IL NUOVO CIMENTO VoL. IV, N. 1 1 o Luglio 1956

On the Energy Loss and Specific Ionization

of a' Relativistic Particle in a Polarizable Medium - I.

F. Bu])INI

isti tuto di Fixica dell' Universith - Trieste

L. TAFFA~ 4

Istituto di Fisica dell' Universitd - Padova

Istituto Nazionale di Fisica Nucleate - Sezione di Padova

(ricevuto il 3 Aprile 1956)

S u m m a r y . - - The average to ta l energy loss of a part icle on passing through a polarizable medium has been calculated by introducing into the theory of Fermi an expression for the dielectric constant which is ap t to represent both the dispersive and absorptive properties of the medium. The average total energy may then be divided into the average energy lost as excitation, (~erenkov radiat ion and pr imary ionization of the atoms of the medium. The effect has been calculated of the polarizabil i ty of the medium on the number of ions generated by the ionizing part icle per uni t length and on the number of events in which an atomic electron has been emit ted with energy greater than a certain threshold energy. These numbers show a logarithmic increase at relativistic energies and a subsequent saturat ion caused by the polarizabil i ty of the medium, both dependent on the threshold energy.

I n t r o d u c t i o n .

The inf luence of t i le p o l a r i z a b i l i t y of t h e m e d i u m on t h e ene rgy loss to

wh ieh~a c h a r g e d p a r t i c l e is s u b j e c t on pa s s i ng t h r o u g h i t , ha s been g iven b y

:FERMI (1,2). H e f o u n d t h a t , due to t h e effect of t h e p o l a r i z a b i l i t y , t h e wel l

k n o w n l o g a r i t h m i c i n c r e a s e of ene rgy loss p o s t u l a t e d b y t h e c lass ica l t h e o r y

(1) E. FER~II: Phys. Rev., 56, 1242 (1939). (3) E. FEnMI: Phys. Rev., 57, 485 (1940).

~4 1 ~. B U D I N I 3~nd L. TAFFARA

of BETHE (3,~), BLOCIt (5,6), WILLIAMS (7) ~nd BOH~ (8) is reduced ~t high energy.

Fe rmi ' s work h~s been subsequent ly extended especially b y the in t roduc t ion of ~ more correct expression for the characteris t ic dielectric constant of the

med ium (,-~4) : these extensions h~ve not changed either the theoret ical scheme nor the quulit~tive results of the original theory.

Fe rmi ' s theory is suitable for describing the ~ver~ge to ta l energy loss of

ionizing p~rticle ~nd, for this problem, it is in s~tisfaetory ~ccord with the exper imenta l results. Difficulties ~rise when the theory of energy loss is in-

voked to explain phenomen~ such ~s the fo rmat ion of tr~cks in cloud chambers

or in nuclear emulsions, in which the exper imenta l informat ion is r epresen ted

b y the number of droplets or gr~ins formed per uni t length of tr~eks. I n this

c~se ~t least three sort of difficulties h~ve to be considered: the . f i r s t concerns

the relat ion between the num ber of ions genera ted b y the ionizing p ~ r t i c l e

~nd the n u m b e r of droplets found in the cloud chamber tr~eks or the n u m b e r

of grMns in emulsion tr~cks. I t is usually ~voided with the help of one of the p~r~meters which ~ppe~r in Fe rmi ' s formula : the ~vera.ge ionization energy. This generally serves ~s ~ normal iza t ion p~ramete r for the theoreticM curve

with respect to the exper imenta l one. The second is connected with the m ~ x i m u m energy t ransferable in close collisions, which in the problem of the to ta l energy loss gi~es rise to the well known problem of f luctuations t r ea ted b y ~ANDAU (15,16)~ ~nd introduces instead, into the ionization problem, ~ p~ramete r which is usual ly

fixed more or less ~rbitr~rily. The thi rd difficulty derives f rom the f~et. t h a t one does not k n o ~ which p~rt of the to ta l loss contr ibutes to the ionizat ion

of the ~toms of the med ium ~nd which p~rt contr ibutes to exci ta t ion ~nd Ccrenkov r~di~tion; this problem h~s been t rea ted in v~rious theoret ical works / bu t wi th v~rious results (1~-~o).

(3) H. BETHE: Ann. der Phys., 5, 293 (1930). (4) H. BETttE: Zeits. /. Phys., 76, 326 (1932). (5) F. BLOCH: Zeits. f. Phys., 81, 363 (1932). (G) F. BLOCH: Ann. der Phys., 16, 285 (1933). (7) E. J. WILLIAMS: Proc. Roy. Soc., A 139, 163 (1933). (s) N. BOHR: Det. Kgl. Dans. Vid. Sels,, 18, 9 (1948). (9) G. C. WICK: Ric. Scient., 11, 273 (1940).

(lo) G. C. WICK: Ric. Scient., 12, 858 (1941). (11) G. C. WICK: Nuovo Cimento, 1, 302 (1943). (13) 0. HALPE~N and H. HALL: Phys. Rev., 73, 477 (1948). (la) R. M. STERNHEI~ER: Phys. Rev., 91, 256 (1953). (1~) G. N. FOWLER ~nd G. M. B. D. JONES: Proc. Phys. Soc., 66, 597 (1953). (1~) L. LAN])AU: Journ. Phys. (USRt2), 8, 201 (1944). (16) K. R. S:~ON: Thesis, Harvard University (1948) (unpublished). (17) M. SCHSNBERG: _Vuovo Cimento, 8, 159 (1951). (~s) M. ScHONB]~,RG: NUOVO Cimento, 9, 372 (1952). (19) M. HUYBR~CH~S and M. SCH5N]~ERG: NUOVO Cimento, 9, 764 (1952). (2o) p. Bu])I~I: Zeits. f. Natur/orseh., 72, 722 (1952):

ON THE E N E R G Y LOSS AND S P E C I F I C I O N I Z A T I O N ETC. - I 25-

In the course of the discussions on this last problem, which followed t h e discovery of relativist ic increase in ionization (31-23), i t has clearly resulted

t ha t i t is not possible to in te rpre t the exper imenta l results b y means of F e r m i ' s theory, if one takes into account the dielectric propert ies of the med ium by

an s(~o) which represents the dispersive propert ies of the med ium and ignores~

the absorpt ive ones. I n fact , in this way, one obtains a relativistic increase

due only to the dispersive propert ies of the med ium i.e. due solely to Cerenkov radia t ion (17).

