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An experimental determination of the resonance broadening constant
Volume 21, number 2
P HY S I C S L E T T E R S
AN E X P E R I M E N T A L
1 May 1966
DETERMINATION OF BROADENING CONSTANT
THE
RESONANCE
J. M. VAUGHAN
Clarendon Laboratory, University of Oxford
Received 21 March 1966 Accurate experimental data on resonance broadening in helium is shown to be in excellent agreement with theoretical values which have recently been corrected.
In the p a s t two y e a r s a l a r g e n u m b e r of t h e o r e t i c a l p a p e r s have been published on the subject of r e s o n a n c e b r o a d e n i n g [1-5]. A r e c e n t a r t i c l e by Omont [6] has s t r e s s e d the s i m i l a r i t y of the u n d e r l y i n g p r i n c i p l e s of m o s t of these c a l c u l a tions. Now that c o r r e c t i o n s have been m a d e [7,8], t h e r e i s good a g r e e m e n t between the t h e o r e t i c a l v a l u e s for the r e s o n a n c e b r o a d e n i n g constant u n d e r the conditions of the i m p a c t a p p r o x i m a t i o n . The p u r p o s e of this l e t t e r is to r e p o r t the r e s u l t s of s o m e a c c u r a t e m e a s u r e m e n t s of line b r o a d e n ing in h e l i u m which a r e in excellent a g r e e m e n t with t h e s e t h e o r e t i c a l v a l u e s . In this work, i n s t e a d of examining the r e s o n a n c e l i n e i t s e l f , which b e c a u s e of its short wavelength (584 ~ ) i s i n a c c e s s i b l e to i n t e r f e r o m e t r i c techniques, v i s i b l e l i n e s leading to the 21P r e s o n a n c e level have been studied. In an e a r l i e r p a p e r [9] Kuhn and Vaughan r e p o r t e d some i n i t i a l r e s u l t s on the line k7281 (21p-31S). T h e s e r e s u l t s have now been c o n s i d e r a b l y r e fined and extended, and the l i m i t s of e r r o r r e duced. The e x p e r i m e n t s c o n s i s t in obtaining a s e r i e s of e m i s s i o n l i n e p r o f i l e s f r o m a gas d i s charge m a i n t a i n e d at different p r e s s u r e s and t e m p e r a t u r e s . The p r o f i l e s a r e scanned with a F a b r y - P e r o t i n t e r f e r o m e t e r and d i r e c t r e c o r d e r . By p r e c i s i o n a n a l y s i s the p r o f i l e s may be a n a l y s e d into t h e i r Doppler and L o r e n t z c o n t r i b u tions. When graphs a r e drawn of Lorentz width v e r s u s gas density the e x p e r i m e n t a l points a r e found to lie on s t r a i g h t l i n e s ; the L o r e n t z i a n s e l f - b r o a d e n i n g c o n s t a n t may thus be a c c u r a t e l y obtained from the slope of such graphs. In the c u r r e n t work a d.c. d i s c h a r g e has been used, cooled with l i q u i d n i t r o g e n or ice. Many p o s s i b i l i t i e s of s y s t e m a t i c e r r o r have been now eliminated. In two l i n e s (k6678 ~ and k7281 ~ ) the o b s e r -
ved b r o a d e n i n g could be a t t r i b u t e d e n t i r e l y to the lower level 21p. The e x p e r i m e n t a l r e s u l t s which have been evaluated over the density r a n g e 1 to 16 × 1017 a t o m s c m - 3 and at two t e m p e r a t u r e s a r e shown in the following table. They a r e exp r e s s e d as whole h a l f - v a l u e widths of L o r e n t z i a n c o m p o n e n t s of p r o f i l e p e r unit of density of 9.66 × 1016 a t o m s c m - 3 (equivalent to 1 m m Hg at 100°K). The quoted e r r o r s a r e e s t i m a t e d total error limits.
