MAGNETIC ROTATION SPECTROSCOPY WITH SYNCHROTRON RADIATION

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MAGNETIC ROTATION SPECTROSCOPY WITH SYNCHROTRON RADIATION

J. Connerade, W. Garton, M. Baig, J. Hormes, T. Stavrakas, B. Alexa

To cite this version:

J. Connerade, W. Garton, M. Baig, J. Hormes, T. Stavrakas, et al.. MAGNETIC ROTATION

SPECTROSCOPY WITH SYNCHROTRON RADIATION. Journal de Physique Colloques, 1982, 43

(C2), pp.C2-317-C2-336. �10.1051/jphyscol:1982225�. �jpa-00221836�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-317

MAGNETIC ROTATION SPECTROSCOPY WITH SYNCHROTRON RADIATION

J.P. Connerade*, W.R.S. Garton*, M.A. Baig, J. Hormes, T.A. Stavrakas* and B. Alexa

*Blaakett Laboratory, Imperial College, London SW7 2AZ, U.K.

Physikalisches Institut, Universitat Bonn, 53 Bonn, F.R.'G.

Résumé : Nous décrivons des expériences récentes où la polarisation linéaire du rayonnement synchrotron dans le plan de l'orbite a été exploitée pour étudier les effets combinés de la biréfringence magnétique circulaire (effet Faraday) et du dichroisme magnétique circulaire (MCD) dans l'ultraviolet. A l'heure actuelle, nos études portent sur les séries principales des alcalino-terreux dans des champs magnétiques allant jusqu'à 4,6 Tesla. Nous avons pu observer des oscillations d'intensité dues à l'effet Faraday jusqu'à n=28 environ. Au-dessus de cette valeur, les structures supplémentaires dues au mélange de 1 en champ intense compliquent l'interprétation du phénomène. A partir de nos mesures, nous avons pu, par une méthode nouvelle, déduire les forces d'oscillateur relatives des transitions atomiques avec une précision qui va jusqu'à + - 0,25%, et qui dépasse donc largement celle des meilleures déterminations antérieures. Notre méthode offre aussi, du moins en principe, l'avantage de permettre l'étude des perturbations de la force d'oscillateur en présence d'un champ magnétique intense. Enfin, elle ne nécessite qu'une optique réfléchissante et peut donc être utilisée dans 1'ultraviolet lointain.

Abstract: We describe recent experiments in which the linear polarisation of synchrotron radiation in the orbital plane has been exploited to study the combined effects of magnetic circular birefringence (Faraday rotation) and magnetic circular dichroism (MCD) in the ultraviolet. To date, our experiments have concentrated on the principal series of alkaline-earths in magnetic fields of up to 4.6 Tesla. We have observed intensity oscillations due to Faraday rotation up to about n=28.

Above this value, additional structures due to 1-mixing in high fields complicate the interpretation of the patterns. From our measurements, we have been able to deduce relative oscillator strengths of atomic transitions by a novel technique.

The precision of these determinations can be as high as + - 0.25% in favourable cases, and is thus considerably higher than in earlier measurements using different techniques. Also, the present method has the advantage that, in principle, one could investigate perturbations of the f-value by intense magnetic fields. Finally, it requires only reflecting optics and can therefore be used in the vacuum ultraviolet.

Introduction. - The present paper is a report on experiments recently performed at the 500 MeV electron synchrotron of the Physikalisches Institut in Bonn, where a laboratory specialised in high resolution vacuum ultraviolet spectroscopy has been set up in collaboration with the Blackett Laboratory, Imperial College. An example of investigations carried out in Bonn using the same source and spectrographic apparatus as the present work is given by Connerade Baig Garton & McGlynn (1980).

In the experiment described below, the high degree of linear polarisation of synchrotron radiation in the orbital plane of the accelerator has been exploited to study the combined effects of magnetic circular birefringence (Faraday rotation) and magnetic circular dichroism from 2300 to about 1600 angstroms.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982225

JOURNAL DE PHYSIQUE

Although dichroism and r o t a t i o n g e n e r a l l y can occur t o g e t h e r , most experiments a r e designed t o i s o l a t e one o r t h e o t h e r o f t h e two e f f e c t s . Thus, t h e p r e s e n t combination i s believed t o be novel and, as w i l l be shown, possesses some s p e c i a l advantages. It d o e s , however, r e q u i r e a new approach f o r t h e i n t e r p r e t a t i o n o f t h e d a t a , and t h i s w i l l be described i n some d e t a i l .

Experimental.

-

The b a s i c experimental arrangement is shown i n Fig.1: r a d i a t i o n from t h e 503 MeV a c c e l e r a t o r is c o l l e c t e d by a c y l i n d r i c a l m i r r o r which focusses only i n t h e h o r i z o n t a l plane, and t h e r e f o r e minimises any admixture o f out-of-plane e l l i p t i c a l l y p o l a r i s e d l i g h t . ' h e l i g h t i s t h u s concentrated on t h e v e r t i c a l entrance s l i t o f a 3-metre vacuum spectrograph designed and constructed a t Imperial College (Learner 1965 )

.

The spectrograph i s equipped with a 5030 l i n e per m holographic g r a t i n g manufactured t o s p e c i a l o r d e r by Jobin-Yvon SA (France). The r e s o l v i n g power o f t h i s combination is believed t o be t h e h i g h e s t c u r r e n t l y a v a i l a b l e on a synchrotron r a d i a t i o n source. We a r e applying t h i s experimental advantage t o t h e study o f a number o f atomic and molecular s p e c t r a which a r e o u t s i d e t h e scope o f t h e p r e s e n t paper (see eg Baig k n n e r a d e and Homes 1982 and r e f e r e n c e s t h e r e i n ) . In t h e work described h e r e , t h e optimum s p e c t r o g r a p h i c r e s o l v i n g power (about 303 000) was n o t achieved, because t h e a p p a r a t u s f u n c t i o n was a f f e c t e d by f a c t o r s independent o f t h e spectrograph (see below). Nevertheless, it was important t o use a high d e n s i t y r u l i n g , because t h e l a r g e tilt (38 degrees a t 2503 angstroms) made t h e o p t i c a l system very s e n s i t i v e t o t h e a n g l e between t h e plane o f p o l a r i s a t i o n o f t h e i n c i d e n t r a d i a t i o n and t h e r u l i n g s on t h e g r a t i n g s u r f a c e . Also, t h e high d i s p e r s i o n enabled i n t e n s i t y o s c i l l a t i o n s much s h a r p e r than t h e a c t u a l l i n e w i d t h t o be observed.

