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Long-range interaction forces between inert gas atoms
Volume 13, number 2
PHYSICS LETTERS
be shown. This is in line with e a r l i e r o b s e r v a tions by Broida and Kane [5] who c a r e f u l l y i n v e s tigated the e m i s s i o n s p e c t r u m of w a t e r vapour in e l e c t r o d e l e s s d i s c h a r g e s and a l r e a d y were able to prove the e x i s t e n c e of an a b n o r m a l d i s t r i b u t i o n over the r o t a t i o n a l l e v e l s of the excited O H - r a d i cal. According to our i n t e r p r e t a t i o n the line at 78 ~ should be due to [6] the t r a n s i t i o n K = 4 ~ K = 3 of the v i b r a t i o n a l level V' = 1. The line at 28 ~ should be due to the t r a n s i t i o n g = 13 ~ K = 12 of V' = 3. F u r t h e r support to this i n t e r p r e t a t i o n is obtained f r o m the fact that a n u m b e r of l i n e s which w e r e o b s e r v e d by Gebbie et al. but which in our e x p e r i m e n t s probably were s u p p r e s s e d due to the use of a q u a r t z - and a KBr-window r e s p e c t i v e l y can also be e a s i l y int e r p r e t e d as belonging to the O H - s p e c t r u m .
LONG-RANGE
INTERACTION
15 November 1964
We gladly acknowledge helpful d i s c u s s i o n s with D r s M. J. Offerhaus and W. C. Nieuwpoort.
References 1. C.K.N.Patel, W.L.Faust, R.A. McFarlane and C.G.B.Gan'ett, Appl.Phys.Letters 4 (1964) 18. 2. C.K.N.Patel, W.L.Faust, R.A. McFarlane, C.G.B. Garrett, Proc. I. E . E . E . 52 (1964) 713. 3. A. Crocker, H.A. Gebbie, M. F. Kimmett and L. E. S. Mathias, Nature 201 (1964) 250. 4. H.A.Gebbie, N.W.B.Stone, F.D.Findlay, Nature 202 (1964) 685. 5. H.P.Broida and W.R.Kane, Phys.Rev. 89 (1953) 1053. 6. G.H. Dieke and N. M. Crosswhite, J. Quant. Spectrosc. Radiat. Transf. 2 (1962) 97.
FORCES
BETWEEN
INERT
GAS ATOMS
J. A. BARKER and P. J. LEONARD Division of Physical Chemistry, C. S.I.R. O. Melbourne, Australia Received 19 October 1964
The l o n g - r a n g e i n t e r a c t i o n potential between a pair of a t o m s is given by s e c o n d - o r d e r p e r t u r b a tion theory [1] as - C a b r - 6 , where the constant Cab can be c a l c u l a t e d f r o m the o s c i l l a t o r s t r e n g t h s of t r a n s i t i o n s f r o m the ground s t a t e s of the a t o m s concerned. Dalgarno and Kingston [2] c a l c u l a t e d v a l u e s of Cab for all p a i r s of a t o m s s e l e c t e d f r o m H, He, Ne and A u s i n g o s c i l l a t o r s t r e n g t h s chosen to s a t i s f y s u m r u l e s [3] which had been evaluated f r o m r e f r a c t i v e index data etc. [4] ; a few theor e t i c a l and e x p e r i m e n t a l r e s u l t s on a b s o r p t i o n s p e c t r a were then a v a i l a b l e as checks. Recently much new i n f o r m a t i o n on i n e r t gas a b s o r p t i o n s p e c t r a has b e c o m e available [5] and e x p e r i m e n t a l o s c i l l a t o r s t r e n g t h s can now be used d i r e c t l y to calculate Cab, with additional use of s u m r u l e s to i n c r e a s e a c c u r a c y . We p r e s e n t here r e s u l t s for like p a i r s of i n e r t gas a t o m s including Kr and Xe *. A l a t e r p a p e r will give r e s u l t s for unlike p a i r s and t h r e e - b o d y i n t e r a c t i o n c o n s t a n t s , as well as d e t a i l s of the o s c i l l a t o r s t r e n g t h s . Available data ** on the a b s o r p t i o n s p e c t r a were collated, the m o s t p l a u s i b l e d i s c r e t e line s t r e n g t h s were chosen and c u r v e s were fitted to the continuous s p e c t r u m b e t w e e n s u c c e s s i v e edges (with r e a s o n a b l e e s t i m a t e s where data
