Part VI: Monte Carlo calculation of the nucleon meson cascade in shielding materials

NUCLEAR INST

S AND METHODS

PAT VI : MONTE CA

32 (1965) 65-69s

NORTH-HOLLAND PUBLISHING CO.

CALCULATION OF THE NUCLEON MESON CASCADE IN SHIELDING NUTERIALS Jr

A. G1"IREL and J. RANFT.'

a

CERN, Geneva

Received 30 October 1964 monte C.`arlo kwlati of a n wn nwts out . The results for c

on the three dimensional development in . ' material were carried initia by incoming protons of

10 and 20 GeV in steel compare favourably with the corresponding experimental results.

production of secondary particles in high energy nucleon-nucleon collisions require extrapolations or guesses. The practical purpose of cascade or shielding calculations is the determination of shielding dimensions necessary for a required attenuation of the flux of incident and secondary high energy particles. From this point of view, local phenomena inside the shielding material are of interest only in so far as they contribute to the propagation of the cascade. All experiments confirm that an incident proton beam with energies in the GeWregion is propagated through thick shields by a highly energetic strongly interacting component with an attenuation length near to the absorption length for nucleons and pious in matter . The electromagnetic showers initiated by neutral mesons do not contribute significantly to the propagation of the cascade and die out fast in all comof secondaries is desired and the inclusion of affects mon shielding materials owing to their relatively short such as multiple Coulomb scattering, elastic uttering, radiation length and are thus neglected in the calculaionization energy loss etc. are to be considered . tion. Strongly interacting particles, the ranges of which The calculations were made for ;steel as shielding became shorter than one absorption length due to material and for incoming protons with energies of energy degradation, are also neglected. 10 and 20 GeV. The experimental conditions ,2) are The remaining high energy component consists of taken into account as closely as possible. The results of baryons and charged mesons . The absorption probathe calculation show a reasonably good agreement with bilities of these different kinds of particles are somewhat different. It is assumed, however, that they alt the corresponding experimental results. have the same absorption length as measured for Basic a mpt entering the calculation protons in matter') . -this rather crude assumption is The nucleon-meson cascade is a complex phenome- suggested by the nearly equal abs _rpti(, n cross-sections non involving mainly (1) protons, neutrons, mesons of high energy protons and pious in complex nuclei . and hy rons, interacting strongly with the nuclei in Protons and pions are the major par', of the high the shielding material ; (2) electrons and photons, which energy strongly interacting collision products . We use interact electromagnetically ; (3) mesons and hyperons in addition only one assumption about the nuraln-, angle and energy distribution of the secondary parfii,vhich decay via the weak interaction process. account complexity requires drastic simplifying assurnpcles produced in each collisior . We take ir 'T'his elastic scattering, multiple Couloc ib s .FF . t - ng, i,)ntions in order to make the calculations manageable. Furthermore, large gaps in our knowledge about the ization energy losses and the &cry of ~;, ons into muons . On leave from Karl-Marx-Universitat, 1-eip ig .

All the calculations so fair made of the development of the particle cascade initiated by multi Gets incident protons in shielding materials treat the one dimensional infinite slab problem, which is related to an incident beam of infinite transverse extension'). Experimental studies of the propagation of the cascade') however, have n made with beams of limited cross-sections . The transverse expansion of the cascade plays in these experiments an important role . This makes it difficult to compare, the experimental results with the calculated predictions . We present here results of Monte Carlo calculations of the three dimensional development of the nucleon meson cascade in steal. The Monte Carlo method is better suited for the calculation of the three dimensional problem than analytical nAetho9ds are. This is specially true if a rather complicated function for the spectrum

65

66

J . A . GEI:Vl3L

AND

J. RANFT

U-Qlc Deft, and in The results ofthe eatcalation will depend essentially the spectrum of on the data: used. OT_ assumptions about in .'high energy collisions . ondaries - produced the underknown and spectra are not yet Very wett . '11Iheoret cal considerations such as the statisticai stood l give resulte) ,vhieh are only a rather crude description of the experimental data. Pion speztra, how-ever. are raxher well described, within ecrta n formula proposed by limits, by the empiri aL 1) : Goccon s~

n

V,

2

ixE

-.~--

~~p~ T

wrp

1

t

T)

