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Vibrational level inversion population of polyatomic molecules, Co2laser
Volume25A. number 2
PHYSICS
LETTERS
c u r r e n t i s t h e n Ieo =Jeo. F e f f . T h e d e p e n d e n c e of t h e p r o b e s a t u r a t i o n c u r r e n t s u p o n t h e p r o b e orientation can now be explained by the simult a n e o u s c h a n g e of t h e e f f e c t i v e p r o b e a r e a F e f f. 3. E v e n a t p o s i t i v e d . c . p r o b e p o t e n t i a l s n o t all electrons entering the sheath are collected by the probe. The electron collection is "impact parameter" limited by the fact that the plane probe is finite and the probe characteristic theref o r e r e s e m b l e s t h a t of a s p h e r i c a l p r o b e [ 8 , 9 ] . These conclusions are confirmed by our experi m e n t a l r e s u l t s [6]. If we d e t e r m i n e d a s b e f o r e the plasma potential by the extrapolated "knee" of t h e c h a r a c t e r i s t i c ( i n t e r s e c t i o n p o i n t of t h e t a n g e n t s to e l e c t r o n d e c e l e r a t i o n a n d s a t u r a t i o n c u r r e n t i n t h e s e m i l o g a r i t h m i c p l o t of t h e e l e c tron current characteristic), but the thermal electron current Ieo from the electron current m e a s u r e d a t p l a s m a p o t e n t i a l , we find t h a t L a n g muir probe and r.f. probe density measurements c o i n c i d e t h e n w i t h i n t h e e r r o r l i m i t s (the r e l a t i v e e r r o r s b e i n g 50% f o r t h e L a n g m u i r p r o b e a n d 5 to 10% f o r t h e r . f . p r o b e i n o u r e x p e r i m e n t ) . T h e a u t h o r w i s h e s to e x p r e s s h i s a p p r e c i a t i o n to P r o f . D r . A. S c h l [ i t e r , D r . G. v. G i e r k e a n d D r . G. M i i l l e r f o r m o s t v a l u a b l e d i s c u s s i o n s and f o r i n t e r e s t in t h i s w o r k . T h e a u t h o r i s g r e a t l y i n d e b t e d to D r . R . S . H a r p f o r i n t e r p r e t i n g t h e f o c u s point, which was observed experimentally by the a u t h o r , a n d f o r s u g g e s t i n g t h e u s e of a low f r e q u e n c y m o d u l a t e d d . c . v o l t a g e i n s t e a d of s e v e r a l fixed d.c. voltages as used before. The author
31 July 1967
t h a n k s D r . A. H e i s e n a n d D r . H. K. W i m m e l f o r helpful discussions.
References 1. K.Takayama. H.Ikegami and S.Miyazaki, Phys. Rev. L e t t e r s 5 (1960) 238. 2. G . P e t e r . G.Miiller and H . H . R a b b e n . Compt. Rend VI e Conf. Int. s u r les Ph~nom~nes d'ionisation dans les gaz. P a r i s . 1963. IV, p.147; H . M . M a y e r . ibid. IV. p.129; G. v. Gierke. G. MUller. G. P e t e r and H. H. Rabben, Z. Naturforschg. 19a (1964) 1107: A . M e s s i a e n , Compt. Rend. 259 (1964) 1710; R . S . H a r p and F.W. Crax~¢ord. J. Appl. Phys. 35 (1964) 3436; J. Uramoto, J. Fujita. H. Ikegami and K. Takayama, I P P J Institute of P l a s m a Physics, Nagoya Univ., Nagoya. Japan, 19 (Dec. 1963); D. Lepechinsky, A. M e s s i a e n and P.Rolland, J. Nucl. Energy C 8 (1966) 165; T. Dote and T . I c h i m i y a , J. Appl. Phys. 36 (1965) 1866. 3. H . K . W i m m e l , Z. Naturforschg. 19a (1964) 1099. 4. P . E . Vandenplas and R. W. Gould, P r o c . V. th Int. Conf. on Ionization phen. in gases. Munich 1961, II, p. 1470; Physica 28 (1962) 357; A. Messiaen and P. E. Vandenplas. Physica 30 (1964) 303; S. M. Levitskii and I. P. Shashurin, Zh. Tehkn. Fiz. 33 (1963) 429. 5. W.O.Schumann, Ann. Physik 43 (1943) 369; F . S c h n e i d e r , Z. Angew. Physik 6 (1954) 456; R.W.Gould, Phys. L e t t e r s 11 (1965) 236. 6. G. P e t e r , T h e s i s , University of Munich (1966), to be published in Z. Naturforschg. (1967). 7. E.Guilino, Phys. Verhandlungen 3 (1963) 87. 8. G.Medicus, J. Appl. Phys. 32 (1961} 2512. 9. S . v . G o e l e r , Ann. Physik 15 (1965) 321.
