IL NUOVO CIMENTO VoL. 3A, N. 3 1 Giugno 1971

Cosmic-Ray Propagation in the Atmosphere (*).

K. O'BRIEN

Health and Sa]ety lSaboralory, U.S . Atomic Energy Commiss ion - New I~orlc, N . Y .

(ricevuto il 27 Ottobre 1970)

S u m m a r y . - - An essentially analytical theory of atmospheric cosmic- ray propagation is developed on the basis of a phenomenological model of hadron-nueleus collisions. This model correctly predicts the sea-level cosmic-ray nucleon, pion and rnuon spectra, the cosmic-ray ionization profile in the atmosphere, and neutron flux and density profiles in the atmosphere. It is concluded that the large-scale properties of atmos- pheric cosmic rays can be accurately predicted on the basis of a purely nucleonic cascade as a result of which all secondaries are mediated by pion production. Implications for energy independence of cross-sections, the recent 70 GeV results from Serpukhov, and nucleonic relaxation rates in the atmosphere are discussed.

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

This paper a t tempts to establish the physics on which the large-scale,

t ime-averaged, one-dimensional properties of galactic cosmic rays in the atmos-

phere depend. The point of departure for this theory is a phenomenological model of high-

energy mlcleon-nueleus collisions which can be applied to analytical t ranspor t

theory. ] 'article spectra, fluxes, densities and ionization calculated from the

theory yield good agreement with measured values indicating the adequacy

of the nuclear model and of the supporting cosmic-ray and geophysical data.

All comparisons are on an absolute basis.

Prel iminary results of this work have already been reported comparing

calculations and measurements of various components of cosmic-ray ioniza-

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction.

34 - 1l Nuovo Cimento A. 521

522 ~. o '~Ri~

t ion in the lower a tmosphere ( < 2.5 k m elevation) ~t a geomaguet ic l a t i tude

of 51 ~ (~).

2. - Atmospheric model.

The a tmosphere is assumed to be a flat slab ]033 g/cm ~ th ick with a con-

s tan t scale height of 6.7 kin. I t is assumed to be composed of a single nuclear species with an a tomic weight of 14.48, a tomic n u m b e r of 7.31, and an ioniza-

t ion poten t ia l of 86.8 V. Because oxygen and ni trogen are so close in the periodic table, this simple assmnpt ion yields the correct nuclear data. The densi ty of the a tmosphere is

(1) o~ ---- r / H ,

where

o is the densi ty in g /em 3,

r is the depth in the a tmosphere in g/era 2, and

H is the scale height in era.

Since the mean free pa th for decay of a charged part icle is

(2) ).a - - ~ cr~ o mac ~ '

where

2q is the mean free p a t h for decay in g/era'-' of a particle of type q,

m~ is the mass of thc part icle in MeV/e -~,

P~ is t.he m o m e n t u m in MeV/e,

c is the veloci ty of light, and

~q is the mean life in the rest f rame, in seconds,

we have the useful result t h a t

(3)

where

2~ -~ 1'~ cr/C~ ,

Cq -~ m a c " H / e % .

3. - Theory of the atmospheric nucleonic cascade.

3"1. The Bol tzmann equations. - I n this paper , a tmospher ic cosmic-ray fluxes will be ob ta ined as analyt ic solutions to an app rox ima te form of the Bo l t zmann equations. The Bo l t zmann equat ions for the nucleonic cascade in the a tmos-

(1) K. O'BRIEN: Journ. Geophys. I~es., 75, 4357 (1970).

C O S M I C - R A Y P R O I ~ A G A T I O N IN TItlE AT~'Y[OSPHEICF, 5 2 ~

phere are

(~.1)

(4.2)

(4.3)

where

r is

E is

s is

~ is at

B~f , ( , . , E , ~2) - S ~ (q - - p, a , =; i = P, n ) ,

B~9g(r, E, s = S~=~ ,

B~q~(r, E, ~ ) = S ~ (~ = y, e; fl = y, e, ,~o, [•

c a

-4 'f ' , 4~ /g'

(qj = np, pn, ,=n, =p, p..'~, e~z, ~ o , eu -~e),

~ . - C ~ = C ~ = C ~ = C . ~ = k ~ - - k ~ o = / % = O ,

the depth in the a tmosphere in g/cm'-',

the particle kinetic energy in MeV,

the uni t vector in the direction of part icle travel,

the part icle flux of a part icle of type q per MeV per second per steradian a depth r with a direclion ~ ,

the to ta l cross-section for absorpt ion of a part icle of type q in cm-"/g,

~,~ is the cross-section for the product ion of particles of type q f rom eolli- sions with, or decay by, particles of t ype j in cm"-/g,

k~ is the stopping power of a charged part icle of type q in air, in MeVcm2/g, and

F~ is the number of particles per MeV per second per steradian at E and D resulting f rom a collision with o1" decay by a part icle of type j at E s and ~ ' .

The subscript ~. implies application to all pions, charged and neutral . The subscript =+ implies application to the charged pions, and ~o to neut ra l pions only.

3"2. iVuvleon-nucleus collisions. - In eq. (4) it has been assumed t h a t atmos- pheric cosmic rays propagate by means of the nucleonic cascade in an exponen- tial a tmosphere (see Fig. 1 of O'BmE~- (~)). Thus all other secondaries result f rom nucleon-nucleus collisions. The following reactions a.re considered:

p + a i r -+ v~p § n + ~ - - ' -t-~o~ ~ ,

(5) =+ ~ ~ .§

7r ~ -* 2~f-+ electromagnetic showers,

--> e + 2 v --> electromagnetic showers,

524 K. O'BRW, X

where vj are the multiplicities of j - t y p e particles result ing f rom the collision

of a nucleon with a nucleus of air. The influence of kaon product ion has earlier been shown to have a small influence on ionization, and it is neglected here, at a considerable saving of compute r t ime (~).

The nucleon-nucleus react ions of eq. (5) will be considered at high energies only, because the ma thema t i ca l fo rm of the approx imat ion to the Bo l t zmann

equat ion to be obta ined is only applicable at energies above abou t 0.1 GeV (*). I t is required (a foreseen ma thema t i ca l result mot iva tes the choice of the

function) t ha t

El G~ = I f ~ U ( E ~ - - W ) ,

(6) G~j -- 2~td0 sin O F~(E~ --~ E, tg) ,

0

~2 = c o s 0 ,

wtmre

G~j is the secondary product ion spec t rum of type-q part icles in tegra ted over

the solid angle,

I~ is an a rb i t r a ry constant depending on q,

1 and n are a rb i t r a ry constants which mus t be the same for all j, q, and

U(x) is the Heavis ide funct ion ( U ( x < 0) = 0, U(x >~ 0) = 1) ,

~q is a lower-energy l imit below which secondary part icle product ion is cut off.

The formula for G,,j is cer ta inly ve ry crude, however it is suitable to rep- resent the behavior of the par t ia l iuelmsticities and multiplicit ies associated with h igh-energy nucleon-nucleus colhsions. Inelast ici t ies are known to va ry quite slowly with energy, and to become essential ly constant at energies of a few 10's of GeV ('0). I f n - - l - t - 1 , t h e n O~ can be r ewr i t t en in t e r m s of a

cons tant par t i a l inelast ic i ty K~ for the product ion of a lype-q part icle

(7) a . = (1 - - Z ) K o ~ U(E~--Vo) �9

(*) Low-energy nucleon transport (<0.1 GeV) is chiefly neutron transport. Low- energy neutron and electromagnetic-shower transport are treated only very roughly here due to the limitations of the transport theory to be described. I t is intended in the near future to apply the S~ method to this problem, and treat it much more generally. The analytic theory presented here is, if less general, quite accurate at high energies, quite simple and rather transparent. (2) Y. FuJi_~o~o and S. IIAxhXAW•: Cosmic rays and high.energy physics, in Ency- clopedia of Physics, Vol. 46/2 (Berlin, 1967), p. 115.