were m i s s i n g , u s u a l l y round 40-100~). The s u m r u l e s were evaluated and the data modified within the e x p e r i m e n t a l e r r o r to i m p r o v e a g r e e m e n t with the t h e o r e t i c a l s u m s ; finally Cab was evaluated with the b e s t e s t i m a t e s of o s c i l l a t o r s t r e n g t h s . The c o n s i s t e n c y of the a b s o r p t i o n d a ~ $* with s u m s d e r i v e d f r o m r e f r a c t i v e indices, Verdet c o n s t a n t s and d i e l e c t r i c c o n s t a n t s [4] i s ~m~/sfying (see table 1) ; the a p p a r e n t inability of t~e a b s o r p tion m e a s u r e m e n t s to account for the o b s e r v e d d i s p e r s i o n [5, 6] is r e s o l v e d by r e c e n t e x p e r i m e n t s . The s u m s S(2) and S(1) depend p r i m a r i l y on the X - r a y s p e c t r u m which does not c o n t r i b u t e a p p r e c i a b l y t o Cab. The t h e o r e t i c a l v a l u e s a r e n o n - r e l a t i v i s t i c [7, 8] ; s i n c e ~these s u m s diverge in r e l a t i v i s t i c theory [8] little attention is paid to them here. F o r He alone an a c c u r a t e t h e o r e t i cal value [7] of S(-1) was used a s a c o n s t r a i n t on the d a t a ; in other c a s e s only S(0), S(-2) and S(-4) were used. The v a l u e s of S(-1) and S(-6) a g r e e * While this work was being completed, a n e w p a p e r by A.E.Kingston on the same subject was announced for publication in Phys.Rev. ** In this letter we have not space to l i s t all sources of data used; complete references will appear in a subsequent paper. 127
Volume 13, number 2
PHYSICS
LETTERS
Table
_____
k=-6 __-__
0.09756
0.09347
0.03228 0.02982
9.995 10
1.897 1.925
0.1798 0.1801
0.07788 0.07826
(a) (b)
18.000 18
4.437 5.257
0.6701 0.6675 2.769 2.773
1.681 I .721
1.349 1.687
I:;
35.999 36
6.350 8.167
4.198 4.198
3.418 3.381
3.713 5.366
I:;
54 .ooo 54
8.807 12.576
6.694 6.830
8.052 7.422
13.090 17.916
I$
> 4.8 x lo4 104000 [8]
1.997 2
8.18 8.167 [7] 330.4 293.7
[8]
Kr Xe
-
(a) Present results. (b) Comparison: S(0) = 2 (Kuhn-Reiche-Thomas
Interaction
(a)
(b)
Present results
._ 1.49 6.43 66.71 130.8 276.7
Dal&no and Kingston [2] 1.46 1121 6.63 68.08
sum rule);
(C)
LennardJones potl. (91
125.7 297.3 608.3
all other S(k) taken from ref. 4 except where marked
Table 2 constant cab in atomic
units.
@I
0.3
Kihara potential [ 91
KirkwoodMffller [ 1 l]
~_..
Smith [lo] using transport properties in addition found for argon cab = 63.7 a.u., in good agreement with our value. The theoretical approximations in columns (e-g) are taken from Salem [12]. The KirkwoodMUller formula is known to give an upper limit for cab, but the extent to which it overestimates was not realised while the L-J potential was considered sufficiently accurate. Most striking is the accuracy of the Slater-Kirkwood approximation, especially for the heavier atoms in which the assumed distinction between “inner” and “outer” electrons is more valid than in the lighter ones.