The o p t i c a l system was s e t up w i t h t h e g r a t i n g r u l i n g s v e r t i c a l . Thus, i n e f f e c t , t h e p o l a r i s e r (or e l e c t r o n a c c e l e r a t o r ) and a n a l y s e r (or s p e c t r o g r a p h ) were crossed, as i n t h e standard configuration f o r observing t h e Faraday e f f e c t .

Of c o u r s e , a g r a t i n g is n o t an i d e a l p o l a r i s i n g element: its behaviour was i n v e s t i g a t e d by a simple experiment i n which a wedge o f c r y s t a l l i n e q u a r t z with t h e o p t i c a x i s running towards t h e apex, a nominal t h i c k n e s s o f 5 mn a t t h e c e n t r e and an a n g l e between t h e f a c e s o f about 1 minute o f a r c was mounted j u s t i n f r o n t of t h e spectrograph s l i t with t h e a x i s a t 45 degrees t o t h e v e r t i c a l . The d i f f e r e n c e between n+ and n-, t h e r e f r a c t i v e i n d i c e s f o r r i g h t - and left-hand c i r c u l a r l y p o l a r i s e d r a d i a t i o n i n c r y s t a l l i n e q u a r t z i s roughly c o n s t a n t a s a f u n c t i o n o f wavelength between 2503 and 2030 angstroms. Therefore, t h e o p t i c a l p a t h d i f f e r e n c e between + and

-

l i g h t expressed i n u n i t s o f wavelengtn v a r i e s a s t h e r e c i p r o c a l wavelength. I f t i s t h e t h i c k n e s s o f t h e c r y s t a l , we have

where n i s an i n t e g e r (274 a t 2190 angstroms), a s t h e c o n d i t i o n f o r r o t a t i n g t h e plane o f p o l a r i s a t i o n through n/2. S i n c e , i n f a c t , we used a wedge, we oDserved a succession o f s l o p i n g f r i n g e s , spaced more c l o s e l y towards s h o r t wavelengths. They a r e i l l u s t r a t e d i n Fig 2. F r m t h e s e and o t h e r d a t a , we e s t i m a t e t n a t tine p o l a r i s i n g e f f i c i e n c y o f t h e g r a t i n g is about 75%.

Between t h e synchrotron and t h e c y l i n d r i c a l m i r r o r , we placed a superconducting magnet capable of producing f i e l d s t r e n g t h s o f up t o 4.7 Tesla over a l e n g t h o f 50 em. A simple wire-wound furnace was used a s t h e a b s o r p t i o n c e l l i n t h e warm bore of t h e magnet. For t h e i n i t i a l experiments r e p o r t e d h e r e , we have i n v e s t i g a t e d t h e vapours o f S r I and & I a t temperatures corresponding to measured p a r t i c l e d e n s i t i e s o f about 1 0 t o t h e 16 cm-3. Further experiments a r e i n hand on elements which possess s i n g l e t p r i n c i p a l s e r i e s and t h e r e f o r e e x h i b i t simple Zeeman s t r u c t u r e s .

/

60 cm long c y l i n d r i c a l a z i n g i n c i d e n c e m i r r o r

erconducting magnet

/ /

500 MeV Electron Synchrotron

S c a l e

o

( 2 metres

F i g 1

Experimental Layout f o r Magneto-optical s t u d i e s a t t h e Bonn 500 MeV electron-synchrotron. For c l - a r i t y , t h e experiments on beam l i n e s 1 , 2 & 3 a r e not shown.

JOUhYAL DE PHYSIQUE

Showing t h e s l o p i n g f r i n g e s observed when a c r y s t a l l i n e q u a r t z wedge is i n t e r p o s e d between t h e c y l i n d r i c a l m i r r o r and t h e entrarice slit o f t h e spectrograph ( s e e t e x t ) . Elnission l i n e s o f CO+ from a hollow cathode s o u r c e were superposed a s wavelength markers.

R e s u l t s and Discussion.

-

A t y p i c a l magneto-rotation p a t t e r n recorded i n t h e p r e s e n t experiments i s shown i n Fig 3. The Zeeman s t r u c t u r e . which c o n s i s t s o f a Lorentz d o u b l e t , can be seen towards t h e c e n t r e o f t h e p a t t e r n . It is surrounded by i n t e n s i t y o s c i l l a t i o n s which a r e s y m n e t r i c a l about t h e f i e l d - f r e e resonance wavelength and which I s h a l l r e f e r t o a s lmagneto-optical b e a t s t . The p h y s i c a l o r i g i n o f t h e s e b e a t s i s as follows: as noted above, t h e p o l a r i s e r and a n a l y s e r i n our experiments were crossed. Also, with t h e combination o f high magnetic f i e l d , long p a t h l e n g t h and high vapour d e n s i t y we used. t y p i c a l atomic f-values r e s u l t e d i n Faraday r o t a t i o n s o f many t u r n s towards t h e c e n t r e o f a b s o r p t i o n l i n e s . Thus, as t h e p r o f i l e o f a l i n e i s scanned i n frequency, with t h e f a s t change i n r o t a t i o n a n g l e a s t h e l i n e i s t r a v e r s e d , t h e emerging l i g h t can have i t s plane of p o l a r i s a t i o n r o t a t e d through d i f f e r e n t i n t e g r a l m u l t i p l e s o f X/2 a t d i f f e r e n t p o i n t s on t h e p r o f i l e , and t h e i n t e n s i t y t h e r e f o r e o s c i l l a t e s between t h e s e p o i n t s . I n o t h e r words, t h e e l e c t r i c v e c t o r o f t h e l i g h t d e s c r i b e s a h e l i x w i t h many t u r n s , and t h e a c t u a l number of t u r n s v a r i e s a s t h e p r o f i l e is scanned.

