BLURRING OF LINES AT THE LIMIT OF THE HYDROGEN SERIES

L. N. K u r o c h k a a n d L . B. R y b k o

An expression is obtained for determining the electron concentration in a hydrogen plasma on the basis ol the last observable line in the series.

The expression is derived for the case of line broadening by the Stark effect (ions + electrons) and the Doppler effect when the observations

are made in the presence of background radiation that reduces the con-

trast of the lines.

One of the most convenient methods for determining the electron concentration in a

plasma is the method based on the use of the number of observable lines in the Balmer

series. This possibility of determining the electron concentration was pointed out by

the Japanese physicist Sugita in 1934 [1]. Somewhat later, Inglis and Teller [2], taking

into account the splitting of the quantum levels of the emitting atoms in the fields of

the ions, obtained a formula giving the connection between the electron concentration n e

(n e = Up) and the number of the last resolved line m s in the hydrogen spectrum:

lg n~ = 23.26 - - 7.5 lg ms. ( 1 )

Although this formula has been widely used, it has nevertheless been frequently

revised [3-11]. As a result of analysis of the observational data, many authors (see,

for example, [12-14] ) have concluded that the Inglis--Teller formula gives overestimated

values of the electron concentrations. There are various reasons for this. One of them

is that the Inglis--Teller iormula is obtained in the framework of the Holtsmark theory.

With the development of quantum,mechanical collision theory (see [15-17]), it became clear

that alongside the line broadening by field ions (molecular Stark effect) it is necessary

to take into account the collision broadening due to electrons. In addition, the Doppler

effect, like the Stark effect (ions + electrons), can lead to an effective blurring of the

lines at the limit of the series and must be taken into account [18]. In addition, ob-

servations of hydrogen emission lines are frequently made on the background of other radi-

ation which sometimes seriously reduces the contrast and reduces the number of observable

lines in the series. For example, in the case of observation of chromospheric flares on the background of the Sun's continuum the lines of the Balmer series are observed to

H14-HII instead of the H28-H20 that would be observed in the pure case. In the present paper, we obtain a formula for determining the electron concentration on the basis of the

last observed line with allowance for the above factors.

According to [18], two neighboring lines with numbers m and m + I of the upper

quantum levels are blurred if the total half-width 2A) 12 of the m-th line is equal to m

the distance Ai between these lines: "n~ m + l

"DlI2 m/A)',., .,+1 = 0.5. ( 2 )

At the same time, the ratio of the total intensity between the lines ~ ],,,, ,,,+I (con-

sisting of the intensities of the wings of the nearby lines) to the total intensity in the

center of the m-th line ~ ~,,, (n is the number of the hydrogen series) is equal to 0.87

on the average. For different values of the electron concentrations n=(n~ ~ up) and ve- locities O~ the atoms ~. {~V 2RT-~.~, ~ is the most probable value of the turbulent ve-

locity), ~ L,~I/_~ ~,~ can take values from about 0.8 to 0.95. The criterion (2) for

blurring of two lines is more stringent than Rayleigh's criterion [19], in accordance with which two spectral components of equal intensity become indistinguishable as a result

of superposition if the ratio of the intensities between them to the maximal intensity be-

comes greater than 0.81.

Astronomic~l 0bservatory,.Kiev University. Translated from Astrofizika, Vol. 15,

No. I, pp. 155-163, January-March, 1979. Original article submitted June 16, 1978; re-

vision submitted November 9, 1978

0 5 7 1 - 7 1 3 2 / 7 9 / 1 5 0 1 - 0 0 9 3 ~ 0 7 . 5 0 �9 1979 Plenum Publishing Corporation 93

When emission lines are observed on the background of extraneous radiation of in- tensity ]o~ we shall, by analogy, assume that two neighboring lines cannot be resolved if

E 1, .,-,-+-S<o~ v / , , --i-- I<oo,

= 0.87. (3)

Since the presence of background radiation lowers the contrast of emission lines, the number

of the last observable line, ray, may differ strongly from the last resolvable line ms+ D if one had ~ = O~ To find the connection between m v and the quantities ne, ~, and ~o~t,

it is necessary to know how the ratio ~ ]~,~+IiE ]m,. changes with decreasing All2m/A). m ..... y

It is impossible to obtain an analytic connection between these quantities, though analysis

of the nature of the superposition of the lines for different n e and ~ enabled us to find that~ to an accuracy of about 5-10%, the following relation is satisfied:

