IL NUOVO CIMENTO

VoL. 4B, N. 1

11 Luglio 1971

Density Dependence in Resonance Broadening (*). 1% G. B I ~

jr.

Del~artment o/ Physics, West Virginia University - Morgantown, W. Va.

(ricevuto il 21 Set~embre 1970)

Summary. - - The purpose of this paper is the explanation of the residual width encountered in resonance broadening. The resonance-broadening problem is treated using the tetradic Liouville formalism and Che Zwanzig projection operator series. It is found that, due to the particular nature of the resonance forces, the half-width of the spectral line contains a constant in addition to a term which is linearly dependent on the density. This not only explains the residual width but it also explains why only linear dependences on density have been observed. Suggestions as to quadratic and higher dependences of the half-width on the density are therefore irrelevant.

1.

-

Introduction.

B y the t e r m (~residual width ~)we m e a n the width of the spectral line which is obtained b y extrapolating the resonance-broadening measurements to zero density. Some time ago K v - ~ and VA~G~A~ (1) discovered t h a t the residual width did not correspond to the n a t u r a l width of the line for certain helium lines. I n t e r e s t i n g l y enough, these authors did find a linear dependence of the half-width on the density to as low a density as t h e y were able to measure in this and in subsequent experiments. Kvm~ and LI~wIs (3)later observed the same p h e n o m e n o n in neon a n d VAVG~A~r (3) observed it in k r y p t o n . I n f o r m a l

(*) (1) (2) (a)

This work was carried out under NASA Grant NGR-49-001-046. H. G. KuH~ and J. M. VAVGgAN: Proe. Roy. Soe., A277, 297 (1963). H. G. KuItN and E. L. Lv.wls: Proc. Phys. Soe., A299, 423 (1967). J . M . VAVGHA~: Phys. Bey., 166, 13 (1968).

1 - ll Nuovo Cimento B.

2

R. G-. BREF~NE jr.

suggestions to the effect t h a t a hitherto unobserved and unconceived quadratic (or higher) dependence of the half-width on the density must come into p l a y at (~low )> densities were advanced, b u t no definite demonstration of t h e m has been forthcoming. ]~ECK, TAF~EBE and MEAD (% among others, have emphasized the necessity for some form of m a n y - b o d y t r e a t m e n t of resonance broadening, this necessity being based on the well-known radiationless exchange effects peculiar to resonance broadening (5). (Interestingly enough H o l t s m a r k ' s original treatm e n t of resonance broadening (6.5) yielded the same density dependence as did t h a t of l~EC~[ et al. Sic transit gloria mundi.) S ~ T ~ and H00PER (~) felt t h a t t h e i r tetradic Liouville formalism was more capable of producing results in the near f u t u r e t h a n was t h e more unwieldy T - m a t r i x formalism of R o s s (8), BEZZEnIDES (9) and ZAIDI (~0), for example. This is doubtless a controversial point, b u t the T - m a t r i x approach has a p p a r e n t l y not y e t produced a specific t r e a t m e n t of resonance broadening. I n any event, it has not furnished a precise description of the observed anomalies in the residual width. I n this p a p e r we ~pply the tetradic Liouville formalism to the resonance broadening. The Liouville formalism, incorporating as an i m p o r t a n t p a r t the Zwanzig projection operator (n), was originally applied to spectral line broadening in a general way b y I~A~o (~2). S~T}t and HOOPE~ cast this formalism in t e r m s of tetrudic operators and specifically applied it to atomic S t a r k broadening. I t was applied to molecular S t a r k broadening b y B~EE~E (~3).

2. - G e n e r a l

considerations.

L e t us recall t h a t casting t h e expression for the radiant intensity in a spect r a l line in the Liouville operator form has the practical effect of eschewing a n y later necessity for t h e introduction of the i n t e r r u p t i o n approximation. This was first done b y F A r o (12). The introduction of the Zwanzig projection operator (n) into t h e problem is of twofold benefit; it has the effect not only of eliminating the necessity for consideration of certain co-ordinates b u t also of introducing a useful p e r t u r b a t i o n series.