COSMIC-RAY PROPAGATION IN THE A T M O S F H E R E 525

IAAOEDOI~ and R A ~ ' ~ (~) have cMculated K~ using the stat is t ical model for

p-p collisions at ]2.5, ]8.8, 30 and 300 GeV/c and it would be quite convenient

to use these resul ts for K , . However , Aa.S~ILLEa and BARISH (4) tiave shown

t h a t the secondary produc t ion spec t rum G~ softens ~dth increasing a tomic weight as a result of the in t ranuclear cascade. This effect has been s imulated in ec[. (7) b y mak ing K~ a funct ion of a tomic weight (5). This is, mffor tunate ly , a t the expense of the conservat ion of energy in high-atomic-weight nuclei,

a l though for air energy is conserved reasonably well. The int ranuclear cascade conserves energy as G~j softens, b y the emission of low-energy particles, bu t

these low-energy part icles are not impor t an t to deep-penet ra t ion calculations a t high energies. Values of K~ in te rpo la ted f rom among the values obta ined earlier arc given in Table I (5).

TABLE I. -- Partial inelasticities lot proton-ai~" collisions.

q Kq

p 0.211

n 0.211

~* 0.180

r=- 0.112

0.180

K + 0.034

K- 0.022

K ~ 0.034

E x p e r i m e n t a l values of par t ia l inelastieities in the energy range 0.1 to 20 TeV have been obta ined in nuclear emulsion f rom the Brawley and I .C.E.F. emulsion stacks (2). The quanti t ies measured were: Koh , the energy t h a t goes into new charged part icles, and hence

(8) K+h= K.+ ~,, K._ ~- KK+-+ K K_ ;

Kv is the energy tha t goes into photon product ion, and so

(9) K~ = K~. ;

(a) R. tIAG~I)OR~ and J. RA.~FT: Suppl. Nuovo Cimento, 6, 169 (1968). (4) R. G. ALSMILLER jr. ariel ,)'. BARISH: OIRNL-3855 (1965). (5) K. O'BI~IEN: SVucl. Instr. Meth., 72, 93 (1969).

526 K. O'BRIEN

K0 is the ene rgy t h a t goes in to long- l ived neu t r a l part icles. I t is a s sumed t h a t

these are neu t ra l kaons. This is a small numbe r , and a n y er ror t h a t m a y arise

here is u n i m p o r t a n t to the s tudy . The energies invo lved in the m e a s u r e m e n t

are h igh c o m p a r e d to t he largest res~ masses invo lved and thus t h e y m a y be

neglected. I n Table I I , t he predic t ions of O'B~IE_~ (5) (which in con junc t i on wi th

eq. (7) will be re fe r red to as t he power law mode l hereaf ter) is c o m p a r e d wi th

these da ta , and wi th t he predic t ions of o the r nuclear models .

TABLE I I . - Partial inelasticities at very high energies.

Experi- Statisti- Power TRB (r Extra- Tril- CKP mental (") eal (b) law (~) model pola- ling (b) (~) ((0.1 --20) model model (20 tion (~) model mod- TeV) (300 (~) GeV) model (~ ) el

GcV) (200 GeV)

/t:= h 0.31 _-4=_. 0.06 0.346 0.303 0.23 0.166 0.21 0.38

/i: v 0.16 0.186 0.157 0.14 0.084 0.09 - -

K o 0.03 0.035 0.029 - - - - 0.01 - -

Total 0.50 =~ 0.07 0.585 0.489 - - - - 0.31 - -

(a) Nuclear emuls ion . (b) Hydrogen . (c) Air , kaons neglected. (d) A l u m i n i u m , kaons neglected.

These o ther nuclear models have all been c o m p a r e d against , and in some cases based upon, acce lera tor t a r g e t yields, m o s t l y a t small angles and a t h igh

s econda ry m o m e n t a . The wel l -known C K P mode l (s), t h e Tril l ing mode l ('),

t h e ex t r apo l a t i on mode l (8), a nd t he T R B m o d e l (9) are considered. RA~'Fr

and BO~AK (~) have modif ied t he fo rmulae used b y TRILLING (~), a.nd this is

re fer red to as the T R B model .

The considerable va r i a t ion is occasion for surprise. The s ta t i s t ica l model ,

t he power law mode l which is an a d a p t a t i o n of it, and the C K P mode l are in

a g r e e m e n t wi th the cosmic- ray emuls ion measu remen t s , t h o u g h all t he models

appea r to agree wi th in a f ac to r of two. This is p r o b a b l y a consequence of the

f~ct t h a t m o s t of the accelera tor t a rge t da ta , and m u c h of t he phys ica l in teres t ,

(~) G. Coecoh~I, L. J. KOESTER and D. H. PERKINS: UCRL-10022, p. 167-190 (1961). (7) G. TmLLI-~G: Lawreuce Radiation Laboratory Report UCID-10148 (1966). (s) T. A. GABRIEL, :[~. G. ALSMILLER jr. and M. P. GUTHRIE: 0RNI,-4542 (1970). (~) J. RANFT and T. BORAK: Improved nucleon.meson cascade calculations, FN-193, 1100.0, National Accelerator Laboratory (1969).

COSMIC-RAY PROPAGATION IN TIlE ATMOSPHERE ~2~

is a t sm~tt f o r w a r d angles ~nd a t large seconda ry m o m e n t a , ~nd this does not:

de t e rmine K~ prec ise ly enough.

The secondary-par t i c l e mult ipl ic i t ies of the power law mode l depend on

t he vah te chosen for l. Since eq. (7) has an (~ in f ra - red ~> divergence, it is no t

sui table for to t a l par t ic le yields. Howeve r , shower par t ic les in an emuls ion

p r o d u c e d b y h igh -ene rgy nucleons have finite lower-energy l imits of 8 0 - ~ e V

for mesons and 500 MeV for p ro tons (,0). Tile power law mode l yields as t he

mul t ip l ic i ty of par t ic les above a lower l imit 1 '

(lo) ~(E~) = [(] - - Z)/Z] K , [ ( E ~ / F ) ' - - 1 ] .

Using the da t a of MEY~n et al. (tt), t h e bes t va lue of l, in t he leas t -squares

sense, was chosen. The p rocedure is descr ibed in a s o m e w h a t more e x p a n d e d

w a y in O~BRIEH (5). I n Table I I I , t h e p red ic t ion of shower par t ic le p roduc-

t ion b y var ious nuclear models are given. The power law resul t is no t real ly

TAn~,E ] [ I . -- Shower particle multiplicities.

E (GeV) Experi- Power Statis- Extrapo- TRB CKP mental (a) law (~) t ie~ (b) l~ ion (~) (a) (,)

model model model model model

12.5 3.8 5.0 3.0 5.5 3.8 4.9

20 5.3 5.9 3.8 - - 4.3 5.5

30 7.3 6.8 4.8 - - 5.0 6.1

200 12 12 - - 6.9 10 9.8

300 13 13 8.5 - - 12 11

1000 17 17 - - - - 20 15

(a) Nuclear emulsion. (b) Hydrogen, total charged-particle yield. (c) Alumirdum, total charged-particle yield, kaons neglected. (d) Air, kaons neglected. (e) Hydrogen, total charged-pion yield.

a p red ic t ion b u t a fit. Since the s ta t is t ica l mode l and the ex t rapo la t ion mode l

c anno t be m a n i p u l a t e d w i t hou t t he p rope r c o m p u t e r codes, the i r r epor t ed

t o t a l charged-par t i c le p r o d u c t i o n is g iven in place of t he shower par t ic le mul t i -

plicities. As the Tri l l ing mode l does no t give b a c k - e m i t t e d part icles cor rec t ly (7)

it has been o m i t t e d f r o m cons idera t ion here.