I n this pape r we will t r y to recast t h e theory b y introducing, f rom the very beginning, ~n s(~o)which represents bo th the dispersive and absorpt iv~ propert ies of the med ium (Sect. i). In this way one can see t ha t the separ- ation of the energy loss into t ha t due to ionization, exci tat ion and ~erenkov

radiat ion m a y be obtained f rom the theory ra ther easily and wi thout ambi -

gui ty (Sect. 2-5). Even in a more correct theory such as this, the energy loss

b y ionization is not, as far as we can see, the mos t suitable one for comparison

with the expe r imen ta l results for the number of ions. The average energy

lost b y ionization depends critically on two paramete r s : the mean energy of ionization of the a toms of the med ium and the m a x i m u m energy which can

be t ransfered to an electron in close collisions. These pa ramete r s can only

be theoret ical ly calculated w i t h great difficulty and therefore usually one does not a t t e m p t their theoret ical determinat ion bu t uses t hem as normalizat ion parameters . However , it is clear tha t , of these two parameters , the second one (the m a x i m u m transferable energy), while impor t an t in the computa t ion of

the energy loss, should not influence the mean number of ions produced pe~ ~: uni t length since collisions in which the energy t ransfer is ve ry large are ex-

t remly rare. For the purposes of obtaining a more suitable expression fo r

comparison with the exper imentul data , we have calculated the direct influence of the polarizat ion on the specific p r i m a ry ionization (Sect. 6). As was to be

expected, the final formula, which predicts a logari thmic increase of the n u m b e r of ions followed b y a sa tura t ion as indicated b y exper imenta l results, does not depend in a critical manner on the m a x i m u m transferable energy.

I t is to th ink (34,35) t ha t the specific num b er of grains observed in emulsions depends on a critical m i n i m um energy necessary for the format ion of a grain,

which in tu rn depends on the type of emulsion and on the degree of develop-

merit. We have a d a p t e d our theory to take account Of this effect, and h~v~

(31) L. VOYVODIC and E. PICKUP: Phys. Rev., 80, 89 (1950). (22) L. VoYvo])Ic: Bristol Con/erence (1951). (23) S. G. GosR, G. M. D. B. JONES and J. G. W. WILSON: Proe. Phys. Soc., 65~

68 (1952). (34) L. M. BROWN: Phys. Rev., 90, 95 (1953). (2~) B. T. PRICE: Report of Progress in Physics, 18, 52 (1955).

~26 L, BUDIlqI & n d L. TAFFARA

discussed the possible quali tat ive consequences of such hypothesis. Quant i ta t ive computa t ions will be given in the second par t of this work. These previsions are susceptible of experimental verification.

1 . - T o t a l E n e r g y Loss .

The to ta l energy loss per unit length suffered b y a particle of charge e in .collisions at impact parameters larger than Q can be calculated, following ;FERm (3), by m e a n s of the flux of the Poynt ing vector coming out from a 4ylinder having its axis on the p~rticle trajectory~ and radius Q i.e. :

co

(1) ~z - - mv - 2 - - fi~ Z*" K ~(z*)K0(z)io dco, o

where

(2) = - = . v / 1 - 9/ ' / )

K ~ Ko are the modified Hankel functions, n is the number of electrons pei" cm a and the other symbols have the usual meaning. In (1) and (2)~ as in the

~ollowing~ the frequencies are measured in units of ~J4:,~ne3/m.

If ~o ~ ~o0 -- atomic radius, the asymptot ic expressions of the t t anke l func- t ions for I~/I<< 1 can be inserted in (1) which then becomes:

co

~We~ _ 4he 4 Re I __ fi2 log o311__ fi3s(o) ] i(o d o , ~(1') ~z mv ~ 0

-where

m y 2

(3) A = ~ and log 7 = 0.577 . . . . uny2~2oe3

In order tha t (1') should give the energy lost by Cerenkov radiation, exci tat ion

and ionization, we have to introduce for e(co) an expression which represents the dispersive as well as the absorptive properties of the medium. As an example we adopt for s(co) the following expression:

o~

{4) s(~o) = 1 +~.= ( ~ - - o 2 + igr + 2~2c~[=1J o~ ' ~ Z co t + ; g o ' 0

ON T H E E N E R G Y LOSS AND SPECIFIC IONIZATION ETC. - I 27

w i t h

I O, f for ~o' < ~ok, o ( ~ ' - ~ ) = /

1 , for of ~ % ,

~aT~(~o') = pho toe lec t r i c cross-sect ion of the a.toms of the m e d i u m and Z ~ tomic

~ u m b e r . T h e re~l ~nd imug in~ry pn r t s of e(~o) c~n nlso be w r i t t e n in the fol lowing

:form (~6).

n / i (gO 2 - - (/)2)

,(5) ~2~e s((J)) = 1 + z ~'==1 ( 0~)2 - - (~~ 2 ~ - g 20)~ +

co

' f ~ ( ~ , ' ) ( ~ o '2 - - o~2), O ( c o ' - - ~o~) I d~ '