A).,,_~ "__' _ 0.17 -7- 0.38 ~ /"' m-, A)<'m. m+ , ~ [m,n

(4 )

If we use this expression and the line blurring criterion ( 3 ) , and also note that

the total intensity in the line center ~ [m differs little in practice from the central

intensity of the actual line ]m,n' we obtain

Akt/2m~ -!-t ~<,,,t = 0.5--0.05 (5)

A k m ~ l , m ~ 2 Imp+l, n

The new criterion for the blurring of two lines with numbers m v + 1 and m v + 2 is more

general than the criterion (2). It follows from (5) that the larger is L,,~t, the smaller

is the value of the ratio A),li2m/A), ,m§ ~ at which the lines become indistinguishable. At

values ~/~%~L,>I0 , two neighboring lines, irrespective of their half-widths, will not be resolved. The line blurring criterion (5) corresponds to the case of the photo-

graphic method of detection of hydrogen emission lines. It is possible that for better methods of detection of the spectrum the coefficients on the right-hand side would be

changed slightly (the first coefficient increasing somewhat and the second decreasing).

Using the criterion (5), we obtain an expression for determining the electron con- centration n e from the number of the last observable line with allowance for the plasma

parameters and the conditions of observation of the emission lines. We use for this the relations [8] relations [8]

A. 3s2 .-. 3:,2 (S) -I- 4,3t2 (D), (6) ~1}2= ~1-1,,2 ~"l/2

A~,~,2'(D) = i I~n27,} , AT:,ii (S) = 4 k m , n ro.10-s ( I +0 .2 y),} c I (7)

Fo = 1 Z ~ 10-~~ s~, kin. n = 5.5"10 - s (mn) t ] m ~ _ n2

1 . 8 5 - n 4 A) ,~+I , ~+2 = - , ( 8 )

R. ( m + l ) ~

where y is the constant of damping due to the collision effect of the electrons [20, 21]:

"l,13F., m [ 4. lO la T) t ( 9 )

(for the function i ~.,~ ~ simple approximate expressions are obtained in [21] for n = I, 2, and 3),,

[ 2 2 ) : We note also that for the upper levels (for n ~ S) the following expressions hold

= ! 0 , D - - - - - 2.6.1016 1 msn2'3e ~/ (10 -:- ~.4 7) 2 . -- ~ t~D -~- 0.7";)2 '

i~9 = 4.4.105n- 2m-2n,7~!3 }, (10)

94

1 0o) = 5"35"10-:3n2~/~a~2 - e r g ' c m - 2 " s e c - l . s r - l . c m - 1 ( 1 1 ) l e r l '

I f t h e i n t e n s i t y o f t h e b a c k g r o u n d r a d i a t i o n 6o, t i s l o w a n d t h e n u m b e r o f t h e l a s t o b s e r v e d

l i n e m v d i f f e r s l i t t l e f r o m t h e n u m b e r o f t h e l a s t r e s o l v a b l e l i n e m S (m S - - m v ~ 5 - 1 0 ) , t h e n i n s u c h a c a s e o n e c a n , b y a n a l o g y w i t h w h a t i s d o n e i n [ 8 ] , t a k e y t o b e t h e s a m e f o r a l l m v (y = 3 . 5 ) , w h i c h i s c l o s e t o r e a l i t y f o r 1 0 ~ e ~ n ~ < ~ ] 0 a~ cm - 3 ( s i n c e y d e p e n d s w e a k l y o n T e , we h a v e t a k e n T e = 1 0 0 0 0 ~ i n t h e c a l c u l a t i o n s ) . I n t h i s c a s e , u s i n g t h e a b o v e r e l a t i o n s , we o b t a i n

Io., ]al, n . = 2.2.10~-3(m~ + 1) -7"s 0.5 -- Q ]i~(T,o ) _ l .25.107n_3(m. , !)-3 ~32 (12)

H e r e

Q =3.4-10 -~s (m, u u 1) ~ n2131110 + ([~D~ 2.4):1 ~a. (13)

F o r m u l a ( 1 2 ) i s f a i r l y g e n e r a l . I f we t a k e ~,,.t = 0 , t h e n t h e n u m b e r o f o b s e r v a b l e l i n e s i n t h e s e r i e s w i l l b e d e t e r m i n e d b y t h e i n f l u e n c e o f o n l y t h e S t a r k a n d D o p p l e r e f f e c t s

