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The equilibrium of ionization in the atmosphere and nuclear combination coefficients
Journal of Atmospheric and Terrestrial Physics, 1956, Vol. 9, pp. 295 to 303. :Pergamon Press Ltd., London
The equilibrium oi ionization in the atmosphere and nuclear combination coefficients P. J. NOLAbT U n i v e r s i t y College, D u b l i n (Received 16 J u l y 1956)
Abstract--The p r o b l e m of ionic e q u i l i b r i u m in t h e a t m o s p h e r e is r e v i e w e d a n d a n e x p l a n a t i o n s u g g e s t e d for t h e e q u i l i b r i u m f o r m u l a w h i c h filvolves t h e s q u a r e root o f t h e n u c l e u s c o n c e n t r a t i o n . A m e t h o d of e s t i m a t i n g t h e c o m b i n a t i o n coefficients of c h a r g e d a n d u n c h a r g e d nuclei b y m e a n s of th," S m o l u c h o w s k i a n d W h i p p l e f o r m u l a e is given. T h e l i m i t s of v a l i d i t y of c e r t a i n a p p r o x i m a t i o n s in e q u i l i b r i u m f o r m u l a e a r e discussed.
I~ the first part of this paper the problem of the equilibrium of ionization in air containing nuclei is reviewed in the light of recent work b y HOLL and MOHLEISEN (1955). In the second part expressions for the combination coefficients of charged and uncharged nuclei are derived. The equilibrium equation for small ions was written in the following form b y NOLAN, BOYLA~~, and DE SACHY (1925): q ~-~ ~ n 2 - - ~ o n N o 4- ~ n N
(1)
I f the ratio N / n is large, the equilibrium equation for large ions gives ~0N0 -- +tN. It is convenient to p u t V/~o = N o / N - - 1 and Z / N = ( N O + 2 N ) / N =l +2 so that
q :
and
q = ~n e 4- b n Z
(3)
where
b = 2l~o/(l 4- 2)
(4)
o:n ~ 4- 2.~nN
(2)
When Z is much larger than n an approximate form of equation (3) m a y be used: NOLAZ¢ and DE SACHY (1927) extended these equations b y making t~ distinction between the n's and V's for positive and negative small ions. In the following discussion we shall neglect this distinction. In experiments with room air these formulae appeared satisfactory. With nuclei decaying in a closed vessel, however, it was found that the coefficient b varied as Z -1/2. Accordingly NOLAN (1929) proposed the empirical equation: q -~ b n Z .
q = ~ n 2 4 - ~ n Z ~12.
(5)
Although the measurements of n and Z made at Glencree from 1928 to 1932 showed better agreement with (5) than with (3) it was obvious that the squareroot formula was inadequate. Actually a formula of the t y p e n Z 1l~ = constant was indicated b y the results. W~m~T (1935) found that for conditions of equa[ visibility a formula n Z ~15 = constant gave the best representation of his observations of small ions and nuclei at Eskdalemuir. The equilibrium of small ions and nuclei in a closed vessel was investigated b y NOLA2~ and GALT (1944) using SCHWEIDLER'S Method I. They established 295
P. J. ~NOLAN
the validity for equilibrium conditions of equation q = an 2 + b n Z and found values of b between 4-0 × 10 -G and 9.0 × l0 -~ cm3/sec. The larger values were obtained with nuclei which had been enclosed for two days and which had grown by coagulation. NOLAN and KEN~A~ (1949) determined equilibrium values of Z / N o for nuclei of various sizes. NOLA~ and DOHERTY (1950) made simultaneous measurements of Z / N o and of the radius of atmospheric nuclei and found t h a t the fraction of charged nuclei was much smaller t h a n t he equilibrium value. Indications of this lack of equilibrium in Dublin air had previously been obtained b y NOLAS" and DEIGXAN (1948). An examination of the effect of this lack of equilibrium on the equations shows t h a t while the equation (3) involving Z m a y hold a p p r o x i m a t e l y over a certain range of nucleus size, equation (2) involving N m a y be seriously in error. BURKE and NOLAN (1950) have found t h a t conditions which favour the accumulation of nuclei near the ground also favour the accumulation of radioactive carriers, and t h a t consequently q m a y v a r y with Z. To summarize we m a y state: 1. E q u a t i o n (3) has been established for equilibrium conditions. 2. The parameters ~0, ~, b and l m a y have a wide range of variation depending on the variation of nuclear size. 3. When the nuclei come under observation a state of equilibrium with small ions m a y n o t have been reached. 4. The equilibrium problem is complicated by the fact t h a t the radioactive content of the lower layers of the atmosphere increases with the nucleus content. The square-root formula has been revived by HOLL and Mi/THLEISEN, who found t h a t their measurements of n and Z could be represented b y the formula n Z ~ = constant where x = 1/2.3. T h e y interpreted this result as follows. The coefficient b is t a k e n as being directly proportional to r the nuclear radius. F r o m JUNGE'S distribution function it is to be found t h a t r oc Z -~/2. These two relations applied to the formula q = b n Z give q oc n Z ~/2. We now proceed to bring forward an alternative i n t e r p r e t a t i o n of the square-root formula. VARIATION" OF COEFFICIENT b WITK NUCLEAR I~ADIUS
We first examine how the coefficient b depends on the radius of the nucleus. F r o m equation (4) it m a y be seen t h a t this involves an examination of the variation with radius of the two parameters ~0 and 1. In this investigation we shall confine our a t t e n t i o n to radii between 1 × 10 -a and 4 × 10 -~ cm. B y so doing we avoid the complication of double-charging which begins at a radius of about 4 × 10 -6 cm. We also almost cover the normal atmospheric size range. F o u r methods of obtaining the desired relations will be discussed and in three of t h e m we require the Whipple formula ~ --~0 = 4~rEw which gives ~0(l -
1) = 4 ~ w
(6)
where E is the electronic charge and w the small-ion mobility. Nolan and K e n n a n
Applying equation (6) to t he values of l obtained b y NOLA~ and K E ~ A N we obtain the variation of V0 with the radius. E q u a t i o n (4) t hen gives us the 296
The equilibrium of ionization in the a t m o s p h e r e a n d nuclear c o m b i n a t i o n coefficients
v a r i a t i o n of b w i t h t h e radius. a n d b oc r 1"36.
W e t h u s find in t h e r a n g e i n d i c a t e d ~0 o~ r l'~7
Boltzmann Distribution Law:
F r o m t h e B o l t z m a n n L a w we c a n deduce (1955) for nuclei w i t h one charge: E2
1 = e 2rkT
.