* * * * *
VIBRATIONAL LEVEL INVERSION POLYATOMIC MOLECULES, B. F . G O R D I E T Z ,
POPULATION CO2LASER
OF
N . N . S O B O L E V , V . V . SOKOVIKOV a n d L. A. S H E L E P I N
P. N. Lebedev Physical Institute of the USSR Academy of Sciences Received 16 June 1967
A method is proposed for a calculation of vibration level populations of polyatomic molecules in the e l e c t r o n i c ground state.
In w o r k i n g m o l e c u l a r l a s e r s t h e s e l e c t i v e e n e r g y p u m p i n g f o r a g r o u p of l e v e l s a s w e l l as the vibration energy relaxation velocity is rather less than the quanta exchange rate in each
m o d e of v i b r a t i o n s [1]. F r o m t h i s a n e x i s t e n c e of the Boltzmann's distribution follows for each m o d e of v i b r a t i o n s " i " h a v i n g t e m p e r a t u r e s T i w h i c h a r e b e i n g d e f i n e d b y t h e t o t a l s t o c k of 173
Volume25A. number 2
PHYSICS LETTERS
vibration quanta in "i". The introduction of vibration temperatures considerably simplifies the problem of population calculations. For an example we shall consider a CO2 laser. The basic physical processes producing inversion and oscillation in molecular lasers were set up in refs. [2]. We shall refer the indexes 1,2,3 to the symmetrical, bending, and asymmetrical mode of vibration respectively. Let the F-number of quanta per second be pumped with the energy of h u3 into the asymmetrical mode of vibrations. We shall restrict ourselves with the calculation of the level population in the non-oscillating system. The relaxation of the vibration energy occurs due to the following most important p r o c e s s e s [3]: hu 3 -~3hv 2 + AE32 , with a probability of W32; hv 3 - ' h v 1 + hv 2 + AF.3(1,2) , with a probability of W3(1,2); h v l -~2hv2 + AE12, with a probability of W12; hv 2 ~ AE20 , with a probability of W20. Here v i a r e frequencies of normal vibrations; AEij a r e corresponding e n e r gy differences t r a n s f o r m i n g into a kinetic one. The radiation p r o c e s s e s may be neglected [4]. The quantity of the vibration energy E i per each mode of vibrations is given by the formulae as follows:
E i=hv i
XiNo
~
2XiN ° Ei = hvi (1 - Xi)------3
i=1,3; (1)
i = 2,
dE3 = hv 3 { F -(W32+ W3(1, 2))X3No}; dt
hV2No I W3(1, 2 ) ( X 3 - X 1 X 2 e x p - A E 3 ( I ' 2 ) )
AE323
+ 3W32(X 3 -X22 e x p - ~ ] +
AEI2"
+ 2W12 (X 1 -X22 e x p - Z ~ - T ~ )
I
+
- W20 { 1 - exp ( - h v 2 / k T ) } (E 2 - E0); dE 1 I dt = hVlNo W3( 1, 2) (Xu\ v - X l X 2 exp
AE3(I,2) ~+ ~-~ /
AE12
- W12 ( X l - X~2 exp - ~ / N
} .
(2)
Here E~. is the bending vibration energy c o r r e s ponding to the gas t e m p e r a t u r e T. The probabilities of inversion p r o c e s s e s a r e determined f r o m the detailed balancing principle. In the stationary 174
':.,:~ ..../.
SO~
..,"
.
,/
/ / i
/
i ~O0
i $'eO
i 6#0
.F*~
Fig. 1. Dependence of laser level population versus gas temperature. case when T 1 and T 2 a r e small enough and when one can neglect an inverse flow of energy into the a s y m m e t r i c a l mode of vibrations, we get the solution of the s y s t e m (2) as follows: / "--k-T-} L~E12\ ~2v~-, X1 = W3(1,2) X3 + exp ~W12
X 2 =X 0 + 3(W32 + W3(1'2)) X 3,
where N o is quantity of molecules in the ground vibration state, X i = exp (- hvi/kTi). Consider balance equations for E l , E2, E3:
dE2
31 July 1967
(3)
w20 F X3 = No( W32 + W3(1,2))' hv2 where x O = exp ( - ~ ) .