I n our experiments, magneto-optical p a t t e r n s have been recorded over a wide range o f n-values and magnetic f i e l d s t r e n g t h s . For example, i n S r I, we have recorded p a t t e r n s froin n

=

11 t o n = 28 ill T i e l d s of 2.5, 3.0, 3.5, 4.0 and 4.6 Tesia. Since t h e s p e c t r a were recorded photographically f o r optimum wavelength r e s o l u t i o n , t h e p a t t e r n s f o r d i f f e r e n t n-values are a l l recorded simultaneously. This h a s t h e following advantages: over t h e d u r a t i o n o f t h e exposure, some o f t h e parameters of t h e experiment (eg t h e magnetic f i e l d s t r e n g t h o r t h e d e n s i t y of absorbers i n t h e f u r n a c e ) could d r i f t o r f l u c t u a t e . The v a r i a t i o n s w i l l , however, be i d e n t i c a l f o r all t h e t r a n s i t i o n s recorded on t h e same p l a t e . Since, as w i l l be explained below, we a r e p r i m a r i l y concerned with t h e determination o f r e l a t i v e f-values from t h e product NflB (where N i s t h e number d e n s i t y o f a b s o r b e r s , f is t h e o s c i l l a t o r s t r e n g t h , 1 i s t h e l e n g t h o f t h e furnace and B t h e magnetic f i e l d s t r e n g t h ) , t h i s simple expedient o b v i a t e s any need t o measure N, 1 o r B a c c u r a t e l y , although an approximate value o f B is i n f a c t necessary t o c a l c u l a t e t h e Zeeman s t r u c t u r e a t t h e c e n t r e . Thus, a l l t h a t i s r e q u i r e d is t o hold t h e s e q u a n t i t i e s s u f f i c i e n t l y c o n s t a n t f o r t h e magneto-optical b e a t s t o be recorded with good c o n t r a s t . SlMll f l u c t u a t i o n s i n any o f them a r e , i n f a c t , accounted f o r i n t h e a n a l y s i s by t h e i n t r o d u c t i o n o f an a p p a r a t u s f u n c t i o n , as w i l l be explained below.

Lorentz Doublet in Absorption MO 'Beats' + side Far Wing Absorption Rotation n/2 Central Région Field-free Wavelength X

MO 'Beats» - side Rotation n/2 Fig. 3 A typical well-resolved magneto-optical pattern (for n = 11 in Sr I).

Far Wing

C2-322 JOURNAL DE PHYSIQUE

I n Fig 3, we have given names t o v a r i o u s p a r t s o f t h e p a t t e r n which w i l l f a c i l i t a t e t h e following d i s c u s s i o n . S e v e r a l approaches a r e h e l p f u l i n t h e i n t e r p r e t a t i o n o f t h e p a t t e r n s , and t h e y can be a p p l i e d i n s t e p s o f p r o g r e s s i v e l y i n c r e a s i n g accuracy:

1. Estimates o f NflB a r e r e a d i l y obtained from well-resolved p a t t e r n s such a s t h e one i n Fig. 3 by applying t h e far-wing approximation o f Mitchell and Zemansky (1971 1, which g i v e s t h e r o t a t i o n a n g l e

9

a s a f u n c t i o n o f t h e detuning Y - 1 from t h e f i e l d - f r e e l i n e c e n t r e :

Clearly, as t h e detuning i n c r e a s e s , t h e r o t a t i o n w i l l tend t o zero. Conversely, moving i n towards t h e l i n e c o r e , t h e r o t a t i o n i n c r e a s e s u n t i l , a t T / 2 , t h e transmission through t h e crossed p o l a r i s e r s is maximum. When t h e detuning is reduced still f u r t h e r , we o b t a i n t h e magneto-optical b e a t s f o r a n g l e s o f x / 2 ,

r ,

3 W 2 , 2n-, e t c . Thus, t h e r o t a t i o n a n g l e s a r e determined a b s o l u t e l y , as s t r e s s e d above, and a r e a c t u a l l y independent o f most n o n - l i n e a r i t i e s i n emulsion response which would a f f e c t r e l a t i v e i n t e n s i t y determinations.

The weakness of t h i s first method o f a n a l y s i s is i l l u s t r a t e d i n Fig 4: The p o i n t s which s a t i s f y t h e far-wing c r i t e r i a b e s t a r e t h o s e which g i v e t h e broadest o s c i l l a t i o n s i n Fig. 3 and a r e t h e r e f o r e t h e l e a s t a c c u r a t e l y determined, while t h o s e which do n o t conform t o t h e approximation g i v e s h a r p s t r u c t u r e s which would be t h e most u s e f u l f o r an a c c u r a t e a n a l y s i s .

2. The second approach i s t o r e t u r n t o t h e theory and seek e x p r e s s i o n s f o r t h e t r a n s m i t t e d i n t e n s i t y which involve none o f t h e far-wing approximations. In p r a c t i c e , t h i s means t h a t both t h e c o n t r i b u t i o n s due t o magnetic c i r c u l a r dichroism and magnetic c i r c u l a r b i r e f r i n g e n c e m u s t be included t o g e t h e r . The e a s i e s t way t o do t h i s i s through standard electromagnetic theory, by d e f i n i n g a complex magneto-optical a n g l e (kckinham 1969 )

.

Several a u t h o r s ( i n p a r t i c u l a r Gawlik e t al. 1979) have considered c l o s e l y r e l a t e d s i t u a t i o n s which a r e a c t u a l l y s p e c i a l c a s e s o f t h e one considered h e r e . I n a forthcoming paper (Connerade 1982), t h e f u l l e x p r e s s i o n s a r e d e r i v e d , and t h e a l g e b r a w i l l t h e r e f o r e n o t be repeated here.

Rather, we s h a l l c o n c e n t r a t e on v a r i o u s numerical approaches we have devised t o c a l c u l a t e t h e p r o f i l e s and r e p o r t on improvements which a r e s t i l l being s t u d i e d .

