Hyperfine structure evolution in an electric field and determination of tensor polarizabilities in He (4 and 5 1D)

Hyperfine ^structure evolution in an electric field and determination of tensor polarizabilities ^{in He} (4 ^{and 5} 1D)

A. Denis, ^Y. Ouerdane, ^{G. Docao and J.} Désesquelles

Laboratoire de Spectrométrie Ionique ^et Moléculaire, ^associé

CNRS,

Université de Lyon I, Campus ^{de la} Doua, 69622 Villeurbanne Cedex, ^France

(Reçu ^{le 5 août} 1986, accept6 ^{le 8 octobre} 1986)

Résumé. 2014 On étudie la variation de la structure hyperfine des niveaux

1D de 3He dans

champ électrique statique. ^Par

méthode de battement quantique

faisceau d’hélium excité par

feuille de carbone, ^{on suit} l’évolution temporelle ^{du taux} d’alignement ^{des niveaux 4 et 51D} pour

série de champs électriques. ^On

^déduit

les valeurs de la polarisabilité tensorielle de

niveaux.

Abstract.

Variations of hyperfine ^{structure in} 3He (n 1D)

calculated

a function of an applied ^electric

field. Using

quantum ^{beat method}

fast helium beam excited and aligned by

thin carbon foil,

observe the time evolution of the alignment ^for

^set of static electric fields. Tensor polarizabilities ^{of levels}

^deduced.

Classification

Physics ^Abstracts

32.60

34.50F

35.10D

1. Introduction.

Beam-foil spectroscopy ^{is now} well known for its efficient measurement of fine and hyperfine ^structures

in many neutral and ionised atoms. Zero field quantum

beat experiments ^on 3He by ^{Brooks et al.} [1] ^{are in fair}

agreement, ^{within 2 % error} bars, ^{with the best avail-}

able theoretical and experimental ^values [1-4] ^{for the}

low lying ^{levels 4 1D and 5 1D.}

Rather less common is the ^{extension of this technique}

to optical polarization measurements of structures when combined with applied electric fields. The study

of electric field induced quantum ^{beats seems to have}

been reduced to hydrogenic species, namely hydrogen [5, 6], charged ^helium [7] ^and doubly ionized lithium

[8]. ^{The case of} non-hydrogenic species ^{has not, as far}

as we know, been studied using this method.

We report ^{here the} observation of intensity ^fluctu-

ation in the emission of Stark-perturbed 2p ’P - ^4d ’D

and 2p ^{’P - 5d ’D} radiations from helium ^atoms formed

by ^the passage ^{of helium} ions through ^a carbon foil.

Comparison ^{of measured} frequencies ^with theoretical calculations using ^{a tensor} operator ^{formalism leads to} the determination of the tensor polarizability ^{of these}

levels.

2. Theory.

To treat the problem ^{of 3He atom in an electric field we}

begin ^{with a} Hamiltonian which includes electrostatic interactions (electron-electron ^and electron-nucleus)

and magnetic interactions (spin-orbit, spin-other ^orbit, spin-spin and nuclear moment-angular moment). ^In 3He the hyperfine coupling energy is ^not negligible compared ^{to the} singlet-triplet splitting ^{and the} mixing

of these two multiplets ^{has to be taken into account in}

the case of the ’D levels. As a consequence, ^a small fraction of the hyperfine ^{structure of the} triplet ^{state is}

transferred to the singlet in which it can be observed.

The field-independent ^{terms of the} matrices have been calculated and are given ^{in tables I and II. These matrix}

elements are similar to those of Brooks et al. [1]

deduced from the work of Lurio et al. [9]. ^{The Fermi}

contact term A (1s, 3He+) ^{and the} singlet-triplet mixing parameter a [10, 11] ^{are taken} equal ^to

-

4 332.8 MHz [12] and 17 527.3 MHz/n3 ^{[1,10] re-}

spectively. ^The magnetic dipole interaction parameter

a is equal ^{to - 0.68 for n}

^{4 and - 0.35 for n}

^{5. D is}

the singlet-triplet separation ; dlg d2 ^and d3 ^{are the fine}

structure splitting ^{of the} triplet ^states.