(1o) U. CAMI~RI~I, 5. H. DAu P. 11. :FowLER, C. F~ANZINETTI, II. MUIRI{EAD, W. O. LOCK, D. II. PERKINS and G. YEKIJTIELI: Phil. Mag., 42, 1241 (1951).

(it) H. MEYER, M. W. TEUCLIER and E. LOtIR~A~N: 2r Cimento, 28, 1399 (1963).

5 2 8 K . O ' B I ~ I E ~

Some of the differences in Table I I I are cer ta inly due to differences in the target nucleus and in the lower-energy limit. Bu t it is clear t ha t the power law model agrees with the exper imenta l da ta as well as the other models.

Multiplicities and inelasticities are averaged quanti t ies re la ted to hadron-

nucleus collisions. Mat ters are different when one considers the fo rm of G,~ predic ted b y the various models. Figure 1 exhibits G=:~(EB, E) for I'J B equal

10 -3 10 -3

~ 10 ~

fi lo -~

10-"

g

lO -5

Xk~ \k\

" - " , , ~ \ \ \ \ \

I EB=IO Gev '" \' \ . , . ~ Es ',,,\\ \ --loo G~v L ,,/ """' \ X ',,/,,/ "'",, I ",, \ \

E(GeV)

l o -~ l I~ li lo -~ 10 0 101 10 2 10 0

Fig. 1. Fig. 2.

k \ E =100,,.GeV

101 10 2 E(GeV)

Fig. 1. - Three models of charged-pion production spectra from proton-air collisions: - - power law model, =++=- ; . . . . . . . . . TRB model, r.+ i-~-; - - - CKP model.

Fig. 2 . - Two models of proton production spectra from proton-air collisions: - - power law model, - - - - - TRB model.

to 10 and 100 GeV protons incident on a.ir calculated using the power law

and T R B prescript ions, a.nd the C K P prescr ipt ion for protons on hydrogen.

The CKP and TICB models agree a t high secondary momen ta , bu t differ else-

where. The greater sophist icat ion of the T R B model can be seen in the graph.

For instance, the inflection point at G~,p(100, 18) corresponds to the transi-

t ion f rom energetic pious result ing f rom isobar decay to low-energy pious emi t t ed isotropieal lyin the center of mass. The power law model over-es t imates pion product ion a t high secondary m o m e n t a bu t underes t imates a t low secon-

da ry m o m e n t a . I n Fig. 2, secondary p ro ton produc t ion predic t ions G(E~, E) are exhibi ted

for protons on air, again for E n equal to 10 and 100 GeV. The re la t iw ~, crudi ty of the power law model is clear. I t underes t imates p ro ton product ion a t high

COS ) I IC-RAY P R O P A G A T I O N IN THE A T M O S P I I E R E 529

secondary m o m e n t ~ ~nd over-es t imates ~t low secondary momen ta . It, is prob-

ably more significant for the cascade ca.lculations to follow t h a t the power

law model should be r ight on the average, t h a n t h a t it should be r ight ~t some par t icu lar secondary energy or angle.

The react ion cross-sec, t ions used in the calculation to follow are assumed

to be cons tant and geometric , i.e. a=~r~L/A , ro --: .1.28 At fm, where Z is Avogadro ' s number . This a.ssumption is val id for low-energy nucleon-nucleus collisions for a tomic weigh% f rom less t h a n 12 to grea ter t h a n 64 (~2), and it is used in wha t follows for all energies and hadrons.

4. - Approximate solution to the Bo l tzmann equations.

4"1. Hadrons and muons. - A solution will be ob ta ined for a sort of (~ Green 's

funct ion >), t h a t is for incident nucleons homogeneous in energy and angle,

of uni t s t rength per s teradian, the integral of which over the cosmic-ray pr imaries yields a tmospher ic cosmic-ray fluxes per s teradian per second.

I f we make the (( s t ra ight -ahead ~> approx imat ion

(.1 - ~2'.~) (11) Foj =: G ~ 2~ '

eq. (4.3) becomes for the hadron component

go

= J d E ~ ( ; o j ( E B , E)%(r, En, f2) (qj ~= np, pn, ~n, ~p) , (12) S,j

where cr is the geometr ic react ion cross-section in cm2/g, and for the muons (~'~)

m.~/m~ C.-~

where t2 is the cosine of the zenith angle of the incident radia t ion and E0 is the energy of the incident nucleon.

The Bol t zmann opera tor of eq. (4.2) for the muon component is then

(].4.1) B~ = t2 ~ + C'~ ~r Pz~r ~E k~.

(r_,) H. W. B]~'riNr: _Phys. ~ev., 188, 1711 (1969). (la) R. G. ALSMIIJLER jr., F. S. AI,S)ItI,L~:R and J. E. MURPHY: 014N1,-3289 (1963).

530 K. O'~RXE~

The neut ron operator is

(14.2)

The remainder are all simplified by the omission of charged-particle stopping:

(t4.3) B, = / 2 ~ -[- a ,

(:1 ~.4) B~ = t2 ~ § a + C~

_ _ _ .

c.r P = r

This omission is not impor tan t for secondaries above about 1 GeV (~4). Compensation for this can roughly be made with the use of the ][eaviside

funet ion of eq. (7), as will be shown. Separat ing the pr imary nueleons from the secondaries produced in the

atmosphere, we obtain

(15) ~ = ~ + ~ ,

where

~ is the ilux of p r imary nucleons, and

~ is the flu:( of secondary nucleons.

This leads to 3 ditlerential equations

c C~ ~ k~ %~-- ~ , ~ r , - E B , ~ , (16.1) .(2 ~r + 1)~r ~cE 2zm~ P,,~r ma

(16.2) / 2 ~ + a % , = 0 ( q = p , n ) ,

(16.3)

E~ax

~-..o - ) o ~ g ( E . - - V . )

( q = p , n, =).

Equa t ion (16.1) is wri t ten in integral form and reduced to quadratures (15).

A 51-point set was found necessary for the integral over space and 7 for the

integral over angle. The solution to eq. (16.2) is

(17.~) ~ = e x p [ - - r / a / 2 ] .

(1,1) R. G. ALSMILLER jr., F. S. AI, SMII, LER and J. ]3. BAItlSH: 0RNI,-3854 (1967). (is) S. A. KRO~RO1): Nodes and Weights o/ Quadrature Formulas, (New York, 1965).

C08MIC-R&Y PIr IN I H E ATMOSPIIERE ~ l

PAssow (~), and ALS_mL~,nn (17) have shown tha t the solution to an integro- differential equat ion of the form of eq. (16.3) wi|,h constant cross-section for incident nucleon flux of energy Eo and zenith direction Y2 is

(17.2) (p.~q = [ ] ~+~,/t"or (1--~)Ko~ ~ U(E0--W)"

�9 r I~ [2 ~/(r/[2) B ( E o , , ~)B(Eo, E)

B(Eo, E) = a ~ (1 - - l ) K t { l n E o - - l n [ E U ( E - - W ) -IF v tU(~ , - -E) ]} ~=D,II

(v = p , n; q = p , n , ~ ) .

Equa t ion (17.2) differs slightly f rom the form obtained by PASSOW (16) and A~S~m.LER (,7) by the inclusion of tile decay t e rm in the solutiom The

reason for this lies in the fact t ha t C~ .~ C, := 0, t ha t G.~ does not appear under the integral of eq. (16.3) and tha t pion product ion and absorption is purely local so tha t the energy independence of thc cross-section required by the

solution can be relaxed. The remaining para, meter ~ is a lower-energy limit beneath which secon-

dary pal%ieles of t ype q are suppressed. Thus, ~ - - - E o , ~n(t ~ = 0. To com- pensate for the neglect, of pro ton stopping, ~/~ is set, equal to 500 MeV.