To compute magneto-optical p a t t e r n s , we s t a r t from an expression f o r t h e t r a n s m i t t e d i n t e n s i t y :

I. P -a+z -a-z

-

^(a++a-

)z

100-P -(a++a-)z

I =

- -

[ ( e F - e T ) + 4 e 2 s i n

91

+

-

^{I,e ( 1 )}

4 100 100

where I is t h e i n t e n s i t y o f t h e i n c i d e n t plane wave, P is t h e p o l a r i s a t i o n e f f i c i e n c y of t h e g r a t i n g , a + and a- a r e t h e a b s o r p t i o n c o e f f i c i e n t s f o r r i g h t and f o r l e f t hand c i r c u l a r l y p o l a r i s e d r a d i a t i o n , z is t h e l e n g t h o f t h e furnace and

9

is t h e Faraday a n g l e .

The q u a n t i t i e s a+, a- and a r e c l e a r l y f u n c t i o n s o f t h e detuning I.'-%and o f t h e magnetic f i e l d s t r e n g t h B. We have t r i e d d i f f e r e n t e x p r e s s i o n s f o r them. The s i m p l e s t t o use i s t h e form g i v e n by standard d i s p e r s i o n theory, namely:

f o r s i n g l e t terms,

ere

^{o l =} eB/4xmc and a l l symbols a r e i n t h e s t a n d a r d n o t a t i o n o f Mitchell and Zemanski (1971). The advantage o f t h i s expression i s t h a t n a r e then r e a d i l y obtained a s :

s o t h a t t h e r o t a t i o n a n g l e

9

i s r e a d i l y c a l c u l a t e d from:

However, it should be borne i n mind t h a t t h e s e e x p r e s s i o n s f o r a + and n- a r e n o t s t r i c t l y a p p l i c a b l e . In f a c t , Doppler broadening dominates under t h e c o n d i t i o n s o f t h e experiment, and t h e a n a l y s i s should be based on a Voigt p r o f i l e . This procedure i s more complicated and w i l l be f u r t h e r discussed below.

g

^(Radians)

Fig. 4

Magnetic Rotation a n g l e a s a f u n c t i o n o f wavelength ( i n angstroms) f o r t h e far-wings o f t h e p a t t e r n i n Fig. 3. ^Notice how t h e p o i n t s move o f f t h e t h e o r e t i c a l curve a s one approaches t h e c e n t r e o f t h e p a t t e r n , where t h e s h a r p e s t o s c i l l a t i o n s w c u r . The accuracy o f any a n a l y s i s based on t h e far-wing formula i s t h e r e f o r e l i m i t e d .

JOURNAL DE PHYSIQUE

I n a d d i t i o n t o t h e expressions (1-4) above, we need t o perform an i n t e g r a t i o n over t h e a p p a r a t u s f u n c t i o n . It i s worth n o t i n g t h a t t h i s is not simply t h e i n s t r u m e n t a l f u n c t i o n o f t h e spectrograph, m i c h could be determined, s a y , by a n a l y s i n g a z e r o - f i e l d p r o f i l e : i n t h e a p p a r a t u s f u n c t i o n , we must i n c l u d e f l u c t u a t i o n s o f e i t h e r t h e magnetic f i e l d o r t h e d e n s i t y o f a b s o r b e r s during t h e exposure. Tne s i m p l e s t (and c r u d e s t ) approach i s t o t r e a t t h i s unknown a s a ' t o p h a t t f u n c t i o n , t h e width o f which is then a d j u s t e d t o reproduce observed magneto-optical p a t t e r n s . A more r e a l i s t i c approach nay be t o use a Lorentzian o f a d j u s t a b l e width. A s we s h a l l s e e , both g i v e very good r e s u l t s .

We begin by c o n s i d e r i n g t h e s i m p l e s t approach. namely (1-4) t o g e t h e r with ' t o p h a t ' s o o t h i n g . Fig. 5 g i v e s two examples o f magneto-optical p a t t e r n s computed i n t h i s way.

P = 70%

Fig. 5

Ccmputed magneto-optical p a t t e r n s f o r n:ll o f S r I, with N f = 8 . 5 8 ~ 1 0 t o t h e 14 and B = 4.2 T e s l a , showing t h e e f f e c t o f _- d i f f e r e n t p o l a r i s e r e f f i c i e n c i e s .

Although n o t s t r i c t l y r e l e v a n t t o t h e p h y s i c s , it may be o f i n t e r e s t t o p o i n t o u t t h a t t h e computations involved i n o b t a i n i n g Fig.5 were simple enough t o be c a r r i e d o u t on a CBM 4032 Miciqoprocessoi- a f t e r some i n t e r n a l modification t o provide a s u i t a b l e f a s t g r a p h i c s d i s p l a y . More e l a b o r a t e c a l c u l a t i o n s (which a r e described below) a r e d e f i n i t e l y n o t i n t h i s c l a s s and r e q u i r e a l a r g e computer.

The simpler theory can t h e r e f o r e be u s e f u l f o r a preliminary a n a l y s i s o f d a t a . A s can be seen from Fig. 5 , t h e p r o f i l e s obtained a r e q u i t e r e a l i s t i c and provide u s with a b a s i s t o a n a l y s e t h e s p e c t r a . Indeed, we can now s e t up a scheme t o compute a l l t h e p r o f i l e s using (1-4), while varying t h e following parameters: P ( t o f i t observed i n t e n s i t y r a t i o s o f t h e magneto-optical b e a t s t o t h e Lorentz components), t h e width o f t h e a p p a r a t u s f u n c t i o n ( t o f i t t h e observed f a l l i n i n t e n s i t y of magneto-optical b e a t s a s t h e i r frequency i n c r e a s e s ) and NfzB ( s e e below). I f n o t measured d i r e c t l y , t h e magnetic f i e l d can be recovered from an a n a l y s i s o f t h e Zeeman s t r u c t u r e (a h i g h l y a c c u r a t e value o f B is n o t r e q u i r e d , s i n c e it w i l l anyway f a c t o r o u t o f t h e r e l a t i v e o s c i l l a t o r s t r e n g t h determinations). The d i s p e r s i o n width

(r/LI-rr)

is taken a s equal t o t h e Doppler width.