+ 2~2e2 o

(6) I m s(co) ---- - - ak(~o) . i=1 ((/)~ - - (/)3)2 .~_ g~o)4 ~- ~7~e2(]) =

W i t h e(co) g iven b y (4), (1') can be i n t e g r a t e d in the complex p l ane on the s ame l ine of I - Ia lpern- t ta l l culculus (13) i.e. f r o m zero t o - i . a , where the

Yrequency a is t h e single pos i t ive roo t of the equa t i on

(7) 1 - - fi2. e(iy) = 0 ,

~nd t h e n f r o m - - i . a to - - i . r (with r v e r y large), a n d f r o m - - i . r to r t h r o u g h

q u a r t e r of ~ circle of rad ius r and cen te r a t t he origin. One thus ob ta ins

(s)

w i t h

~ Wq. 2ze~n [ A fi2 ] - - ~ [ lOgl__~fi2 + a ( 1 - f i 2 ) - § ,

(9)

r

J ~ dy 1 - - ~ l o g r

I ~ ,r co r

~ / ~ 2 + y 2 _ giY 27~'e~ k=l oJ' + y - - gy ~i=l ./ a oJlz a

[ = - ~ log [(o~ + a~)S,I~ �9 (~o~ + a2) s'-I~ ,.. ( ~ + a2) fnl~] - -

I I lm +f [ 2 2~2e%~1 k(~o') log [ o / 2 + a 2] dco ' .

~k

(2~) A. H. CO~PTON and S. K. ALLISON: X-Rays in Theory and Experiment (1948).

2 8 P . B U D I N I t~I t ( l L . T A F I e A R A

The integral on the right hand side of (9) can be calculated exact ly when the analytical expressions of ak(oJ') are known; however, in general, one can write:

(10) J = - log [(co~ + a2) s'i'. ( ~ o ~ - - a 2 ) s:l' ... (o,)~ -l- a~) s"/'] - -

- - log [ ( ~ + a~)t .... ( ~ + ,~)t , ] ,

where Nk represents a weighted mean value which, in general, is very near ,~o the frequencies of ionization, oJk and ]~ are given by

(11)

Put t ing:

(12)

and:

co

m e [" 1 7,, - 2 ~ e ~ J ~ ( ~ ) d ~ ' .

o) k

gO 1 a ) " (% + ... (~J. + a~y ,,,',

i 09 2

i = 1 t c=1

one gets :

(14) 8 W o o 27~e~n log , ~z mv~ (~o *~ ~ * ~ - - i7)5 + a~(1--/3)~--/J,].

Fur ther taking int6 account tha t :

(13') (51 ~ ~2 = 1 ,

the limiting value for / ~ - + 1 (a -+ co) is given by:

(15) ~W~o (v : c) -- 2~e4nlog m c ~ ~z m e 2 7 ~ n y ~ e 2 '

tha t is, it is independent ol s(co), in agreement with the result of Fermi. Ir~ this way one obtains an expression very similar to tha t obtained by 1--Ialpern- Hall bu t in which the frequencies ~o* and oJy, referring respectively to. the discrete spectrum (principally responsible for dispersion and Cerenkov radiation) and to the continuous (principally responsible for ionization) of the atoms, of the medium, are separated. Already in this formula one can separate two

O N T H E E N E R G Y L O S S A N D S P E C I F I C I O N I Z A T I O N E T C . - I 29

contributions proportionM to 81 and d2 referring respectively to the discrete and continuous spectrum.

As is known, the total energy loss (1) contributes both to the 0erenkov radiation and to the excitation and ionization of the atoms of the medium~ we now proeede to the separation of these contributions.

2 . - Cerenkov Radiation.

The 0erenkov radiation is, by definition, formed by' energy sent at large distances from the path of the primary particle; thus, choosing ~ = 01 >> ~0, and inserting in (1) the asymptotic expressions of the Hankel functions for large arguments, one obtains for it

(16) 8W o, (0er.) = 8z

Cerenkov

where the integration must be made over those frequencies which s'atisfy the condition

,(17) /~2 Re e ( ( ~ ) - 1 > 0 .

I t is more convenient, for our purpose, to separate the integration over the frequencies co < coo from the integration over the frequencies ~o > 09o with ~oo -----lowest frequency of ionization and therefore, taking account of (6), (16) -can also be written

8W~, (16') (Cer.) :

8z

- k.(o,)[.j ~176 C.erenkov < t o o

0erenkov > w o

where Imoc c e(~o) represents only the first term of (6).

Clearly, because of the dependence of (16) on ~1, it is impossible to define, in a general way, the 0erenkov radiation observed at distances greater than ol without speeifing the particular experimental situation. On the other hand,

3 0 P. BUDIINI D A l d L. T A F F A R A

(16) would be independent of Q~ only if the medium were rigorously non-

absorbing [ Im s ( c o ) - 0]. Now, if the ideal condition of a rigorously non- absorbing medium can be approximated in real i ty in the region of discrete spectrum (low density, temperature , ...) where dispersion is the characterist ic feature of s(co), this is not possible in the region of continuous spectrum where absorption characterizes in an essential way the dielectric behaviour of the medium. This means tha t it is admissible to pu t gi = 0 bu t we must leave in the integral par t of the dielectric constant g r 0; in this way, while in the first t e rm of (16') the exponential reduces to one, in the second te rm it is unchanged. Therefore, in the limit ~1 ---> c~ and with g~ = 0 in (16'), we ob- tain the following expression for the max imum energy sent as Cerenkov radia t - ion at very large distances from the pa th :

(18) ~W~z lae~nmv 2 J [ ; I r i s - ~,w~l ] (C .......... )-- ~ co dco.

(~erenkov <: (o o

3 . - Excitation and Ionization.