(~) G. P. RECK, H. TAKEBE and C. A. MnAD: Phys. Rev., 137, h 683 (1965). (5) R. G. B~ENE jr.: The Shi]t aug Shape o] Spectral Lines (Oxford, 1961). (e) J. HOLTSMARK: Zeits. Phys., 34, 722 (1925). (7) E. W. SMIT~ and C. F. HOOPER jr.: Phys. Bey., 157, 126 (1967). (8) D. W. Ross: Ann. o] Phys., 36, 458 (1966). (9) B. BEZZm~IDES: Phys. Rev., 159, 3 (1967). (10) H. R. ZAIDI: Phys. l~ev., 173, 123 (1968). (11) R. ZwANzm: Journ. Chem. Phys., 33, 1338 (1960). (12) U. FANO: Phys. l~ev., 131, 259 (1963). (18) R. G. BREEZE jr.: Journ. Mot. Speet., 26, 465 (1968).

DENSITY

D]~P]~NDENCE

IN :RESONANCE

BI~0ADENING

The dilute approximation led F A r o to a product form for the density matrix~ his thermal-bath and emitter co-ordinates being untangled by it. In order to arrive at essentially the same result, let us consider the following ttamiltonian for a collection of emitters:

-tt= ZKi@ Z-Ui@ ZVi,,

(1)

where K{ is the translational kinetic energy of the i-th atom~ H{ the Itamiltonian for the isolated i-th atom and Vi~ the pair interaction potential. W h e n we make the dilute approximation V i j : 07 the density m a t r i x takes the form

(2)

Y=

T~

~,

where ya refers to the internal co-ordinates of the various atoms, :~ to their translational co-ordinates. We now construct essentially the same projection operator which has been common to previous t r e a t m e n t s

(3)

Pi

= r' ~r, {_~},

where P is our projection operator and M is an arbitrary operator. The subsequent development is identical to what has been previously detailed (7.13) so t h a t we simply write down the resulting line shape expression: (da)

I(o) = z-1 ~:r~{~. [ o ) - ]-I(Yo ~)},

(db)

= Trt{T(co)Yt},

(4c)

[~o -

]

= o~ - - ~ - i - - ~ - ~ ~ < L ~ [ K ~ s

-- P)(~/~)]->

--

{A~ - - {--i <-~0> - - {--1 - - {--2 < L 1 K 0 ( ( X ) ) ( 1 - - P ) "~I> 9

In eqs. (4) the Liouville operator L has been broken up into a perturbed (L1) and an unperturbed portion (L0). The Liouville operator is generally written as

(5)

L=H|174

where H is given by eq. (1). K~

is the unperturbed resolvant co

(6)

K~

= -- i f e x p [io~t]exp [-- iLo tlt~] at 0

R. G. BREElq~

jr.

and the averaging indicated b y the brackets in eqs. (4) is over the translational co-ordinates. Finally, we r e m a r k t h a t the series of eq. (de) for n > l is the standard f o r m for a p e r t u r b a t i o n in which Z~ is the p e r t u r b a t i o n and K~ is the Green's function. SmTH (14) and MORSE and FESCttBACH (15) have discussed the convergence of this series, and we base our cut-off for n > 2 on their work.

3. - Wave functions and broadening potential. We shall hypothesize a collection of n identical two-state atoms, one of which is in its u p p e r state, t h e other n - 1 of which are in their lower states. The wave functions for the lower and u p p e r states will t h e n be

(Ta) (Tb)

~

=

~,o(1) ~o(2) y'r~

... ~ o ( n ) ,

P~+(1) % ( 2 ) . . . %(n)

w h e r e / ~ is the p e r m u t a t i o n operator without change of sign, the substripts 0 and + indicating lower and u p p e r atomic states, respectively. The interaction H a m i l t o n i a n leading to the resonance broadening is

(8)

v = ~ r(~, j ) . $:>i=1

The form of V(i, j) is not of basic importance to the p h e n o m e n a peculiar to resonance broadening as we shall see. W h a t leads to the hitherto anomalous results is the fact t h a t s u m m a t i o n is carried out over all pairs of atoms. The specific f o r m of the pair interaction which we shall use is the dipole-dipole result which was developed b y MARGV,~IAU(16) long ago (17) ~2

(9)

Vii -~ ~ (YiY~ + z i z j - - 2xix~) ,

where r~. is the interatomie distance and the x~, etc. are internal atomic co-ordinates. Equations (7) give the wave functions relating to the internal co-ordinates; each of these must be multiplied b y a translational function. Suppose our trans-

(14) ]~. W. SMITH" Thesis, University of ~lorida (1966). (15) p. M. MORS~ and H. ~ESCHBACH: Methods o] Theoretiea~ Physics (New York, 1953). (le) H. MARG~I~AII: Phys. Bey., 40, 387 (1932). (17) In Appendix A we show that the Margenau result may effectively be obtained for hydrogen by applying the viewpoint we take herein for the long-range case.