(4 L,ondon
(9) [ 111
Slater Kirkwocd [ 111
_____
55.5 [ll] 102.5 222.8
as well as could be expected with the interpolated and extrapolated values of ref. 4. Cur results for Cab are given in table 2; we estimate an error of * 3% for Xe and considerably less for the other gases. Agreement with ref. 2 is satisfactory. Comparison with empirical estimates from second virial coefficients [9] confirms recent evidence [9, lo] that the two-parameter Lennard- Jones potential is insufficiently flexible ; values from the three-parameter Kihara potential are much closer to ours. Barker, Fock and
128
k=-4
k=-2
0.3478 0.3463
Ne
He Ne A Kr Xe
k=-1
k=O
0.7524 0.7525 [ 71
120 .o 121.336 [7]
He
A
1
_~ k=l
k=2
15 November 1964
1.70 12 135 295 730
1.29 4.88 57.9 112 243
1.74 8.10 67.0 125 259
We are extremely grateful to Dr. J. A. Ft. Samson, Dr. Anne Pery- Thorne, Dr. K. Madden and Dr. P. H. Metzger for their kindness in supplying us with unpublished data and essential information. 1. See e.g. T.Kihara, Adv. Chem. Phys. 1 (1958) 268. Proc. Phys.Soc. 2. A.DaIgarno and A.E.Kingston, 78 (1961) 607. 3. A.DaIgarno and N.Lynn, Proc.Phys.Soc.A70 (1957) 802. 4. A.DaIgarno and A. E. Kingston, Proc. Roy. Sot. A259 (1960) 424. 5. K. L. Wolf and K. F. Herzfeld, Geiger-Scheel Handb.d. Phys. 20 (1928) 490. 6. A. Pery-Thorne and W. R.S. Garton, Proc. Phys, Sot. 76 (1960) 833. Phys.Rev.115 (1959) 1216. 7. C.L.Pekeris, Phys.Rev. 135 (1964) 8. K.R.Piech and J.S.Levinger, A332 and J. M. Prausnitz, 3. Chem. Phys. 9. A.E.Sherwood 41 (1964) 429. 10. J.A. Barker, W. Fock and F.Smith, Phys. Fluids 7 (1964) 897. 11. L . SaIem , Mol. Phys. 3 (1960) 441. 12. Cf. also a value of 1.540 a.~. obtained by 0. Sinanoglu, Adv.Chem. Phys. 6 (1964) 409, using a non-empirical many-electron theory.
PHYSICS LETTERS
be shown. This is in line with e a r l i e r o b s e r v a tions by Broida and Kane [5] who c a r e f u l l y i n v e s tigated the e m i s s i o n s p e c t r u m of w a t e r vapour in e l e c t r o d e l e s s d i s c h a r g e s and a l r e a d y were able to prove the e x i s t e n c e of an a b n o r m a l d i s t r i b u t i o n over the r o t a t i o n a l l e v e l s of the excited O H - r a d i cal. According to our i n t e r p r e t a t i o n the line at 78 ~ should be due to [6] the t r a n s i t i o n K = 4 ~ K = 3 of the v i b r a t i o n a l level V' = 1. The line at 28 ~ should be due to the t r a n s i t i o n g = 13 ~ K = 12 of V' = 3. F u r t h e r support to this i n t e r p r e t a t i o n is obtained f r o m the fact that a n u m b e r of l i n e s which w e r e o b s e r v e d by Gebbie et al. but which in our e x p e r i m e n t s probably were s u p p r e s s e d due to the use of a q u a r t z - and a KBr-window r e s p e c t i v e l y can also be e a s i l y int e r p r e t e d as belonging to the O H - s p e c t r u m .
LONG-RANGE
INTERACTION
15 November 1964
We gladly acknowledge helpful d i s c u s s i o n s with D r s M. J. Offerhaus and W. C. Nieuwpoort.
References 1. C.K.N.Patel, W.L.Faust, R.A. McFarlane and C.G.B.Gan'ett, Appl.Phys.Letters 4 (1964) 18. 2. C.K.N.Patel, W.L.Faust, R.A. McFarlane, C.G.B. Garrett, Proc. I. E . E . E . 52 (1964) 713. 3. A. Crocker, H.A. Gebbie, M. F. Kimmett and L. E. S. Mathias, Nature 201 (1964) 250. 4. H.A.Gebbie, N.W.B.Stone, F.D.Findlay, Nature 202 (1964) 685. 5. H.P.Broida and W.R.Kane, Phys.Rev. 89 (1953) 1053. 6. G.H. Dieke and N. M. Crosswhite, J. Quant. Spectrosc. Radiat. Transf. 2 (1962) 97.