As shown in a previous work (27) one can quite generally separate f rom

the total energy loss (1) the energy spent in pr imary excitat ion and ionization of the atoms of the medium:

co

(19) 0Wqo (exc. Jr ion.) = A /72 Re s(co) az m v ~ J Is(co) l s 1 - .

0

In order to integrate (19) it is more convenient to pu t it into the fo rm:

(20)

with :

(21)

(21')

(2r')

~W~176 ( e x c . + ion.) = 4e4--n [P1 q- P2 + Q] ~z mv ~ '

co

P1 = �89 Be ;ico dco log co211-- #~(co)] '

9

P2 -- �89 l~e t ico dco log J e(co)

0

A

co~[1 - - # ~ * ( c o ) ]

co

Q fi~ R co Re ~(co) do~ ~(co)

0

(27) p. BUDINI: Nuovo Cimento, 9, 236 (1953).

O N T t ~ E E N E R G Y L O S S A N D S P E C I F I C I O N I Z A T I O N E T C . - I 31

and to procede separately to the integration of the single terms.

3"1. Integration o] P~ and P~. - For this purpose we observe tha t :

1) The singularities of [s(e))] -~ lay on the upper half of the complex plane. These singularities, as will be shown, are simple poles.

2) In the lower half of the complex plane there is, at most, one sin- gularity of log (e)2[1--fi2s(e))]) and this falls on the immaginary axis for e) = - - - i . a with a defined by:

(22) 1 - - p s ( - - ia) = 0 .

3) l~oting tha t :

(23) 1 - - fi~s(iy) = 1 - - fi2s*(-- i y ) ,

it follows from 2) tha t the only possible singularity of log (e)~[1--fi2s*(e))]} in the upper half of the complex plane lays on the immaginary axis at ~o = i . a .

Therefore it is convenient to calculate P~ integrating first on the immaginary axis from zero to - - i . a and f r o m - i . a t o - i . r (later r will go to infinity) and then on a quarter of circle from - - i . r to r. P~ is to be integrated sym- metrically to P~ on th e upper half of the complex plane i.e. from zero to i. a~ from i . a to i . r and from i . r to r; owing to 1) the rcsidua of the integrand a t the poles of [e(m)] -~ will contribute to P~.

We have then:

I I ! III (24) P1 = P 1 + P 1 + P 1 ,

where:

(25) - - i a

lie) do) P'I = �89 l~e j - ~ - log

0

A e ) ' [ 1 - - p~(e)) ] '

(25') /~ie) de) A

P ; : �89 log ] ,

- - i a

(25")

and

(26)

_P~" = �89 Re f i e ) de) log ~o~[ 1 _ fl2s(e))],

H- F~ -?/~2' ~- ~ Re Residua iv) A

32 P. BUDINI ~nd L. TAFFARA

-/r

;~ where:

P~ (27)

, , , , (27')

(27"/

- } a

i a

P'2 = �89 Re rio &o log A J ~(~)) ~ ' [1 - - fl~s*(~o)] '

0

i r

P~' = �89 Re fie)do) log A J s(co) o~[1 - - fl~e*(~)] '

i a

" d i m d o A G = �89 ~ e .T ~ log ~o~[~-- ~*(o~)T

i e

l~ow, as one can easily verify, P : --P'~ = 0

because the integr~nds ~re imm~ginary inside the interval of integration.

P : + P~ is a monotonically decreasing func- t ion of a; its max imum value, obtMnedf or a = 0, is given by:

(28)

co

~ k

while it vanishes for a--> co. Therefore, owing to the very small values of /r i!

g~ and g, it is possible to neglect P1 4 P2 without appreciable error. /1l i l l

In P~ ~-P2 we pu t o - - r exp [i~s] and obtain:

o

)'" T" fl r ~ exp [ 2 i ~ log t l + 2 = - - d~ s(r,q~)

r

A

o

A

r~ exp [2i~][1 - - p~z*(r, e)]

7I

f fr~ exp [2i~] = dF[ e(r,q))

.2 0

A log r 2 exp [2i~][1 - - fl~s*(r, q~)]

r: exp [ - - 2i~] log A _~ ~(r, - - ~) r~ exp [-- 2i~][1 - - f i ~ ( r , - - ~)]J '

ON T I I E E N E R G Y L O S S A N D S F E C I F I C I O N I Z A T I O N E T C . - t 3 3

~ow since r is ve ry large we have

l~e e(r, ~v) = ]~e s(r, --- q~) = Re s*(r, ~) ,

Im e(r, ~) = I m s*(r, ~) = - - I m e ( r , - - ~v) .

'Then the two te rms of the in tegrand are complex conjugate to each other and

n e + P:') = o .

In conclusion:

~(29) A

3"2. Integration o] Q. - I n this case we mus t t ake into account t ha t the

'singularities of the in tegrand are the poles of [s(~o)] -1 which lie, as a l ready

ment ioned, in the upper half of the complex plane a n d t h e poles of Re s@)

corresponding to o ) ~ - - ~ 2 igw = 0 which obviously lie in the lower half.

F o r this reason it is convenient to in tegra te (21") in the same way as for P~. 'Then one can easily demons t ra te t ha t all the contr ibutions of the integrat ion

on the immag ina ry axis and on the circle vanish identically and therefore

only the residua of the in tegrand funct ion remain which are a t the poles of Is(w)] =1 in the upper half of the complex plane i.e. :

,(30)

Taking into account (29) one finds the following expression for the energy loss for p r i m a r y exci tat ion and ionization:

(31) ~W~. (exc. ~- ion.) =

~z

2e4n Re ~ [t~esidua [ ico I A l}] - - mv 2 ~ e - ~ log 09~[ 1 _ fl2e.(co) ] 2fl 2 Re s(o~) .