[ 8 ] . We h a v e m,, ~ m S + D , a n d

n~ 7.7.10o.~(ms,D _) 1)-V.S l.25.10=n_3(ms+D + 1)-~ ,32 (14)

If in addition n e = 0, then in this case m v = m D and then [18]

l g ( m D @ l ) = 3 . 5 1 _ l g n 2 3 - - l g ~ 1~3- . (15)

For ~ = 0 and n e ~ 0, using (14), we obtain the Inglis--Teller formula corrected for the

influence of electron collisions [8]:

lgn, ~ 22 .9- - 7.5 lg (ms -~ i). (16)

An a n a l o g o u s f o r m u l a c a n b e o b t a i n e d i f o n e u s e s t h e f a c t t h a t t h e c e n t r a l i n t e n s i t y o f t h e l a s t r e s o l v a b l e l i n e d i f f e r s l i t t l e f r o m t h e c o n t i n u u m i n t e n s i t y a t t h e s e r i e s

limit, i.e., ] ~ , ~ = ~,2(ke), [22]. In this case,

2~3 = 2.6.1016m-S[(10 ~_ 2.4 7)j + =( ~ ~ 0.77)~]-~ ~ ( 1 7 ) d e , ~ O

If we can ignore the line broadening due to the Doppler effect (B D << i) and, as before,

take y = 3.5, then we obtain

lg ne = 22.7-- 7.5 Ig ms. (18)

This formula coincides with formula (16) to within the accuracy of the theory and

the method of determining n e. In the third edition of Astrophysical Quantities, Allen,

referring to [23], recommends its use for determination of the electron concentration.

If in the expression (7) for ~ = 0 we take for the line half-width A),ll2(S) the value

Y = 0 (the case corresponding to Holtsmark theory), we essentially obtain the Inglis--

Teller formula [8] :

lgn~= 23.3-- 7.5 lg (m~ ~ 1). ( 1 9 )

The main difference of this formula from the Inglis--Teller formula (i) is that here, in

the second term on the right, we have (m S + I) whereas in (i) we have m S . These differ-

ences are unimportant, particularly since Inglis and Teller regarded their formula as

approximate.

When determining the electron concentration by means of formula (12) on the basis

of the last observed line my, one must know the following quantities: the most probable 21 ~--3/2

velocity ~ of the atoms and the generalized emission measure n,ti~ , which characterize

the line intensities of an emitting plasma, and also the intensity of the background

radiation ]~o,t" The first two parameters can be determined from the profiles of the upper

terms of the hydrogen series, and ~o~t by direct measurement. The first approximation for

n e can be obtained in accordance with Eq. (16). This value is certainly overestimated.

A more accurate value of n e is given by formula (14). It can be used to find Q, which is

needed to calculate n e by means of (12) by successive approximation.

95

\k rt4

kk

\k , ,

i \ \ \ \ \

,I J- , \ , ~0 20 30 40 50

m"D~

Fig. I.

Note that in the given case the variability of the damping constant 7 along the

line profile [17] is not taken into account. In the wings of the lines, the importance

of electron collision broadening decreases and the lines begin to reproduce the quasi-

static broadening like the ions. For lines whose upper level is determined by a large

quantum number (m > I0), the ions and the electrons give rise to the same quasistatic

broadening already in the half-widths. Since the values of y for the last observable

lines are nest 3 or 4 units, we can, using the relations (7), readily see that the half-

widths of the lines hardly differ for collision or quasistatic broadening by the electrons.

At the same time, the line profiles will be undoubtedly different. However, in the given

calculations we are not interested in the line profiles but only their half-widths. There

is no particular point in aiming for high accuracy in calculations of the line half-widths;

this is both Because of the limited accuracy of the theoretical calculations [15] and also

the fact that the accuracy in the determination of n e by the considered method is re-

stricted by the multiplicity of fixing of the value of m v-

To illustrate how important it is tO take into account all the factors considered

above in the determination of ne, we give Fig. I. The dashed curve in Fig. I is the

dependence of the last resolvable line of the Balmer series on the electron concentration

constructed in accordance with the Inglis--Teller formula. The dependence is significantly

altered if one takes into account the broadening effect of the electrons on the line pro s

files (upper continuous curve) The dependence is considerably changed if one takes into

account the additional broadening of the lines by the Doppler effect.