41

exp

~p

(1}

RF

`hey a-sum:d for the average multiplicity n. a relation nx ;:z Eo~ which is in good agreement with measurements in the cosmic rey energy region . The inelasticity K was assumed to be independent of Eas the energy of the im., ming protons . With 7o, = E'EQ they obtained for the mean energy of the pions T ;r. Fi. At energies below 30 GeV the relation nR ~--. Et holds better'), therefore we use formu a (l) with ?', Et. We use the parameters poc =0.18 GeV and T(30 GeV) = 3.75 GeV and obtain. dzN d EdQ

E° t 1 75(ED/30)1

e 3 .75

E

(1 /30)* _ . _+ (2) ( .E

e,-.p

0

) I

.

200

curves for Ec between 0' and

30' .

10 and 30 GeV and 8 between

We use the formula also

for

production

outside the 0 and E range for which it waa fitted to the existing; expesimenta °. data . certainly

larger

In

out the

this

region

agreement

Lhe

with

the

The dependence of the total production integrated

l0"and 80' which follows from (2) is in agreement with dN(dOs0 -2 . ;.

reproduced

witn

an

error

of

half

an

order

'

) of

magnitude .

neAr a

depth

moon-,

near

to

the

primary

energy .

Therefore

v!e

high energy behaviouf of (2). The mean midtiplicity of the strongly interacting high,

depends

on the

For EO < 3 GeV

energy the

EQ

curves

of the incoming particles. given by

Powell,

Fowler

and Perkins"" were used to give

But the low energy particles

important for the cascade . gives a

of 2000 lcm2 arises frorn lush enemy (contributing to the track density) .

think it is unnecessary to make any correction for the

Formula (2) gives y production which is too low in n :ompanson with experimental curves wait small secondary cn-ergie.S E.

26M Qf*mt

energy particles (with minimum ionization) produced

The spectra. measured at 45', 60' and 90°, ref. are

200

density, the stag density in the beam axis and at $, 10 and 20 cm outside the beam axis . The change in tl°c slope of the curves

energies

behaviour' . ') :

I

few

over all momenta as a function of the angle 6 between

measured

1 4015

1000

cawade initiated b~- a narrow, well collimated incoming proton beam of EO = 20 GeV `n stee1 . The curves give the integaated scar

errors

existin data. i s tolerable, .

the

600

Fig. 1 . The fon itudinlJ development or the star density for a

Fomula (2) agrees reasonably well with the measured

are

a

non-vanishing

pion

are not very

Furthermore, the formula production

n s = 0 . 555 E o _- 0 .17 (Eo

Ge 1 `)

Between 3 and 30 GeV the expermaental results fat well with formula')

at the energy

rt

of the primary panicles . We use the forniala not only

~- 0 .85 FA( .

(4)

for the pion production but for the production of all

The total

secondaries . One expects outgoing proteins which hav% .

assumcd to be 50'," of the ab ,; orption cross- section 3 ) .

i-oss-section

for clastic scatterivs au,, .,) . is

ST UDICS

IN STEEL--PART

67

VI

secondary energies Eo and E in steps of < 0.5 CïeV. The second programme calculated for a sample of primary protons all secondary and highex order high energy particles. It stored the coordinates of the stars produced by them . The stars of the primary particles are atl assumed to be at X = Y = Z = 0. The list corresponds in principle to the distribution function of secondary and higher order stars which result from the interaction of one primary particle at the point 0. Energy, direction and flight path of all particles produced, as well as their multiple Coulomb and elastic scattering are randomly determined. The ionization energy losses were taken into account . The third programme used these results to construct the three dimensional star density which results from a ntonoenergetic beam of primary particles with a given intens!ty distribution. All the stars which derive from every primary particle are distributed with exponentially decreasing weights (accc:ding to the nuclear mean

â

I T tt .v ,r vel Townt of the tar density far a initiated by a narrow well collimated incoming proton of 20 GeV in st l. "I` paru te;rs give the depths in jcina .

e u : for the angular distribution caf the elastically scattered protons in the laboratory system the expression") d where

&ßt .1,

z

C,acet .p

3nh'

r

Here r is the nuclear radius r = Z4 -h1m$ c. This distribution function is in reasonably good agreement with expetimental data for pp") and p-Pb scattering") . The Rossi formula") was used for the multiple Coulomb scattering. Four the ionization energy loss and range calcuLtions we use the FORTRAN routines by Rosncr and Mellwain") . The decay kinematics for the n- p + v decay was calculated using a slightly modined form of the FORTRAN programme of Salmeron and S onley" ;.
rG?