In order to calculate (31) explicitely it is necessary to localize the zeros of s(co). We make the simplifying a s s u m p t i o n :

g i = g = go

i t can be shown tha t this hypothesis is not essential and th~/t the following

a rguments can be extended to the general ease gi # go. I n order to unify

3 - I I N u o v o Cimenlo.

34 P. BUDINI and L. TAFFA~A

the calculations for the discrete and continuous spectrum, and considering tha t J~(o)') are wellbeh~ved functions, we put

(32) ,. s((~) = 1 q- ~7 co~ - - (o 2 + igoo9 '

with Ao)~ = 1 for the discrete and J~Ao)~ ~ J~(co')d~o' ~-(mcl2~e~)ak(oY)dw' as. A~o~ tends to zero, for the continuous.

Then one can e~sy see that , in each band of the complex plane parallel to the immaginary axis and delimited by the frequencies o), and (o,.+1 there is a simple zero of s((o) at the point

i (33) ~ = x~ + ~ go,

with x, with good approximation only depending on o~, ], and given b y

(34) x~ = ~o~-- + LAo->~, (*) �9

The residua which appear in (31) can be then easily calculated; one finds o~ taking into account tha t Re e(N~) = �89

(35) _ ~ (exc. ~_ion.)= 2e~n Re 2 z ~ [ A fi2s,(N~)l =

- - ~o; - - j ~ A % ) log = m v 2 ~ ] i A ( D i / ( C O ~ _ _ - - ~ 2 (~[1 --/32e*(~,,)] i

-- 2~e~nmv ~ Re ~,. Ldco,, log N~[1 - - fl2e*(~)]

The last step is a good approximation for the discrete spectrum and is exact for the continuous which is of interest lot the present work. Going to the limit for A~o~ -+ 0 in (35) and separating tile residua in the discrete spectrum from those in the continuous, one finally obtains:

(36)

to multiply ]~ A,

n [ A __ fi~s,(~5~)l + ~W~ (ext. + ion.) -- .2~e~nmv ~ Ige~=l ~" ]~ log ~E1 --/~2s*(~D]

c o

2'~e4n Re ~(co') de)' log ,2 ~ - + mY 2 ~=1 ~ o~ LI-- fi ~*(~')] 3"~e*(~') "

go k

J ! (*) A further approximation brings about a factor (l+Z~,) -i with Z~ given by (3% in (34) (see also note on page 38).

ON THE E N E R G Y LOSS AND S P E C I F I C I O N I Z A T I O N ETC. - I 35

Clearly the first t e r m on the r igh t h a n d side represents the energy spent

in exci ta t ion, and the second t e r m in ionizat ion. F o r a more deta i led s tudy

we will discuss the two pa r t s of (37) separate ly .

4. - E x c i t a t i o n .

F r o m (36) we see t h a t t he b e h a v i o u r of the energy loss in exc i ta t ion as

well as in ion iza t ion a t re lat ivis t ic ene rgy depends, eriticMly on t he va lues of -

e*((o) a t the po in t s hS~ where e(eo) vanishes. F r o m the discrete p a r t of the spec-

t r u m o n e ' c a n easily get, inser t ing (33) in s*(e)):

0,~(] + G) ~ o~(1 + ~)~ s * ( ~ ) = 1 + 0~(1 + Z'~) 2 + i 1 + 0~(1 + 2~) 2 ' (37)

wi th

(as)

und

2goa)~ 0~ - (*),

h

fj_/J 0)# (39) Z,', --: Z 2 (a~'

J # v COy

which are of the same fo rm as,(41) and (46) of reference (27) (+).

The energy spen t in exc i ta t ion is t hen g iven b y

(4i) ~Wo~ (exc.) ~z

27~e4n ~ [ = my 2 .~1~ log

A

{I 1 #~o? ( I + G ) + ( 1 +

_ f12 0~(1 + Z~) 3 g i #30 , (1 ~- Z ; ) ] 1 + 0~(1 + 2~) ~ + ~ - - t g - ] " " 2 x i 1 + Oi + ( 1 + ' 2 2 2 2 ~ ) f l O~ " J

(*) In the general case (g~ ~ go) one would have obtained

2gio~ (40) 0~ -

Pi

(+) Actually the Oi in (38) differ from the corresponding of refere0ce (3:) by a factor 2. This depends on the method developed for the calculation in (~7). In fact this metho4 consisted in evaluating the integral (34) with the mean value theorem and taking the integral function in square brackets of (34) at the maxima of ai/[eI 2. Now, in this evaluation the errors are of the order of gi/(o~ as one can see, for example, on comparing (40) of (27) with (33) of this paper. This gives the factor �89 in the 0~ as compared with the exact calculation followed here. In any case this factor is unimportant in the general discussion and in the final formula it would imply a factor 2 in the arguments of the logarithm in (53) of (27).

36 V . B U D I N I 3 A l d L . T A F F A R A

The impor tan t difference between (41) and the corresponding expression

of reference (~7) lays in the fact t ha t in (41) i ]~ < 1; t ha t is in (41) propel" i = l

account is taken of the fact t ha t exci tat ion refers to the discrete spect rum

of the atoms. The t e rm in tan -~ in (41) is ve ry smM1 and can be dropped. For the discussion on the energy spent in exci tat ion i t then follows, as in

Sect. 4 of reference (~7), t ha t the relativistic increase of this energy is in com- pet i t ion w i t h the energy spent in Cerenkov. radiat ion and depends on the rat io 0~ (which contains gddensity of the medium); for 0~ << 1 one has no

relativistic increase in excitat ion and (41) becomes

i V 2 (42) ~We' (ext.) -- 27~e~n i ]i log ~ o ~ e 2

~z i v 2 i=~ '

5 . - I o n i z a t i o n .