Table 1 Pressure broadening and shift in helium. Transition
k7281 ~ (21p-31S) k6678~
(21p-31D)
T ~ 80°K T ~ 280°K broadening shift broadening shift (10-3cm-I) (10-3cm-1) (10-3cm-1)(10-3cm-1) 7.85 + 0.4
0 :~0.1
8.1 :~ 0.5
0 -~0.2
7.8 ~-0.4
0 -~0.05
7.9 :b 0.5
0 • 0.2
In t h e s e r e s u l t s the a b s e n c e of line shif~ and the c o n s t a n c y of the b r o a d e n i n g at the two t e m p e r a t u r e r a n g e s , verify the existence of p u r e r e s o n a n c e i n t e r a c t i o n . This would be expected in view of the l a r g e s t r e n g t h s of the r e s o n a n c e t r a n s i t i o n . The m e a n value for the b r o a d e n i n g constant 5 = 7.9 + 0.5 × 10-3 cm -1, i s much m o r e a c c u r a t e than the i n i t i a l value r e p o r t e d in ref. 9, and within the l i m i t s of e r r o r of the l a t t e r . The e x p e r i m e n t a l value may be conveniently c o m p a r e d with theory, with the following e x p r e s sion _
e2 _
f J J ' N cm "1 -
-
5 = /~aj, 8mc2 ~'JJ' w h e r e f j a , and ~ j j , a r e the absorption o s c i l l a t o r s t r e n g t h and wave n u m b e r of the r e s o n a n c e line. N i s the n u m b e r density of the a t o m s p e r c m - 3 , and k j j , i s a n u m e r i c a l factor containing the s t a t i s t i c a l weights. 153
Volume 21, number 2
PHYSICS LETTERS
Schiff and P e k e r i s [10] have c a l c u l a t e d with very great accuracy a theoretical oscillator s t r e n g t h fo r the helium r e s o n a n c e line. This v a l u e m u s t now be r e g a r d e d a s r e l i a b l e , e s p e cially in v i e w of its a g r e e m e n t with many other c a l c u l a t i o n s . H e l i u m thus o f f e r s a unique opportunity for the c o m p a r i s o n of t h e o r y with e x p e r i m e n t . Using t h e i r f - v a l u e of 0.2761 + 0.0001 in the above e x p r e s s i o n k j j , ( e x p e r i m e n t a l ) is found to be 1.44 + 0.09. A f t e r c o r r e c t i o n of e r r o r s the following r e s u l t s for k j j , ( t h e o r e ti c a l) a r e obtained: Bryon and F ole y [1] 1.33; Watanabe [4,7] 1.44; Ali and G r i e m [5,8] 1.41. Omont [6] has the va l ue 1.45 which he c o n s i d e r s should be a c c u r a t e to ± 0.04. In o r d e r to explain an a p p a r e n t d i s c r e p a n c y with t h e i r t h e o r y , Ali and G r i e m have d i s c u s s e d a n u m b e r of o b j e c t i o n s to the i n t e r p r e t a t i o n of the e x p e r i m e n t a l work d e s c r i b e d in ref. 9. Following the e l i m i n a t i o n of s o m e e r r o r s in t h e i r c a l c u l a t i o n s this d i s c r e p a n c y no l o n g e r e x i s t s . C o n s i d e r a t i o n of the r a n g e of gas density, t e m p e r a t u r e and the width of p r o f i l e s etc. shows that the c u r r e n t e x p e r i m e n t s a r e well within the l i m i t s of the i m p a c t a p p r o x i m a t i o n . Additionally, eff ect s such as ' c o l l i s i o n a l n a r r o w ing' can be shown to be n e g l i g i b l e . The a g r e e m e n t between the e x p e r i m e n t a l value and the l a t e s t t h e o r e t i c a l v a l u e s is thus e x c e l l e n t . One may conclude that the t e c h n i q u e of m e a s u r e m e n t of r e s o n a n c e b r o a d e n i n g now o f f e r s a m e a n s for the a c c u r a t e d e t e r m i n a t i o n of o s c i l l a t o r s t r e n g t h s of r e s o n a n c e l i n e s . On the other hand at e x t r e m e l y low d e n s i t i e s , w h e r e the b r o a d e n i n g is of the s a m e o r d e r as the n a t u r a l width, t h e o r y does not a g r e e with e x p e r i m e n t . C o n t r a r y to what had been a s s u m e d in r e f . 9, the l i n e a r e x t r a p o l a t i o n of b r o a d e n i n g plots to z e r o density does not give the c o r r e c t value f o r the n a t u r a l width of the f r e e a t o m . A q u a l i t a t i v e
154
1May 1966
explanation of this s u r p r i s i n g fact has r e c e n t l y been given in t e r m s of a doubling of the t r a n s i tion p r o b a b i l i t y f r o m the s y m m e t r i c state of coupled a t o m s [11]. The d i s c r e p a n c y between the t h e o r e t i c a l o s c i l l a t o r s t r e n g t h of Schiff and P e k e r i s and the l a r g e r value d e r i v e d by l i n e a r e x t r a p o l a t i o n to z e r o p r e s s u r e [9] must now be c o n s i d e r e d as s t r o n g e v i d e n c e for this e n h a n c e m e n t effect due to coupling. T h i s effect, which is i m p o r t a n t only at v e r y low d e n s i t i e s , is not included in c u r r e n t t h e o r i e s of p r e s s u r e b r o a d e n ing. It would not significantly a l t e r the slope of b r o a d e n i n g plots obtained o v e r a l a r g e r a n g e of density. I wish to thank P r o f e s s o r A. Omont for showing us his l a t e s t p a p e r b e f o r e publication, and a l s o P r o f e s s o r H. R. G r i e m for c o m m u n i c a t i n g the r e s u l t s of an a m e n d e d calculation. I am indebted to Dr. H. G. Kuhn for his constant i n t e r e s t in this work.