When an electric field perturbation ^is added, ^we obtain a Stark Hamiltonian [13] ^which splits ^{into two} parts : HStark

Hr, ⁺ Ht. The scalar contribution HS produces ^an overall shift of the levels and the tensor

contribution Ht splits ^the different Mf ^magnetic

sublevels.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802022700

Table I.

Matrix elements of the total system hamiltonian for Mf I

1/2. Lx., ^and ati ^{are related to} singlet ^states

and (IS3 ^and (It3 ^{are related to} triplet ^{states. The matrix has to be} completed by symmetry.

i) ^{For a} singlet :

E

is the electric field strength. as (J) and a t ( J ) ^{are the scalar and tensor} polarizabilities ^{of the J level and are}

defined as follows :

Table II.

Matrix elements of ^{the total} system Hamiltonian for I Me I

3/2, ^{with the same notations as in table 1} for scalar and tensor polarizabilities. ^{The matrix has to be} completed by symmetry.

The sum is over all J’ levels which lie near the J level. wo is ^the energy of ^{the J} level and w’ the energy of ^a close J’. (Jllp, 11 V’) ^{is the electric dipole matrix element.}

ii) ^{For a} triplet,

It can be seen from these expressions ^{that the scalar} polarizability plays ^{a role} only ⁱⁿ diagonal ^{elements and}

that the tensor polarizability contributes to M

M’

transitions because of

3 j coefficient » rules.

The addition of Stark perturbation ^{allows us to}

determine the entire matrices displayed in tables I and II.

Diagonalisation of these matrices gives ^the energy of all the levels together with their evolution as a function of the electric field.

In figure ^{1 we} illustrate the evolution of singlet ^states

alone. Because only ^{the 41} I Mf ^{= 0 terms are coupled,}

just ^two frequencies appear :

and

These two frequencies ^{were the} subject ^{of our} exper- imental work.

Fig. ^1.

Variations of hyperfine magnetic ^sublevel energies

as a

function of squared electric field in 3He (4 ’D).

3. Experiment.

To measure the energy differences ^{between n} ’D sub- levels of 3He and their evolution as a function of an

electric field, ^the _quantum ^beat technique ^{has been}

used on fast-foil excited beam.

The ion source of a 350 keV Danphysik accelerator is filled with a mixture of helium-3 and helium-4, ^the

L’ L ^L Z/J ^{000 - ClJ’}

latter being ^{used for} precise calibration of the atom

velocity. After acceleration up ^to about 100 keV, ^the 3He’ beam is sent through ^{a thin} (10 ug/cm2) ^self-

supporting ^{carbon foil and} through ^{an electric field} parallel ^{to the} helium beam. The field is produced by ^a

set of two circular metallic plates ^{biased at} potential

+ V and 0 respectively. ^{The carbon foil is} placed ^{on the}

first electrode (Fig. 2).

Fig. ^2.

Experimental arrangement. ^The plates ^{used to} apply the electric field

140

in diameter, ^are separated by

distance of 3

and have a circular aperture ⁴

ⁱⁿ diameter d : diaphragm, ^{1: focus} lenses, p : linear polarizer,

if : interference filter,

slit, P.M. : EMI 9635Q Peltier cooled photomultiplier, ^{F.C. :} Faraday cup.

The light ^{emitted at 90° to the} beam passes through

an optical system that includes a lens, ^{a linear} polarizer

and an interference filter. A second lens focusses the transmitted light ^{on a slit} placed in front of the detector

(P.M. ^{EMI 9635} Q). ^The decays ^of 4 1D and 5 1D levels

are observed at 4 922 A and 4 388 A respectively. ^At

the _energy we chose to use, both amplitude ^{and the}

total light ^{are more} intense than at higher energies [14].

Moreover, ^the range of studied frequencies (80-

250 MHz) ^{does not} require ^a long displacement ^{of the} target ^for exploration of several periods of modulation.

On the other hand, ^{the lifetime of the} target ^{is short at}

such a low velocity ^{and we are} obliged ^{to use} very weak beam intensities ( ~ ¹⁰⁰ nA).