Most o f t h e parameters a r e held c o n s t a n t from one p r o f i l e t o t h e n e x t f o r d i f f e r e n t p r o f i l e s recorded on t h e same p l a t e , s i n c e t h e experimental c o n d i t i o n s a r e i d e n t i c a l (We have, however, n o t i c e d a v a r i a t i o n o f t h e e f f i c i e n c y o f t h e g r a t i n g a s a p o l a r i s e r over a range o f about 4Q3 angstroms). Tnus, t h e problem reduces t o a determination of NfzB f o r each p r o f i l e . Attempts along t h e s e l i n e s have l e d u s t o y e t a t h i r d approach, which we regard as t h e most a c c u r a t e t o d a t e .

3. The Magneto-Optical Vernier (MOV) Technique: The b a s i s of t h i s approach is a study of t h e behaviour o f computed p a t t e r n s a s NfzB is v a r i e d . A s i t t u r n s o u t , t h e r e a r e d i f f e r e n t rates o f v a r i a t i o n o f t h e p r o f i l e s over d i f f e r e n t p a r t s of t h e r o t a t i o n curve: i n t h e far-wing, t h e p a t t e r n v a r i e s slowly, moving s l i g h t l y o u t from t h e l i n e c e n t r e a s NfzB i n c r e a s e s i n accordance w i t h t h e far-wing formula, while r e l a t i v e i n t e n s i t i e s remain s u b s t a n t i a l l y unchanged, provided t h e magneto-optical b e a t s are w e l l o u t s i d e t h e width o f t h e Lorentz components. Moving in towards t h e l i n e c e n t r e , where t h e r o t a t i o n a n g l e s become much l a r g e r , one f i n d s t h a t t h e r e l a t i v e i n t e n s i t i e s i n t h e p a t t e r n f l u c t u a t e r a p i d l y as a f u n c t i o n of NfzB.

This behaviour is i l l u s t r a t e d i n Fig. 6. which shows j u s t t h e c e n t r a l p a r t o f t h e computed p a t t e r n s a s a f u n c t i o n o f Nfz

.

In what follows. we r e f e r t o one period of t h e f l u c t u a t i o n a s a magneto-optical cycle. The e x i s t e n c e o f t h e c y c l e s w a s a l r e a d y demonstrated by Gawlik e t a l . (1979) f o r t h e s p e c i a l c a s e o f f i x e d energy photoexcitation a t t h e zero-field resonance frequency, i n which c a s e t h e dichroism terms c a n c e l , l e a v i n g o n l y t h e fast-varying c o n t r i b u t i o n due t o b i r e f r i n g e n c e : they were a b l e , by studying t h e b e a t s as a f u n c t i o n o f magnetic f i e l d s t r e n g t h , t o o b t a i n an a c c u r a t e measurement of t h e r e l a t i v e f-values of t h e sodium D l i n e s . By c o n t r a s t , in t h e p r e s e n t method, t h e f i e l d B is held c o n s t a n t and t h e p r o f i l e is scanned in frequency. This h a s a number o f advantages. ( a ) We can use t h e o u t e r part o f t h e p r o f i l e s t o determine NfzB c o a r s e l y , s a y , t o w i t h i n one magneto-optical c y c l e , and then use t h e f a s t v a r i a t i o n a t t h e c e n t r e of t h e p r o f i l e t o determine NfzB a c c u r a t e l y . This is t h e MOV method r e f e r r e d t o above, and is capable of an optimum accuracy o f +- 0.25%. (b) Holding t h e f i e l d c o n s t a n t allows u s t o work a t very high f i e l d s and, by performing measurements a t s e v e r a l d i f f e r e n t values o f B, it should be p o s s i b l e t o study t h e i n f l u e n c e of t h e magnetic f i e l d s t r e n g t h i t s e l f on t h e r e l a t i v e o s c i l l a t o r s t r e n g t h s . ( c ) By working with high f i e l d s , one a c h i e v e s a high s e n s i t i v i t y a s w e l l a s a high accuracy, and r e l a t i v e o s c i l l a t o r s t r e n g t h s where one o f t h e f-values is 10 t o t h e minus 4 or l e s s can be determined t o an accuracy which i s s t i l l o f t h e o r d e r of t l Wh.

A s pointed o u t above, t h e r e remains some p o s s i b i l i t y o f s y s t e m a t i c e r r o r s i n an a a l y s i s based on t h c MOV technique when t h e simple d i s p e r s i o n formulae (2-4) a r e used. Nevertheless, we have obtained r e s u l t s which, p l o t t e d on t h e 'renormalised' graphs of quantum d e f e c t t h e o r y ( S t a r a c e 1976) a r e more c o n s i s t e n t a t high n than r e s u l t s obtained by t h e 'hook1 technique (Parkinson Reeves and Tomkins 1976) and join more smoothly with t h e p h o t o i o n i s a t i o n c r o s s s e c t i o n . This is i l l u s t r a t e d i n Fig.7.

I n o r d e r t o determine t h e s i g n i f i c a n c e o f t h e s y s t e m a t i c e r r o r s mentioned above and a l s o t o e x p l o r e p o s s i b l e improvements i n t h e d e t a i l s o f t h e p r e d i c t e d pdLLernu, we h a v e &so per-Cormed c a i c u l a t i o n s u s i n g Voigt p r o f i l e s , convoivea wltn a Lorentzian instrument f u n c t i o n . The computations were performed by using a simple algorithm (Hui Amstrong and Wray 1978) f o r t h e complex e r r o r f u n c t i o n , t h e r e a l and imaginary p a r t s o f which y i e l d t h e absorption c o e f f i c i e n t s and r e f r a c t i v e i n d i c e s . Many more p o i n t s were used (about 1000) i n o r d e r t o d i s p l a y well-resolved p a t t e r n s . These c a l c u l a t i o n s were c a r r i e d o u t on t h e CDC 76CK) computer a t Imperial College. A t y p i c a l p r o f i l e is displayed i n E'ig.8.

A s can be seen from Fig. 8 , t h e new c a l c u l a t i o n s using Voigt p r o f i l e s and t h e more reasonable a p p a r a t u s f u n c t i o n a r e a s i g n i f i c a n t improvement over e a r l i e r ones based on simple d i s p e r s i o n t h e o r y and a ' t o p h a t 1 smoothi.ng procedure: whi1.e

t h e o v e r a l l f e a t u r e s a r e t h e same, t h e Lorentz components a t t h e c e n t r e o f t h e p a t t e r n a r e c l o s e r t o t h e experimental l i n e s h a p e s , and, with t h e f i n e r mesh used i n t h e more e l a b o r a t e c a l c u l a t i o n s , more d e t a i l e d s t r u c t u r e can be reproduced.

