons are extracted to pro-duce a muon beam which is about 80% spin polarized ina direction longitudinal to the beam path. In the tech-nique known as�SR, application of a magnetic fieldin a direction perpendicular to the beam path charac-terizes muon spin rotation experiments, a longitudinalmagnetic field pertains in muon spin relaxation obser-vations and the major technique of ALC, avoided level-crossing resonance. These techniques are fully describedelsewhere.1 Having a zero magnetic moment, the pionitself cannot serve as the basis of similar experimentalmethods.

Positive muons have a lifetime of 2.2µs, and whenstopped in a sample can form muonium atoms, Mu, beforedecay. Muonium acts like a light radioactive isotope ofH and can react to produce muonium-substituted radicalsor lodge in diamagnetic environments. On muonium for-mation, 50% of the spin polarization is lost although itis restored under an applied magnetic field. With radical

* Correspondence to: B. Webster, Chemistry Department, University ofGlasgow, Glasgow G12 8QQ, UK; e-mail: brian@chem.gla.ac.uk

formation more than 50% of the initial polarization van-ishes. Longitudinal field repolarization measurements onmuons in rhombic sulfur indicate that at low temperaturesboth interstitial muonium and muonium radical formationoccur, with about 10% of the muons rendered unreactivein diamagnetic sites.2 At low temperatures below 100 Kan ALC spectrum shows a weak signal attributed to a rad-ical with a hyperfine coupling constant of 233š 5 MHz,the sign of the coupling constant being undetermined.3

Interestingly, this radical is not observed using transverse-field�SR at temperatures between 20 and 320 K.4,5 Above200 K only two diamagnetic entities are observed, oneseemingly formed from a paramagnetic precursor possiblymuonium.

In a theoretical study of muonium behaviour in sulfur,it was found that at certain positions of Mu with respect toan S8 ring optimization of the geometry led to ring openingwith the production of an SMu radical.3,6 Figure 1 showsthe electron density change as this SMu radical emergesfrom the sulfur ring. The view is from above the ring, andthe Mu located between next-nearest neighbour S atomsextracts the S below.

The calculated value for the muonium isotropic hyper-fine coupling constant of�234 MHz matches well, per-haps fortuitously, with the ALC observation. However,the ready formation of sulfanyl radicals in the vapourphase at room temperature by a surface reaction of H withsulfur indicates that cleavage of an S8 ring can occur.7

Other bridging locations examined for muonium lead tothe generation of an open-chain Mu—S7—Sž with theodd electron localized on an S atom at the opposite endof the chain to Mu. This present short study concernsthe 33S hyperfine interaction in33SMu, since experimentsinvolving 33S-enriched sulfur could prove useful in elimi-nating SMu from further consideration, and looks to othersites in S8 for plausible explanations of the behaviour ofmuons in sulfur.

Copyright 2000 John Wiley & Sons, Ltd. Magn. Reson. Chem.2000;38: S16–S19

CHARACTERIZATION OF THE MUONIUM RADICAL IN SULFUR S17

Figure 1. Calculated electron density map showing thefragmentation of an S—S bond with the formation of anSH or SMu 2 radical.

RESULTS AND DISCUSSION

A sulfanyl radical is a degenerate open-shell species.Aligned along thez-axis an odd electron occupies a 2�x or2�y orbital with a configuration 1�22�23�21�44�25�22�2

x

2�1y for SH 2. As noted elsewhere, one can preserve

the degeneracy of the� levels by the use of equivalence-restricted open-shell SCF theory.6 As the concern hereis with the hyperfine interaction in33S X, 2, X D D,H and Mu, and not details of energy, this configura-tion is taken as the reference for all single and doubleexcitations (CISD) in a configuration interaction calcu-lation performed using the program MELD.8 Taking a6–31GŁŁ basis, the calculated equilibrium distanceRe is133.7 pm for SH2 compared with an experimental valueof 134.06630 pm.7

The total energy and isotropic hyperfine coupling con-stant,Aiso, were calculated over a range of values for theinternuclear distance. After fitting the energies to a Morsepotential function, the vibrational energies were evaluatedand the Morse wavefunction was used to calculate vibra-tionally averaged values for the quantities of interest withthe molecule in the lowest vibrational level.

Table 1. Morse parameters for isotopologues of 33SH 2from a CISD calculation, equilibrium internuclear distanceRe, dissociation energy wavenumber De, fundamentalvibrational wavenumber ωe, anharmonicity wavenumberωexe and the zero-point vibrational energy Ezp

33SX Re (pm) De (cm�1) ωe (cm�1) ωexe (cm�1) Ezp (aJ)

Mu 133.5 34345 8279 499 0.080H 2784 56 0.027D 1974 28 0.019

Figure 2 shows the potential energy curve pertinentto SMu with the minimum of the curve at�0.867 aJand eight vibrational levels below the dissociation limit.Table 1 contains the Morse parameters for the isotopo-logues of SH and values for the zero-point energyEzp,defined by 1

2ωe� 1

4ωexe. For 33SH the value ofEzp is

0.027 aJ, with 25 levels in the potential curve. As shownin Fig. 2, the first vibrational level for33SMu lies higherin the potential curve with a value forEzp of 0.08 aJ. Forsuch reasons muonium radicals are often found to havea muonium bond that executes large-amplitude vibra-tions, simply due to the light mass of the muon. Thisis clear here in the inset of Fig. 2, which shows the prob-ability distribution functionD(R) for the coordinateR.This function represents the probability of observing theSMu internuclear distance in the range dR at the pointR. In the zero level SMu vibrates significantly furtherfrom its equilibrium value for R than does SH. In thepresent case the expectation valuehR� Rei is 1.47 pm for33SMu compared with 0.55 pm for33SH and 0.39 for33SD.Such behaviour often leads to large isotope effects forMu in comparison with H or D. The pertinent questionshere are the following: does this enhanced vibrationalbehaviour have any significant effect on the33S hyper-fine parameters for33SMu as compared with their valuesfor 33SH?; what are the values of these parameters?; andwhat is the value of the applied field required for detec-tion of am D 0 transition from33SMu, if it is formed,

Figure 2. Morse potential energy curve from a CISD calculation for an SMu 2 radical. The inset shows the probabilitydistribution D(R) for the lowest vibrational level.

