HAL Id: jpa-00247669
https://hal.archives-ouvertes.fr/jpa-00247669
Submitted on 1 Jan 1992
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Measurement of the ratio of the squares of the reduced magnetic dipole matrix elements for the 876 nm and
648 nm transitions in atomic bismuth
M. Macpherson, R. Warrington, D. Stacey, K. Zetie
To cite this version:
M. Macpherson, R. Warrington, D. Stacey, K. Zetie. Measurement of the ratio of the squares of the reduced magnetic dipole matrix elements for the 876 nm and 648 nm transitions in atomic bismuth.
Journal de Physique II, EDP Sciences, 1992, 2 (4), pp.749-755. �10.1051/jp2:1992163�. �jpa-00247669�
Classification Physics Abstracts
32.60 32.70F 35,10
Measurement of the ratio of the squares of the reduced magnetic dipole matrix elements for the 876 nm and 648 nm transitions in
atomic bismuth
M. 3. D. Macpherson, R. B. Warrington, D. N. Stacey and K. P. Zetie
Clarendon Laboratory, Parks Road, Oxford, OXI 3PU, G-B-
(Received 19 December 1991, accepted 8 January1992)
Abstract. Using Faraday spectroscopy, we have nleasured the ratio K of the squares of the reduced magnetic dipole nlatrix elenlents for the 876 nm (~S~j~-~D~j~) and 648 nm
(~S3 ~- ~D5j~) transitions in the 6p~ ground configuration of atomic bismuth. We find K
=
8.75(25), in good agreenlent with theory.
1 Introduction.
We have studied magnetc-optical (Faraday) rotation II, 2] in the vicinity of the magnetic dipole (Ml) transitions 876 nm and 648 nm in atomic bismuth; the work gives a determination of the ratio ~ of the squares of the corresponding Ml transition matrix elements. Faraday
spectroscopy is particularly useful in this case because the 648 nm transition is in a region of
significant molecular absorption, making a measurement of ~ by a comparison of the atomic
absorption spectra of the two lines impossible.
The work forms part of our programme to test electroweak theory at very low energy by studying parity non-conserving (PNC) optical rotation in atoms. The bismuth atom is an
experimentally favourable system for such a study, and measurements have now reached a precision of 2 il in the 876 nm transition [3]. The results can be used within the framework of the Standard Model to determine the mass of the Z° vector boson [4]. The limiting factor in the determination, however, is the accuracy of the atomic calculations which need to be carried
out on the bismuth atom; the quantity required is the ratio R = Im(El/Ml), where El is the PNC electric dipole amplitude and Ml the parity conserving Ml amplitude for the transition.
Results with an uncertainty approaching the 10 ~ level [6] have been reported, and efforts
are being made to improve the accuracy. Such efforts are assisted by the existence of reliable experimental data for other properties of the bismuth atom, particularly those associated with the 876 nm and 648 nm transitions on which PNC work has been carried out.
750 JOURNAL DE PHYSIQUE II N°4
The 6p~ ground configuration of bismuth exhibits intermediate coupling, but is considerably
closer to the jj scheme than to the LS; coupling coefficients based on the experimental prop-
erties of the levels have been deduced [6, 7]. If one uses these coefficients and assumes 100 ~ overlap between the p1j2 and p3j2 wave-functions, values of the Ml reduced matrix elements of the transitions can be simply calculated. The results are in good agreement with recent ab
initio calculations [8] carried out as part of the same programme which gave the theoretical result for R mentioned above [6]. However, there is a need for an experimental test, and this provided the motivation for the present work.
2 Theory.
The Faraday rotation spectrum in both lines is quite complex; there are many hyperfine com- ponents (bismuth has nuclear spin 1 = 9/2) and although the transitions are predominantly
of Ml character, there is an appreciable electric quadrupole (E2) component in each. In [Ii, expressions were derived giving the frequency dependence of the optical rotation produced by a longitudinal magnetic field in the vicinity of an absorption line of mixed Ml and E2 character in a dilute atomic vapour. The transition was assumed to be subject to Doppler and collisional
broadening, and the treatment was valid for the case in which the Zeeman splitting is much less than the separation of the hyperfine components. The radiation field was assumed weak
enough for saturation effects to be negligible. We denote the total angular momentum of a
hyperfine level by F, and use the subscripts I, j to distinguish the lower and upper levels of the transition respectively. The field affects the refractive index of the vapour in two ways:
through changes in the energies of the sublevels and through mixing of the states. The former
gives rise to an optical rotation which for any given hyperfine component is symmetrical about the unperturbed position wo (F;, l§) while the latter leads to an antisymmetric contribution.