Thus, t h e more e l a b o r a t e scheme w i l l , it is thought, provide a b e t t e r framework f o r e x p l o i t i n g t h e MOV method. A t t i m e o f w r i t i n g , t h i s a s p e c t o f t h e a n a l y s i s is s t i l l being developed, which i s why t h e r e s u l t s of Fig. 7 a r e based on simple d i s p e r s i o n theory. In p a r t i c u l a r , we a r e s t i l l e x p l o r i n g t h e most a p p r o p r i a t e

0.23% change in Nfz 2.8% Change in Nfz

Absorption Fig. 6 A full magneto-optical cycle for n = 11 in Sr I, as derived from calculations using dispersion theory. The existence of fast changes at the centre of the patterns while the far-wing pattern changes slowly with Nfz is the basis of the Magneto-Optical Vernier (MOV) technique described in the text.

Photon Energy E (eV)

F i g . 7

P l o t o f renormalised o s c i l l a t o r s t r e n g t h s a s a f u n c t i o n of energy f o r t h e p r i n c i p a l s e r i e s o f S r I. showing t h e smooth j o i n which is obtained between t h e p r e s e n t d a t a (rectangl-es) and t h e p h o t o i o n i s a t i o n c r o s s s e c t i o n o f S r I a s measured by Hudson e t a1.(1969), scaled up by 1.5. This i s compared with 'hook d a t a ' by Parkinson Reeves and Tankins (1976-open c i r c l e s ) , who suggest a s c a l i n g f a c t o r of 1.9 f o r t h e d a t a o f Hudson e t a1.(1969). A s c a l i n g f a c t o r o f 1.7 was suggested by Lut jens ( 1972 )

.

' h e 'hook' values f o r n= I I t o 15 a r e very c l o s e t o t h e p r e s e n t values and a r e omitted f o r c l a r i t y . A t high n.

t h e p r e s e n t measurements s t i l l show some d e p a r t u r e s from a s t r a i g h t l i n e and t h i s is discussed i n t h e t e x t .

apparatus f u n c t i o n t o use. One might, f o r example, p r e f e r a Gaussian shape, but t h e matter is not a simple one, because t h e width o f t h e a p p a r a t u s f u n c t i o n i s i n f a c t dominated by f l u c t u a t i o n s i n furnace d e n s i t y and magnetic f i e l d s t r e n g t h , r a t h e r than j u s t o p t i c a l e f f e c t s .

Tnus, before deciding on t h e f i n a l method o f i n t e r p r e t a t i o n , we wish t o s t u d y t h e i n f l u e n c e o f various p o s s i b l e choices on t h e r e l a t i v e f-values obtained f o r a given s e t o f d a t a . This should a l s o provide u s e f u l information on t h e accuracy of our technique.

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Fig. 8

Calculated magnetorotation pattern for n=ll in. Sr I. The parameters are NE1 = 8.38 x 1014, H = 4,2 Tesla. A Voigt profile was assumed for the field-free absorption coefficient and the full expression including both circular birefungence and dichroism is then smoothed over a Lorentzian apparatus function.

Another a s p e c t o f t h e p a t t e r n i n Fig.3 which h a s n o t been d i s c u s s e d s o f a r i s t h e a s p n e t r y with r e s p e c t t o l i n e c e n t r e which i s apparent i n t h e Zeeman components. A first temptation i s t o seek its cause i n one of s e v e r a l processes which l e a d t o antisymmetric Faraday e f f e c t s ( c f Fortson and Wilets 1980). These a r e , i n o r d e r o f s i g n i f i c a n c e as t h e magnetic f i e l d s t r e n g t h is i n c r e a s e d , t h e Back-Goudsmit e f f e c t which breaks t h e coupling of J and I i n hyperfine s t r u c t u r e , and t h e Paschen-Back e f f e c t , which breaks LS coupling and t h u s mixes s i n g l e t and t r i p l e t s t a t e s . I n p r i n c i p l e , i f both t h e l a t t e r causes o f asymmetry could be removed, eg by e x t r a p o l a t i n g t o zero f i e l d s t r e n g t h , one would be l e f t with t h e Weinberg-Salam e f f e c t f o r atoms, which, however, is much t o o small t o be d e t e c t e d by our approach.

All t h e e f f e c t s a s s o c i a t e d with antisymmetric r o t a t i o n a c t simultaneously on t h e a b s o r p t i o n c o e f f i c i e n t s and r e f r a c t i v e i n d i c e s , and t h e r e f o r e a f f e c t both t h e magneto-optical b e a t s and t h e Lorentz components a t t h e c e n t r e . Now, we f i n d t h a t t h e magneto-optical b e a t s i n well-resolved p a t t e r n s such a s t h e one i n Fig.3 a r e i n f a c t symmetrical about t h e c e n t r e of t h e p a t t e r n s t o within experimental e r r o r

( f o r p a r t i a l l y resolved p a t t e r n s , t h e s i t u a t i o n is s l i g h t l y d i f f e r e n t , as discussed below). Ekperimentally, t h e asymmetry only a f f e c t s t h e i n t e n s i t y of t h e Zeeman components. This s u g g e s t s t h a t t h e cause, a t l e a s t f o r S r I a t t h e f i e l d s t r e n g t h s we have s t u d i e d , is not connected with any antisymnetry o f t h e Faraday e f f e c t .