For the continuous one obtains for e*(~') appearing in (37), dropping terms

of the order g2/o)~ with respect to 1,

(43)

where

co

s'*(~d') = 1 + Z" (o ' ) + -~2 __ c~,2 __ 2igoY ' 7e= I J

~k

(44) ~"(~o') = ~ ~o~- ~o '~ '

and therefore for ~o'-- o~ >~ g (which is the only impor tan t case in applications) -

(45) 11-- #~* (~ ' ) I = { [ 1 - - #~ ~e s(~,')]~ + [#~ I m ~(o/ ) ]~} - - 11-- #~(o~') I �9

We get also the following formula for the ionization loss:

(46) ~W~176 (ion. prim.) = ~z

co

_ ) l o g

ta x

with I 1 - - fl2s*(bV)l g iven by (45). Considering tha t the integrand of (46) is a wellbehaved function of o)' and

ON THE E N E R G Y LOSS AND S P E C I F I C I O N I Z A T I O N ETC. - I 37

that 1~(~o') is rapidly decreasing in the interval o f integration~ one c~n apply the mean value theorem and so obtains~ taking account of (45),

2~e4n ~ _ A f12 Re e(~) (47) ~Wq. (ion. prim.) -- ~ ~ t ~ log(~l 1 fl2ei(Tj~)l

where co~ is an opportune mean value. The saturation limit for f12 = 1 is given by

2~e4n s I A 1 (47') ~Wq~ (ion. prim.) (//2 : 1) 7~-~v2. ~ 1k= log (~k[l__ s(e~k) I Re s(~k) .

We can conclude then that, in the case of ionization loss there always exists a relativistic increase whose maximum value depends essentially on t l - - s (~) I which, in the general case, is very different from 1.

The analytical integration of (46) (or exact calculation of ~ ) is difficult in the general case owing to the complicated form of the integrand. An attempt to integrate (46) for the simplest cases and numerical results will be given in a future paper.

6. - Connection with Energy Loss for Close Collisions.

The preceeding considerations, based on the statistical description of the medium, are valid for ~o (collision parameter) larger than the radius of the atoms of the medium. For smaller collision purameters one can ignore the polarizability of the medium and then the energy loss with energy transfers smaller than T is given by (s)

(48) 3W~z <~ 27~e4nmv 2 log mT~2722~ 2

This can be added to (14) giving rise to the total energy loss suffered by a relativistic ionizing particle in ~ medium, i.e. :

iV 2~e4n [ B. T ] -- ~ - [log (a).~, "r __//2) q- a~(1 - - / j 2 ) _ [/2 ]

where

l/~ 2V2.

(50) B ~ 2r ~ .

:Now, from the formulas of the preceeding paragraphs one has that, as far

38 p . B U D I N I a l i a L . T A F F A R A

as the dependence o f the ene rgy loss on Qo is concerned, we cun wri te (*)

(51) ~W__~z t>q. (ext.) -- 2ze4nmv 2 - - - - (~1 log ~ § te rms independen t of ~o,

(52) --3Woz >~o (ion. prim.) -- 2:~e~nmv 2 -- - - 62 log ~Oo 2 ~- t e rms i ndependen t of ~o �9

I f we sum (48) wi th (51) and (52) and r e m e m b e r (13'), the t e rms con-

ra in ing ~o e l iminate and we ob ta in t h a t the e n e r g y to t a l ly spent in exc i t a t ion

and ionizat ion in collisions wi th any: collision p a r a m e t e r does no t depend on ~0

as is to be expec ted and as results f rom (49) t ak ing in to accoun t t h a t the

l~erenkov energy loss does no t depends on ~0.

S u m m i n g (36) and (48) we thus ob ta in

BT _ ] (53) 0Woz (exe. + ion.) -- 2ne*nmv 2 R e i ~=1 ]~ log ~ [ 1 - - D2e*(~,)] fi2e*(~*) +

co

2~e~n R e ~ ) 'f ,(co') d<o' BT _ _ _ _ R 2 e * ( ~ + ~ ] log ~ ' ~ [ 1 - ~2~*(G)] " ' '] �9

I n order to separa te the con t r ibu t ions of exc i ta t ion and ioniza t ion f r o m

this expression let us also mul t ip ly and divide the a r g u m e n t s of the loga r i thms

(*) If one takes account of the correction referred in the note at p. 34 then in place of ~1 and ~2 in (51) and (52) should appear:

a~ = ~ (1 + z:~) 2 and

r

eo/c

Now, it can be easily seen that Z ' are small and of alternate signs such that the property 6~+ 6'~ = 1 is maintained with good approximation. Thus, the sum of (51) and (52) joins continuously with (48) for small Pc. On the other hand, the energy spent at small ~o in excitation and ionization or in excitation of a single line has a Qo-dependenee which, at first sight, should be not expected from the theory of Bohr (for ~o o < radius of the atoms of the medium) where the polarizability of the medium is not taken into account. In fact; tlie energy absorbed from the atoms at small dis- ~ances is independent of eo but there are grounds to expect that the subsequent repar- ~ition of this energy in excitation of a single line and ionization, does depend on the polarizability of the medium (20). This is still an open problem.

O N T H E E N E R G Y L O S S A N D S P E C I F I C I O N I Z A T I O N E T C . - I 39

in each term of the first sum in (53) by I~, the ionization of the electron to which the ], refers (*).

We obtain

- ~ W 2~e~n ~ [ F6~[1 --B'/j~e*(~)]I'~ ] .(54) (exc. q- ion.) -- Re _ ]~ log --/3~e*(~) q- mY2 i l l

co

23ze n - - T 27~e4n ~ " B T q- m-~- _ ~ / i l o g ~ q- - - - R e /,~(a)')dr log

.= m v 2 ~_~ .~ r - - fl2s*(~')] mlc

- ~ * ( ~ ' i ] �9

Clearly then, only the first term is to be interpreted as energy spent in excitation t~s it implies collisions with energy transfers smaller than the ioniz- ation energies of the various shells. The second sum contributes to ionization so long as T ~ Ik, as it implies collisions with energy transfers larger than ~he ionization energy (+).