B y m e a n s of e q u a t i o n s (6) a n d (4) v a l u e s of ~]0 a n d b for v a r i o u s v a l u e s of t h e r a d i u s can be c o m p u t e d . I n order to c o m p a r e w i t h our o t h e r d e r i v a t i o n s we e x p r e s s t h e r e l a t i o n s in t h e f o r m s ~o ¢c r u, b oc r v. T h e B o l t z m a n n f o r m u l a gives c o n s i d e r a b l e c h a n g e of u a n d v w i t h t h e radius. F o r t h e r a n g e 2 × 10 6 to 3 × 10 -6 c m t h e v a l u e s are u = 1.71 a n d v ~ 1.26. Junge
JUDGE (1954) considers t h a t t h e simple f o r m u l a 70 = 47rdr is s a t i s f a c t o r y , w h e r e d is t h e diffusion coefficient of t h e small ion. U s i n g t h e }Vhipple f o r m u l a he o b t a i n s : 1 = -7- ~ 7o
dr~-wE1 dr
~ _ WE __. dr
H e e v a l u a t e s wE/d b y i n s e r t i n g e x p e r i m e n t a l values of w a n d d. T h e r e is so m u c h u n c e r t a i n t y a b o u t t h e s e v a l u e s t h a t we consider it p r e f e r a b l e t o use t h e welle s t a b l i s h e d f o r m u l a w / e d = 1/]cT = 2.5 × 10 la (for t e m p e r a t u r e 16.8°C) T h u s 5.67 × 10 -6 1 ~ ] _c T a k i n g ~]0 oc r a n d e q u a t i o n (4) we g e t b o c r °'76. r
Bricard
F r o m BRICARD'S p a p e r on ionic e q u i l i b r i u m (1948) we d e d u c e t h a t V0 ocr °s a n d b oc r °'Ss in t h e size r a n g e w i t h which we are dealing. BRICARD'S figures do n o t agree w i t h t h e W h i p p l e f o r m u l a ; t h e y give 70(1 - - 0.75) = c o n s t a n t . F o r t h e coefficient u in t h e r e l a t i o n ~]0 o c r ~ we h a v e o b t a i n e d b y t h e four m e t h o d s of a p p r o a c h t h e v a l u e s 1.77, 1.71, 1.00, a n d 0.80. F o r t h e coefficient v in t h e r e l a t i o n b o c r ~ we h a v e o b t a i n e d t h e v a l u e s 1.36, 1.26, 0.76, a n d 0.55. RELATION BETWEEN Z AND r E x p e r i m e n t s on t h e d e c a y of nuclei in a large closed vessel i n d i c a t e t h a t d u r i n g c o a g u l a t i o n t h e t o t a l v o l u m e of t h e nuclei r e m a i n s a p p r o x i m a t e l y cons t a n t . T h e slight decrease in v o l u m e a c t u a l l y o b s e r v e d c a n be a c c o u n t e d for b y diffusion loss a t t h e b o u n d a r y (NOLA~ a n d KUFFEL, 1955). This process of g r o w t h b y c o a g u l a t i o n m a y o c c u r in t h e a t m o s p h e r e if for e x a m p l e a large air m a s s p o l l u t e d in a c i t y is carried a w a y t o t h e c o u n t r y . I f t h e air m a s s is sufficiently l a r g e for diffusion losses a t t h e b o u n d a r y to be negligible, a n d if t h e r e is no mixing, t h e i b r m u l a Z r 3 ----c o n s t a n t m a y a p p l y . I f r oc Z -1/3 a n d b oc r ~ t h e n b oc Z -vIa. E q u a t i o n b n Z ~ c n Z ~ gives b oc Z x-1 v so t h a t x : 1 - - 3" I f we t a k e v ---- 1.36 we o b t a i n t h e s q u a r e - r o o t f o r m u l a a p p r o x i m a t e l y . W e 297
P. J. NOLAN
consider t h a t this t r e a t m e n t gives a s a t i s f a c t o r y e x p l a n a t i o n o f t h e original d e c a y e x p e r i m e n t s in a closed vessel (1929). This r e l a t i o n will a p p l y to t h e a t m o s p h e r e o n l y u n d e r t h e conditions specified. I t c a n n o t a p p l y for e x a m p l e w h e n t h e r e is a t e m p e r a t u r e i n v e r s i o n a n d t h e nuclei are r e t a i n e d n e a r t h e source. T h e J u n g e a n d B r i c a r d v a l u e s of v give v a l u e s of x o f a b o u t 0-8. To e x p l a i n t h e Glencree v a l u e x ~ 0.3 it is n e c e s s a r y to bring i n t o a c c o u n t a v a r i a t i o n of q w i t h Z. T h e a r g u m e n t g i v e n a b o v e will n o t be seriously d i s t u r b e d if we include sizes larger t h a n 4 × 10 -6 c m r a d i u s a n d t a k e m u l t i p l e charges into a c c o u n t . T h e Table 1. Values of Radius
1 × 10.6 cm 2 × 10-6 cm 3 × l0 -6 em 4 × 10-6 cI]~
Experimental
BoL~rZSfAN~
9'5 3.4 2.20 1.74
I7.8 4.22 2.61 2.05
JUNGE
BRICARD
6.76 3.88 2.92 2.44
4.(t0 2.44 2.00 1.72
effect will be to increase t h e v a l u e of v for t h e l a r g e r radii. F o r e x a m p l e w i t h t h e B o l t z m a n n f o r m u l a t h e values of v are 1.98, 1.26, 1.09 for r a n g e s 1-2, 2-3, 3 - 4 × 10 - G c m . I f we include d o u b l e charges t h e v a l u e of v for r a n g e 2-7 × 10 6 c m is 1.36 (1.45 for r a n g e 2 - 3 × 10 -6 decreasing to 1.25 for r a n g e 6-7 × 1 0 - 6 c m ) . T h e v a l u e s of l for t h e f o u r m e t h o d s are g i v e n in T a b l e I. I t m a y be n o t e d t h a t t h e e x p e r i m e n t a l v a l u e s are d e d u c e d f r o m e q u a t i o n 2/I ~ - Z / N o - I. T h e p e r c e n t a g e e r r o r in large values of l is c o n s e q u e n t l y large. CRITICAL ]~ADII FOR ~[ULTIPLE C]=[ARGES
T h e B o l t z m a n n d i s t r i b u t i o n law gives for m u l t i p l y - c h a r g e d nuclei: p2 ~2
N-2 = 1 ~rt:~' N~ w h e r e p is t h e n u m b e r of u n i t charges. F o r t h e s a m e v a l u e of N o / N . the radii for p charges are p r o p o r t i o n a l to p2. F o r radius 3 × 10 -6 cm the d o u b l y - c h a r g e d nuclei are 2 . 2 % of t h e t o t a l nuclei a n d for radii 4 x 10 -6, 5 × 10 -6 t h e figures are 5 . 4 % a n d 8.6%. JUNGE e x t e n d s t h e W h i p p l e f o r m u l a to t h e case of like charges on nucleus a n d ion. H e t h u s o b t a i n s t h e critical radius for d o u b l e charging: r : : w e / d = 5.76 X l0 -6 cm. F o r larger v a l u e s of p t h e critical radius will be p - 1 t i m e s this value. F r o m t h e B o l t z m a n n law a n d t h e W h i p p l e f o r m u l a we o b t a i n t h e following r e l a t i o n for ~0 ~o e2r~T-- 1 ==47feW.
)
W h e n r is v e r y large, ca. 30 × 10 -6 cm, this b e c o m e s rio = 4 7 f e w .