The observed rotation at frequency w is a sum of terms, two for each combination of F;, lj;
~~ ~~' ~~ '~~ (21+1)(2J;
+ 1) ~~~~ ~ ~~~~ ~ ~~~~~~ ~~~~~
x / B(z)N(z)dz
x D'(wo (F;, F;) w) (la)
~~ ~~' ~ ~)~ (21+ )(2J;
+ 1) ~~~ ~ ~~~ ~ ~~~~~ ~~~~~
x / B(z)N(z)dz
x D IwoIF;, l§ w) jib
In these expressions, the coefficients Umn, Vmn are complicated combinations of 6j func- tions and the Landd gj-factors of the upper and lower levels. The line-shape functions D (woIF;, 1§ w) and lY (wo(F;, F; w) are obtained by convolution of a Gaussian func- tion representing Doppler broadening with a dispersion profile and its derivative respectively.
The Umn, Vmn and the line-shape functions are given explicitly in [ii. The rotation also de-
pends on the line integral of the magnetic field B(z) multiplied by the atomic number density N(z) over the path of the light beam; we denote this integral by F in the following. The thec- retical profiles used for fitting the experimental data in the present experiment were obtained
by summing the contributions expressed in (la) and (16) over all hyperfine components. The theoretical profile has been verified to high accuracy in experiments reported in [Ii and [2], where the validity of the approximations involved in the theory is also discussed.
The absorption spectrum can be written in terms of the absorption coefficient k(w) :
~~°'~ ')~
(21+
)(2J;
+ 1)
~~~ ~ / ~~~~~~
~ £ ((Ki (l§, ( + X~If2 (F;, l§ A(wo(F;, ( w)) (2)
F,, q
where A(wo(F;, F; w) is the absorption profile of a single component, obtained by convolv-
ing a Lorentzian with a Gaussian, and the Iii (F;, l§ ), K2 (F;, 1§ are numerical coefficients
given explicitly in [ii. We use these expressions to explain our experimental procedure in section 3.
3. Experiment.
3. I APPARATUS AND THE MEASUREMENT OF SMALL ANGLES. The experiments were car-
ried out on
a modified version of the apparatus developed to study PNC optical rotation in bismuth [9, 3]. It is shown in figure I. Two tunable lasers Ll and L2 provided continuous-wave
single mode light, Ll giving about 100 mW at 876 nm, L2 about 60 mW at 648 nm. The
light was matched into single mode polarization preserving fibres Fl and F2. The emerging
beams were superposed on a beam-splitter B, so that both followed the same path through
the polarimeter; as a result, measurements could be made using either light source in rapid
succession simply by blocking off the unwanted beam. The polarimeter consisted of two crossed
polarizers PI and P2 between which were a cylindrical oven O and a Faraday modulator M.
The oven contained two open-ended parallel ceramic tubes; one contained bismuth, and was electrically heated to about 1600 K so that there was bismuth vapour in the central region.
The pressure of the helium buffer gas under these conditions was about 170 mbar.
The oven could be moved sideways (without any movement of optical components, notably oven windows) in order to allow the light to travel though either tube. This allowed rotation spectra to be recorded with and without bismuth in the light path so that background rotations and absorption could be measured. A longitudinal magnetic field could be applied to the bismuth
vapour by passing a current through a coil wound on the outside of the oven; unwanted fields
were excluded by a mu-metal shield outside the whole oven assembly.
Ll Fl B PI M O P2 D
1S76nm
~~
~~~fi )164Snm
L2 F2
Fig. i. Schematic diagram of the apparatus. The key is given in the text.
The Faraday modulator was a solid glass cylinder in a field coil; it was used to cycle the direction of the plane of polarization of the light through angles A, -A and 0. The rotation
752 JOURNAL DE PHYSIQUE II N°4
caused by the bismuth vapour (together with any other contribution, due for example to slight uncrossing of the polarizers) is given in terms of the signals S at the detector D as follows [2]:
~ S(A)~~(-~~ ~S(0) ~~~
As (3) shows, the modulation angle A (~ l mrad) gives the calibration of the polarimeter.
It was determined at 876 nm to a precision of 0.5 ~ in connection with our measurements of PNC optica1rotation [3]. The value of A at 648 nm is different because the Verdet constant of the modulator glass is wavelength dependent. It was found as follows. The analyser P2 was
physically rotated through a small angle (with no bismuth in the light path). This angle was
first measured by operating the polarimeter with 876 nm light. Once the angle was known, the calibration of the polarimeter at 648 nm could be found by operating the polarimeter at this wavelength. This procedure was repeated for a series of angles.