We obtain thus for energy spent in excitation

2z~e4n ~ [ B . IT~ f l 2 R e s , ( ~ ) ] (55) ~ w (exc.) - ~ ,=21]/ l o g ~ o ~ l l _ ~ , ( ~ i ) I

where e*(~) is given by (39). The energy spent in ionization with energy transfers smaller than T i~

given by:

(56) OW (ion. prim.) 2z~e4n ~ ]i log T ~z -- my2 ,=1 ~ +

co

q- m y 2 ,,--~. (~o') do)' log o9,211 __/32e,(bS,) I mk

co

--2:~e~nmv 2 log ~ + = ((o')do' logco,2 I

mk

where e*(~') is given by (43) and i0 by

_/~2 Re e*(~')] =

B I k fi~ Re e*(~')]}

(57) ~o = I~;.Ii; ... I~;,

(*) Rigorously ]i as well as ~o i should Carry two indices i.e. ]i~ and o)i~, the one referring to the various absorption lines of a single electron and the other referring -to t-he various electrons. In order to avoid this formal complication we keep a single index for ]i and wi but one shonld take account of the fact that the indices k of Ik vary only when ]i and co i in the sum refer to different electrons.

(+) If T ~ I~ it would contribute negatively to excitation but in the following we ~vfll consider only the case T > Ik.

40' P. BUDINI and L. TAFFARA

where

(58)

and J~l, Jk2, ]~3, .../k~ are the oscillator strength of the absorption lines referr ing to the K th electron. Clearly then:

(59) ~ ]'~ = c~1 @ ~2 = 1 .

In applications one may be interested in the energy lost in collisions in which the electrons come out f r o m the ionized atom with energy larger t h a n a

minimum energy 3. One has than simply to substi tute L0 q- 3 for the ioniz- ation energy Ik and thus obtains

27ce4~g T (60) 0W (ion. prim.) --

~z ~ log (I0 + 3) +

where

(61)

. c o

+ ~ T v ~ (co')d~o' log

a) k + ~:/fi

B(I,, + "~)

Io + 3 = (I1 + 3)72(I2 + z)T; ... (I. + 3)7;

~2 Re s*(~5')t ,

7. - C a l c u l a t i o n of t h e N u m b e r of I o n s .

Until now we have only t rea ted the problem of the energy loss of the ionizing particle. The formulas of the energy loss are, however, not the most convenient for comparison with experimental results where the number of ions is the most impor tan t datum. In fact, one can easily verify tha t the max-

imum transferable energy T plays in (56) a criticN role especially for those elements for which dl << d2, tha t is for all bu t the lightest ones. Now, if it is to be expected tha t tile rare collisions with large energy transfer T p lay

an impor tan t role in the problem of the to ta l energy loss and give rise to the well known problem of fluctuations (15), these collisions are certainly not

determinant for the problem of the number of ions, and so tha t parameter T

should not enter critically in the formula to be compared with specific ionization. We will therefore t ry to give a method to be applied directly to the c a t

culation of the number of ions. Wi th this aim in view we start from the fol- lowing expression

c o

4e4n tr [~i do) [ A /~. ] ~N~176 (ion. -- exe.) = ~ eJe-~)pogco2!]_f124co) ] Ree(~o) , (62) 0z

0

ON THE ENERGY L088 AND SPECIFIC IONIZATION ETC. - I ' 41

obtained by dividing the integrand of (19) by Cho, and which gives t h e number of events (ionization + excitation) produced by a relativistic particle in collisions with collision-parameter larger than ~0. Put t ing (62) in the form=

(63)

with :

~2V~.~z (ion. + ext.) = ~4e4n [-P~ 4- P2 4- Q] ,

(64) P1

(6i ')

(6u

co

= �89 Re Js(o~) log 0

A co~[1 --9~s(~.)] '

co

f/aco ~,. = �89 Re j ~ log

0

A

co211 ~*(co) ] '

co

O = - ~~ R e f i 0

R e e(co) dco e(co)

We proceed separately to the integration in the complex plane of t h e single terms (64) on the same line as for the energy loss: this is possible because the integrands of (64) present the same poles as the integrands of (21). In this way, maintaining the same meaning for the symbols as in the integration of energy loss, we can also in this case write

(6,~)

i ii iii P* = P* 4- P1 4-P1 ,

[ (5 p~ = Ps 4- P~ 4- P~" + �89 Re ~ Residua log

One obtains

(66) i i Ii ii

P 1 -}- -P2 -}- P 1 -}- P 2 ~ -

r . / : ~_1]i giY dy log A i i~O~ 4- y2)_g~y2 iy.,[l_fi,,.e,(iy)] I 4-

o

co r

o~ k o

In order to calculate the order of magnitude of these terms, we will sire-

4 2 P. BUDINI and L. TAFFARA

plffy the expression for s*(y) by put t ing it in the following form:

e*(O),-- 1 (67) e*( iy) = 1 +

l § ~

In this way, taking into account tha t the integrand function has poles for 1--fl2s*(iy) - - 0 i.e. at the point

[fl,~*(0) - = [ ~_~fl~ ~l ~ , 1

,(66) becomes . . . . . . " g , [ A

(68) P1 ~- P2 -u ~1 -~ P2 : li~]i=1 ~2.[10g fl~,(o) 1 "~ § a 2 log §

r

1 ~ . f l /c(" ,) d . j [1~. ~4 , 2 log + ~ ~' '~ ! ~ l~'~ f l2e*(O) - - 1 ~ a 2

e9 k

I t will result obvious later tha t this term is to be neglected with respect ~o the other because the smMlness of the constants g~ and g.