2rkT E2
298
- - 87r d r
The equilibrium of ionization in the atmosphere and nuclear combination coefficients
It is interesting to note t h a t HARt~EI~'S formula (1935) ~0 ~ 67rdr(1 -- e ~) gives V0 ----6wdr for large values of the radius and t h a t NOLA~ and KEI~'NAN found t h a t their results could be reconciled with the Harper formula provided the 6w were changed to 12w. I t m a y also be noted t h a t the relation ~7o oc r(1 --e- ~ ) c ~ 0-25 × 10 Ggives ~o Qc r x'77 in the range r ~ 1 >4 l0 -6 cm to r ~ 4 >; 10 %m. COMBI:NATIOI~ COEFFICIEI~'TS OF C]-[ARGED AI~D UI~'CHARGED ~ U C L E I
KENNEDY (1916) found t h a t about half the nuclei derived from a Bunsen flame were charged and t h a t the average charge per nucleus varied with experimental conditions between two and six electronic charges. He measured the coagulation coefficient and decided t h a t it was the same whether the nuclei were charged or uncharged. The experiments with uncharged nuclei were performed by passing the air from the Bunsen through an electric field which removed all the charged nuclei. Now although at the beginning of the ensuing decay experiment there were no charged nuclei present they were formed during the course of the experiment by combination of nuclei with " n a t u r a l " small ions. An examination of the figures enables an estimate of the time constant, of this charging process to be made. This indicates t h a t during a large portion of the decay the charged fraction was small. We shall denote the nuclear combination coefficient by fl and indicate the nnclei involved by suffixes, 0 for uncharged, 1 for positive charge and 2 for negative charge. We neglect the subsequent history of the doubly charged nucleus formed by a fill or a /~.~2 combination. Since positive and negative large-ion mobilities are equal it is obvious t h a t ill0 ~ fi2o and f i H - fi22- The electrical attraction between a charged and an uncharged nucleus is unlikely to increase appreciably the collision frequency especially if the nuclei are large. We will assume then t h a t fi00 ---fi~0 and note t h a t the error involved in this assumption decreases with increasing nuclear size. We shall show later t h a t KENR-EDY'S coagulation results with charged and uncharged nuclei afford experimental evidence for the approximate equality of the two coefficients. I f we apply the Whipple theory to the collisions between nuclei we obtain
/3~ --/3~o + 8~1+~;
/3,~ -/31o - s ~ W
(7)
where W is the large ion mobility. NOLAN and KEI~NAN (1949) made simultaneous measurements of the diffusion coefficient D and of the coagulation coefficient ~, of nuclei derived from hot platinum. They established the validity of Ss~oLvcgows~I'S formula y - - 8 w r D . Now in these experiments on mixed charged and uncharged nuclei the average combination coefficient is shown later to be fl00 or fllo. Consequently we m a y put: fi00 =/710 = 2y = 16~rD
(S)
The factor 2 arises from the circumstance t h a t the combination of two nuclei results in the loss of one nucleus. B y means of formulae (7) and (8) we are able to calculate values of the fl 299
P. J. NOLAN coefficients for nuclei of different sizes. These are s h o w n in Table 2. The diffusion coefficient is o b t a i n e d f r o m t h e radius b y a p p l i c a t i o n of the Millikan f o r m of STOK~S'S law. The m o b i l i t y of t h e large ion is o b t a i n e d f r o m the relation
W ~ eD/kT. Table 2 r x 10s cm
0-5 1.0 2.0 3.0 4.0 5.0 10.0 20.0 30.0
I [
i
!
D × 106 cm2sec-1
floo × 109
508 131 34.8 16.5 9.82 6.67 2.17 0.82 0.49
12"8 6"6 3.5 2"48 1'96 1"68 1 "08 0'82 0'74
CIII3SeC
~ STreW ~'< 10u !
fl12 × 109
¢2II13S~C: 1
(~II13S(~C 1
1
'
fit~ :: 10~ i
(q133S(q2 -1
l ' i
73.
!
19'0
I
i I
5.04 2.39 1.41 0.965 0.314 0.119 0.072
86"0 25'6 8"54 4'87 3'37 2"64 1"39 0"94 0"81
0"09 0"55 0"72 0"77 0"70 0"67
The values for radius 0.1 v,, 10 -6 cm are fl00 = 62 × 10 9, 8~EW = 1796 × 10 -9 a n d i l l 2 = 1.86 × 10 -6. F o r smaller values of the radius fi00 is negligible comp a r e d w i t h 87raW a n d we h a v e fi12 = StreW, a f o r m u l a w h i c h corresponds t o LA~GEVI~'s f o r m u l a for the r e c o m b i n a t i o n of small ions ~----47re(w 1 q-w2). LA?ca~vI~'s f o r m u l a gives a v a l u e for e a b o u t t h r e e times the o b s e r v e d value 1.4 × 10 -6, a n d similarly our v a l u e of fl12 for r a d i u s 0.1 × 10 .6 is o b v i o u s l y t o o large. The theories we h a v e used will n o t b e a r e x t r a p o l a t i o n to such small v a l u e s of t h e radius. I t is of interest to note the f o r m u l a we h a v e o b t a i n e d for flll
(;5)
flll = 16~rrD - - 87raW = 16rrD r - -
This gives t h e critical radius for double c h a r g i n g ~ c2/2k, T = 2.88 × 10 -6 c m a n d t h e r a d i u s of t h e r e s u l t a n t nucleus = 3.63 × 10 -6 am. The critical radius for the d o u b l y c h a r g e d nucleus f o r m e d in this w a y is t h e n a p p r o x i m a t e l y the same as t h e critical radius for t h e small-ion d o u b l e - c h a r g i n g process. The Boltzm a n n law includes m u l t i p l y c h a r g e d nuclei p r o d u c e d in b o t h w a y s b u t t h e n u m b e r of d o u b l y c h a r g e d ions p r o d u c e d b y t h e fl~ or fi22 process is n o r m a l l y m u c h less t h a n t h e n u m b e r p r o d u c e d b y the V~I or ~22 process. The ~11 coefficient is of t h e o r d e r one t h o u s a n d times t h e fl~l coefficient for t h e same size nucleus so t h a t p r o v i d e d N / n is n o t v e r y large f111NIN2 will be m u c h smaller t h a n ~]117]12V1 The curious v a r i a t i o n o f fl1~ w i t h radius m a y be noted. The v a l u e increases f r o m zero a t 2-88 × 10 -8 em to a m a x i m u m at 8.0 X 10 .8 a n d t h e n decreases. W i t h increasing r a d i u s t h e values o f t h e coefficients fl00, file, flll come t o g e t h e r ; the influence of t h e charge on t h e collision f r e q u e n c y decreases as t h e nuelei b e c o m e larger. 300
T h e e q u i l i b r i u m of ionization in t h e a t m o s p h e r e a n d n u c l e a r c o m b i n a t i o n coefficients I{ELATION
BETWEEN
AND THE
THE
COAGULATION
COMBINATION
COEFFICIENT
COEFFICIENTS
For the coagulation of a mixture of charged and uncharged nuclei when Z --N O +N 1 +
N 2 we m a y write :
2)' Z2 = flooNo 2 @ fl~N12 @ 3222\ 2" @ 2fi~oNiNo @ 2fi20N2Xo @-/)12-' 1~) 2" Putting N, =N 2 =N, : 2fi1 o We get
fin =fi22,
fiio ==fi2o a n d
a p p l y i n g relM, ion /~,t + fi12
27Z2 = dooNo 2 + 4fiaoN;Yo @ 4fito N2 giving, if we a s s u m e filo = floo, 27Z2 ---fioo(No @ 2N) 2.