D~.NSXTY DEP~XDENe~ ~X ~ESONaNCE Ba0XD~NG lational s t a t e is ]a}. T h e n

(~0a)

]~> = Ip~> l p # ... lp~>,

O0b)

Iv,> = ~

1

exp [ i ~ , . p , / ~ ] ,

where V is the normalizing volume, p ; t h e m o m e n t u m of t h e ~-th particle.

4. - R e q u i r e d m a t r i x

elements.

F i r s t let us s i m p l y write down t h e m a t r i x e l e m e n t of K~ erence. F r o m eq. (6) it is obviously

for f u t u r e ref-

r

(11)

(Ko)~,~.~ ~ = - - i j e x 9 [i[(o - - o ) ~ - - (%,] t] d t . 0

The m a t r i x e l e m e n t over t h e lower a t o m i c s t a t e will, of course, be zero:

while t h a t over t h e u p p e r will be (13a)

<~*~1~v~,lu~'> =

cd~[~5~>'>~

-~ ,

where (13b)

e2 h ],n C~ ----- - -

8 ~ 2m~:o

in t h e hydrogenic case. H e r e ]~ is t h e oscillator s t r e n g t h for a dipole t r a n sition b e t w e e n t h e two atomic levels, vo t h e f r e q u e n c y of such a transition. W e shall require t h e foliowing m a t r i x e l e m e n t combinations:

-I- l n ( n - - 1)(n - - 2)(n - - 3) (~[rL"l~'} (~'IrT[[~} n-=.

L e t us r e m a r k t h a t the first t e r m on t h e right of eq. (14) concerns a twoparticle interaction, t h e second a three-particle a n d t h e t h i r d a four-particle. The simpler diagonal e l e m e n t of eq. (13a), of course, leads only to a two-

1%. G. ]B~]~]~N~] jr.

6

particle result:

(15) First, let us simplify eqs. (14) and (15) b y noting that, for n--n--l--n--2=n--3. Then

(16/

1 n
~ fdrr~

2-17

2Vz n f d r~,

~t~1019,

= 2~rN_P~,

o

where we transformed the co-ordinate origin to the i-th particle. Here N is the particle density, I"2 a constant. The integral w h i c h / ' 2 represents diverges logarithmically as it stands although upper and lower cut-offs could certainly be introduced on physical grounds. Since this will not affect our result, we do not do so. ~ e x t let us consider the two-particle m a t r i x element from eq. (14): 1

(17)

<~lr~l~'> = ~

f f exp[ir(p~ -- q J] exp [irj .(pj-- qr 1 f

= V

dri drj ir.l ~

dr

exp [i~. (Ap~-- @~)] 7 ~ '

where we have again transformed the co-ordinate origin to the i-th atom. In this case, however, the m o m e n t u m difference must subsequently include t h a t of the i-th atom. Taking the polar co-ordinate as the angle appearing in the scalar product, we observe t h a t integration leads to

<~Ir~l~'> =

(18)

4~r F dz 4• V J s i n z ~z~ = y r~ ,

where z = ]Ap~-- App.]r, hp~ = p ~ - - q~. Equation (18) would likewise require a lower limit which would logically introduce a dependence on p and ~ such t h a t 1-'3=-P2(Ap~, Ap~). The three-particle matrix d e m e n t would differ from the two one only in t h a t /~(Ap~, Apj) would be replaced b y F~'(p~, Apj).

5. -

The resonance

line width.

We must evaluate the various terms in eq. (de). Let us first evaluate t h a t portion of the last t e r m in this equation which includes the projection operator P. Using eq. (3) we find (19)


---- Tr~ (L1K~

~Tr~{L1/rt}} =
.

DENSITY

DEPEND~NCE

IN I ~ E S O N A N C E BI~,0ADENING

7

It is now quite straightforward to show, using methods which we sh,~ll upply below, thut eq. (19) is egneelled by the fonr-p~rticle portion of
(2o)

where the ]v gre ~he diggongl elements o~ the density m ~ r i x :

hS

(21b)

](P') = V(2~mkr)~ exp [--

p~12m~T].