FORCES
BETWEEN
INERT
GAS ATOMS
J. A. BARKER and P. J. LEONARD Division of Physical Chemistry, C. S.I.R. O. Melbourne, Australia Received 19 October 1964
The l o n g - r a n g e i n t e r a c t i o n potential between a pair of a t o m s is given by s e c o n d - o r d e r p e r t u r b a tion theory [1] as - C a b r - 6 , where the constant Cab can be c a l c u l a t e d f r o m the o s c i l l a t o r s t r e n g t h s of t r a n s i t i o n s f r o m the ground s t a t e s of the a t o m s concerned. Dalgarno and Kingston [2] c a l c u l a t e d v a l u e s of Cab for all p a i r s of a t o m s s e l e c t e d f r o m H, He, Ne and A u s i n g o s c i l l a t o r s t r e n g t h s chosen to s a t i s f y s u m r u l e s [3] which had been evaluated f r o m r e f r a c t i v e index data etc. [4] ; a few theor e t i c a l and e x p e r i m e n t a l r e s u l t s on a b s o r p t i o n s p e c t r a were then a v a i l a b l e as checks. Recently much new i n f o r m a t i o n on i n e r t gas a b s o r p t i o n s p e c t r a has b e c o m e available [5] and e x p e r i m e n t a l o s c i l l a t o r s t r e n g t h s can now be used d i r e c t l y to calculate Cab, with additional use of s u m r u l e s to i n c r e a s e a c c u r a c y . We p r e s e n t here r e s u l t s for like p a i r s of i n e r t gas a t o m s including Kr and Xe *. A l a t e r p a p e r will give r e s u l t s for unlike p a i r s and t h r e e - b o d y i n t e r a c t i o n c o n s t a n t s , as well as d e t a i l s of the o s c i l l a t o r s t r e n g t h s . Available data ** on the a b s o r p t i o n s p e c t r a were collated, the m o s t p l a u s i b l e d i s c r e t e line s t r e n g t h s were chosen and c u r v e s were fitted to the continuous s p e c t r u m b e t w e e n s u c c e s s i v e edges (with r e a s o n a b l e e s t i m a t e s where data
were m i s s i n g , u s u a l l y round 40-100~). The s u m r u l e s were evaluated and the data modified within the e x p e r i m e n t a l e r r o r to i m p r o v e a g r e e m e n t with the t h e o r e t i c a l s u m s ; finally Cab was evaluated with the b e s t e s t i m a t e s of o s c i l l a t o r s t r e n g t h s . The c o n s i s t e n c y of the a b s o r p t i o n d a ~ $* with s u m s d e r i v e d f r o m r e f r a c t i v e indices, Verdet c o n s t a n t s and d i e l e c t r i c c o n s t a n t s [4] i s ~m~/sfying (see table 1) ; the a p p a r e n t inability of t~e a b s o r p tion m e a s u r e m e n t s to account for the o b s e r v e d d i s p e r s i o n [5, 6] is r e s o l v e d by r e c e n t e x p e r i m e n t s . The s u m s S(2) and S(1) depend p r i m a r i l y on the X - r a y s p e c t r u m which does not c o n t r i b u t e a p p r e c i a b l y t o Cab. The t h e o r e t i c a l v a l u e s a r e n o n - r e l a t i v i s t i c [7, 8] ; s i n c e ~these s u m s diverge in r e l a t i v i s t i c theory [8] little attention is paid to them here. F o r He alone an a c c u r a t e t h e o r e t i cal value [7] of S(-1) was used a s a c o n s t r a i n t on the d a t a ; in other c a s e s only S(0), S(-2) and S(-4) were used. The v a l u e s of S(-1) and S(-6) a g r e e * While this work was being completed, a n e w p a p e r by A.E.Kingston on the same subject was announced for publication in Phys.Rev. ** In this letter we have not space to l i s t all sources of data used; complete references will appear in a subsequent paper. 127
Volume 13, number 2
PHYSICS
LETTERS
Table
_____
k=-6 __-__
0.09756
0.09347
0.03228 0.02982
9.995 10
1.897 1.925
0.1798 0.1801
0.07788 0.07826
(a) (b)
18.000 18
4.437 5.257
0.6701 0.6675 2.769 2.773
1.681 I .721
1.349 1.687
I:;
35.999 36
6.350 8.167
4.198 4.198
3.418 3.381
3.713 5.366
I:;
54 .ooo 54
8.807 12.576
6.694 6.830
8.052 7.422
13.090 17.916
I$
> 4.8 x lo4 104000 [8]
1.997 2
8.18 8.167 [7] 330.4 293.7
[8]
Kr Xe
-
(a) Present results. (b) Comparison: S(0) = 2 (Kuhn-Reiche-Thomas
Interaction
(a)
(b)
Present results
._ 1.49 6.43 66.71 130.8 276.7
Dal&no and Kingston [2] 1.46 1121 6.63 68.08
sum rule);
(C)
LennardJones potl. (91
125.7 297.3 608.3
all other S(k) taken from ref. 4 except where marked
Table 2 constant cab in atomic
units.