References 1. F.W. Byron Jr. and H.M.Foley, Phys. Rev. 134 (1964) A625. 2. A.Omont. J. de Physique. 26 (1965) 26. 3. G.P.Reck, H.Takebeand C.A.Mead, Phys. Rev. 137 (1965) A683. 4. T.Watanabe, Phys. Rev. 138 (1965)A1573. 5. A . W . A l i a n d H . R . G r i e m , Phys. Rev. 140 (1965) A1044. 6. A.Omont, Comptes Rendus, 262 (1966) B190. 7. T.Watanabe, Phys. Rev. 140 (1965) AB5. 8. A. W. Ali and H. R. Griem~ private communication. to be published. 9. H.G.Kulm and J.M.Vaughan, Proc. Roy. Soc. (London) A277 (1964) 297. 10. B.Schiff and C . L . P e k e r i s , Phys. Rev. 134 (1964) A638. 11. H.G.Kuhn, E.L.Lewis and J.M.Vaughan, Phys. Rev. Letters 15 (1965) 687.
P HY S I C S L E T T E R S
AN E X P E R I M E N T A L
1 May 1966
DETERMINATION OF BROADENING CONSTANT
THE
RESONANCE
J. M. VAUGHAN
Clarendon Laboratory, University of Oxford
Received 21 March 1966 Accurate experimental data on resonance broadening in helium is shown to be in excellent agreement with theoretical values which have recently been corrected.
In the p a s t two y e a r s a l a r g e n u m b e r of t h e o r e t i c a l p a p e r s have been published on the subject of r e s o n a n c e b r o a d e n i n g [1-5]. A r e c e n t a r t i c l e by Omont [6] has s t r e s s e d the s i m i l a r i t y of the u n d e r l y i n g p r i n c i p l e s of m o s t of these c a l c u l a tions. Now that c o r r e c t i o n s have been m a d e [7,8], t h e r e i s good a g r e e m e n t between the t h e o r e t i c a l v a l u e s for the r e s o n a n c e b r o a d e n i n g constant u n d e r the conditions of the i m p a c t a p p r o x i m a t i o n . The p u r p o s e of this l e t t e r is to r e p o r t the r e s u l t s of s o m e a c c u r a t e m e a s u r e m e n t s of line b r o a d e n ing in h e l i u m which a r e in excellent a g r e e m e n t with t h e s e t h e o r e t i c a l v a l u e s . In this work, i n s t e a d of examining the r e s o n a n c e l i n e i t s e l f , which b e c a u s e of its short wavelength (584 ~ ) i s i n a c c e s s i b l e to i n t e r f e r o m e t r i c techniques, v i s i b l e l i n e s leading to the 21P r e s o n a n c e level have been studied. In an e a r l i e r p a p e r [9] Kuhn and Vaughan r e p o r t e d some i n i t i a l r e s u l t s on the line k7281 (21p-31S). T h e s e r e s u l t s have now been c o n s i d e r a b l y r e fined and extended, and the l i m i t s of e r r o r r e duced. The e x p e r i m e n t s c o n s i s t in obtaining a s e r i e s of e m i s s i o n l i n e p r o f i l e s f r o m a gas d i s charge m a i n t a i n e d at different p r e s s u r e s and t e m p e r a t u r e s . The p r o f i l e s a r e scanned with a F a b r y - P e r o t i n t e r f e r o m e t e r and d i r e c t r e c o r d e r . By p r e c i s i o n a n a l y s i s the p r o f i l e s may be a n a l y s e d into t h e i r Doppler and L o r e n t z c o n t r i b u tions. When graphs a r e drawn of Lorentz width v e r s u s gas density the e x p e r i m e n t a l points a r e found to lie on s t r a i g h t l i n e s ; the L o r e n t z i a n s e l f - b r o a d e n i n g c o n s t a n t may thus be a c c u r a t e l y obtained from the slope of such graphs. In the c u r r e n t work a d.c. d i s c h a r g e has been used, cooled with l i q u i d n i t r o g e n or ice. Many p o s s i b i l i t i e s of s y s t e m a t i c e r r o r have been now eliminated. In two l i n e s (k6678 ~ and k7281 ~ ) the o b s e r -