From st~ndurd definitions of tetr~dic operators (22)

(z,~),,~,,~,.,,~,,~.= a~,;.- <~1 Vl~fl'> a~,~,.9

Equations (11), (21) ~nd (22), together with the fuct that the matrix element over the lower utomic stute is zero, yield

(23)

For the two-purticle cuse the summution in eq. (23) may be written

(2~) {p, q) = {(p, q):p ~ , q~c~'}. Since (%~,= ~.(p~--q~)/2m~ and by eqs. (21), (24) becomes

where

t

(25)

- ~ ~ (p~- ~)t12~

I<~I~:?~I~':>I~]-[i(p~)

1 c ~ ( ~ - 1 ) ~: ~ qlq$

9e x p

[--~(p,"~--q~)tl2m~JJ(p~)](pj)2

H Z J(p~),

8

1~. ~. ]~m~EN]~ jr.

where the K r o n e c k e r delta p r o d u c t arises from the m a t r i x element. normalization assures us t h a t

Now

H ~ l(p~) -- i so eq. (25) becomes 2

2u C~

~ . ~ V~ ~ T,2](Pi)](PJ) exp[--~(p~--q~)t/2m~] exp[--i(p~--q~)t/2m~].

(26)

~JqJ

I n t h e limit as the v o l u m e goes to infinity V

Using eqs. (21/)) and (27) in eq. (26) yields the two-particle result

~(mkT).h 8

exp [-- i(ibt + a)p ~] exp [-- ibtq ~]dp dq

a = (2mkT) -1 ,

(28b)

,

b = (2m~) -1 .

The three-particle portion of eq. (23) m a y be similarly evaluated with the result 2~ 89C~N (29)

[-- ibtq ~]dp d q . h3(mkT)~ f f P ~ exp V-- (ibt + a) p~] exp

W e have thus obtained the following result:

where

C~

(30b)

7~(mkT)aha ,

C~

hS(mkT)~ ,

(30c)

.~I(T)

(30d)

i~(T) =fexp [i[~- ~]t]dt~f~ exp [-- (ibt + ~)~] exp [-- ~t~]dpdq.

Using precisely the same techniques it is quite straightforward to obtain the following results for the as y e t u n e v a l n a t e d terms remaing in eq. (de):

(315)

]g-~ <~5,>~.~ =

]~

DENSITY

DEPENDENCE

IN I ~ E S O N A N C E

9

BROADENING

The spectral line shape now results from substituting eqs. (30) and (31) into eq. (da) and rendering the denominator of the resulting expression real. It is apparent that such a procedure yields the following expression for the spectral line width: a = ~e~ F~(T) + ~r

(32)

iV~(T);V,

wherein we specify the real part of F1 by F~.

6. -

Remarks.

I t is obvious from eq. (32) that the width of the resonance-broadened spectral line depends not only on the first power of the atomic density but also on a constant. Such a result readily explains the apparent problem with the residual width. For all practical purposes, measurement of resonance width at varying density will yield a linear variation of width with density down to however low a density we choose. At the same time, however, these widths can never extrapolate to the natural line width because of the first term in eq. (32). Therefore our result yields the residual width, which differs from the natural width, and demonstrates that no low-density changes in half-width density dependence need be conjectured. In order to obtain specific numerical results it would, of course, be necessary to eliminate divergences such as those arising from the lower limit in eq. (18). It may be that this can be accomplished in specific instances by including exponential terms of the type discussed in Appendix A, although an electronic data-processing machine program will probably be required. Meanwhile, there is the rough indication from eq. (32) that the natural-width--residual-width difference may vary as T -~. If we take Zo= [Api--Apjlro as a lower cut-off, we find /"2 = 1 + Ci(zo), where Ci(z0) is the cosine integral of lower limit Zo. There is thus a momentum dependence in Ci(z0) which, during the averaging process, will introduce an additional temperature dependence unless Ci(zo) is small compared to one. For zo> 0.5, Ci(zo) remains smaller than 0.5, attaining this value only at about 2.0. Therefore, it would appear possible for experiment to reveal a rough dependence of the natural-width--residual-width on T -s.

10

~. G. B ~ , ~

jr.