Using the probabilities [3, 5], we evaluated level populations. The dependences of the 00°1 and 10°0 level populations N upon gas t e m p e r a t u r e a r e presented in fig. 1. The continuous curves correspond to the pumping of F = 2.1 × 1018, the dotted ones to F = 2.1 × 1017; the dot-dash line gives a dependence of the equilibrium 10°0 level population. Addition of He lowers the t e m p e r a t u r e s T 1 and T2, since the probability of W20 is increasing. The population calculation technique considered above can be extended to other polyatomic molecules as well. The p r e l i m i n a r y estimates point out to the following m i x t u r e s which a r e expected to be promising f r o m the point of view of obtaining inv e r s e population H2Se-N2, C2N2-N2, NH3-C2H2, COS.
Volume25A. number 2
PHYSICS
LETTERS
References
31 July 1967
3: K.F.Herzfeld. Disc. Faraday Soc.. N 33 (1965) 22. 4. H.Statz, C.L.Tang and G.F.Koster, J. Appl. Phys. 37 (1966} 4273. 5. T.L.Cottrell et al., Trans. Faraday Soc.. 62 (1966} 2655.
1. E.V.Stupoehenko, S.A. Loser, A . J . Osipov, Relaxation Processes in Shock Waves. "Nauka', Moscow, 1965. 2. N.N.Sobolov, V.V.Sokovikov. JETP Letters 4 303 (1966}; 5 (1967) 122. *****
THE
SHIFT
AND WIDTH OF ISOLATED THE ADIABATIC LIMIT
ION
LINES
IN
D. E. ROBERTS and J . DAVIS
Physics Department, Imperial College, London. S. W. 7. Received 17 June 1967
Calculations of the shift and width of isolated ion lines in the adiabatic limit are reported.
R e c e n t p r e s c r i p t i o n s f o r the c a l c u l a t i o n s of S t a r k widths of i s o l a t e d ion l in e s [1,2] include only the e f f e c t s of i n e l a s t i c p e r t u r b e r c o l l i s i o n s . H o w e v e r , it has b e e n indicated [3] that e l a s t i c c o l l i s i o n s can c o n t r i b u t e significantly to the widths of s o m e lines even when h y p e r b o l i c p e r . . t u r b e r o r b i t s a r e used for c a l c u l a t i n g the in e l a s t i c c r o s s s e c t i o n s . In addition, shifts a r e caused mainly by e l a s t i c c o l l i s i o n s and cannot t h e r e f o r e be e s t i m a t e d using e x i s t i n g p r e s c r i p tions u n l e s s they a r e i n f e r r e d f r o m a d i s p e r s i o n r e l a t i o n r e l a t i n g the shift and the width [4]. We have c a l c u l a t e d the width, W, and shift, d, of an i s o l a t e d line u s i n g the adiabatic t h e o r y of Lindholm and F o l e y [5] by employing h y p e r bolic r a t h e r than s t r a i g h t line p e r t u r b e r o r b i t s . Using a p a r a m e t r i c r e p r e s e n t a t i o n f o r the o r bit [6] we can e x p r e s s the p h a s e shift caused by a single p e r t u r b e r a s : -
T/(z, k) = 2k3
; arctan (k))
- ~ - + 1)+
,(1)
w h e r e p 3 = C 4 / v , k = P/Po and z = e 2 / m P o v2. v is the v e l o c i t y and p the i m p a c t p a r a m e t e r of the p e r t u r b e r . Po is the W e i s s k o p f r a d i u s and z is a m e a s u r e of the r a t i o 'of p e r t u r b e r potential e n e r gy to kinetic e n e r g y . C 4 i s the q u a d r a t i c Stark coefficient. W and d can now be e x p r e s s e d in the f o r m :
Idl j w, d
(:units
of Z~'Nevpo~)
~-
w Idl
I~,, .
.
.
4T ~
" t.e Do 0.4
NI 8
D.~ 6 t-o 4 o.,
.2
.
0.6 . . . . . .
.
.
I I I
w_.
I I !