The main experiment necessitates the measurement of the light intensity ^at the exit of the polarizer ^turned alternately parallel (III) ^and perpendicular (I..L.) ^to

the helium beam, repeating ^the measurements many times for each foil position. ^{The same} operation ^is repeated ^{for about} twenty ^foil positions separated by ^a regular interval. From a series of recordings, ^the

7,, ^-11

variation of

A’ = I II + 2 ; ..L. ^can be deduced as a

function of the distance to the target (Fig. 3b), ^the

electric fied being ^{fixed at a} given ^{value. We use this} parameter ^{A’ to free ourselves of} all absolute intensity

variations when the distance to the foil changes (expo-

nential damping ^of oscillation amplitude) ^{or as a}

function of time (target ageing).

Fig. ^3.

Modulation of the polarization ratio of the

41D 3He level as

function of distance to the target ^when

627 V/cm field is applied (3b). ^{Quantum beat} pattern ^{for the} 4He line at 3 888 A recorded before (3a) ^{and after} (3c) ^the 3He

(3b). ^{From these curves} 3a and 3c the evolution of the velocity from va

^1.987 to Vc

1.938 mm/ns is deduced and consequently vb =1.962 mm/ns. The

3b is an

example ^of measurement obtained when the applied ^electric

field is such as the frequencies 5/2, 3/21 - 3/2, 13/21 ^{and 5/2,} 1/2 ( - 3/2, 1/2) ^{are too close to} be resolved.

A least square analysis ^{of the sinusoid} gives ^the wavelength of the beat corresponding ^to the difference of energy between ^two sub-levels for a given ^electric

field strength. ^{In the} present ^{case, because four levels} selectively coupled ^two by ^two by ^{the field are involved,}

the recording ^{includes two sinusoids} rather than just

one. However, ^the modulation corresponding ^{to the}

difference in energy between ^the 5/2, 11/21 ^{and 3/2,} 1/2 sub-levels only ^{reaches a} significant amplitude ^for

field strenghts ^{above 1 kV/cm. At lower field} strengths, only ^{the beats due to the} 5/2, 13/21

3/2, 13/21 ^transi-

tion are observed.

In order to deduce these frequencies ^{we need to}

know a precise value of the emergent ^beam velocity.

Therefore, ^{we recorded the} modulation curve of the

4He line at 3 888 A whose frequency ^{is known to be}

658.55 MHz [15], ^without changing ^the accelerator volt- age. The ^curves given ⁱⁿ figures ^{3a and 3c} correspond

to the modulated curve recorded before and after the main experiment (curve 3b). They ^{show a} systematic

decrease in velocity (typically ^{of the order of 3 %} during ^{a whole} sequence) ^{due to} thickening ^{of the} foil, which is also responsible ^{for the} growing ^of intensity ⁱⁿ

curves 3a and 3c (the ^{cross section increases when the}

energy decreases). ^The adopted velocity ^{is the mean}

value of the two extreme velocities. An additional shift of velocity ^{is due to the} fact that the energy ^{loss is not} the same for helium-3 and helium-4 ions submitted to the same accelerating voltage. Using ^{the curves of}

energy ^loss of Ziegler [16], correction of velocity ^{can be}

made : it is of the order of 1 % for a 10 ug/cm2 target thickness and an incident _{energy of} 100 keV

Another interesting result obtained from these re-

cordings ^{is the} target position ^which coincides with the

zero phase of beat. The reproductibility ^{of measure-}

ments allows us to fix the foil position ^{with a} precision

of 50 _tim.

4. Analysis and results.

For each electric field, ^the experimental ^{data are fitted}

to a sinusoid or a sum of sinusoids with the help ^{of a}

least square programme.

Fig. ^4.

Evolution of the two frequencies 5/2, 332 1 - ^3/2, 3/2) ^and 5/2, 1/2 - 3/2, 1/2

function of squared

electric field strength Solid lines are theoretical curves calcu-

lated from matrices of tables I and II with best value of

at.

The error on each frequency is estimated to be of the order of 5 MHz when uncertainties in the beam vel-

ocity, target position and statistics are taken into account. Figure ⁴ illustrates the frequency ^measure-

ments made for electric fields ranging ^{from 0 to}

1.3 kV/cm.

