Hyperfine Interact (2012) 209:25–28DOI 10.1007/s10751-011-0526-y

Density shift and broadening of dipole transitionsin antiprotonic helium

Dimitar Bakalov

Published online: 12 January 2012© Springer Science+Business Media B.V. 2012

Abstract We present the numerical values of the density shift and broadening slopesof laser-stimulated dipole transitions in antiprotonic helium atoms at temperature1.5◦K and 5.4◦K, evaluated in the semiclassical approach with a realistic interatomicpotential calculated ab initio. Compared to our 2000 paper, the present results coverthe much wider set of transition lines that have been subject of precision spectroscopysince then.

Keywords Antiprotonic helium · Density shift and broadening · Laser spectroscopy

1 Introduction

At helium gas target pressure of the order of 1 bar the density shift and broadeningof the spectral lines corresponding to dipole transitions in antiprotonic helium atomshave been shown [1] to be comparable with the leading relativistic corrections and,therefore, important to be taken into account in the interpretation of the precisionlaser spectroscopy experiments PS205 and AD3 performed at CERN [2, 3] and inthe extraction of improved accuracy values of the fundamental characteristics ofthe antiproton. At such low target densities the shift of the resonance transitionfrequency δν and its broadening δ� are, to a good approximation, linear functions ofthe number density n: δ� = n.α, δν = n.β. The slopes α and β have been evaluatedin [1] for a few specific transition lines of experimental interest with a modifiedversion of the semi-classical approach of Anderson [4] using the interatomic potentialcalculated ab initio in [1]. Table 1 compares the experimental results at temperatures

The work was partially supported by Grant DO 02-288 of the Bulgarian Scientific Fund.

D. Bakalov (B)Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of sciences,Tsarigradsko ch. 72, Sofia 1784, Bulgariae-mail: dbakalov@inrne.bas.bg

26 D. Bakalov

Table 1 Slopes of thecollisional shift β andof the broadening α

(in 10−21 GHz.cm3).A: experimental values;B: semiclassical results;C: quantum estimatesfrom the elastic scatteringphase shifts

Transition A B C

λ, nm β α β α β α

597 −4.05(7) 0.30(15) −4.05 0.36 −3.94 0.36470 −1.50(10) <1.0 −1.47 0.08 −1.26 0.06372 −0.4(1) −0.22 0.03 −0.09 0.01296 0.4(1) 0.35 0.02 0.44 0.01726 −3.8(2) −3.42 0.25 −4.07 0.38617 3.8(5) −2.07 0.11 −2.53 0.17714 −3.3(8) −3.75 0.30 −4.48 0.45

Table 2 Slopes α and β of the broadening and shift of favored transitions (n, l) → (n − 1, l − 1) inp̄4He, in the form β(α) (in units 10−21 GHz.cm3), evaluated at T = 5.4◦K or T = 1.5◦K

n T, ◦K l + 1 l + 2 l + 3 l + 4 l + 5l

31 5.4 0.46(0.02)1.5 0.52(0.01) 0.34(0.01)

32 5.4 0.35(0.02) 0.07(0.02)1.5 0.40(0.01) 0.15(0.01) −0.26(0.01)

33 5.4 0.18(0.02) −0.22(0.03) −0.84(0.04)1.5 0.26(0.01) −0.11(0.01) −0.68(0.02) −1.51(0.07)

34 5.4 −0.04(0.03) −0.59(0.03) −1.47(0.08) −2.71(0.18)1.5 0.05(0.01) −0.46(0.02) −1.25(0.05) −2.38(0.16) −3.95(0.40)

35 5.4 −0.37(0.03) −1.16(0.06) −2.35(0.15) −4.05(0.36) −6.50(0.87)1.5 −0.24(0.01) −0.97(0.04) −2.05(0.12) −3.55(0.33) −5.66(0.82)

36 5.4 −0.82(0.05) −1.94(0.12) −3.57(0.30) −5.94(0.74)1.5 −0.66(0.03) −1.68(0.09) −3.15(0.27)

37 5.4 −1.50(0.09) −3.07(0.23) −5.35(0.61)38 5.4 −2.50(0.17) −4.70(0.49)39 5.4 −3.96(0.37)

about 5.4◦K (column A) with the theoretical values, calculated in the semi-classicalapproach of [1] (column B) and from the quantum estimates of the elastic scatteringphase shifts of [5] (column C).

The theoretical values were found to be in agreement with the experimentalresults except for the 617 nm line; the calculations of the collisional quenching ratefor the initial state (36, 34) also failed. The possible explanation of this have beenbriefly discussed in [6].