,,v
o-oD
o.~
I I I
I IO
Fig. 1. 175
PHYSICS
LETTERS
c u r r e n t i s t h e n Ieo =Jeo. F e f f . T h e d e p e n d e n c e of t h e p r o b e s a t u r a t i o n c u r r e n t s u p o n t h e p r o b e orientation can now be explained by the simult a n e o u s c h a n g e of t h e e f f e c t i v e p r o b e a r e a F e f f. 3. E v e n a t p o s i t i v e d . c . p r o b e p o t e n t i a l s n o t all electrons entering the sheath are collected by the probe. The electron collection is "impact parameter" limited by the fact that the plane probe is finite and the probe characteristic theref o r e r e s e m b l e s t h a t of a s p h e r i c a l p r o b e [ 8 , 9 ] . These conclusions are confirmed by our experi m e n t a l r e s u l t s [6]. If we d e t e r m i n e d a s b e f o r e the plasma potential by the extrapolated "knee" of t h e c h a r a c t e r i s t i c ( i n t e r s e c t i o n p o i n t of t h e t a n g e n t s to e l e c t r o n d e c e l e r a t i o n a n d s a t u r a t i o n c u r r e n t i n t h e s e m i l o g a r i t h m i c p l o t of t h e e l e c tron current characteristic), but the thermal electron current Ieo from the electron current m e a s u r e d a t p l a s m a p o t e n t i a l , we find t h a t L a n g muir probe and r.f. probe density measurements c o i n c i d e t h e n w i t h i n t h e e r r o r l i m i t s (the r e l a t i v e e r r o r s b e i n g 50% f o r t h e L a n g m u i r p r o b e a n d 5 to 10% f o r t h e r . f . p r o b e i n o u r e x p e r i m e n t ) . T h e a u t h o r w i s h e s to e x p r e s s h i s a p p r e c i a t i o n to P r o f . D r . A. S c h l [ i t e r , D r . G. v. G i e r k e a n d D r . G. M i i l l e r f o r m o s t v a l u a b l e d i s c u s s i o n s and f o r i n t e r e s t in t h i s w o r k . T h e a u t h o r i s g r e a t l y i n d e b t e d to D r . R . S . H a r p f o r i n t e r p r e t i n g t h e f o c u s point, which was observed experimentally by the a u t h o r , a n d f o r s u g g e s t i n g t h e u s e of a low f r e q u e n c y m o d u l a t e d d . c . v o l t a g e i n s t e a d of s e v e r a l fixed d.c. voltages as used before. The author
31 July 1967
t h a n k s D r . A. H e i s e n a n d D r . H. K. W i m m e l f o r helpful discussions.
References 1. K.Takayama. H.Ikegami and S.Miyazaki, Phys. Rev. L e t t e r s 5 (1960) 238. 2. G . P e t e r . G.Miiller and H . H . R a b b e n . Compt. Rend VI e Conf. Int. s u r les Ph~nom~nes d'ionisation dans les gaz. P a r i s . 1963. IV, p.147; H . M . M a y e r . ibid. IV. p.129; G. v. Gierke. G. MUller. G. P e t e r and H. H. Rabben, Z. Naturforschg. 19a (1964) 1107: A . M e s s i a e n , Compt. Rend. 259 (1964) 1710; R . S . H a r p and F.W. Crax~¢ord. J. Appl. Phys. 35 (1964) 3436; J. Uramoto, J. Fujita. H. Ikegami and K. Takayama, I P P J Institute of P l a s m a Physics, Nagoya Univ., Nagoya. Japan, 19 (Dec. 1963); D. Lepechinsky, A. M e s s i a e n and P.Rolland, J. Nucl. Energy C 8 (1966) 165; T. Dote and T . I c h i m i y a , J. Appl. Phys. 36 (1965) 1866. 3. H . K . W i m m e l , Z. Naturforschg. 19a (1964) 1099. 4. P . E . Vandenplas and R. W. Gould, P r o c . V. th Int. Conf. on Ionization phen. in gases. Munich 1961, II, p. 1470; Physica 28 (1962) 357; A. Messiaen and P. E. Vandenplas. Physica 30 (1964) 303; S. M. Levitskii and I. P. Shashurin, Zh. Tehkn. Fiz. 33 (1963) 429. 5. W.O.Schumann, Ann. Physik 43 (1943) 369; F . S c h n e i d e r , Z. Angew. Physik 6 (1954) 456; R.W.Gould, Phys. L e t t e r s 11 (1965) 236. 6. G. P e t e r , T h e s i s , University of Munich (1966), to be published in Z. Naturforschg. (1967). 7. E.Guilino, Phys. Verhandlungen 3 (1963) 87. 8. G.Medicus, J. Appl. Phys. 32 (1961} 2512. 9. S . v . G o e l e r , Ann. Physik 15 (1965) 321.