Thus the coagulation coefficient for the mixture of charged and uncharged nuclei is the same as for uncharged nuclei alone. I t is true that in KENNEDY'S experiment multiply charged nuclei were present. Nevertheless we consider that this result affords support for the hypothesis that fioo and filO are nearly equal. VALIDITY
OF
APPROXIMATION IN-
EQUILIBRIUM
EQUATION
H a v i n g o b t a i n e d a n e s t i m a t e of t h e m a g n i t u d e of t h e coefficient, fl12 we are n o w able to discuss a n i m p o r t a n t , a s p e c t of e q u i l i b r i u m e q u a t i o n s (1) a n d (2). T h e e q u a t i o n for p o s i t i v e large ions is
dN1/dt = ~]aOnlNO - - t]21n2N1 - - - f i 1 2 ~ 7 1 / ' V 2 W i t h e q u i l i b r i u m , o m i t t i n g d i s t i n c t i o n of sign of ions: •]nN -- ~]onNo -- fi12N 2
~ '~o
_No fi12N -
-
N
(9)
~]0n
I n t h e p r e v i o u s t r e a t m e n t we h a v e a s s u m e d t h a t fl12N/~on is negligible c o m p a r e d w i t h No/N. W e n o w e x a m i n e t h e v M i d i t y of this p r o c e d u r e . Table 3 r 5< 106 C~Ill
fl12 >~- 1()~ CIYI3Se,C 1
q0 ": 106 (,III3S(%? 1
fi12 ;': ll(~ )]1)
fl12 N I]Olt
1 1 3 4
25.6 8.54 4.87 3.37
2.28 1.0 2.0 3.2
91 s-5 2.4 1.1
0-23 11.12 0-0S 0-06
I n T a b l e 3 we h a v e i n s e r t e d values of ill2 f r o m T a b l e 2 a n d values of ~(, f r o m NOLA>- a n d KE>-XAN (1949). T a k i n g q ---- 10 a n d Z = 25,000 we e s t i m a t e v a l u e s of N/n which give t h e figures for fil~N/rjon in t h e last column. Since the values of No/N f r o m w h i c h these m u s t be s u b t r a c t e d to give ,l/rlo are 9.5, 3.4, 2.2 a n d 1.74, it is seen t h a t t h e " c o r r e c t i o n s " are of t h e o r d e r 4°/:o- F r o m t h e a p p r o x i m a t e f o r m u l a q --- bnZ it m a y be seen t h a t for a g i v e n q t h e v a l u e of N/n is p r o p o r t i o n a l :301
P. d. ~OLA~
to Z e. Hence if Z = 100,000 the corrections are of the order 60 % and the equation 7/'1o = N o / N is not valid. NOLAN and K~N~A~ made 74 simultaneous measurements of Z / N o and of t h e radius. We have examined their results to ascertain the error involved ill deducing ~/~0 from the measured N o / N . The median value of Z was 14,000, in 51 experiments Z was less t h a n 25,000 and in only 10 experiments was Z greater t h a n 35,000. We conclude t h a t the error involved in deducing ~/~1o is less t h a n the experimental error. (It must be a d m i t t e d t h a t in two experiments with Z = 100,000 and Z = 70,000 there is no evidence of the large differences between N o / N and ~/~0 indicated by our theory). We obtain equation (9) in another form if we eliminate n by means of the a p p r o x i m a t e form of equation (2) No--~](1-~ N ~]0
2fl12 ~_~T2) -q "
(10)
I t might be argued t h a t the fact t h a t N o / N in Dublin air is found to be larger t h a n expected is not caused by lack of equilibrium but is to be explained b y equations (9) or (10) F r o m the Table given by NOLAX and DOHERTr showing th e variations of Z / N o with Z in atmospheric air we select two cases. For Z -50,000 the measured value of N o / N was 8.0 and the equilibrium value of NOLaN and KENYA>" was 2-3. As we have seen, this equilibrium value can be t aken as equal to ~q/~0. Application of equation (10) gives N o / N = 2.3 x 1-03 = 2-4. This calculation shows t h a t there exists u n d o u b t e d l y lack of charge equilibrium. A similar conclusion can be drawn from the rest of the Dublin observations ex cep t those t a k e n during fog and t e m p e r a t u r e inversion conditions. The coneentration is t hen v e r y large and the size of the nuclei is also larger t h a n normal. Fo r Z = 120,000 the measured value of N0/N was 2.5. Application of equation (10) gives in this case N o / N = 1.7 × 1.5 -~ 2.55. A p p a r e n t l y under these conditions equilibrium charge distribution was present with Z / N o ..... 1.S and a value Z / N o = 2.2 to be expected if the =¥2 t erm in equation (10) were neglected. When Z is v e r y large, equations (1), (2) and (3) are not legitimate approximations. E q u a t i o n (2) for example becomes: q -- ~n 2 + 2,qnN -+- fl12N2 Using our values of/31~ for t he range 1 x 10 - 6 c m to 4 × 10 - 6 c m we find t h a t th e t e r m fi12N 2 is about 0.4, t h a t is 4 per cent of the normal value of q, for Z = 40,000. The ~n 2 t e r m is usually omitted when nuclei are plentiful. It is of interest to insert here a similar calculation for the lower limit of Z for this approximation. I f a = 1 " 4 × 10 -6 the t e r m ~n 2 is 0.4 for n = 5 3 5 and we estimate t h a t this corresponds to nucleus concentrations 40,000, 15,000, 9,000 and 6,000 for radii 1, 2, 3, 4 × 10 -s cm. Hence the appr o xi m at e formula q ~--2~TnN is applicable with less t h a n a 4~o error only over a range from Z = 10,000 to Z = 40,000 for ordinary atmospheric nuclei. A c k n o w l e d g e m e n t - - T h e a ut hor is indebted to Dr. E. F. FAHY, who suggested t h e application of the B ol t z m ann law to the equilibrium of charges on nuclei. 302
The equilibrium of ionization in the atmosphere a n d nuclear combination coefficients
~EFERENCES BRICARD Z. BURKE T. a n d NOLAN J . Jo HARPER \V. R . HOLL ~V. a.nd ~{LTHLEISEN R. JUNGE (!. KENNEDY H . NOLAN J . J., BOYLAN R . K., a n d DE SACHY- G. P. NOLAN J . iL a n d DE SACHY G. P. NOLAN P. J . NOLAN P. g. NOLAN P. J . a n d DEIGNAN J . NOLA.~,T t ). J . a n d DOHERTY D. J . NOLAN P. ,J. a n d GALT R . I. NOLAN P. J . a n d KE~NAN E . L. NOLAN I ). J . a n d KUFFEL E. WHIPPLE F. J . W . WRIGHT H . L.
1949 1950 1935 1955 1955 1916 1925 1927 1929 1955 1948 1950 1944 1949 1955 1933 1935
J. Geophys. Res. 54, 39. 1)roc. Roy. Irish Acad. 53, 145. Phil. ~lag. 20, 740. Geofis. pura appl. 31, 115. Jour. Meteor. 12, 13. Proc. Roy. Irish Acad. 33, 58. Proc. Roy. Irish Acad. 37, 1. Proc. Roy. Irish Acad. 37, 71. Proc. Roy. Irish Acad. 38, 49. G . R . P a p e r 42, U.S. A i r F o r c e C a m b r i d g e R e s e a r c h C e n t e r 113. Proc. Roy. Irish Acad. 51, 239. Proc. Roy. Irish Acad. 53, 163. Proc. Roy. Irish Acad. 50, 51. Proc. Roy. Irish Acad. 52, 171. Geofis. pura. appl. 81, 97. I?roc. Phys. Soc. 45, 367. Quart. J. R. Met. Soc. 61, 93.