ved b r o a d e n i n g could be a t t r i b u t e d e n t i r e l y to the lower level 21p. The e x p e r i m e n t a l r e s u l t s which have been evaluated over the density r a n g e 1 to 16 × 1017 a t o m s c m - 3 and at two t e m p e r a t u r e s a r e shown in the following table. They a r e exp r e s s e d as whole h a l f - v a l u e widths of L o r e n t z i a n c o m p o n e n t s of p r o f i l e p e r unit of density of 9.66 × 1016 a t o m s c m - 3 (equivalent to 1 m m Hg at 100°K). The quoted e r r o r s a r e e s t i m a t e d total error limits.
Table 1 Pressure broadening and shift in helium. Transition
k7281 ~ (21p-31S) k6678~
(21p-31D)
T ~ 80°K T ~ 280°K broadening shift broadening shift (10-3cm-I) (10-3cm-1) (10-3cm-1)(10-3cm-1) 7.85 + 0.4
0 :~0.1
8.1 :~ 0.5
0 -~0.2
7.8 ~-0.4
0 -~0.05
7.9 :b 0.5
0 • 0.2
In t h e s e r e s u l t s the a b s e n c e of line shif~ and the c o n s t a n c y of the b r o a d e n i n g at the two t e m p e r a t u r e r a n g e s , verify the existence of p u r e r e s o n a n c e i n t e r a c t i o n . This would be expected in view of the l a r g e s t r e n g t h s of the r e s o n a n c e t r a n s i t i o n . The m e a n value for the b r o a d e n i n g constant 5 = 7.9 + 0.5 × 10-3 cm -1, i s much m o r e a c c u r a t e than the i n i t i a l value r e p o r t e d in ref. 9, and within the l i m i t s of e r r o r of the l a t t e r . The e x p e r i m e n t a l value may be conveniently c o m p a r e d with theory, with the following e x p r e s sion _
e2 _
f J J ' N cm "1 -
-
5 = /~aj, 8mc2 ~'JJ' w h e r e f j a , and ~ j j , a r e the absorption o s c i l l a t o r s t r e n g t h and wave n u m b e r of the r e s o n a n c e line. N i s the n u m b e r density of the a t o m s p e r c m - 3 , and k j j , i s a n u m e r i c a l factor containing the s t a t i s t i c a l weights. 153
Volume 21, number 2
PHYSICS LETTERS
Schiff and P e k e r i s [10] have c a l c u l a t e d with very great accuracy a theoretical oscillator s t r e n g t h fo r the helium r e s o n a n c e line. This v a l u e m u s t now be r e g a r d e d a s r e l i a b l e , e s p e cially in v i e w of its a g r e e m e n t with many other c a l c u l a t i o n s . H e l i u m thus o f f e r s a unique opportunity for the c o m p a r i s o n of t h e o r y with e x p e r i m e n t . Using t h e i r f - v a l u e of 0.2761 + 0.0001 in the above e x p r e s s i o n k j j , ( e x p e r i m e n t a l ) is found to be 1.44 + 0.09. A f t e r c o r r e c t i o n of e r r o r s the following r e s u l t s for k j j , ( t h e o r e ti c a l) a r e obtained: Bryon and F ole y [1] 1.33; Watanabe [4,7] 1.44; Ali and G r i e m [5,8] 1.41. Omont [6] has the va l ue 1.45 which he c o n s i d e r s should be a c c u r a t e to ± 0.04. In o r d e r to explain an a p p a r e n t d i s c r e p a n c y with t h e i r t h e o r y , Ali and G r i e m have d i s c u s s e d a n u m b e r of o b j e c t i o n s to the i n t e r p r e t a t i o n of the e x p e r i m e n t a l work d e s c r i b