L

28G

5 .72

857

110

1429 MFr'

Fig. 3. The longitudinal development of the siia . density for a cascade initiated by a reiat,vely broad incorning protor: hearn of F'0 - 20 f-ieV in steel . The profile jf the inco .ning bearu is given ry the curve belongirg to the depth of ®.() g,,,rnh in fig. A .

= Comparison 10 the incoming the WV 20CI-0 'i pail electron usdl, Irdasonable correspondim~ aan are and Zg the free in reache, flux 2111 calculations is A incoming uniform flux in recent of 11 diflerently calculated The arrangement and Coulomb fcm2 g/cni' in me about was given 20 rotationally and values of proton block for the the gives probability with for cale-uhation -was other xabs path, density the of at to restdt GeV tracks, Ithe neutral given protons steel transition performed nuves Ithe of star greater the Tvo necessary shielding in the muons shielding, half ?a!dft mm awith statistics These to density the are using rats beam proton primary 146 fig The Bering separate the ofthis in initial narrow in were densities star experiniet Itan longitudinal (narrow multiplicities, which forms increase given maxinium particles, lais of tLe 4gjcml at is figs for In shielding errors nuclear symmetric depth, resulting was (broad only order with region gaps density distribution i-ot R,,, with performed this to beam material) calculation transition value experiment same this contribute particlas of 1-4 in aad and Iprogramme find and beam) possible data and the do Iwith measured prcwedure Possible of incoming The the table xmean beam) mainly decay and were arrangements The, 20 of material value broad not cOrreSPORdift but magnitude the ifrom and figures elastic the Ione density density such etc Cep' cross-section the in and mm region for in possible Iinclude were is effects treated free to not inside radiatior only the was carried There neutrons, which transversal length of the reasons the at pion acompared beam the experimentally the derive proton the scattering used in The direction the path to the star taken distribution A to the attenuation used second which same at of the systernatic action are decay the statistical show GEIBEL was The the traversed enhanocdepth of of axot density athe out for muons for nuclew beatns density length of differ cross which deptli This gaps into track steel E steel first with one this on the ab-is an th,, of of isLor at =-a AND &, Jthe transition 4-account cur%* methods RANFT The influence may extension stiniulm'ug 20 agreemcnt The may are initiated statistical and is already muons authors AGeN present (~O) elsewhere") for value gives region a not teansvemw no quantities be which track the the used incomparison in region of by resulting the be show between steel becomes of the of depth same In error the between adiscussions in density) the was correct this slope relatively this calculations I'lit developmew, the order may too isnuclear of order Lot together of cakuiation pararneiers in region experiments from Al of 0significant thvnk proton much the adse of general the accounted is of The the broad #/cm2 and the of given calenlated ± during mean wt, -n-calculated denb-tcausc maguitudc of ho-m background with background Dr 1 In muon Xkv % have incoming gives 1000 pgood the to At fig +free The L for the further the the cannot incoming vthe star curves N flux GeV the of Idecay Hoffmann and dcpths in Difierences path course experimental the A, depth density the proton protfic the ext-4 is perimental measured radiation details radiation has influence U be a0ter possible used neutrou in(on calculaproton and of jiferrtz where taker of for beam z1this been (M f4e for the the for inat of

J.

68 M the multiple the au~unt. distribution . The estimated increased pracedure i4ith 5.

.

.

.

. .

experfinenta; "rhe

CT-RN of w" incomin,,jy, The shieldinf4; We sorption ;. x in was "on broader which Track behave f.ram me,-7kn .anew of are density contribwe density . The densities 20 errors ai,ad of errors . Some i-exults the ment int~,gra$ed aF,,~,ain

Ir

.

.

. 1U,fi

. .

Fig . . cascade of 'rho

. .

.

.

.