The neglect of charged-particle stopping, the straight-ahead approximation, and the constant geometric cross-section make eq. (17.2) increasingly shaky as secondary-particle energies go below 1 GeV and fails completely by 100 MeV. At, high energies, eq. (17.2) should become and remain quite accurate as long as t and a can be t rea ted as constants. I t may seem tha t the neglect of hadron product ion by incident pions should f~il at high energy since )..~ (eq. (3)) can become very long�9 However, as argued by A])AIR (~8), any pion which interacts with a nucleus can be t r ea ted as an absorption. This arises because of the eombinat, ion of the relat ively low pion inelasticity with the steepness of the nucleonic energy spectrum. Only rarely will a pion be emit,ted from a pion- nucleus collision with a.n energy near tha t of the incident nucleon (see Fig. 1 for the predictions of CKP and TI~B at high secondary energies). The steep- ness of the energy distr ibution then causes the number of pions resulting from pion-nucleus collisions to be small compared with those produced direct ly in

nucleon-nucleus collisions. The restr ict ion on the form of G~j imposed by the power law model would

seem a priori to bc the most serious l imitat ion on the use of eq. (17.2). I t is

(1~) C. PAssow: Phenomenologische Theorie zur Berechnung ei~er K~kade aus schweren Teilchen (Nukleonenkaskade) in der Materie, DESY Notiz A 2.85 (1962). (17) F. S. AI.SmLLER: 0RNL-3746 (1965). (ls) R. K. ADAIIr Phys. Rev., 172, 1370 (1968).

532 K. O'BRn,:N

then wor th point ing out t ha t eq. (17.2), sui tably modified, has been ~pplied

to accelerator beam measurements in iron for p ro ton energies f rom 1 to

18 GeV (~,~9), and an 8 GeV pion b e a m on t in (~o) with excellent results.

Once it is es tabhshed tha t the pa r ame te r s under lying the power law model

are correct and t ha t the cross-sections ~re correct, more general methods can

be used and m a n y of the approx imat ions made here to s tay within the bounds

of Passow's ma thema t i ca l f r amework (~6,17) can be abandoned. The relat ive

ease and c lar i ty of Passow's approx imat ion make it of value in itself, however.

4"2. Photons and electrons. - Elect romagnet ic-shower p ropaga t ion is not

so problemat ic as nucleon t ranspor t . Essent ia l ly exact Monte Carlo t r ea tmen t s exist and have been tes ted against exper imenta l da ta (~1.._,~)..These calculations

~re difficult to car ry out over the range of depths and energies requfi'ed, and

as the goal a t this t ime is the e s t abhshment of sufficient conditions to deter-

mine the ~tmospher ie flux, the p ropaga t ion of the e lectromagnet ic cascade

is t r e a t ed ve ry pr imit ively .

Since the mean life of the neu t ra l pion is 0.9] .]0 -~ s, C,~o/Pr is huge com-

pa red with ~ (C~o == 3.3.10 ~~ GeV), it decays immedia te ly into 2 photons. The

m u c h decay probabilfl ,y is ve ry much less (C~ == ].1 GeV) and is of signifi- cance only beh)w about 10 GeV. Energy deposit ion is calculated f rom the assumpt ion t ha t the to ta l energy of the neut ra l pion produced per g r am of

a,ir (from eq. (] 7.2)) is absorbed ~t the point of product ion, and ~ the to ta l energy of the decaying muon (from cq. (16.1)) is absorbed ~t the point of decay (*).

This assumpt ion will deter iorate inversely to the geomagnet ic lat i tude. As the geomagnet ic cut-off rises toward ]7 GV, the neutral-pion product ion spec t rum

will become harder , und as the radia t ion length of air is of the same order of

ma,gnitude as the nucleonic collision mean free pa th , neglect of t r anspor t will become increasingly serious. However a,t higher lat i tudes, the error will be seen to be tolerable.

5. - Solar and tellurian modification of the galactic cosmic-ray spectrum.

5"1. Solar act iv i ty and the in terplanetary medium. - I t is well known t h a t

var ia t ion of solar act ivi ty , th rough the agency of the solar wind, modula tes

(19) T. W. ARMSTRONG and R. G. ALSMILLI.;R jr.: The nucleon-meson cascade in &on induced by 1 and 3 GeV p~'otons, in ORNL-RSIC-25, ANS-SD-9; Shielding Benchmark Problems, edited by A. E. PROFIO, Oak Ridge Nt~tional Laboratory (1970). (20) K. O'BRIE.~: Nucl. Instr. Meth., in press (1970). (21) C. D. Zl~mnv aud II. S. ~NOR,~N: ORNL-3320 (1962). (22) tI. :BECK: USAEC Report HASL-213 (1969). (*) liowever, at t.his time, an attempt is being made to apply the electrolnagnctie- shower code CASCADE (22) to this problem to improve the treatment of this important component of atmospheric cosmic rays.

COSMIC-RAY PROI'&GATIO-N IN THE AT~OSPIIEICE 5 3 ~

the cosmic-ray spec t rum found along the E a r t h ' s orbit. This modula t ion is a eonsequence of cosmic-ray t r anspor t th rough the in te rp lane ta ry med ium and it is formal ly the same as t ha t which would be produced by a heliocentric electric tield having a magni tude at the E a r t h ' s orbit of about 100 MV at

solar m i n i m u m and about 1000 5,IV at solar m a x i m u m (~3,~.4).

The electric-field model is a useful computa t iona l tool for represent ing

solar effects as the po ten t ia l is the only adjustable pa ramete r . This represen-

ta t ion is for convenience only. I t is not asserted here t ha t a heliocentric poten-

t ial of this size exists and is responsible for solar modulat ion. The electric-field model or the modula ted cosmic-ray ilux is (25)

n(E) = n0(T) [P(E!/S [ ~ r LP(-T).I W(1~)3 '

T - E + ZU~ (18)

P(x) --__ _l ~/x~ T 2A m~c"x , C

W(x) - x + Am~ c 2 ,

where

no is the unmodu la t ed galactic spec t rum of a tomic weight A, and atomic number Z, per s teradian per cm 2 per s per MeV, having an energy of E MeV, and

U is the solar po ten t i a l in MV.

For the calculations to follow, the unmodu la t ed spec t rum is t aken f rom F I ~ R and WA]PDrXGT0.~ (2a.~6) below :tO GeV per nucleon. Above t h a t energy the spec t rum is t aken f rom PE'rE~s (27) with which i~ has a smooth overlap.

Calculations of a tmospher ic ionization will be made and compared with

some of the measu remen t s pe r fo rmed b y ~-EII.ER (28). These measurements have been analysed to yie ld the ineident-l)rot(m cosmie-ray spect rum. This is compared to the predict ions of the electric-field model for U-~ 200 MeV in Fig. 3. The measuremen t s were pe r fo rmed during the course of several months in 1965~ and are near a solar minimum~ bu t a m in imum not as deep

(23) L. J. GLEESO.~ and W. I. AxvoIu): Can. Journ. Phys., 46, $937 (1968). (24) p. S. FR~:IEI~ ~nd C. J. WAm)I~GTO~: Space Sci. Rev., 4, 313 (1965). (25) A. EIIMART: Proceedings o] the International Con/erence on Cosmic Rays (Moscow, 1959) Vol. 4, 140 (1960). (~6) p. S. FREI~,u and C. J. WADDI~G'rO~-: Proceedings o] the Ninth International Con- ]erence on Cosmic Rays (London, 1965); Vol. 1, 176 (1966). (~7) B. P]~TERS: Cosmic rays, ill llandbook o] Physics, edil~d by E. U. COSDON and II. 0mSHAW (New York, 1958), p. 9. (~s) It. V. ~N]~IIF.R: Journ. Geophys. l~es., 72, 1527 (1967).