Much progress has been achieved in the spectroscopy of antiprotonic helium sincethat time. Many more transitions lines have been measured, including two-photon [7]and magnetic hyperfine ones [8], and transitions in p̄3He. The experimental accuracyhas been significantly improved by, among other, performing the measurementsat lower helium gas target densities and temperatures, thus reducing some of thesystematic uncertainties. Though at gas pressures of the order of a few mbar thedensity effects are suppressed, they still give a visible contribution as comparedto the drastically reduced experimental uncertainties. This has urged the repeatedcalculation of the slopes α and β for a wider set of E1-transitions in both p̄3He andp̄4He and for different temperatures. In the present paper we give their numericalvalues, which could be used in the extrapolation of the experimental resonance

Density shift and broadening of p̄He 27

Table 3 Slopes α and β of the broadening and shift of unfavored transitions (n, l) → (n + 1, l − 1)

in p̄4He, in the form β(α) (in units 10−21 GHz.cm3), evaluated at T = 5.4◦K or T = 1.5◦K

n T, ◦K l + 1 l + 2 l + 3 l + 4l

31 5.4 −0.82(0.04)1.5 −0.87(0.03)

32 5.4 0.47(0.03) 0.31(0.03)1.5 −0.52(0.02) 0.19(0.01)

33 5.4 −0.05(0.03) −1.06(0.06) 2.39(0.14)1.5 −0.05(0.01) 0.88(0.03) 2.09(0.12)

34 5.4 −0.76(0.05) −2.07(0.11) −3.75(0.30) 5.87 (0.68)1.5 0.60(0.02) 1.80(0.09) 3.32(0.28) 5.19(0.68)

35 5.4 −1.74(0.09) −3.42(0.25) −5.54(0.61) −8.20(1.32)36 5.4 −3.06(0.21) −5.18(0.54) −7.83(1.20)37 5.4 −4.82(0.47) −7.50(1.10)38 5.4 −7.13(1.00)

Table 4 Slopes α and β of the broadening and shift of favored transitions (n, l) → (n − 1, l − 1) inp̄3He (in units 10−21 GHz.cm3, in the form β(α))

n T, ◦K l + 1 l + 2 l + 3 l + 4 l + 5l

31 5.4 0.38(0.02) 0.07(0.02)1.5 0.43(0.01) 0.16(0.01)

32 5.4 0.20(0.02) −0.23(0.02) −0.91(0.04)1.5 0.27(0.01) −0.11(0.01) −0.74(0.02)

33 5.4 −0.05(0.02) −0.66(0.04) −1.60(0.08) −2.97(0.19)1.5 0.06(0.01) −0.51(0.02) −1.37(0.06) −2.61(0.16)

34 5.4 −0.39(0.03) −1.27(0.06) −2.57(0.15) −4.45(0.38) −7.21(0.93)1.5 −0.26(0.01) −1.07(0.04) −2.25(0.13) −3.92(0.35) −6.24(0.87)

Table 5 Slopes α and β of the broadening and shift of unfavoured transitions (n, l) → (n + 1, l − 1)

in p̄3He (in units 10−21 GHz.cm3, in the form β(α))

n T, ◦K l + 1 l + 2 l + 3 l + 4l

31 5.4 0.48(0.03)1.5 −0.53(0.02)

32 5.4 −0.09(0.03) −1.18(0.05)1.5 −0.02(0.01) −0.98(0.03)

33 5.4 −0.86(0.04) −2.28(0.12) −4.10(0.31)1.5 0.69(0.02) 1.99(0.09) 3.64(0.29)

34 5.4 −1.93(0.09) −3.76(0.26) −6.07(0.64) −8.98(1.32)1.5 1.66(0.07) 3.32(0.25) 5.38(0.64) 7.91(1.39)

28 D. Bakalov

Table 6 Temperature dependence of the slopes β, α of the shift and broadening of the (39, 35) →(38, 34) transition line (in units 10−21 GHz.cm3)

T, ◦K 1.5 2.0 2.5 3.5 4.5 5.4 6.0 7.0 8.0 10.0 12.0 15.0 20.0

β −3.55 −3.68 −3.78 −3.91 −3.99 −4.05 −4.08 −4.15 −4.21 −4.33 −4.44 −4.58 −4.83α 0.33 0.34 0.35 0.36 0.36 0.36 0.37 0.37 0.37 0.38 0.39 0.41 0.44

frequencies to zero target density. The density effects in two-photon and magneticM1-transitions, which require a substantial extension of the semi-classical approachused here, will be considered in future papers.

2 Numerical results

The values of the slopes of the density shift, β, and density broadening, α, evaluatedfor a set of E1 transitions in p̄4He and p̄3He for temperatures T = 5.4◦K and 1.5◦Kusing the method presented in [1], are given in Tables 2, 3, 4 and 5. We distinguishthe cases of favored transitions with �n = �l = ±1 and unfavored transitions with�n = −�l = ±1 because of the different dependence on the principal and orbitalmomentum quantum numbers n, l of the initial metastable state of p̄He.

The temperature dependence of the slopes α and β has been studied in somemore details for the (39, 35) → (38, 34) line (see Table 6). The dependence is rathermonotonous down up to 1.4◦K: both the shift and the width slopes decrease smoothlyas the temperature is lowered. This dependence holds, of course, for very lowdensities only where critical phenomena do not take place.

Acknowledgements The author is grateful to the EXA2011 organizers for their support, and to hisco-authors in Ref. [1] and to Dr. B. Obreshkov for their contribution at earlier stages of the work.

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