303
The equilibrium oi ionization in the atmosphere and nuclear combination coefficients P. J. NOLAbT U n i v e r s i t y College, D u b l i n (Received 16 J u l y 1956)
Abstract--The p r o b l e m of ionic e q u i l i b r i u m in t h e a t m o s p h e r e is r e v i e w e d a n d a n e x p l a n a t i o n s u g g e s t e d for t h e e q u i l i b r i u m f o r m u l a w h i c h filvolves t h e s q u a r e root o f t h e n u c l e u s c o n c e n t r a t i o n . A m e t h o d of e s t i m a t i n g t h e c o m b i n a t i o n coefficients of c h a r g e d a n d u n c h a r g e d nuclei b y m e a n s of th," S m o l u c h o w s k i a n d W h i p p l e f o r m u l a e is given. T h e l i m i t s of v a l i d i t y of c e r t a i n a p p r o x i m a t i o n s in e q u i l i b r i u m f o r m u l a e a r e discussed.
I~ the first part of this paper the problem of the equilibrium of ionization in air containing nuclei is reviewed in the light of recent work b y HOLL and MOHLEISEN (1955). In the second part expressions for the combination coefficients of charged and uncharged nuclei are derived. The equilibrium equation for small ions was written in the following form b y NOLAN, BOYLA~~, and DE SACHY (1925): q ~-~ ~ n 2 - - ~ o n N o 4- ~ n N
(1)
I f the ratio N / n is large, the equilibrium equation for large ions gives ~0N0 -- +tN. It is convenient to p u t V/~o = N o / N - - 1 and Z / N = ( N O + 2 N ) / N =l +2 so that
q :
and
q = ~n e 4- b n Z
(3)
where
b = 2l~o/(l 4- 2)
(4)
o:n ~ 4- 2.~nN
(2)
When Z is much larger than n an approximate form of equation (3) m a y be used: NOLAZ¢ and DE SACHY (1927) extended these equations b y making t~ distinction between the n's and V's for positive and negative small ions. In the following discussion we shall neglect this distinction. In experiments with room air these formulae appeared satisfactory. With nuclei decaying in a closed vessel, however, it was found that the coefficient b varied as Z -1/2. Accordingly NOLAN (1929) proposed the empirical equation: q -~ b n Z .
q = ~ n 2 4 - ~ n Z ~12.
(5)
Although the measurements of n and Z made at Glencree from 1928 to 1932 showed better agreement with (5) than with (3) it was obvious that the squareroot formula was inadequate. Actually a formula of the t y p e n Z 1l~ = constant was indicated b y the results. W~m~T (1935) found that for conditions of equa[ visibility a formula n Z ~15 = constant gave the best representation of his observations of small ions and nuclei at Eskdalemuir. The equilibrium of small ions and nuclei in a closed vessel was investigated b y NOLA2~ and GALT (1944) using SCHWEIDLER'S Method I. They established 295
P. J. ~NOLAN
the validity for equilibrium conditions of equation q = an 2 + b n Z and found values of b between 4-0 × 10 -G and 9.0 × l0 -~ cm3/sec. The larger values were obtained with nuclei which had been enclosed for two days and which had grown by coagulation. NOLAN and KEN~A~ (1949) determined equilibrium values of Z / N o for nuclei of various sizes. NOLA~ and DOHERTY (1950) made simultaneous measurements of Z / N o and of the radius of atmospheric nuclei and found t h a t the fraction of charged nuclei was much smaller t h a n t he equilibrium value. Indications of this lack of equilibrium in Dublin air had previously been obtained b y NOLAS" and DEIGXAN (1948). An examination of the effect of this lack of equilibrium on the equations shows t h a t while the equation (3) involving Z m a y hold a p p r o x i m a t e l y over a certain range of nucleus size, equation (2) involving N m a y be seriously in error. BURKE and NOLAN (1950) have found t h a t conditions which favour the accumulation of nuclei near the ground also favour the accumulation of radioactive carriers, and t h a t consequently q m a y v a r y with Z. To summarize we m a y state: 1. E q u a t i o n (3) has been established for equilibrium conditions. 2. The parameters ~0, ~, b and l m a y have a wide range of variation depending on the variation of nuclear size. 3. When the nuclei come under observation a state of equilibrium with small ions m a y n o t have been reached. 4. The equilibrium problem is complicated by the fact t h a t the radioactive content of the lower layers of the atmosphere increases with the nucleus content. The square-root formula has been revived by HOLL and Mi/THLEISEN, who found t h a t their measurements of n and Z could be represented b y the formula n Z ~ = constant where x = 1/2.3. T h e y interpreted this result as follows. The coefficient b is t a k e n as being directly proportional to r the nuclear radius. F r o m JUNGE'S distribution function it is to be found t h a t r oc Z -~/2. These two relations applied to the formula q = b n Z give q oc n Z ~/2. We now proceed to bring forward an alternative i n t e r p r e t a t i o n of the square-root formula. VARIATION" OF COEFFICIENT b WITK NUCLEAR I~ADIUS
We first examine how the coefficient b depends on the radius of the nucleus. F r o m equation (4) it m a y be seen t h a t this involves an examination of the variation with radius of the two parameters ~0 and 1. In this investigation we shall confine our a t t e n t i o n to radii between 1 × 10 -a and 4 × 10 -~ cm. B y so doing we avoid the complication of double-charging which begins at a radius of about 4 × 10 -6 cm. We also almost cover the normal atmospheric size range. F o u r methods of obtaining the desired relations will be discussed and in three of t h e m we require the Whipple formula ~ --~0 = 4~rEw which gives ~0(l -
1) = 4 ~ w
(6)
where E is the electronic charge and w the small-ion mobility. Nolan and K e n n a n
Applying equation (6) to t he values of l obtained b y NOLA~ and K E ~ A N we obtain the variation of V0 with the radius. E q u a t i o n (4) t hen gives us the 296
The equilibrium of ionization in the a t m o s p h e r e a n d nuclear c o m b i n a t i o n coefficients
v a r i a t i o n of b w i t h t h e radius. a n d b oc r 1"36.
W e t h u s find in t h e r a n g e i n d i c a t e d ~0 o~ r l'~7
Boltzmann Distribution Law:
F r o m t h e B o l t z m a n n L a w we c a n deduce (1955) for nuclei w i t h one charge: E2
1 = e 2rkT
.