e d in ref. 9. Following the e l i m i n a t i o n of s o m e e r r o r s in t h e i r c a l c u l a t i o n s this d i s c r e p a n c y no l o n g e r e x i s t s . C o n s i d e r a t i o n of the r a n g e of gas density, t e m p e r a t u r e and the width of p r o f i l e s etc. shows that the c u r r e n t e x p e r i m e n t s a r e well within the l i m i t s of the i m p a c t a p p r o x i m a t i o n . Additionally, eff ect s such as ' c o l l i s i o n a l n a r r o w ing' can be shown to be n e g l i g i b l e . The a g r e e m e n t between the e x p e r i m e n t a l value and the l a t e s t t h e o r e t i c a l v a l u e s is thus e x c e l l e n t . One may conclude that the t e c h n i q u e of m e a s u r e m e n t of r e s o n a n c e b r o a d e n i n g now o f f e r s a m e a n s for the a c c u r a t e d e t e r m i n a t i o n of o s c i l l a t o r s t r e n g t h s of r e s o n a n c e l i n e s . On the other hand at e x t r e m e l y low d e n s i t i e s , w h e r e the b r o a d e n i n g is of the s a m e o r d e r as the n a t u r a l width, t h e o r y does not a g r e e with e x p e r i m e n t . C o n t r a r y to what had been a s s u m e d in r e f . 9, the l i n e a r e x t r a p o l a t i o n of b r o a d e n i n g plots to z e r o density does not give the c o r r e c t value f o r the n a t u r a l width of the f r e e a t o m . A q u a l i t a t i v e
154
1May 1966
explanation of this s u r p r i s i n g fact has r e c e n t l y been given in t e r m s of a doubling of the t r a n s i tion p r o b a b i l i t y f r o m the s y m m e t r i c state of coupled a t o m s [11]. The d i s c r e p a n c y between the t h e o r e t i c a l o s c i l l a t o r s t r e n g t h of Schiff and P e k e r i s and the l a r g e r value d e r i v e d by l i n e a r e x t r a p o l a t i o n to z e r o p r e s s u r e [9] must now be c o n s i d e r e d as s t r o n g e v i d e n c e for this e n h a n c e m e n t effect due to coupling. T h i s effect, which is i m p o r t a n t only at v e r y low d e n s i t i e s , is not included in c u r r e n t t h e o r i e s of p r e s s u r e b r o a d e n ing. It would not significantly a l t e r the slope of b r o a d e n i n g plots obtained o v e r a l a r g e r a n g e of density. I wish to thank P r o f e s s o r A. Omont for showing us his l a t e s t p a p e r b e f o r e publication, and a l s o P r o f e s s o r H. R. G r i e m for c o m m u n i c a t i n g the r e s u l t s of an a m e n d e d calculation. I am indebted to Dr. H. G. Kuhn for his constant i n t e r e s t in this work.
References 1. F.W. Byron Jr. and H.M.Foley, Phys. Rev. 134 (1964) A625. 2. A.Omont. J. de Physique. 26 (1965) 26. 3. G.P.Reck, H.Takebeand C.A.Mead, Phys. Rev. 137 (1965) A683. 4. T.Watanabe, Phys. Rev. 138 (1965)A1573. 5. A . W . A l i a n d H . R . G r i e m , Phys. Rev. 140 (1965) A1044. 6. A.Omont, Comptes Rendus, 262 (1966) B190. 7. T.Watanabe, Phys. Rev. 140 (1965) AB5. 8. A. W. Ali and H. R. Griem~ private communication. to be published. 9. H.G.Kulm and J.M.Vaughan, Proc. Roy. Soc. (London) A277 (1964) 297. 10. B.Schiff and C . L . P e k e r i s , Phys. Rev. 134 (1964) A638. 11. H.G.Kuhn, E.L.Lewis and J.M.Vaughan, Phys. Rev. Letters 15 (1965) 687.