Another p o s s i b i l i t y i s some e l l i p t i c i t y of t h e i n c i d e n t r a d i a t i o n . Synchrotron r a d i a t i o n i s o n l y plane-polarised i n t h e o r b i t a l plane o f t h e a c c e l e r a t o r . A s one i n c r e a s e s t h e acceptance a n g l e out o f t h e plane o f t h e o r b i t ( o r i f , f o r some reason, t h e o r b i t o f t h e e l e c t r o n s migrates s l i g h t l y from t h e t r u e p l a n e ) t h e r a d i a t i o n w i l l , i n f a c t , be e l l i p t i c a l l y p o l a r i s e d . In e f f e c t , we can r e p r e s e n t t h i s a s a s l i g h t admixture o f c i r c u l a r l y p o l a r i s e d r a d i a t i o n i n t h e i n c i d e n t beam. Now, such a n admixture does n o t a f f e c t t h e r o t a t i o n p a t t e r n s a t a l l , s i n c e t h e r e is no p r e f e r r e d plane i n c i r c u l a r l y p o l a r i s e d l i g h t . However, t h e Zeeman p a t t e r n i s a f f e c t e d , because c i r c u l a r l y p o l a r i s e d l i g h t o f a given sense (say p o s i t i v e ) w i l l be absorbed by t h e Lorentz component on one s i d e o f t h e p a t t e r n , b u t n o t by t h e corresponding Lorentz component on t h e o t h e r s i d e .

We have included t h i s e f f e c t i n our c a l c u l a t i o n s . and we f i n d t h a t good r e s u l t s a r e indeed obtained f o r a n admixture o f c i r c u l a r l y p o l a r i s e d l i g h t o f a few percent i n t h e i n c i d e n t beam. This i s shown i n Fig.9. The admixture of c i r c u l a r l y p o l a r i s e d l i g h t f o r Fig.9 is perhaps a l i t t l e h i g h e r than one would expect, but i t could conceivably be due t o o p t i c a l i m p e r f e c t i o n s , f o r example to a g r a z i n g incidence r e f l e c t i o n on one s i d e o f t h e beam l i n e tube.

Fig. 9

Calculated p r o f i l e i n c l u d i n g an admixture of 15"/0 r i g h t hand c i r c u l a r l y p o l a r i s e d l i g h t , t h e n e t e f f e c t o f which is t o i n c r e a s e t h e i n t e n s i t y t r a n s m i t t e d through t h e crossed p o l a r i s e r s a t a l l wavelengths except i n t h e 0 peak, a t t h e c e n t r e of which r i g h t hand c i r c u l a r l y pol&ised l i g h t is s t r o n g l y absorbed. An asymnetry is thereby introduced i n t h e p a t t e r n , but i t w i l l be noted that t h i s does n o t extend t o t h e magneto-optical b e a t s . Also, t h e c e n t r e o f t h e p a t t e r n has t h e same appearance a s i n t h e p r o f i l e o f Fig.8, which was c a l c u l a t e d u s i n g otherwise i d e n t i c a l parameters.

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So f a r , t h e d i s c u s s i o n has c e n t r e d on well resolved magneto-optical p a t t e r n s , i e on p a t t e r n s f o r which t h e m e t o - o p t i c a l b e a t s s t a n d w e l l c l e a r o f t h e Zeeman s t r u c t u r e . A proper understanding o f t h e s e is c l e a r l y necessary t o t a c k l e t h e more complex s i t u a t i o n where t h e s t r u c t u r e s o v e r l a p i n energy, which we s h a l l r e f e r t o as p a r t i a l l y r e s o l v e d p a t t e r n s .

P a r t i a l l y resolved p a t t e r n s a r e encountered a s one progresses t o higher n-values up a Rydberg s e r i e s f o r a given magnetic f i e l d s t r e n g t h . With decreasing f-value, t h e a v a i l a b l e r o t a t i o n d e c r e a s e s a l s o , and e v e n t u a l l y t h e magneto-optical b e a t s move i n t o t h e l i n e w i d t h o f t h e i n d i v i d u a l Lorentz components, so t h a t it becomes d i f f i c u l t t o a t t a c h p r e c i s e l a b e l s t o t h e peaks a s was done i n Fig.3.

Examples o f p a r t i a l l y resolved p a t t e r n s a r e given i n Fig.10. The first question one might a s k is whether t h e MOV method is still a p p l i c a b l e under such c o n d i t i o n s . F o r t u n a t e l y , a c l e a r answer is provided by t h e experiment i t s e l f : as can be seen i n Fig.10, t h e o u t e r r e g i o n s o f t h e p a t t e r n s change slowly with i n c r e a s i n g n-value, whereas t h e i n t e n s i t y a t t h e c e n t r e (along t h e dashed c u r v e i n t h e f i g u r e ) changes f a s t . This f a c t demonstrates t h a t t h e MOV method c o n t i n u e s t o provide enhanced accuracy i n t h e determination o f f-values under p a r t i a l l y resolved c o n d i t i o n s .

I n a d d i t i o n , t h e methods o f p r o f i l e computation described above a l s o remain a p p l i c a b l e under p a r t i a l l y resolved c o n d i t i o n s , which is a n o t h e r important advantage over t h e method o f a n a l y s i s based on t h e far-wing approximation.

Examples o f p r o f i l e s computed under p a r t i a l l y resolved c o n d i t i o n s a r e presented in Fig. 11 and a r e seen t o reproduce t h e observed p a t t e r n s well.

Fig1 1

Examples o f computed p r o f i l e s under p a r t i a l l y resolved c o n d i t i o n s . The corresponding experimental p r o f i l e s a r e shown i n Fig.10.

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The p l o t o f o s c i l l a t o r s t r e n g t h s given i n Fig. 7 was obtained by using d i s p e r s i o n theory and t h e simpier of t h e two computer codes described above.

Although t h e t r e n d o f t h e d a t a a t high n i s much more s a t i s f a c t o r y than t h e one obtained from e a r l i e r measurements by t h e 'hook' technique (Parkinson Reeves and Tankins (1976), t h e r e a r e s t i l l d e p a r t u r e s from t h e s t r a i g h t l i n e i n Fig. 7 which, we do n o t understand, i n p a r t i c u l a r ( i ) around n = 18 and ( i i ) f o r n = 25, 26, 27 and 28.