B y m e a n s of e q u a t i o n s (6) a n d (4) v a l u e s of ~]0 a n d b for v a r i o u s v a l u e s of t h e r a d i u s can be c o m p u t e d . I n order to c o m p a r e w i t h our o t h e r d e r i v a t i o n s we e x p r e s s t h e r e l a t i o n s in t h e f o r m s ~o ¢c r u, b oc r v. T h e B o l t z m a n n f o r m u l a gives c o n s i d e r a b l e c h a n g e of u a n d v w i t h t h e radius. F o r t h e r a n g e 2 × 10 6 to 3 × 10 -6 c m t h e v a l u e s are u = 1.71 a n d v ~ 1.26. Junge
JUDGE (1954) considers t h a t t h e simple f o r m u l a 70 = 47rdr is s a t i s f a c t o r y , w h e r e d is t h e diffusion coefficient of t h e small ion. U s i n g t h e }Vhipple f o r m u l a he o b t a i n s : 1 = -7- ~ 7o
dr~-wE1 dr
~ _ WE __. dr
H e e v a l u a t e s wE/d b y i n s e r t i n g e x p e r i m e n t a l values of w a n d d. T h e r e is so m u c h u n c e r t a i n t y a b o u t t h e s e v a l u e s t h a t we consider it p r e f e r a b l e t o use t h e welle s t a b l i s h e d f o r m u l a w / e d = 1/]cT = 2.5 × 10 la (for t e m p e r a t u r e 16.8°C) T h u s 5.67 × 10 -6 1 ~ ] _c T a k i n g ~]0 oc r a n d e q u a t i o n (4) we g e t b o c r °'76. r
Bricard
F r o m BRICARD'S p a p e r on ionic e q u i l i b r i u m (1948) we d e d u c e t h a t V0 ocr °s a n d b oc r °'Ss in t h e size r a n g e w i t h which we are dealing. BRICARD'S figures do n o t agree w i t h t h e W h i p p l e f o r m u l a ; t h e y give 70(1 - - 0.75) = c o n s t a n t . F o r t h e coefficient u in t h e r e l a t i o n ~]0 o c r ~ we h a v e o b t a i n e d b y t h e four m e t h o d s of a p p r o a c h t h e v a l u e s 1.77, 1.71, 1.00, a n d 0.80. F o r t h e coefficient v in t h e r e l a t i o n b o c r ~ we h a v e o b t a i n e d t h e v a l u e s 1.36, 1.26, 0.76, a n d 0.55. RELATION BETWEEN Z AND r E x p e r i m e n t s on t h e d e c a y of nuclei in a large closed vessel i n d i c a t e t h a t d u r i n g c o a g u l a t i o n t h e t o t a l v o l u m e of t h e nuclei r e m a i n s a p p r o x i m a t e l y cons t a n t . T h e slight decrease in v o l u m e a c t u a l l y o b s e r v e d c a n be a c c o u n t e d for b y diffusion loss a t t h e b o u n d a r y (NOLA~ a n d KUFFEL, 1955). This process of g r o w t h b y c o a g u l a t i o n m a y o c c u r in t h e a t m o s p h e r e if for e x a m p l e a large air m a s s p o l l u t e d in a c i t y is carried a w a y t o t h e c o u n t r y . I f t h e air m a s s is sufficiently l a r g e for diffusion losses a t t h e b o u n d a r y to be negligible, a n d if t h e r e is no mixing, t h e i b r m u l a Z r 3 ----c o n s t a n t m a y a p p l y . I f r oc Z -1/3 a n d b oc r ~ t h e n b oc Z -vIa. E q u a t i o n b n Z ~ c n Z ~ gives b oc Z x-1 v so t h a t x : 1 - - 3" I f we t a k e v ---- 1.36 we o b t a i n t h e s q u a r e - r o o t f o r m u l a a p p r o x i m a t e l y . W e 297
P. J. NOLAN
consider t h a t this t r e a t m e n t gives a s a t i s f a c t o r y e x p l a n a t i o n o f t h e original d e c a y e x p e r i m e n t s in a closed vessel (1929). This r e l a t i o n will a p p l y to t h e a t m o s p h e r e o n l y u n d e r t h e conditions specified. I t c a n n o t a p p l y for e x a m p l e w h e n t h e r e is a t e m p e r a t u r e i n v e r s i o n a n d t h e nuclei are r e t a i n e d n e a r t h e source. T h e J u n g e a n d B r i c a r d v a l u e s of v give v a l u e s of x o f a b o u t 0-8. To e x p l a i n t h e Glencree v a l u e x ~ 0.3 it is n e c e s s a r y to bring i n t o a c c o u n t a v a r i a t i o n of q w i t h Z. T h e a r g u m e n t g i v e n a b o v e will n o t be seriously d i s t u r b e d if we include sizes larger t h a n 4 × 10 -6 c m r a d i u s a n d t a k e m u l t i p l e charges into a c c o u n t . T h e Table 1. Values of Radius
1 × 10.6 cm 2 × 10-6 cm 3 × l0 -6 em 4 × 10-6 cI]~
Experimental
BoL~rZSfAN~
9'5 3.4 2.20 1.74
I7.8 4.22 2.61 2.05
JUNGE
BRICARD
6.76 3.88 2.92 2.44
4.(t0 2.44 2.00 1.72
effect will be to increase t h e v a l u e of v for t h e l a r g e r radii. F o r e x a m p l e w i t h t h e B o l t z m a n n f o r m u l a t h e values of v are 1.98, 1.26, 1.09 for r a n g e s 1-2, 2-3, 3 - 4 × 10 - G c m . I f we include d o u b l e charges t h e v a l u e of v for r a n g e 2-7 × 10 6 c m is 1.36 (1.45 for r a n g e 2 - 3 × 10 -6 decreasing to 1.25 for r a n g e 6-7 × 1 0 - 6 c m ) . T h e v a l u e s of l for t h e f o u r m e t h o d s are g i v e n in T a b l e I. I t m a y be n o t e d t h a t t h e e x p e r i m e n t a l v a l u e s are d e d u c e d f r o m e q u a t i o n 2/I ~ - Z / N o - I. T h e p e r c e n t a g e e r r o r in large values of l is c o n s e q u e n t l y large. CRITICAL ]~ADII FOR ~[ULTIPLE C]=[ARGES
T h e B o l t z m a n n d i s t r i b u t i o n law gives for m u l t i p l y - c h a r g e d nuclei: p2 ~2
N-2 = 1 ~rt:~' N~ w h e r e p is t h e n u m b e r of u n i t charges. F o r t h e s a m e v a l u e of N o / N . the radii for p charges are p r o p o r t i o n a l to p2. F o r radius 3 × 10 -6 cm the d o u b l y - c h a r g e d nuclei are 2 . 2 % of t h e t o t a l nuclei a n d for radii 4 x 10 -6, 5 × 10 -6 t h e figures are 5 . 4 % a n d 8.6%. JUNGE e x t e n d s t h e W h i p p l e f o r m u l a to t h e case of like charges on nucleus a n d ion. H e t h u s o b t a i n s t h e critical radius for d o u b l e charging: r : : w e / d = 5.76 X l0 -6 cm. F o r larger v a l u e s of p t h e critical radius will be p - 1 t i m e s this value. F r o m t h e B o l t z m a n n law a n d t h e W h i p p l e f o r m u l a we o b t a i n t h e following r e l a t i o n for ~0 ~o e2r~T-- 1 ==47feW.
)
W h e n r is v e r y large, ca. 30 × 10 -6 cm, this b e c o m e s rio = 4 7 f e w .