In practice, the signals S were normalized using signals from other detectors, not shown in the diagram, to remove the effects of laser intensity and frequency fluctuations [3]. A single angle measurement took about 10 ms; during this period the oven heating was switched off to avoid effects due to the magnetic field of the heater. Angle measurements were made at 200
points across the frequency range scanned, which was about 10 GHz for both fines. There is
no need to record the entire hyperfine structure pattern of either line, and in fact it would be impracticable to do so in 648 nm because the components are spread over a range of 60 GHz due to the large splitting of the upper level. The absorption spectrum of the vapour was also recorded for 876 nm; it is derived simply from the denominator of (3), which is proportional to the oven transmission, so that the same detector readings gave both rotation and absorption profiles,
3.2 MEASUREMENT OF ~. Rotation spectra of 876 nm and 648 nm were recorded under
as nearly as possible the same oven conditions. It is clear from equations (I) that if the line
integra1F is the same for both, the ratio of the matrix elements can be deduced from the data without knowledge of the distribution of either the magnetic field or the bismuth vapour density. Measurements were made (16 in all) on each line alternately, the rotation spectrum of each line being recorded twice in succession. In each case, the data were fitted with the
theoretical profile given in equation (I) (Sect. 2), giving a determination of the factor [Ml (~F
for that line. The line-shape factors were also determined. We note particular features of the measurements made on the two lines in the following.
In 648 nm, the large upper level hyperfine splitting causes the components to be grouped according to the particular upper hyperfine level involved; in preliminary experiments we in-
vestigated each group. The molecular absorption is significantly larger than that due to the atoms in this spectral region, and although the molecular Faraday rotation is small it is clearly
observable. The groups least affected are the 6 and 7 groups (I.e., ( = 6 and 7); the lat- ter was chosen for quantitative measurements. A typica1scan of10 GHz over this group is
shown in figure 2; it includes the F;
= 6, F; = 7 (mainly Ml) and F; = 5, F; = 7 (pure E2)
components, denoted by (6, 7) and 16, 7) respectively. Despite the goodness of the fit, the residual curve indicates the presence of a small amount of optical rotation due to molecules,
since there are features well away from the atomic components. In fitting these spectra, we determined also the ratio x of the E2 to the Ml reduced matrix element for the transition;
we obtained -0.617(1) from the 7 group, and carried out measurements also on the 6 group whicfi gave -0.612(3), where the errors quoted are entirely statistical. However, further work is required, particularly on the other groups, before the level of systematic error introduced
by the background molecular rotation can be reliably estimated. Earlier measurements, also carried out by Faraday spectroscopy, have given values of -0.60(2) [Ii and -0.636(10) [10].
Fortunately, the value of ~ derived from our data is very insensitive to the assumed value of x for either transition.
300
,
5 o 5
frequency (GHz)
4
~4
5 o 5
Fig. 2. Faraday rotation at 648 nm in bismuth. The (lj
= 6, lj = 7) hyperfine component is at the
zero of the frequency scale, with the (5, 7) pure E2 component 2.9 GHz away towards lower frequency.
The experimental points are shown, together with the best-fit theoretical profile (the continuous curve);
the residual curve (experiment theory) is shown on an enlarged scale. Despite the goodness of the fit, there
are systematic effects above the level of the noise. They are attributed at least partly to molecular Faraday rotation, since there are features in the far wings where there is negligible rotation due to the atoms.
For 876 nm, the 10 GHz scan range always included the (6, 6) and adjacent components
(the complete profile is shown in Fig. 5 of [2]). Molecular Faraday rotation is negligible in this
wavelength region. As mentioned earlier, the absorption spectrum was recorded simultaneously
with the rotation spectrum; it was mainly of use during the running of the experiment itself.
The spectra were displayed in real time so that a continous check could be kept on the maximum
optical depth of the bismuth vapour, which occurs at the peak of the (6, 6) component. To allow convenient operating conditions for both lines, this depth was maintained at around 4, giving a corresponding maximum (atomic) optical depth at 648 nm of about 0.6. The analysis of the absorption spectra was also of interest; it allowed a check of the line-shape parameters
derived from the Faraday fits, and in addition it permitted a separate estimate of the two factors appearing in F. An accurate value of the optical depth at 876 nm was obtained for each spectrum from the fitting procedure, giving a determination of (M1876(~ f N(z)dz. Since F was known from the corresponding Faraday fit, it was possible to estimate the magnetic
field acting on the bismuth, assuming that it was approximately uniform over the hot region
of the oven. The result obtained, 7 x 10~~ T, is consistent with the value calculated on the
basis of the field coil geometry, but a critical comparison is not possible because of the effect of the mu-metal shield.