~ K. O'BllIEN

as t h a t of 1954 (28), so the value of U is reasonable. The agreement is quite

good, and this spec t rmn will be used for the ionization calculations to follow.

Some neut ron calculations will require different values of U. These will be

ta.ken, where possible, f rom .FI~EI]~:I~ and WADm~'G~0~" (~.6).

6O

50 ~astern honizon g ~oL \

20

V-----c---_4 \ lo - ~ - - - - _ ~ "-.% \

w~st~ ~ ~

0 10 20 30 40 50 60 70 80 geomQgnetic lQtituQ'e (ctegrees)

A 90

Fig. 3. - The East-West variation of geomagnetic cut-off rigidity as a function of geomagnetic latitude compared with the effective cut-off rigidity for an isotropic detector, o effective rigidity, RmH'• et aI.

5"2. The Ear th ' s magnetic field. - The magnet ic field of the E a r t h deflects incoming cosmic rays depending on their r igidi ty and angle of incidence, so

t ha t for each angle of incidence there is a critical rigidity below which the incom-

ing part icle cannot in terac t wi th the E a r t h ' s a tmosphere . I n the calculations

which follow, a single cut-off r igidi ty will be applied. The p r i m a r y spec t rum

will be assumed unchanged in angle and energy above the cut-off, and vanish

below it. RICIITNEYER et al. (29) have calculated the effective cut-off r igidi ty

seen b y an isotropic detector exposed to the p r i m a r y spec t rum at the top of

the a tmosphere and this cut-off will be used here.

I n Fig. 4 the cut-off for the eas tern a.nd western horizons and the zenith, f rom LE3~AITRE and VALLa~r (80) are shown along with the values obta ined

(29) F. K. RICIITMEYEII, E. H. KENNARD and T. LAWRITS]~: Introduction to Modern Physics (5th Edition) (New York, 1955), p. 566. (ao) G. LE-~AITRE and M. S. VALLAI~TA: Phys. Bey., 50, 493 (1936).

COS.~C-RA~ raO~'AGATIO~ ~S ~ n AT.~OSe~I~E 535

b y RIClI~rEY>nr et al. (..9) as a funct ion of geomagnet ic lat i tude. I t is evident t ha t the assumpt ion of i so t ropy of the radia t ion near cut-off is not justified

a t la t i tudes below 40 ~ to 45 ~

l0 -3

10--~

" j -

10 -5

f f l

E 10 -6 ,

10-7 t

i0 -a

lo -2

o

t I 1 I l L I I

10 -1 10 ~ E(Oev)

1 I I

10 ~ 10 2

Fig. 4. - Comparison of electric-field model calculations of the incident cosmic-ray proton spectrum ( - - - - - U ~ 200 MV) with Neher's measurements (o).

This, in combinat ion with the assumptions made with respect to electro-

magnet ic shower t ranspor t , will p robab ly cause the calculations based on an isotropically incident spec t rum to fail s t low lat i tudes. Consequently, at this

stage in the deve lopment of the calculations, the exper imenta l comparison will be l imited to higher lat i tudes.

6. - C o m p a r i s o n w i t h e x p e r i m e n t .

6"1. Sea-level part ic le spectra. - The first step lit producing integral quan-

t i t ies such as ionization or neu t ron densi ty is the calculation of the differen-

t im energy-angle part icle distributions.

As a tes t of eq. (17.2), the ver t ical component of the cosmic-ray spectra has

been calculated for a geomagnet ic la t i tude of 57 ~ Equa t ion (17.2) with 9 = 1

and r ---- 1033 g /cm ~-, was in tegra ted over the source spectrum. Seventy percent

of the source spec t rum was assumed to be composed of free protons, and 30 per-

cent of bound neutrons and protons all having the energy distr ibution given

b y eq. (18), with U ~ 200 MV. Fo r geomagnet ic purposes bound nuclei were t r ea t ed as bound, bu t for the purpose of a tmospher ic t r anspor t t r e a t ed as free, i.e. an r162 was assumed to behave exact ly like 2 free neutrons and

536 ~. O'BEIE~

2 f ree p r o t o n s . I n F ig . 5, t h e c a l c u l a t e d v e r t i c a l c o m p o n e n t of t h e c o s m i c - r a y

n u c l e o n s p e c t r u m of one cha rge s t a t e (neu t rons or p r o t o n s ) is c o m p a r e d w i th

t h e e x p e r i m e n t a l sea - lcvc l p r o t o n s p e c t r u m of BROOKE a n d WOT,~'E~DALE (3~)

a n d t h e sea - l eve l n e u t r o n s p e c t r u m of Astr ro~- a n d COATS (32).

10 - 6

I' 1o ,

I0 -'~

X

10 :2 _ u3

~E t)

-la 10 "--

f

1 o -~s !

\ \

\

10 - a

--7 lO

- 8 lO

> 10--s ~ X

m

~ 10 - :~ _

-:1 lO

1~ I

10 -13

I 0 -~

I

I 10 o 10 2 10 4 10 c 10 ~ 10 2

E(GeV) E(GeV)

Fig. 5. Fig. 6.

Fig. 5. - The vertical component of the cosmic-ray nucleon flux of one charge state a t sea-level as measured by : �9 BROOKE and WOI,FENDAT.E (31), protons, �9 AslrroN and COATS (a2), neutrons, and . . . . as calculated.

Fig. 6. - The vertical componeltt of the cosmic-ray pion flux as mea,sured by BROOKE et al. (o) (33), and as calculated ( - - - - ) .

(31) G. BROOKE and A. W. WOLFENDALE: Pr0c. Phys. Soc., 83, 843 (1964). (3~) F. ASnTO-~" and R. B. COATS: Journ. Phys. A, 1, 169 (1968). (33) G. BROOKE, M. A. MEYER and A. W. WOLrE~'DAL~: Proc. Phys. Soc., 83, 871 (1964),

COSMIC-RAY PROPAGATION- IN TIIE ATMOSPHERE 537

- j

10 6 ~ o

10-

10 B

i T~ io -~

X

~ I0-'~

10 -~:

10 -!~ ~_

10-" i

�9 j I o - " L - - i I i . . . . . . I .

10 103 10~ 102 103 )0" E(GeV)

Fig. 7 . - The vert ical componell t of the cosmic-ray much flux as measured by: 0 0 w ] ~ and ~'VILSO~ (34), = HOI.~ES et al. (ss), ~, GAI{D~X>m et al. (3s), O IIAYxAI~ and WOIa,'ENDAI, E (37), and as calculated - - Some exper imenta l points below 10 GeV have been omit ted for clarity.

(.~4) ]3. G. 0wE-~" and J. G. WILSOn: Proc. Phys. Soc., 68, 409 (1955). (35) j . E. R. IIoLM~S, B. G. Owing" and A. L. ]~ODGERS: Proc. Phys. Soc., 78, 505 (1961). (38) .~[. C-cARD]~NEI~, D. G. JONES, F. E. TAYLOR and A. W. WOLP]'~NDALE* 1)'roe. Phys. Soc., B0, 697 {1962). (~7) F. J . HAYr~AN and A. W. WOLFV~'DAL~: Proc. Phys. Soe., 80, 710 (1962).

35 - I1 N~ovo Cimento A.

538 K . O'BRIEN

At this a tmospher ic depth, eq. (17.2) predicts near ly equal numbers of

neu t ron and protons, and so bo th exper imenta l and theoret ical da ta were

combined. Agreement is ve ry good over 4 decades of energy and 10 of in-

tensi ty . The calculated ver t ical componen t of the sea-level pion spec t rum is com-

pa red in ]~'ig. 6 wi th the measuremen t s of BlCOOKE et al. (s3). Agreement is

good over mos t of the range of comparison.