2rkT E2
298
- - 87r d r
The equilibrium of ionization in the atmosphere and nuclear combination coefficients
It is interesting to note t h a t HARt~EI~'S formula (1935) ~0 ~ 67rdr(1 -- e ~) gives V0 ----6wdr for large values of the radius and t h a t NOLA~ and KEI~'NAN found t h a t their results could be reconciled with the Harper formula provided the 6w were changed to 12w. I t m a y also be noted t h a t the relation ~7o oc r(1 --e- ~ ) c ~ 0-25 × 10 Ggives ~o Qc r x'77 in the range r ~ 1 >4 l0 -6 cm to r ~ 4 >; 10 %m. COMBI:NATIOI~ COEFFICIEI~'TS OF C]-[ARGED AI~D UI~'CHARGED ~ U C L E I
KENNEDY (1916) found t h a t about half the nuclei derived from a Bunsen flame were charged and t h a t the average charge per nucleus varied with experimental conditions between two and six electronic charges. He measured the coagulation coefficient and decided t h a t it was the same whether the nuclei were charged or uncharged. The experiments with uncharged nuclei were performed by passing the air from the Bunsen through an electric field which removed all the charged nuclei. Now although at the beginning of the ensuing decay experiment there were no charged nuclei present they were formed during the course of the experiment by combination of nuclei with " n a t u r a l " small ions. An examination of the figures enables an estimate of the time constant, of this charging process to be made. This indicates t h a t during a large portion of the decay the charged fraction was small. We shall denote the nuclear combination coefficient by fl and indicate the nnclei involved by suffixes, 0 for uncharged, 1 for positive charge and 2 for negative charge. We neglect the subsequent history of the doubly charged nucleus formed by a fill or a /~.~2 combination. Since positive and negative large-ion mobilities are equal it is obvious t h a t ill0 ~ fi2o and f i H - fi22- The electrical attraction between a charged and an uncharged nucleus is unlikely to increase appreciably the collision frequency especially if the nuclei are large. We will assume then t h a t fi00 ---fi~0 and note t h a t the error involved in this assumption decreases with increasing nuclear size. We shall show later t h a t KENR-EDY'S coagulation results with charged and uncharged nuclei afford experimental evidence for the approximate equality of the two coefficients. I f we apply the Whipple theory to the collisions between nuclei we obtain
/3~ --/3~o + 8~1+~;
/3,~ -/31o - s ~ W
(7)
where W is the large ion mobility. NOLAN and KEI~NAN (1949) made simultaneous measurements of the diffusion coefficient D and of the coagulation coefficient ~, of nuclei derived from hot platinum. They established the validity of Ss~oLvcgows~I'S formula y - - 8 w r D . Now in these experiments on mixed charged and uncharged nuclei the average combination coefficient is shown later to be fl00 or fllo. Consequently we m a y put: fi00 =/710 = 2y = 16~rD
(S)
The factor 2 arises from the circumstance t h a t the combination of two nuclei results in the loss of one nucleus. B y means of formulae (7) and (8) we are able to calculate values of the fl 299
P. J. NOLAN coefficients for nuclei of different sizes. These are s h o w n in Table 2. The diffusion coefficient is o b t a i n e d f r o m t h e radius b y a p p l i c a t i o n of the Millikan f o r m of STOK~S'S law. The m o b i l i t y of t h e large ion is o b t a i n e d f r o m the relation
W ~ eD/kT. Table 2 r x 10s cm
0-5 1.0 2.0 3.0 4.0 5.0 10.0 20.0 30.0
I [
i
!
D × 106 cm2sec-1
floo × 109
508 131 34.8 16.5 9.82 6.67 2.17 0.82 0.49
12"8 6"6 3.5 2"48 1'96 1"68 1 "08 0'82 0'74
CIII3SeC
~ STreW ~'< 10u !
fl12 × 109
¢2II13S~C: 1
(~II13S(~C 1
1
'
fit~ :: 10~ i
(q133S(q2 -1
l ' i
73.
!
19'0
I
i I
5.04 2.39 1.41 0.965 0.314 0.119 0.072
86"0 25'6 8"54 4'87 3'37 2"64 1"39 0"94 0"81
0"09 0"55 0"72 0"77 0"70 0"67
The values for radius 0.1 v,, 10 -6 cm are fl00 = 62 × 10 9, 8~EW = 1796 × 10 -9 a n d i l l 2 = 1.86 × 10 -6. F o r smaller values of the radius fi00 is negligible comp a r e d w i t h 87raW a n d we h a v e fi12 = StreW, a f o r m u l a w h i c h corresponds t o LA~GEVI~'s f o r m u l a for the r e c o m b i n a t i o n of small ions ~----47re(w 1 q-w2). LA?ca~vI~'s f o r m u l a gives a v a l u e for e a b o u t t h r e e times the o b s e r v e d value 1.4 × 10 -6, a n d similarly our v a l u e of fl12 for r a d i u s 0.1 × 10 .6 is o b v i o u s l y t o o large. The theories we h a v e used will n o t b e a r e x t r a p o l a t i o n to such small v a l u e s of t h e radius. I t is of interest to note the f o r m u l a we h a v e o b t a i n e d for flll
(;5)
flll = 16~rrD - - 87raW = 16rrD r - -
This gives t h e critical radius for double c h a r g i n g ~ c2/2k, T = 2.88 × 10 -6 c m a n d t h e r a d i u s of t h e r e s u l t a n t nucleus = 3.63 × 10 -6 am. The critical radius for the d o u b l y c h a r g e d nucleus f o r m e d in this w a y is t h e n a p p r o x i m a t e l y the same as t h e critical radius for t h e small-ion d o u b l e - c h a r g i n g process. The Boltzm a n n law includes m u l t i p l y c h a r g e d nuclei p r o d u c e d in b o t h w a y s b u t t h e n u m b e r of d o u b l y c h a r g e d ions p r o d u c e d b y t h e fl~ or fi22 process is n o r m a l l y m u c h less t h a n t h e n u m b e r p r o d u c e d b y the V~I or ~22 process. The ~11 coefficient is of t h e o r d e r one t h o u s a n d times t h e fl~l coefficient for t h e same size nucleus so t h a t p r o v i d e d N / n is n o t v e r y large f111NIN2 will be m u c h smaller t h a n ~]117]12V1 The curious v a r i a t i o n o f fl1~ w i t h radius m a y be noted. The v a l u e increases f r o m zero a t 2-88 × 10 -8 em to a m a x i m u m at 8.0 X 10 .8 a n d t h e n decreases. W i t h increasing r a d i u s t h e values o f t h e coefficients fl00, file, flll come t o g e t h e r ; the influence of t h e charge on t h e collision f r e q u e n c y decreases as t h e nuelei b e c o m e larger. 300
T h e e q u i l i b r i u m of ionization in t h e a t m o s p h e r e a n d n u c l e a r c o m b i n a t i o n coefficients I{ELATION
BETWEEN
AND THE
THE
COAGULATION
COMBINATION
COEFFICIENT
COEFFICIENTS
For the coagulation of a mixture of charged and uncharged nuclei when Z --N O +N 1 +
N 2 we m a y write :
2)' Z2 = flooNo 2 @ fl~N12 @ 3222\ 2" @ 2fi~oNiNo @ 2fi20N2Xo @-/)12-' 1~) 2" Putting N, =N 2 =N, : 2fi1 o We get
fin =fi22,
fiio ==fi2o a n d
a p p l y i n g relM, ion /~,t + fi12
27Z2 = dooNo 2 + 4fiaoN;Yo @ 4fito N2 giving, if we a s s u m e filo = floo, 27Z2 ---fioo(No @ 2N) 2.