754 JOURNAL DE PHYSIQUE II N°4
4. ltesults.
Our final result is ~
= 8.76(26). The uncertainty is made up of three contributions, all of
comparable magnitude. First, the measurements made on each line individually showed that the size of the rotation (and hence the value of F) changed typically by a few per cent from
one recording to the next. Although much of this was due to a slow drift in oven conditions which could be allowed for, the effect contributed an uncertainty of1 ~. Secondly, the value of ~ obtained varied depending on whether the line profile analysis was carried out with the Gaussian widths of the two lines allowed to float independently or constrained to remain in the inverse ratio of their wavelengths (the ratio to be expected theoretically). The two methods gave results which differed systematically by ~, much larger than the purely statistical
uncertainty, probably because of slight misanalysis of 648 nm profiles due to the background
molecular rotation. Finally, the calibration procedure was subject to an uncertainty of 2 ~.
Table I shows that the result is in good agreement with the theoretical predictions. The ah initio calculations of [8] treat the residual electrostatic interaction using second-order perturbation theory, with semi-empirical higher order corrections. The comparison with experiment shows
that these corrections are indeed necessary.
Table I. Values of ~, the ratio of the squares of the Ml matrix elements for the 876 nm and 648 nm transitions.
Present work 8.76(26)
Theoretical results based on empirical coupling coefficients.
Using those of [6]: 8.72 Using those of 8.70 Ah initio calculations [8]:
To second order in the residual electrostatic interaction 8.07
With semi-empirical higher-order corrections
5. Conclusion.
Using Faraday spectroscopy, we have measured the ratio of the squares of the reduced magnetic dipole matrix elements in the 876 nm and 648 nm transitions in the 6p~ ground configuration
of atomic bismuth. It is very satisfactory that our result agrees with that derived from ah initio calculations which were primarily carried out to predict the PNC optical rotation in the transitions. It would clearly be j more stringent test of the theory to make absolute
measurements of the transition probabilities of these lines, but it would be difficult to achieve
a worthwhile precision.
Acknowledgements.
We are grateful to Mr. G. Quelch, for expert technical assistance, and to Mr. C Goodwin, who
supplied optical coatings. The work was supported by the Science and Engineering Research Council, which also provided a Studentship for KPZ. M3DM held a Research Fellowship at
Christ Church, Oxford, and RBW holds a Commonwealth Scholarship.
Finally, we are delighted to be given this opportunity to congratulate Professor P. 3acquinot
on his many contributions to atomic and optical physics during the course of a long and
distinguished career, and to wish him well in the future.
References
Ill ROBERTS G. J., BAIRD P. E. G., BRIMICOMBE M. W. S. M., SANDARS P. G. H., SELBY D. R.
and STACEY D. N., J. Phys. B13 (1980) 1389.
[2j TREGIDGO K. M. J., BAIRD P. E. G., MACPHERSON M. J. D., PALMER C. W. P., SANDARS P.
G. H., STACEY D. N, and THOMPSON R. C., J. Phys. B 19 (1986) 1143.
[3] MACPHERSON M. J. D., ZETIE K. P., WARRINGTON R. B., STACEY D. N, and HOARE J. P., Phys. Rev. Lett. 67 (1991) 2784.
[4] SANDARS P. G. H., J. Phys. B 23 (1990) L655.
[5] DZUBA V. A., FLAMBAUM V. V. and SUSHKOV O. P., Phys. Lett. 141A (1989) 147.
[6j LANDMAN D. A. and LURID A., PAYS. Rev. Al (1970) 1330.
[7]DEMBCZYNSKI J., ARCIMOWICZ B., STACHOWSKA E. and RUDNICKA-SZUBA H., Z. PAYS. A 310 (1983) 27.
[8j DZUBA V. A., FLAMBAUM V. V., SILVESTROV P. G. and SUSHKOV O. P., PAYS. Rev. A 44
(1991) 2828.
[9j MACPHERSON M. J. D., STACEY D. N., BAIRD P. E. G., HOARE J. P., SANDARS P. G. H., TREGIDGO K. M. J, and WANG GUOWEN, Europhys. Lett. 4 (1987) 811.
[10] BOGDANOV Yu. V., KANORSKII S. I., SOBEL'MAN I. I., SOROKIN V. N., STRUK I. I. and YUKOV E. A., Opt. Specktrosk. 61 (1986) 281 [English translation: Opt. Spectrosc. 61 (1986) 446].