The sea-level muon spec t rum for a zenith angle of 0 ~ is shown in Fig. 7

compared with measuremen t s b y OWE~ and WrLSO~ (34), HoL_~ES eta l . C~S), GA~D~,;~ et al. (~) and Hu and WOL~E~DALE (~7). Agreement with experi-

m e n t is quite sa t is factory over the range f rom about 1 to about 1000 GeV.

All measuremen t s and calculations were for 57 ~ geomagnet ic lat i tude.

I n Fig. 8, the sea-level muon spec t rum is calculated for a zenith angle of 75 ~ and compared with the measu remen t s of STEFAh'SKI et al. (3s). The lowest

expe r imen ta l point is above the m a x i m u m cut-off for this la t i tude, 52 ~ (see

Fig. 4).

10 -~

10 -a

10 -9

T~

:E

L~lo~O ~E

o

10 -12

1 0 - 1 3 _ _ _ _ _ . I t - - - -

10 o 101 10 2 10 3 E(GeV)

Fig. 8. - The muon spectrum at a zenith angle of 75 ~ as measured by STEFANSKI et aL (o) (~s) and as calculated ( ).

(as) R . J . S~EFANSKg R. K. ADAIR and II. KASHA: l~hys. Rev. Ze#., 20, 950 (1968).

COSMIC-RAY Fs IN TIIE ATMOSPHERE 53~}

As has been observed earlier, the assumpt ions t ha t lead to eq. (17.2) become increasingly shaky below about 1 GeV. This affects all the charged-part icle dis tr ibut ion, including the muon energy distr ibution, which, of course, depends on the pion distr ibution, and can be seen clearly in Fig. 5, 6 and 7, where the calculated fluxes rise above the measured fluxes in every case.

6"2. Cosmic-ray ionization in the atmosphere. - Ioniza t ion f rom protons, pious and muons is calculated b y mul t ip ly ing the energy dis%ributions b y the

appropr ia te s topping powers as described earlier (1). The lower-energy l imit for

the pro ton , chargcd-pion and neutra l -p ion energy distr ibutions is ]00 ]~feV,

below which the theory fails. The muons are allowed to slow down to 10 MeV,

below which ve ry little is cont r ibuted to She ionization. I t was found necessary empirical ly to use an upper l imit of 104 GeV to

include all significant contr ibutors to the ionization. I n Fig. 9, the calculated ionization (in units of I , the number of ion pairs

per cm 3 of ~NTP air) a.t a geomagnet ic la t i tude of 55 ~ is compared with the measuremen t s of h'E~mR (~.s), and la ter da ta as repor ted b y G~OI~(~E (39) down

to 600 g /cm -~. To complete the curve, the resul ts of L0WDER and BECK (4o)

f rom 600 g /cm 2 to sea-lcvcl measured at 51 ~ geomagnet ic la t i tude a t about

the same t i m e are included.

1~

i-

o 2;o ~ o' 800 1 O0 400 600

clepth in atmosphere(g/crn 2)

Fig. 9. - The cosmic-ray ionization profile at 55 ~ as measured by: �9 ~'~II~R (2s), �9 LOWD~R and B~CK (4o), und as calculated - - ; - - - - - - ~NEH]~R, aS reported by GEORGE.

(39) ~r GEORG)]: Journ. Geophys. Rev., 75, 3693 (1970). (40) W. M. LowD~I~ and II. L. BECK: Journ. Geophys. Res., 71, 4661 (1966).

K . O ' B R I E N

Overall ag reement is seen to be within 20 % with the except ion of the region near 600 g/era ' , where the d isagreement is nearer 40%. The compar ison is absolute, i t mus t be emphasized. The composi t ion of the to ta l ionization is

shown in Fig. 10. Because the secondary fluxes in terac t differently with the

a tmosphere each componen t has a not iceably different profile. The k ink in

the electron curve about 850 g /em ~ is a consequence of the t ransi t ion f rom

shower product ion originating in neutra l -p ion decay a t low depths to shower

product ion resul t ing f rom muon decay a t larger depths.

103[

I 0 2 1 ~

0 400 800 1200 ^

depth in atmosphere(g/cm ~)

,~ l

/ 101

i ~ o

o

10 ~ 0

Fig. 10. Fig. 11.

200 . . . . . 8;o o ; o - 400 600 1 depth in atmosphere ( g/cm 2)

Fig. 10. - The composition of cosmic-ray ionization in the atmosphere at 55 ~

Fig. i I . - The cosmic-ray ionization profile at 44 ~ as measured by: [] N~HER (-"8), o G E O R G E (39 ) ; _ _ as calculated.

A similar comparison is shown in Fig. 11, where the calculation is carried out a t 4~ ~ and the da ta again are t aken f rom N ~ E ~ (~s). To complete the

curve, the da ta of G].:ORGE (39) are included f rom 188 g/em: to sea-level. The

d isagreement is typical ly 20 % with higher values a t near 800 g /cm 2 and 50 g /cm 2.

The la t te r is p robab ly a result of the depar ture of the incident p r i m a r y cosmic-

r ay tlux h 'om isot, 'opy, and the hardening of the pho ton product ion spec t rum tha t results f rom the higher average cut-off (see Fig. 4). George's (30) measure-

ments were carried out during 1968 near a solar m a x i m u m , and hence the addi t ional modula t ion, if removed, would make the d isagreement worse.

6"3. Cosmic-ray neutrons in the atmosphere. - The lower-energy l imit of the cascade calculations described above is 100 MeV, and the ag reement with

C O S M I C - R A Y s I N T H E A T M O S I ? H ] ] R E 5 4 1

measurement indicates tha t this cut-off, which is forced on the calculations by the l imitations of the anglytical theory, is adequate ly high. This is consequence of charged-particle stopping which limits the number of charged particles at low energies. 1N'entrons are uncharged however, and extend all the way down to the rmal energies. In order' to account for neut ron fluxes below ]00 MeV, the cosmic-ray neu t ron spectrum repor ted by HEss et al. (4,) a* sea-level and 44 ~ geomagnetic la t i tude has been pa tched onto the calculated

differential spectrum at 100 MeV. This ~pprogch is ra ther rude, and fails at small depths as it cannot account for the diffusion hardening which takes place near a vacuum boundary .

10 1

! A A ,0~ l~ i ~

~ 0 - 3 t -_ I i I _ i I ' . _ l I I

0 400 800 1200 ctepth in atmosphere(g/crn z)

Fig. 12. - The cosmic-ray neutron flux in the atmosphere as measured by: [] BOEI~r,A et al. (4~), 2 = 46.5~ a BOELLA et al. (As), Z= 42~ o YAMASIIITA et al. (44), Z= 44~ . . . . as calculated: I) A=46.5 ~ , U=400MV; lI) A=42 ~ , U=300MV.

This is clearly seen in Fig. 12, where the neu t ron flux measurements of BOEI~I~A st al. (4~,43), and YAMASHITA et al. (4,) are compared with calculations.

At depths greater t han 200 g/cm 2 the agreement is reglly ra ther good. The

(41) W. N. IIEss, E. H. CANFIELD and R. E. LINGENFELTER: Journ. Geophys. l~es., 66, 665 (1961). (42) G. ]3OELLA, G. DI'~GLI ANTONI, C. DILWORTH, G. GIANN~LLI, E. RICCO, L. SCARSI a~td D. SHAPIRO: NUOVO Cimento, 29, 103 (1963). (43) G. BOELLA, G. D]~GLI A~TO~N'I, C. DILWOWrII, l~[. PA-~ETTI, L. SCARSI ansi D. S. INTRILI~ATOIr Journ. Geophys. /~es., 70, 1019 (1965). (4~) M. YAMASnlTA, L. D. STIll'HENS and H. W. PATTERSON: Journ. Geophys. Bes., 71, 3817 (1966).