* * * * *
VIBRATIONAL LEVEL INVERSION POLYATOMIC MOLECULES, B. F . G O R D I E T Z ,
POPULATION CO2LASER
OF
N . N . S O B O L E V , V . V . SOKOVIKOV a n d L. A. S H E L E P I N
P. N. Lebedev Physical Institute of the USSR Academy of Sciences Received 16 June 1967
A method is proposed for a calculation of vibration level populations of polyatomic molecules in the e l e c t r o n i c ground state.
In w o r k i n g m o l e c u l a r l a s e r s t h e s e l e c t i v e e n e r g y p u m p i n g f o r a g r o u p of l e v e l s a s w e l l as the vibration energy relaxation velocity is rather less than the quanta exchange rate in each
m o d e of v i b r a t i o n s [1]. F r o m t h i s a n e x i s t e n c e of the Boltzmann's distribution follows for each m o d e of v i b r a t i o n s " i " h a v i n g t e m p e r a t u r e s T i w h i c h a r e b e i n g d e f i n e d b y t h e t o t a l s t o c k of 173
Volume25A. number 2
PHYSICS LETTERS
vibration quanta in "i". The introduction of vibration temperatures considerably simplifies the problem of population calculations. For an example we shall consider a CO2 laser. The basic physical processes producing inversion and oscillation in molecular lasers were set up in refs. [2]. We shall refer the indexes 1,2,3 to the symmetrical, bending, and asymmetrical mode of vibration respectively. Let the F-number of quanta per second be pumped with the energy of h u3 into the asymmetrical mode of vibrations. We shall restrict ourselves with the calculation of the level population in the non-oscillating system. The relaxation of the vibration energy occurs due to the following most important p r o c e s s e s [3]: hu 3 -~3hv 2 + AE32 , with a probability of W32; hv 3 - ' h v 1 + hv 2 + AF.3(1,2) , with a probability of W3(1,2); h v l -~2hv2 + AE12, with a probability of W12; hv 2 ~ AE20 , with a probability of W20. Here v i a r e frequencies of normal vibrations; AEij a r e corresponding e n e r gy differences t r a n s f o r m i n g into a kinetic one. The radiation p r o c e s s e s may be neglected [4]. The quantity of the vibration energy E i per each mode of vibrations is given by the formulae as follows:
E i=hv i
XiNo
~
2XiN ° Ei = hvi (1 - Xi)------3
i=1,3; (1)
i = 2,
dE3 = hv 3 { F -(W32+ W3(1, 2))X3No}; dt
hV2No I W3(1, 2 ) ( X 3 - X 1 X 2 e x p - A E 3 ( I ' 2 ) )
AE323
+ 3W32(X 3 -X22 e x p - ~ ] +
AEI2"
+ 2W12 (X 1 -X22 e x p - Z ~ - T ~ )
I
+
- W20 { 1 - exp ( - h v 2 / k T ) } (E 2 - E0); dE 1 I dt = hVlNo W3( 1, 2) (Xu\ v - X l X 2 exp
AE3(I,2) ~+ ~-~ /
AE12
- W12 ( X l - X~2 exp - ~ / N
} .
(2)
Here E~. is the bending vibration energy c o r r e s ponding to the gas t e m p e r a t u r e T. The probabilities of inversion p r o c e s s e s a r e determined f r o m the detailed balancing principle. In the stationary 174
':.,:~ ..../.
SO~
..,"
.
,/
/ / i
/
i ~O0
i $'eO
i 6#0
.F*~
Fig. 1. Dependence of laser level population versus gas temperature. case when T 1 and T 2 a r e small enough and when one can neglect an inverse flow of energy into the a s y m m e t r i c a l mode of vibrations, we get the solution of the s y s t e m (2) as follows: / "--k-T-} L~E12\ ~2v~-, X1 = W3(1,2) X3 + exp ~W12
X 2 =X 0 + 3(W32 + W3(1'2)) X 3,
where N o is quantity of molecules in the ground vibration state, X i = exp (- hvi/kTi). Consider balance equations for E l , E2, E3:
dE2
31 July 1967
(3)
w20 F X3 = No( W32 + W3(1,2))' hv2 where x O = exp ( - ~ ) .