For 4 1D level we compare the experimental ^data

with theoretical curves obtained with different values of the tensor polarizability. This is done for both trans-

itions. The best X2 value is obtained with:

for the

for the

and with:

for both transitions simultaneously.

For the 5 1D level with similar curves the result is :

In table III our present ^{results are} compared ^with

theoretical and other experimental ^{values. Previous}

beam-foil data were obtained in 4He using ^different geometries [17,18]. ^The level-crossing data of Szostak et al. [19] ^are also available for comparison. The pre- cision of the measurements is about 1 %. Agreement ^is

also found with calculations using ^{the Coulomb} approxi-

mation of Bates and Damgaard [20] ^{for the} dipole

matrix elements and the table of Martin [21] ^{for the}

level energies.

Table III.

Tensor polarizabilities (kHz/(V/cm)2) ^{of 4 and 51D levels in helium.}

References

[1] BROOKS, ^R. L., STREIF, ^{V. F. and} BERRY, ^H. G., Nucl. Instrum. Meth. 202 (1982) ^113.

[2] ^BESSIS, N., LEFEBVRE-BRION, ^{H. and} MOSER ; C. M., Phys. ^{Rev. A 135} (1964) ^957.

[3] DESCOUBES, ^J. P., ^Thèse d’Etat, ^Paris (1967) ; DESCOUBES, ^J. P., ⁱⁿ Physics of

^{and two-}

electron atoms (Elsevier : North-Holland, Amsterdam) ^1969.

[4] LIAO, ^P. F., FREEMAN, ^R. R., PANOCK, ^{R. and} HUMPHREY, ^L. M., Opt. Commun. 34 (1980)

195.

[5] ANDRÄ, ^H. J.,, Nucl. Instrum. Meth. 90 (1970) ^343.

[6] ALGUARD, M. J. and DRAKE, ^C. W., ^{Nucl. Instrum.}

Meth. 110 (1973) ^311.

[7] BOURGEY, J., DENIS, ^{A. and} DÉSESQUELLES, J., ^J.

Physique ³⁸ (1977) ^1229.

[8] PINNINGTON, ^E. H., BERRY, ^H. G., DÉSESQUELLES,

J. and SUBTIL, ^J. L., Nucl. Instrum. Meth. 110

(1973) ^315.

[9] LURIO, A., MANDEL, ^{M. and} NOVICK, R., Phys.

Rev. 126 (1962) ^1758.

[10] DEROUARD, J., LOMBARDI, ^{M. and} JOST, R., ^J.

Physique ⁴¹ (1980) ^819.

[11] ^{MILLER, T. A. and FREUND, R.} S., Adv. Mag. ^{Res. 9} (1977) ^50.

[12] SCHUESSLER, H. A., FORTSON, ^{E. N. and} DEHMELT, H. G., Phys. ^{Rev. 187} (1969) ^5.

[13] ANGEL, J. R. P. and SANDARS, ^{P. G.} H., ^{Proc. R.}

Soc. London Ser. A 305 (1968) ^125.

[14] PERCIVAL, ^{I. C. and} SEATON, ^M. J., ^{Philos Trans. R.}

Soc. London A 251 (1958) ^114.

[15] WIEDER, ^{J. and} LAMB, ^{Jr. W.} E., Phys. ^{Rev. 107} (1957) ^125.

[16] ^{ZIEGLER, J.} F., ^Helium stopping powers and ranges in all elements (Pergamon, ^New York) 1977,

Vol. 4.

[17] OUERDANE, Y., DENIS, A. and DÉSESQUELLES, J., J. Physique ^{Lett. 44} (1983) ^L-871.

[18] OUERDANE, Y., DENIS, ^{A. and} DÉSESQUELLES, ^{J. to}

be published ⁱⁿ Phys. ^{Rev. A} (1986).

[19] SZOSTAK, D., ^VON OPPEN, ^{G. and} PERSCHMANN,

W. D., Phys. Lett. 76A (1980) ^376.

[20] BATES D. R. and DAMGAARD, A., Philos Trans. R.

Soc. London A 242 (1949) ^101.

[21] ^{MARTIN, W.} C., ^J. Phys. ^Chem. Ref. ^{Data 2} (1973)

257.