542 K. O'BRIEN

lack of i so t ropy of the incident flux near cut-off and the absence of diffusion

hardening lead to an over-es t imate at small depths. The measu remen t s of

BOELLA et al. (42) and YAMASE[TA et al. (44) are both ground-]evel measu remen t s

r a the r than free-air measurements , bu t ye t are seen to fall on the curve. The

effect of the air-ground interface on the calculat ions has not been evaluated.

10 ..6

o ~176176176176176176176176

o o

o~

- -7

1 0 - 8

%

9 0 o

o o o

~ o o %

% o

o o o

o o

l~ 9 I I l I I I i _ I I 0

200 400 600 800

clepth in efmosphene ( g/cm 2) I I 1000 1200

Fig. 1 3 . - The cosmic-ray neutron density in the atmosphere as measured by: - - YuA~ (45). A=52 o, [] GOLJ) (46), ).=53: - - as calcu/atcd, ).=52 ~ U--600 MV.

I n Fig. ]3, the neu t ron densi ty measu remen t s of YuA~" (4~) and GOLD (As)

have been compared with calculations on the same basis. Again~ the agree-

m e n t be tween calculations and measurements is good except at small depths,

wi th the except ion of the ground-level value of GOLD (46). GOLD (As), recognizing

t ha t his values were quite high compared with balloon measuremen t s in free air (47), a t t r ibu ted this to the interface with the ground, to which the neu t ron appears much more sensit ive than the neu t ron flux.

(45) C. L. YUAN: Phys. Rev., 81, 175 (1951). (4~) :R. GOLD: _Phys. Rev., 165, 1411 (1968). (47) R. F. MIL~S: Journ. Geophys. Res., 69, 1277 (1964).

COSMIC-RAY PROPAGATION IN THE ATMOSPHERE 543

Yuan ' s da ta appea r to have a marked ly different slope f rom the measure- ments . This can also be seen in BOELLA et al. (~3) and :Fig. 12. This effect m a y be a consequence of operat iug near the threshold of in s t rumen t sensit ivity, since the calculation is in good a~ :eement with the deeper measurements ,

all the way down to sea-level both in Fig. 5 and 12. Such an effect will lead

to r a the r long repor ted re laxat ion lengths.

~o-6

u~

t ) v

o o o

o ~ o

o ~ o

10_7 ~ Oo ~ ~

~ ~

~176 ~176 ~

1 0 - ' o o

0 200 400 600 800 1000 depth in cztmospher'e (g/cm 2)

I I 120o

Fig. 1 4 . - The cosmic-ray neutron density in the tttmosphere as measured by MILES (ds), and as calculated ( - - - , 2=41 ~ U=400 MV). o HAYMES, NYU flight 91, A HAYnes, NYU flight 93.

Figure 14 shows the da ta of MILES (48). I n this case a genera! underes t imate

appears below 200 g /cm 2. The identical calculations can be applied to the da ta of H A v ~ s (49) and So~ng_~A~ (.~0) which have been conver ted to neu t ron

densi ty and adjus ted b y MILES (48) for differences in la t i tude and t ime so tha t

compar ison with his own results might be made. Above 100 g /cm 2, the da ta

(as) R. F. MILV:s: Thesis, California Institute of Technology (1963). (4~) R. C. HAYMES: .Phys. Iiev., 116, 1231 (1959). (so) R. K. SOB~RMAI~: Phys. Bey., 102, 1399 (1956).

$ 4 4 K. O ' B R I E N

of S0SEn]~A~- (~o) in Fig. 15 are seen 1,o be in excellent agreement with calcula-

tion. At smaller depths, the neglect of leakage, which was not, t aken into account

in the low-energy model, causes the calculat ion to be too high. Al though much

more sca t te r appears in r measu remen t s of HAv~ES W) (Fig. 16), ag reement

is d e a r l y reasonably good except again a t small depths.

10 -~ 10- 6 E

I

L . . . . . . ~ . . I ] "~0-8 [ ' ~ .I I r

J

lO-So 100 200 300 400 500 0 100 200 300 400 500

cleDth in atrnosphere(g/cm 2) cleDth in atrnosphece(g/cm 2)

Fig. 15. Fig. 16.

Fig. 1 5 . - The cosmic-ray neutron density in the atmosphere a~ measured by SOB~R~" (~o), and adapted by MILES (ds) (o, NYU flights 66 and 67), and as calculated ( - - - - , 2 - -41 ~ , U=400MV).

Fig. 1 6 . - The cosmic-ray neutron density in the atmosphere as measured by HAYMF.S (~9) and adapted by MILr (4s) (o 2 - 41~ and ~s calculated ( - - - - - - )~ - -41 ~ U- - 400 MV).

o

7 . - D i s c u s s i o n .

Analyt ica l calculations of the secondary energy dis tr ibut ion of cosmic rays

in the a tmosphere , cosmic-ray ionization, and neu t ron flux and densi ty have

been performed. The model of luuclcolu-nucleus collisions on which the calcula-

tions depend is based on the colustant par t i a l inelasticit, ies of T~ble J, and cons tant geometr ic react ion cross-sections. The t r anspor t c~lculations assume

a pure ly nucleonic eascade, and consider (eq. (5)) only protons, neutrons, pions,

electrons and photons. I n t)artimflar, the muon ionization and the high-zenith-angle energy distribu-

t ions calculated on the basis of 7:~-->~ decay, are in agreement with measure- m e n t (Fig. 8) and support the conclusion of STE~t_~-sIr et al. (3s) tha t there

COSMIC-RAY PROPAGATION IN THE ATMOSPHERE 5 4 5

is no o ther impor t an t contr ibut ion to the muon flux below 300 GeV such as would account for the Utah muon measurements (s~).

ICA.~-vr and ]t0~AX (~) observe tha t the mcasm'cments of negat ive pions a t 70 GeV of BcsHh-I.~ et al. (5~-) are in s t rong d isagreement with the T R B model

of hadron-nuclcus collisions and with the stat is t ical inodel of s trong interac-

tions (s). The ext rapola t ion model disagrees b y an order of m~gni tude with the same da ta (s). I f the measuremen t s are correct, t hey cannot imply a change

in the appropr ia te par t ia l inelasticities. As has been shown, the power law model

used in the cah;ulations of the fluxes in ~ig. 5-8 is in agreement with thesc mod-

els. Thus ~ large errol �9 in the assigned v~lues of vj or K~ would lead to quite

large errors in the differential nucleon, p ion and muon fluxes s tar t ing somewhere be tween 30 and 70 GcV. However , the measuremen t s are only ~t small angles ( < 1 5 m r a d ) and high secondary m o m e n t a ( > 45 GeV/c), and it is possible t h a t average quant i t ies such as mult ipl ic i ty and inelast ici ty are not much

affected b y what happens in this region. I t has been suggested t h a t nucleon-nucleon and meson-nucleon cross-sec-

t ions m a y vanish a t infinite energy (5~). These calculations, pc r formed with

constant crogs-scctions, indicate tha t hadron-nuclcus cross-sections ~re essen-

t ial ly constant and geometr ic out to ] 04 GcV. For instance, ~ 10 ~o pe r tu rba t ion

of the nucleon-nucleus cross-section will cause a change in the nucleonic flux

in the region of (103+104) GeV of 150% . This woutd I)ut theory and experi- men t out of agreement in Fig. 5. But , such a discrepancy could be accounted for in t e rms of errors in the p r i m a r y spec t rum used, or errors in the sea-level

measurements . Much larger changes in the cross-section however, would lead to ve ry big changes in the sea-level flux which would not be rcconciliablc wi th the d%ta.