Thus the coagulation coefficient for the mixture of charged and uncharged nuclei is the same as for uncharged nuclei alone. I t is true that in KENNEDY'S experiment multiply charged nuclei were present. Nevertheless we consider that this result affords support for the hypothesis that fioo and filO are nearly equal. VALIDITY
OF
APPROXIMATION IN-
EQUILIBRIUM
EQUATION
H a v i n g o b t a i n e d a n e s t i m a t e of t h e m a g n i t u d e of t h e coefficient, fl12 we are n o w able to discuss a n i m p o r t a n t , a s p e c t of e q u i l i b r i u m e q u a t i o n s (1) a n d (2). T h e e q u a t i o n for p o s i t i v e large ions is
dN1/dt = ~]aOnlNO - - t]21n2N1 - - - f i 1 2 ~ 7 1 / ' V 2 W i t h e q u i l i b r i u m , o m i t t i n g d i s t i n c t i o n of sign of ions: •]nN -- ~]onNo -- fi12N 2
~ '~o
_No fi12N -
-
N
(9)
~]0n
I n t h e p r e v i o u s t r e a t m e n t we h a v e a s s u m e d t h a t fl12N/~on is negligible c o m p a r e d w i t h No/N. W e n o w e x a m i n e t h e v M i d i t y of this p r o c e d u r e . Table 3 r 5< 106 C~Ill
fl12 >~- 1()~ CIYI3Se,C 1
q0 ": 106 (,III3S(%? 1
fi12 ;': ll(~ )]1)
fl12 N I]Olt
1 1 3 4
25.6 8.54 4.87 3.37
2.28 1.0 2.0 3.2
91 s-5 2.4 1.1
0-23 11.12 0-0S 0-06
I n T a b l e 3 we h a v e i n s e r t e d values of ill2 f r o m T a b l e 2 a n d values of ~(, f r o m NOLA>- a n d KE>-XAN (1949). T a k i n g q ---- 10 a n d Z = 25,000 we e s t i m a t e v a l u e s of N/n which give t h e figures for fil~N/rjon in t h e last column. Since the values of No/N f r o m w h i c h these m u s t be s u b t r a c t e d to give ,l/rlo are 9.5, 3.4, 2.2 a n d 1.74, it is seen t h a t t h e " c o r r e c t i o n s " are of t h e o r d e r 4°/:o- F r o m t h e a p p r o x i m a t e f o r m u l a q --- bnZ it m a y be seen t h a t for a g i v e n q t h e v a l u e of N/n is p r o p o r t i o n a l :301
P. d. ~OLA~
to Z e. Hence if Z = 100,000 the corrections are of the order 60 % and the equation 7/'1o = N o / N is not valid. NOLAN and K~N~A~ made 74 simultaneous measurements of Z / N o and of t h e radius. We have examined their results to ascertain the error involved ill deducing ~/~0 from the measured N o / N . The median value of Z was 14,000, in 51 experiments Z was less t h a n 25,000 and in only 10 experiments was Z greater t h a n 35,000. We conclude t h a t the error involved in deducing ~/~1o is less t h a n the experimental error. (It must be a d m i t t e d t h a t in two experiments with Z = 100,000 and Z = 70,000 there is no evidence of the large differences between N o / N and ~/~0 indicated by our theory). We obtain equation (9) in another form if we eliminate n by means of the a p p r o x i m a t e form of equation (2) No--~](1-~ N ~]0
2fl12 ~_~T2) -q "
(10)
I t might be argued t h a t the fact t h a t N o / N in Dublin air is found to be larger t h a n expected is not caused by lack of equilibrium but is to be explained b y equations (9) or (10) F r o m the Table given by NOLAX and DOHERTr showing th e variations of Z / N o with Z in atmospheric air we select two cases. For Z -50,000 the measured value of N o / N was 8.0 and the equilibrium value of NOLaN and KENYA>" was 2-3. As we have seen, this equilibrium value can be t aken as equal to ~q/~0. Application of equation (10) gives N o / N = 2.3 x 1-03 = 2-4. This calculation shows t h a t there exists u n d o u b t e d l y lack of charge equilibrium. A similar conclusion can be drawn from the rest of the Dublin observations ex cep t those t a k e n during fog and t e m p e r a t u r e inversion conditions. The coneentration is t hen v e r y large and the size of the nuclei is also larger t h a n normal. Fo r Z = 120,000 the measured value of N0/N was 2.5. Application of equation (10) gives in this case N o / N = 1.7 × 1.5 -~ 2.55. A p p a r e n t l y under these conditions equilibrium charge distribution was present with Z / N o ..... 1.S and a value Z / N o = 2.2 to be expected if the =¥2 t erm in equation (10) were neglected. When Z is v e r y large, equations (1), (2) and (3) are not legitimate approximations. E q u a t i o n (2) for example becomes: q -- ~n 2 + 2,qnN -+- fl12N2 Using our values of/31~ for t he range 1 x 10 - 6 c m to 4 × 10 - 6 c m we find t h a t th e t e r m fi12N 2 is about 0.4, t h a t is 4 per cent of the normal value of q, for Z = 40,000. The ~n 2 t e r m is usually omitted when nuclei are plentiful. It is of interest to insert here a similar calculation for the lower limit of Z for this approximation. I f a = 1 " 4 × 10 -6 the t e r m ~n 2 is 0.4 for n = 5 3 5 and we estimate t h a t this corresponds to nucleus concentrations 40,000, 15,000, 9,000 and 6,000 for radii 1, 2, 3, 4 × 10 -s cm. Hence the appr o xi m at e formula q ~--2~TnN is applicable with less t h a n a 4~o error only over a range from Z = 10,000 to Z = 40,000 for ordinary atmospheric nuclei. A c k n o w l e d g e m e n t - - T h e a ut hor is indebted to Dr. E. F. FAHY, who suggested t h e application of the B ol t z m ann law to the equilibrium of charges on nuclei. 302
The equilibrium of ionization in the atmosphere a n d nuclear combination coefficients
~EFERENCES BRICARD Z. BURKE T. a n d NOLAN J . Jo HARPER \V. R . HOLL ~V. a.nd ~{LTHLEISEN R. JUNGE (!. KENNEDY H . NOLAN J . J., BOYLAN R . K., a n d DE SACHY- G. P. NOLAN J . iL a n d DE SACHY G. P. NOLAN P. J . NOLAN P. g. NOLAN P. J . a n d DEIGNAN J . NOLA.~,T t ). J . a n d DOHERTY D. J . NOLAN P. ,J. a n d GALT R . I. NOLAN P. J . a n d KE~NAN E . L. NOLAN I ). J . a n d KUFFEL E. WHIPPLE F. J . W . WRIGHT H . L.
1949 1950 1935 1955 1955 1916 1925 1927 1929 1955 1948 1950 1944 1949 1955 1933 1935
J. Geophys. Res. 54, 39. 1)roc. Roy. Irish Acad. 53, 145. Phil. ~lag. 20, 740. Geofis. pura appl. 31, 115. Jour. Meteor. 12, 13. Proc. Roy. Irish Acad. 33, 58. Proc. Roy. Irish Acad. 37, 1. Proc. Roy. Irish Acad. 37, 71. Proc. Roy. Irish Acad. 38, 49. G . R . P a p e r 42, U.S. A i r F o r c e C a m b r i d g e R e s e a r c h C e n t e r 113. Proc. Roy. Irish Acad. 51, 239. Proc. Roy. Irish Acad. 53, 163. Proc. Roy. Irish Acad. 50, 51. Proc. Roy. Irish Acad. 52, 171. Geofis. pura. appl. 81, 97. I?roc. Phys. Soc. 45, 367. Quart. J. R. Met. Soc. 61, 93.
303