We find further

.(69)

~nd

(70)

~hus

471)

tit Ill P1 + P ~ = 0 ,

8No' (ion. + ext.) = 8z

2e~n f . ~ ( A [Residua , - - - - log

= mvq~ ]~e i [ re ( , ) , 2 1 1 - - fi=e*(-)]

Taking into account (33) and (34), (71) becomes:

27~e~n A ]~ 8N~176 (ext. + ion.) - - ~v-~v~i=.~ I "---7" ~72) ~z

[ A - / l o g - ~ , ~ , [ " m~V/(1 - - fl'(O~/1 +0~)(1+2~.)2) 2 _L (f12(O,/1 -r 0~)(1 -~- S, ))~

co

.+ mv=h,~= , j -~; l o g . , ~ ] l _ _ fi2e*(b~')] m k

where s*(~5') is given by (43).

O N T H E E N E R G Y L O S S A N D S P E C I F I C I O N I Z A T I O N ]ETC. - I 43

Clearly the first te rm on the right hand side represents the number of exci ta ted atoms and the second te rm the number of ions produced per unit length by the pr imary particle at a distance larger than ~o from its t ra jectory. The second te rm is of importance for comparison with experimental da ta on ~onization and therefore we will concentrate our a t tent ion on it in the following.

Here also we have to connect formula (72) with the number of events ge- ne ra ted by the particle at close distances. For this purpose we differentiate

,(48) with respect to T and obtain:

(~W ) 2~e~n 1 {73) d -~z <e. -- mv~ T dT"

This is the well known formula (s) representing the mean energy loss per cm per in terval d T of the transfered energy T for large values of T (so long as the energy of the ionizing particle >> T). In order to re-obtain (48), we have to in tegrate (73) between the limits To and T with:

2h'~ (74) s - -

m~oo~7 ~

We thus obtain a relat ion between Qo and To. I4emembering tha t To is the minimum transferable energy in close collisions we can take for To the mean ionization energy of the atom, or be t te r still for each electron its ionization energy.

In order to obtain the number of ions created in collisions at close distances (< ~o) we have only to divide (73) by T and to integrate it between To and T:

T

az<+0= = To

eo can now be el iminated also from (72) with the substi tut ion (74). Summing with (75) the formula obtained in this way, and put t ing in each term of the sum To = Ik, ionization energy of the corresponding electron (correspondingly we must subst i tute ~ ]-2/I~ for l/To in (75)), we obtain:

k

~N (76) ~ (ion.) -- +

co

[!(") d.'[ + j ~ j log

(o k

]J

4 4 g . BUDINZ D~Ild L. TAFFARA

In general, T >> ik and therefore the number of ions will be independen~ of the maximum transferable energy T as is to be .expected .

In part icular eases the integral appeuring in (76) can be explieitely integ- ra ted; in general the mean value theorem can be applied and one obtains:

~N (ion.) - 2=e*n ~ f ] ~ - - 1 (77) ~z ~ ~ tE Y +

]~ log ~ ,~ I + ~ co~ LI - ~ (o3~) i

This formula shows tha t the number of ions presents a logarithmic increase and a subsequent saturat ion because of the polarizabili ty of the medium.

t n this ease also one may be interested in the number of ions in which the electron comes out with energy larger than a minimum energy 7. One has

then simply to substi tute in the preeeeding formula the ionization energy L~

by I~: + 7:

(78) oN (ion.)

~t(co') dco'[ B. (L + ~)

As T increases~ the terms with the integrals will, in general decrease faster than the corresponding terms in the sum since ](co') behaves us co -~ where n > 1 i on the other hand the s(co) inside the logari thm will be calculated for larger values of the argument co at which s(co) will become very near to one; thus the integral t e rm will present a more prolonged logarithmic increase as fl --~ 1. In other words with in'creasing 7 the logarithmic increase should become

f l a t t ened (predominance of the finite over the integral terms) bu t reach la te r saturat ion as f i -+ 1. This behaviour will be iUustrated in numerical applic- ations in the next par t of this work and could be tested~ for example~ b y the dependence of the logarithmic increase on the degree of development in nuclear

emulsion (~,~s,~7). For 7 ve ry large the integral t e rm vanishes and if T }} I~ we obtain be-

cause of (59):

(,79) d N (ion.) > _ 2=e'n (1 1) d-7 mv~ ~ - - ~ '

tha t is the number of events presents no relativistic increase and they are ge-

nerated only in close colhsions.

ON THE ENERGY LOSS AND SPECIFIC IONIZATION ETC. - I 45

R I A S S U N T O

Si calcola la, perdita di energia totale media che una particella subisce nell 'at tra, versare un mezzo polarizzabile introducendo nella teoria di Fermi un'espressione della r dielettriea ehe rappresenti correttamente sia le propriet~ dispersive che quelle di assorbimento del mezzo. L'energia totale media si pus cosl separare in energia media spesa in eceitazione, radiazione Cerenkov e ionizzazione primaria degli atomi del mezzo. Si calcola l'effetto della polarizzabflith del mezzo sul numero di ~ ioni generati dalla parti- celia ionizzante per unit~ di lungh6zza e sul numero di eventi riei quali g stato emesso un elettrone atomieo con energia maggiore di una certa data che potrebbe venir iden- ~ificata come energia di soglia per la formazione di un granulo in emulsione nucleare. All 'aumentare dell'energia della particella ionizzante questi humeri presentano un aumento logaritmico seguito da saturazione. Sia l 'andamento dell 'aumento che il valore Taggiunto alla sa~urazione dipendono in modo critico dall'energia di soglia.