I n these calculations, all hadron-nucleus cross-sections are equal a.nd geo-

metric. I t is known tha t pion-nucleon cross-sections are -~ of the nuclcon-mlcleon

cross-sections (5~). As ADAIR (~s) points out, pion-nuclcus and nucleon-nucleus

cross-sections will differ b y less t han this ~s a consequence of the in t ranuclear cascade. H e calculates the rat io to be 0.77. IdA~]~T ~nd I~O~A],-(9) ob ta in 0.83 f rom publ ished exper imenta l data . The eitect on pion and muon spectra of a 20 ~ error in the react ion cross-section would be much less t h a n in the

nucleon ca.se, as pions are locally produced, and locally absorbed.

(st) II. E. BER(iESON, ,I. W. KEUI,'FEL, M. 0. I,A~SO-~', G. W. MAso.~ and J. L. ()SI~ORN~: Phys. I~ev. Letl., 21, 1089 (1968). (52) y . B. BcsII.~I.~, S. P. DENISOV, S. V. DONSKOV, A. F. DVNAITS~,:V, Y. P. GOI~IN, V. h. KACIIANOV, Y. S. KHODIREV, V. I. KOTOV, V. 3'[. KUTYIN, A. I. I)ETRUKI=IIN, xl'-. D. PROKOSIIKIN, I~. A. ~{AZUVAI,]V, I'{. S. SIIUVALOV, D. A. STOYANOVA, J. V. ALLABY, P. DIXON, A. N. DIDDENS, P. DIJTEIL, C,-. GIACOMELLI, R. M~UNIEIr J.-P. PEIGNEUX, K. SCHLIJPMANN, M. SPIGIIEI,, C. A. STAIILBRANDT, if.d ). STROOT gild A. M. WETtiERELL: Phys. Le$t., 29B, 48 (1969). (ss) A. M. WETIIEmCLL: Selected Topics in Particle Physics, edited by J. STF~INBERGEIt (New York, 1968).

5 ~ K . O ~ B R I E I ~

Apparen t ly one-dimensional nucleonic cascades depend only weakly on the

angular and energy behavior of the secondary product ion spectra, bu t s t rongly

on the par t ia l inelastieitics, multiplicit ies and cross-sections. Tile calculation

described here which is based on a s t ra ight ahead power law approx ima t ion

to the doubly differential product ion spec t rum yields good agreement with

the sea-level differential fluxes of nucleons, pions and muons, wi th the cosmic-

r ay ionization a t all depths in the a tmosphere , and with the neu t ron fluxes

and densities in the a tmosphere . Last ly , it m a y be observed tha t in Fig. 12, 13, and 14, the neu t ron calcula-

t ions appeal" to re lax more rap id ly t h a n some sets of measurements . Ye t in

Fig. 5, and the sea-level values of Fig. 12, this would appear to represent

exper imenta l error, possibly a consequence of opera t ing near the threshold of exper imenta l sensit ivi ty. Too rapid a re laxa t ion ra te of the nucleon spec t rum

would also result in the calculated pion spec t rum of Fig. 6 being to() low, which

it is not, as the pions are produced locally f rom nucleon-nucleus collisions.

The neu t ron a t t enua t ion is not ac tual ly exponent ia l a l though its depar tu re

f rom cxponent ia t ion is not large (5~). I n Table IV, re laxat ion lengths ob ta ined

f rom the neu t ron densi ty calculations at 2 .= 41 ~ and with U = 400 MV are compared with those of MILES (47). Agreement is poor, bu t the calculated neu t ron densities agree well wi th those of HAYM-ES (49) and SORERMAN (~0) when

they arc reduced to the same conditions and are never less t h a n half of those of MILES (4s). I n addit ion, those of Sxmos-A'rI D],; FRITZ and Clccm_~I (54) have

measured cosmic-ray neut ron a t tenua t ion lengths in air at 25 ~ (55) as a func- t ion of a tmospher ic depth which are included for comparison. I n this case ag reement is quite close, and this suggests t h a t exponent ia l re laxat ion lengths

are not well de termined b y exper imenta l a t t enua t ion data , as re laxa t ion ra tes

are not cons tant with height and the t rue var ia t ion is m a r r e d b y poorer qual i ty at grea ter depths and at lower intensities.

The success of the preceding calculations rests pr imar i ly on the phenomeno-

logical model of hadron-nueleus collisions prcsented here. The description of

TABLE ]•. -- Co~parison o] calculated neutror~ attenuation lengths /or i t= 41 ~ and U = 400 MY with measure~n~ents.

Atmospheric depth ( g / t i n 2) Calculated (g/cm '~) Measured (g/era-")

300 149 155 (a)

500 129 125 (a), 165 ~ 20 (b)

1033 113 l l5 (~)

(a) Jl ~ 25 ~ (~') . (b) -t ~ ~i ~ (,7).

(54) N. A. S1MIOI~ATI DE FRITZ and A. A. CICCHINI: ~Vuovo Cimento, 40 B, 220 (1967). (55) A. G. McNxsn: Terrest. Magnetism Atmospheric Elec., 41, 37 (1936).

COSMIC-I~AY PROFAG&TION IN TIIE AT3IOSlaIIEI~13 5~7

t h e e x c l u s i v e l y nuc l eon ic ca scade c o m b i n e d w i t h t h e neg lec t of k a o n s is a

s imp l i f i ca t i ou w i t h which m o r e s o p h i s t i c a t e d a n d e x p e n s i v e ca l cu l a t i ons m a y

d i spense . O t h e r , m o r e obv ious , a p p r o x i m a t i o n s a re b e i n g d i s c a r d e d as t h e

s t u d y p rogresses .

R.eal i m p r o v e m e n t on t h e fo rms of t h e d o u b l y d i f f e ren t i a l p r o d u c t i o n s p e c t r a

m u s t p r o b a b l y a w a i t p r o g r e s s in st, ud ies b e i n g c a r r i e d ou t a t O a k R i d g e a n d

e l sewhere . The a d e q u a c y of t h e p o w e r l aw n u c l e a r m o d e l for ca l cu l a t i ons of

t h i s t y p e is c lear f r o m t h e r e su l t s shown he re , a n d t h e a v e r a g e p r o p e r t i e s ,

p a r t i c u l a r l y t h e p a r t i a l i I l e l a s t i c i t i e s a n d c ross -sec t ions , as a p p l i e d to a i r ,

a p p e a r to b e we l l e s t a b l i s h e d .

�9 R I A S S U N T 0 (*)

Si svfluppa mm tcoriu assenzi~hnente an~litie~ della prop~g~zione atinosfcrica dei raggi cosmiei sulla base di un modelh) fenomcnologieo della collisioni ~drone-nuclco. Questo modello prcveda aorratt~mcnte gli spet t r i della eomponanti mttonica, pioniea c nueleonica dei raggi cosmici ul livello del mare, il profilo di ionizzazionc dei raggi cosmici nall'&tmosfara cd il flusso a i profili di dansi~t dei ncutroni neli '~tmosfera. Si conclude alto lc proI)ricth a grande scala dai raggi cosmiei possono assere previste ~ccuratamente slfll~ b~sa di Ulm c~se~t~ puramentc nucleonic~ in conseguenz~ della quMe tu t t i i secondari hamlo per intermcdi~rio la produziona di pioni. Si 4iscutono lc implicazioni per l ' indipendcnza dMl'energi& della sazioni d 'ur to, i reccnti r isul ta t i a 70 GeV ot tclmti a Serpukhov ed i rappor t i di rilazsamento nucleonico nell 'atmosfera.

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