@I
0.3
Kihara potential [ 91
KirkwoodMffller [ 1 l]
~_..
Smith [lo] using transport properties in addition found for argon cab = 63.7 a.u., in good agreement with our value. The theoretical approximations in columns (e-g) are taken from Salem [12]. The KirkwoodMUller formula is known to give an upper limit for cab, but the extent to which it overestimates was not realised while the L-J potential was considered sufficiently accurate. Most striking is the accuracy of the Slater-Kirkwood approximation, especially for the heavier atoms in which the assumed distinction between “inner” and “outer” electrons is more valid than in the lighter ones.
(4 L,ondon
(9) [ 111
Slater Kirkwocd [ 111
_____
55.5 [ll] 102.5 222.8
as well as could be expected with the interpolated and extrapolated values of ref. 4. Cur results for Cab are given in table 2; we estimate an error of * 3% for Xe and considerably less for the other gases. Agreement with ref. 2 is satisfactory. Comparison with empirical estimates from second virial coefficients [9] confirms recent evidence [9, lo] that the two-parameter Lennard- Jones potential is insufficiently flexible ; values from the three-parameter Kihara potential are much closer to ours. Barker, Fock and
128
k=-4
k=-2
0.3478 0.3463
Ne
He Ne A Kr Xe
k=-1
k=O
0.7524 0.7525 [ 71
120 .o 121.336 [7]
He
A
1
_~ k=l
k=2
15 November 1964
1.70 12 135 295 730
1.29 4.88 57.9 112 243
1.74 8.10 67.0 125 259
We are extremely grateful to Dr. J. A. Ft. Samson, Dr. Anne Pery- Thorne, Dr. K. Madden and Dr. P. H. Metzger for their kindness in supplying us with unpublished data and essential information. 1. See e.g. T.Kihara, Adv. Chem. Phys. 1 (1958) 268. Proc. Phys.Soc. 2. A.DaIgarno and A.E.Kingston, 78 (1961) 607. 3. A.DaIgarno and N.Lynn, Proc.Phys.Soc.A70 (1957) 802. 4. A.DaIgarno and A. E. Kingston, Proc. Roy. Sot. A259 (1960) 424. 5. K. L. Wolf and K. F. Herzfeld, Geiger-Scheel Handb.d. Phys. 20 (1928) 490. 6. A. Pery-Thorne and W. R.S. Garton, Proc. Phys, Sot. 76 (1960) 833. Phys.Rev.115 (1959) 1216. 7. C.L.Pekeris, Phys.Rev. 135 (1964) 8. K.R.Piech and J.S.Levinger, A332 and J. M. Prausnitz, 3. Chem. Phys. 9. A.E.Sherwood 41 (1964) 429. 10. J.A. Barker, W. Fock and F.Smith, Phys. Fluids 7 (1964) 897. 11. L . SaIem , Mol. Phys. 3 (1960) 441. 12. Cf. also a value of 1.540 a.~. obtained by 0. Sinanoglu, Adv.Chem. Phys. 6 (1964) 409, using a non-empirical many-electron theory.