APPENDIX

W e shall begin b y a p p e a l i n g to t h e Coulomb i n t e r a c t i o n b e t w e e n two hydrogen atoms: e2

e2

e~.

e~

rij

fix

ri i

where r ~ is t h e s e p a r a t i o n of t h e i - t h electron f r o m t h e J - t h nucleus, r , t h e interelectron distance a n d R t h e internuclear separation. W e shall now show t h a t , for t h e long-range ease, eq. (A.1) yields t h e Margenau result. F r o m s t a n d a r d definitions of t e t r a d i c operators

(A.2)

(.~)~,,,j~a, =
O~,e,-- a~o, = -- Ifi>,

so t h a t we n e e d consider initially only t h e difference of eq. (A.1) over t h e u p p e r a n d lower states. :First of all it is a p p a r e n t t h a t t h e integral of t h e last t e r m on t h e right of eq. ( A . 1 ) - - t h e nuclear r e p u l s i o n - - w i l l b e t h e s a m e over t h e u p p e r a n d lower states so t h a t t h e difference zeroes out. As our only f u r t h e r specific consideration let us n e x t d e t e r m i n e t h e general f o r m for t h e m a t r i x element over t h e first t e r m on t h e right of eq. (A.1), t h e a t t r a c t i o n b e t w e e n t h e i - t h electron a n d t h e j - t h nucleus. W e o b t a i n e2

Since we h a v e a one-particle o p e r a t o r only identical excitations in t h e first integral will yield nonzero results. T h e n for, say, i = 1 we would obtair~ (A.3)

ff

... ~0+(1) ~ ~p+(1)~po(2)~po(2)... ~po(n)~po(n) dv~ ... dv,~ +

d- n

"'" ~po(1)~ ~po(1)~p+(2)~p+(2) ... d~h ... d~:,~-t- ...

qf f

...... = ~

~oo(1)~ ~,o(1) ~po(2)~po(2) ... dTx ... d~:, =

y+(1) - - ,p+(1) d~-x + - 7 -

~Tow since n - - l - - n (A.r

rla-

rij

~po(1) - - yoO) d~-~ rl~

_f

~,o0) - - ~,o0) d r , .

for n of t h e order of 1019~ this becomes n

~p+(1) ~ V,+(1) dye.

rxj

DENSITY DEP]~ND]~NCE IN RESONANCE BROADENING

11

I n like fashion we deal with the remaining t e r m s in eq. (A.1) so t h a t

e2

§ Z n ( ( V + ( ) %(~) - - W+(i)V0(i) dT, dzr §

w0(i) V+(i) dTi dTi

The second t e r m on the right of eq. (A.5) is a Coulomb, the third a hybrid integral. The methods of evaluation for such integrals are standard ones involving re-expression in ellipsoidal co-ordinates and application of the Legendre expansion. Therefore, details are not called for here. A complete evaluation yields an extremely complex expression involving a number of exponential functions of the internuclear separation. We shall restrict ourselves to relatively low densities and long ranges so t h a t the exponential terms m a y be dropped. For the first t e r m on the right of eq. (A.5) a simple transformation to ellipsoidal co-ordinates yields f ~f+(i) rt,/~/)_~(i) eA aT i =

(A.6a)

Rei2j ,

where Rir is t h e separation of the i-th and j-th nuclei. Using the Legendre expansion in the evaluation of the hybrid integral leads to a result which is exclusively composed of exponential functions which we drop. The ellipsoidal expansion m a y again be used to obtain the Coulomb result: (A.6b)

f f ~p+(i) ~o(i) ~e 2 yJ+(i) %(i) 4% d ~ ---- ~e ~ ~-

3 "22e 2

j~-.

Substitution of eqs. (A.6) into eq. (A.5) results in

Ulu>-

e~ =

3.23 .

which is essentially the dipole-dipole result which MAICGE~AU(16) obtained.

9

RIASSUNTO

(*)

Si vuole spiegare l'ampiezza residua nell'allargamento della risonanza. Si tratta il problems dell'allargamento della risonanza col formalismo tetradico di Liouville e le serie degli operatori di proiezione di Zwanzig. Si trova the, per la natura partieolare delle forze di risonanza, la semiampiezza della linea spettrale contiene una costante oltre ad un termine dipendente linearmente dalla densitY. Ci6 spiega non soltanto l'arapiezza residua ma anche perch~ si sono osservate solo dipendenze lineari dalla densitY. Sono quindi irrilevanti le ipotesi su dipendenze dalla densit~ della semiampiezza, quadratiche e di ordine pi~ alto. (*)

Traduzione a eura della •edazione,

12

]~. G. ~ E N E

jr.

3aBHCHMOCTI, ynmpemm pe3oHaHca OT II.YIOTHOCTH.

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