C m s i d e r i n g ( i ) f i r s t , i t is i n t e r e s t i n g t h a t t h e 'hook' data ( a l s o shown i n Fig. 7 ) e x h i b i t a jump between n = 17 and n = 18 which is even more pronounced t h a n i n t h e p r e s e n t d a t a . Our e r r o r e s t i m a t e is +I -8% a t n = 18, and t h e d e p a r t u r e from l i n e a r i t y i n our d a t a t h e r e f o r e seems s i g n i f i c a n t . Esherick (1977 Fig. 3 ) has presented a quantum d e f e c t p l o t based on t h e e a r l i e r d a t a by Garton and Codling (1968) which a l s o shows a d e p a r t u r e from l i n e a r i t y a t n = 18, although t h e r e seems t o be no p e r t u r b e r nearby. With t h i s i n mind, we have r e - i n v e s t i g a t e d t h e upper members o f t h e p r i n c i p a l s e r i e s i n S r I up t o n = 85 in zero f i e l d . Our new d a t a (Fig. 12) show no s i g n of p e r t u r b a t i o n s i n t h e quantum d e f e c t , o r o f any t r a n s i t i o n s t o t r i p l e t s t a t e s , even at t h e h i g h e s t values o f n recorded.

We have t h e r e f o r e searched f o r a p o s s i b l e source o f s y s t e m a t i c e r r o r i n t h e f-value measurements around n = 18. Between n = 17 and 1 8 , t h e r e occurs a n impurity l i n e a t 2201 -42

1

which is due t o 7p1P, i n C a I. _In t h e presence o f t h e f i e l d , t h i s a b s o r p t i o n l i n e d i s a p p e a r s and is replaced by a p a i r o f magneto-optical i n t e n s i t y maxima, t h u s confiming t h a t t h e impurity i s o n l y p r e s e n t within t h e a b s o r p t i o n column i n s i d e t h e high-field region. From t h e known f-value o f t h e Ca I t r a n s i t i o n (Parkinson Reeves and Tankins 1976). we c a l c u l a t e a Ca/Sr r a t i o i n t h e vapour o f -10-4. The a d d i t i o n a l r o t a t i o n due t o t h i s l i n e might conceivabl-y be a s o u r c e o f e r r o r i n o u r measurements, s i n c e its f-value i s about 30 times g r e a t e r than t h a t o f t h e n = 18 l i n e i n S r I, but f u r t h e r work is needed t o confirm t h i s p o s s i b i l i t y .

In Fig. 13, we demonstrate t h a t t h e d e p a r t u r e from l i n e a r i t y a t n = 18 is a r e a l e f f e c t by g i v i n g both experimental and c a l c u l a t e d p r o f i l e s .

Turning now t o ( i i ) , it w i l l be n o t i c e d that t h e values around n = 25, 26, 27 and 28 i n Fig. 7 l i e below t h e ' b e s t ' s t r a i g h t l i n e through t h e d a t a p o i n t s , and t h a t t h e discrepancy i n c r e a s e s s y s t e m a t i c a l l y with i n c r e a s i n g n-value. Our e s t i m a t e s s w e s t that t h e e r r o r s a r e t3.7'70, 5.677, 6.7% and 8.3% r e s p e c t i v e l y f o r t h e s e t r a n s i ' t i o n s and t h a t t h e d e p a r t ~ ~ r - e from l i n e a r i t y is b a r e l y s i g n i f i c a n t . Nevertheless, t h e s y s t e m a t i c t r e n d i n Fig. 7 does suggest some underlying reason f o r it.

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A t time of w r i t i n g , we can o n l y s p e c u l a t e t h a t t h i s t r e n d might be due t o p e r t u r b a t i o n s i n t h e f-value produced by t h e a p p l i e d magnetic f i e l d . It would seem q u i t e reasonable t h a t such p e r t u r b a t i o n s should become important a t n-values around 25: t h e d a t a o f Garton and Tcmkins (1969) show t h a t 1-mixing s a t e l l i t e s appear around n = 30 i n f i e l d s o f 40kG and, indeed, our own s p e c t r a show (Fig. 1 4 ) t h a t t h e d a t a become u n i n t e r p r e t a b l e above n = 28 because o f high f i e l d e f f e c t s . It is an open question what f u r t h e r information could be derived from our d a t a a t high n-values. We a r e i n possession of d a t a recorded a t s e v e r a l f i e l d s t r e n g t h s which would allow u s i n p r i n c i p l e t o s e a r c h f o r s y s t e m a t i c changes i n f-value a s a f u n c t i o n o f t h e a p p l i e d magnetic f i e l d . Our a n a l y s i s is s t i l l i n p r o g r e s s , but first r e s u l t s i n d i c a t e t h a t it may be necessary t o r e p e a t our experiments a t still higher f i e l d s t o r e v e a l t h e t r e n d , and t h i s w i l l r e q u i r e an upgrade o f our superconducting magnet.

One o f t h e advantages o f holding t h e magnetic f i e l d c o n s t a n t a s we have done i n our experiment is t h a t h i g h f i e l d s a r e r e a d i l y obtained and e f f e c t s due t o t h e f i e l d a r e e a s i l y probed. 'his opens up e x c i t i n g prospects f o r t h e a p p l i c a t i o n o f t h e MOV technique t o s t u d i e s o f high f i e l d e f f e c t s .

Conclusion.

-

We have demonstrated a novel technique f o r t h e measurement o f atomic o s c i l l a t o r s t r e n g t h s , which achieves a higher accuracy and s e n s i t i v i t y than o t h e r previously known methods. The new approach r e q u i r e s f u r t h e r development, both experimental and t h e o r e t i c a l t o be a p p l i e d t o f u l l advantage.

For example, i n its p r e s e n t form, t h e t h e o r y has only been worked o u t i n d e t a i l f o r s i n g l e t t o s i n g l e t t r a n s i t i o n s . Also, it may be more a p p r o p r i a t e t o use a l a s e r r a t h e r than a synchrotron r a d i a t i o n s o u r c e , and sub-Doppler techniques could probably be a p p l i e d with advantage. The new method holds g r e a t promise f o r t h e study o f p e r t u r b a t i o n s i n f-value due t o high e x t e r n a l magnetic f i e l d s .

Acknowledgements.

-

The p r e s e n t work received f i n a n c i a l support from t h e S.E.R.C.

(UK) and t h e B.M.F.T. (West Germany). We a r e a l s o g r a t e f u l t o t h e Argonne National Laboratory ( I l l i n o i s , USA) f o r t h e l o a n o f t h e superconducting magnet.

Fig. 14 - Showing the uppermost series members of Sr 1 in a field of 4.2 T. Note the increase in complexity of the spectrum above n = 28 which is attributed to 1-mixing.

JOURNAL DE PHYSIQUE

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