Copyright 2000JohnWiley & Sons,Ltd. Magn.Reson.Chem.2000;38: S16–S19

S18 B. WEBSTER

in an ALC experiment on muons in an enriched sulfursample?

Hyperfine parameters

Table 2 lists values for the33S isotropic hyperfine couplingconstant of33SX, X D Mu, H, D, averaged over thezero-point vibration. Looking first at33SH, the calculatedvalue for Aiso of 27.5 MHz compares reasonably withan experimental result of 32.6 MHz obtained asbC c/3from the magnetic parameters reported by Ashworth andBrown.7

Figure 3 shows the variation of the isotropic33Shyperfine coupling constant with internuclear distance for33SMu. The vibrational effect of the muon is slight, asone anticipates for this radical. WithAiso D 343.47�.33S/,where�.33S/ is the spin density in atomic units at the sul-fur nucleus, the calculated value forAiso is 27.2 MHz in33SMu. Table 2 lists the values calculated forAiso for theisotopologues of33SH.

For an ALC observation,B0, the value of the field atresonance, is specified by the equation

B0 D �∣∣A� � AS

∣∣. � � S/

� �∣∣A� C AS

∣∣2 e

.1/

where �, S and e are the gyromagnetic ratios for themuon, sulfur nucleus and electron, respectively. Taking avalue of�233 MHz forA� andC32.29 MHz forAS, thisbeing the experimental value of the isotropic hyperfinecoupling constant of33S in 33SH with a correction of

Table 2. Vibrationally averagedvalues from a CISD calculationof the isotropic 33S couplingconstant Aiso for the isotopo-logues of 33SH

33SX Aiso (MHz)

Obs.a 32.59H 27.49D 27.54Mu 27.19

a Ref. 7.

Figure 3. Variation with internuclear distance R (pm) ofthe 33S isotropic hyperfine coupling constant Aiso (MHz)for 33SH 2 from a CISD calculation.

Figure 4. Muonium bridging site in S8.

�0.3MHz for muonium substitution,one finds a valuefor B0 of 1.0T. An ALC spectrumfor muons stoppedin an enrichedsample33S8 having a resonancearoundthis applied longitudinal field could well confirm thatthe elusive radical formed in S8 is a muonatedsulfanylradicalSMu.

However,thereis thealternativepossibility thatA� hasa positive sign. The following result is presentedas anillustrative exampleof how this sign changecould occur;it is not in any way definitive. Figure 4 showsMu in asymmetricalbridgesite of an S—S bondof S8.

It happensthat a UHF calculationsusing a 6–31GŁŁ

basisyields a value of C229MHz for A� when Mu isat this site 131pm distant from adjacentS atomsof thepuckeredring. The value for the spin operator OS2 hereis 0.79. The 33S hyperfinecoupling constantscalculatedfor the S atomslabelled1 and 2 in Fig. 4 are 317 and94MHz, respectively.

An ALC spectrumfor theenrichedsampleshouldshowevidenceof resonancesat applied longitudinal fields of0.32and0.50T arisingfrom 33S nucleiat thesepositions.Naturally this result hassomebasisset dependence,butthisshouldnotsignificantlyaltertheconclusion.Themaindifficulty is that with an isolatedspeciesoptimizationofthe geometryusually leadsto ring openingto either anopenchainor the SMu radical.Perhapsin the condensedphase there is more stabilization of bridge bonds. Ifopen-ring radical products are formed, any muons insuch a locality should swiftly depolarize.Sulfur is thedepolarizingmaterialpar excellence, althoughthereasonsfor this effect remain unknown. ALC experimentswitha 33S-enrichedsamplecould illuminate the matter andunmaskthe radicalpresentat low temperatures.

Acknowledgements

I thank Dr R. M. Macrae for some Mathematicav3.0 coding andDr K. L. McCormackfor her assistance.

REFERENCES

1. LeeSL, Kilcoyne SH,Cywinski R. MuonScience:Muonsin Physics,Chemistry,and Materials. Institute of PhysicsPublishing:London,1999.

2. Cox SFJ, Cottrell SP, Hopkins GA, Kay M, Pratt FL. HyperfineInteract. 1997;106: 85.

Copyright 2000JohnWiley & Sons,Ltd. Magn.Reson.Chem.2000;38: S16–S19

CHARACTERIZATION OF THE MUONIUM RADICAL IN SULFUR S19

3. Cox SFJ, Reid ID, McCormack KL, Webster BC.Chem. Phys. Lett.1997;273: 179.

4. Cox SFJ, Reid ID.Appl. Magn. Reson.1997;12: 227.5. Reid ID, Cox SFJ.Physica Bin press.6. Webster B, McCormack KL, Macrae RM.J. Chem. Soc., Faraday

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7. Ashworth SH, Brown JM.J. Mol. Spectrosc.1992;153: 41.8. Davidson ER. QCPE, Program 580. Quantum Chemistry Group:

Indiana University, Bloomington, Indiana, 1991.

Copyright 2000 John Wiley & Sons, Ltd. Magn. Reson. Chem.2000;38: S16–S19