Using the probabilities [3, 5], we evaluated level populations. The dependences of the 00°1 and 10°0 level populations N upon gas t e m p e r a t u r e a r e presented in fig. 1. The continuous curves correspond to the pumping of F = 2.1 × 1018, the dotted ones to F = 2.1 × 1017; the dot-dash line gives a dependence of the equilibrium 10°0 level population. Addition of He lowers the t e m p e r a t u r e s T 1 and T2, since the probability of W20 is increasing. The population calculation technique considered above can be extended to other polyatomic molecules as well. The p r e l i m i n a r y estimates point out to the following m i x t u r e s which a r e expected to be promising f r o m the point of view of obtaining inv e r s e population H2Se-N2, C2N2-N2, NH3-C2H2, COS.
Volume25A. number 2
PHYSICS
LETTERS
References
31 July 1967
3: K.F.Herzfeld. Disc. Faraday Soc.. N 33 (1965) 22. 4. H.Statz, C.L.Tang and G.F.Koster, J. Appl. Phys. 37 (1966} 4273. 5. T.L.Cottrell et al., Trans. Faraday Soc.. 62 (1966} 2655.
1. E.V.Stupoehenko, S.A. Loser, A . J . Osipov, Relaxation Processes in Shock Waves. "Nauka', Moscow, 1965. 2. N.N.Sobolov, V.V.Sokovikov. JETP Letters 4 303 (1966}; 5 (1967) 122. *****
THE
SHIFT
AND WIDTH OF ISOLATED THE ADIABATIC LIMIT
ION
LINES
IN
D. E. ROBERTS and J . DAVIS
Physics Department, Imperial College, London. S. W. 7. Received 17 June 1967
Calculations of the shift and width of isolated ion lines in the adiabatic limit are reported.
R e c e n t p r e s c r i p t i o n s f o r the c a l c u l a t i o n s of S t a r k widths of i s o l a t e d ion l in e s [1,2] include only the e f f e c t s of i n e l a s t i c p e r t u r b e r c o l l i s i o n s . H o w e v e r , it has b e e n indicated [3] that e l a s t i c c o l l i s i o n s can c o n t r i b u t e significantly to the widths of s o m e lines even when h y p e r b o l i c p e r . . t u r b e r o r b i t s a r e used for c a l c u l a t i n g the in e l a s t i c c r o s s s e c t i o n s . In addition, shifts a r e caused mainly by e l a s t i c c o l l i s i o n s and cannot t h e r e f o r e be e s t i m a t e d using e x i s t i n g p r e s c r i p tions u n l e s s they a r e i n f e r r e d f r o m a d i s p e r s i o n r e l a t i o n r e l a t i n g the shift and the width [4]. We have c a l c u l a t e d the width, W, and shift, d, of an i s o l a t e d line u s i n g the adiabatic t h e o r y of Lindholm and F o l e y [5] by employing h y p e r bolic r a t h e r than s t r a i g h t line p e r t u r b e r o r b i t s . Using a p a r a m e t r i c r e p r e s e n t a t i o n f o r the o r bit [6] we can e x p r e s s the p h a s e shift caused by a single p e r t u r b e r a s : -
T/(z, k) = 2k3
; arctan (k))
- ~ - + 1)+
,(1)
w h e r e p 3 = C 4 / v , k = P/Po and z = e 2 / m P o v2. v is the v e l o c i t y and p the i m p a c t p a r a m e t e r of the p e r t u r b e r . Po is the W e i s s k o p f r a d i u s and z is a m e a s u r e of the r a t i o 'of p e r t u r b e r potential e n e r gy to kinetic e n e r g y . C 4 i s the q u a d r a t i c Stark coefficient. W and d can now be e x p r e s s e d in the f o r m :
Idl j w, d
(:units
of Z~'Nevpo~)
~-
w Idl
I~,, .
.
.
4T ~
" t.e Do 0.4
NI 8
D.~ 6 t-o 4 o.,
.2
.
0.6 . . . . . .
.
.
I I I
w_.
I I !
,,v
o-oD
o.~
I I I
I IO
Fig. 1. 175