.-;

.

..

:star .

.

.

. ., .tal .

. .The

.

.

The niany Nvork" .

20

.

25

35

30

40

45 .

.0

irwomiAg

.. Finally, transition The vAues errors The curves the component, tion . steel which into of longitudinal this curves Therefore, An energies reported the

IS

10

. .iity

. .1,

;0.

.

:x . .

. .

.

.

. . .

. to

.

.

SHIELLJING S1 U1:1ES IN STEEL--PART V1 Corn

69

T: f °- n of alcutaaed and measured2) results

Nuclear man im path used in the calculaw t (!jcrn 2) Numhec of primary particles in

loulation

Transition effft A

1 .8

Transition effect U

400 175

kxl

efe

2)

3)

4)

axis

i

Transition effect A

Cale. exp. tracks exp. stars

Transition efl t U

Ca1c. (S/Cm : ) exp. tracks exp. stars

110

Cale. Wcm 2) exp . tracks exp . stars

120

C. Pa ow, DESY Notiz A 2 .85 (19,!-,1) unpublished . R. G. Alsmiller Jr., l»" . S, Alsmillwr and J. E.. Murphv, ORNL 3289 0962), ORNL 3365 (1963), ORNi.. 3412 (1%3). R. G. Alsmiller Jr. and J. E. Murphy, ORNI. 3367 (1963). S. J. Lindenbauri, Ann4lal Rev. Ntcl. Set, 11 (1961) 213 . C. M. Fischer, NIRL/R 40 (1%3). R. G. Alsmiller Jr., ORNL 3370 (1964). J. Geibel, K. U . Reich and J. Seetaen, Nuce. Instr . and Meth., 32 (1965) 45. .t. Citron, 1 . Hoffmann, C. Passow, W. R. Nelson and M . Whitchced, Nucl. Instr . and Meth., 32 (1965) 48. R. C. Childers, C. D. Zerby. C. M, Fischer and R. Thomas, Nucl . Instr. and Meth ., 32 (1965) 53. J. Baarh, K. Goebel and A. 14. Sullivan, Nucl. Instr. and Meth., 32 (1965) 57. L. Hoffmann and A. H. Sullivan, Nucl. tnstr . and Meth., 32 (1965) 61 . A . Citron, L. Hof ann and C. Passow, Nucl. Instr. and Meth., 14 (1961) 97 A. Ashmore, G . C or i, A. N. Diddens at,d :A M. Wetherell, Phys. Rev . Lett., 5 (1960) 576. J. V. Behr and R. Hagvdorn, CERN 60-20 (196B) . .

3.5 390 165

1 .25

5)

"~ e) 9)

15) ab)

1 .8 400

1

175

1 .75

119 119

i

2.6 530 190

!~

1 .35

200

150

128

130

3.5 3.5 620 780 185 185 L25 1 .3 155 155 120 120

2.6 530 195

2.4 320 136

ta . Cocconi, L. J. Koes'er and D. I-i . Perkins, UCRL 1002:'., p. 16i. P. K. Malkutra, Bombay, Tata Institute, private communication 1962. K. H. Hanson, Inst. for Theor. Phys. Universty- of Copenhagen, Denmark, private communication . J D. Dowell, R. H. Good, B. Leontic, E. Lundby, R . Meunier and J.-P . Stroot, CERN 62-27 (1962). V. L. Fitch, S. L. M.-yer and P. A. Piroue, Phys. Rev., 126 (1962) 1849. C. F. Powell, t'. H. Fowler pnd D. H . Perkins, The Study of Elementary Particles by the Photographic Method (Pergarnor, Press, 1959). H. G6ing, Z. Naturf., 18a (1963) 1182. A. N . DiMens, E. Lillethuc, Ct, Ma gning, A . E. Taylor, T G . Walker and .A . M. Wetherell, Ffiys. Rev . Lett., 9 (1962) 108 and 111 . B. Rossi, Hi~~h Energy Particles (1956). J. L. Rosner, R. L. McIlwain, PPAD 484 D. Princeton University, 1963. R. A. Salmeron, CERN 63-20 (1963). N Spoonl,-y and R. A. Salmeron, private communicmion J. Ranft, CERN 64 (1964).