In Fig. I, the curve tangent to the upper thin continuous curve corresponds to the

line blurring when allowance is made for the random velocities of the atoms (~ = 2"106

cm/sec~ ~o~ ~ 0). At small values of ne, the line blurring is due to the Stark and Doppler

effects. With increasing ne, the Stark effect plays an ever more important part in the

line blurring. The point of merging of the two curves indicates that at :the given ~ the

line blurring is due solely to the Stark effect. The heavy continuous curves give the

dependences of m v on log n e for some values of the intensity of the background radiation

for ~ = 20 km/sec. It can be seen from Fig. i that the dependences in which we are in-

terested change appreciably when background radiation is present, and this must he taken

into account in the determination of n e. Since the dependences of m v on n e become weaker,

the accuracy in the determination of the electron concentration in the presence of back-

ground radiation decreases. In this case, exact knowledge of the quantities ~, n~[~ -3'2 ,

and ~o~ becomes particularly important. In addition, if there are large differences be-

t~en m s and mv, it is expedient to determine ne, not at the fixed value y = 3.5, but

rather by calculating a corresponding y for each line. In this case

4.8.10~i0.5__ o ~o,,t ]3,2 )3,~ ~ = t 1 , ~ {)-o) 2 . 8 - 1 0 ~ ~ - , ( 2 0 )

(1 = 0.2 ~)3:'* (m,, -~ 1) v's (1 + 0.2 7)~'~2 n3 (m. 1 ) ~

96

Q= 1.q.10-~(rn~_ I)~ z : 3 r , l n o (21) ' m t~ ~u T 2 . 4 " i ) - u - r- (}D + 0 . 7 " ; V ] ~-~

Using Eq. (20), we calculated the values of ~o,t;' ]r n (),~} (Table i) for which the lines

of the Balmer series with the numbers m v become the last observable lines at fixed n e and

(see also Fig. I). The values of the relative intensity of the background radiation

given in Table 1 are of practical interest. For comparison, we point out that the Balmer

lines of solar protuberances, which are characterized by a small value of the emission

97

m e a s u r e , a r e o b s e r v e d on a b a c k g r o u n d o f s c a t t e r e d s k y l i g h t w i t h ~.:.~: ]~:~(k0) = 2 - 2 0 0 , and t h e o b s e r v a t i o n o f s o l a r f l a r e s , w h i c h a r e p r o j e c t e d o n t o t h e s o l a r d i s k , a r e made f o r ~o, t /~0~(;0] = 1 0 - 4 0 0 . T h e r e i s no d o u b t t h a t a t l a r g e v a l u e s o f t h i s r a t i o t h e S t a r k e f f e c t no l o n g e r p l a y s a d o m i n a n t r o l e i n t h e b l u r r i n g o f t h e f a r members o f t h e s e r i e s and t h e n t h e d e t e r m i n a t i o n o f n e b e c o m e s i m p o s s i b l e . And i n t h e c a s e s when i t i s p o s - s i b l e t o d e t e r m i n e t h e e l e c t r o n c o n c e n t r a t i o n , i t m u s t b e b o r n e i n mind t h a t t h e a c c u r a c y i n t h e c a l c u l a t i o n o f n e d e p e n d s s t r o n g l y on t h e a c c u r a c y w i t h w h i c h t h e o t h e r p a r a m e t e r s i n t h e e m p l o y e d f o r m u l a s a r e known.

We a r e s i n c e r e l y g r a t e f u l t o I . S . K o n d r a s h o v a y a , L. A. S t a s y u k , and V. A. O s t a p e n k o for assistance rendered during the present work.

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Moscow ( 1 9 6 3 ) . 17 . V: S . L i s i t s a , Usp . F i z . Nauk , 122, 449 ( 1 9 7 7 ) . 18 . L . N. K u r o c h k a , A s t r o f i z i k a , 2 , 131 ( 1 9 6 6 ) . ! 9 . R a y l e i g h , P h i l o s . M a g . , 8 , 261 ( 1 8 7 9 ) . 20~ Lo Ao M i n a e v a a n d I . I . S o b e l ' m a n , J . Q. S. R. T . , 8 , 783 ( 1 9 6 8 ) . 21 . L. Ao M i n a e v a , A s t r o n . Z h . , 45 , 578 ( 1 9 6 8 ) . 22 . L. N. K u r o c h k a , S o l n . D a n n y e , No. 6 , 90 ( 1 9 7 7 ) . 23~ Lo N. K u r e c h k a and L. B. ~ a s l e n n i k o v a , S o l . P h y s . , 1_~1, 33 ( 1 9 7 0 ) .

98