Electron-photon cascade calculations and neutron yields from electrons in thick targets

NUCLEAR

INSTRUMENTS

AND METHODS

48 (t967) ~o9-It6, ,© N O R T H - H O L L A N D

PUBLISHING

CO.

E L E C T R O N - P H O T O N CASCADE C A L C U L A T I O N S AND N E U T R O N YIELDS F R O M E L E C T R O N S IN T H I C K TARGETS* R G A L S M I L L E R , Jr a n d H S M O R A N

Oak RMge National Laboratory, Oak Rtdge, Tennessee, USA Received 22 J u n e 1966 T h e electron-photon cascades induced in cylindrical targets o f various sizes a n d materials by electrons in the energy range of 30 to 200 M e V have been studied a n d c a l c u l a t m n s of the resulting

n e u t r o n ymlds are presented These calculated yields are cornpared with experimental ymlds a n d a p p r o x i m a t e a g r e e m e n t is obtained

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

The code allows for bremsstrahlung production by charged particles, for the continuous slowing-down, small-angle multiple Coulomb scattering and for largeangle Coulomb scattering of electrons and positrons. Pair production, Compton scattering and photoionization are the only physical processes allowed for

When a high-energy electron beam enters a thick target, an electron-photon cascade is initiated and the photons of" the cascade interact with the nuclei of the target to produce photoneutrons Since these photoneutrons are often used as an experimental source of low-energy neutrons, ~t is of some ulterest to study the cascade and the resulting neutron yield as a function of the parameters of the target and electron energy In this paper studies of the cascades induced in cylindrical targets of various sizes and materials by electrons m the energy range 30 to 200 MeV are discussed and calculations of the resulting neutron yields are presented In section 2 the electron-photon cascade is described very briefly In section 3 results on the energy deposition in the target by the cascade and on the energy-angle distribution of the photons escaping from the target are given and discussed In section 4 the photon track lengths are considered, and neutron yields as a function of various parameters are given Comparisons of the calculated ymld with experimentally measured yields are also shown

Io 3

r=O TO 025'cm g

tOZ

i ~0 ~

S >o ~_ z

~ o

~

The electron-photon cascade calculations were carried out using Monte Carlo methods and an IBM7090 computer code written + by Zerby and Moran a -3) Since an extensive discussion of the differential cross sections used and the details of the code are given in ~- 3) only a brief quahtatlve description will be given here.

• 0i 75 cm

_

cascade

* R e s e a r c h s p o n s o r e d by the U S A t o m i c Energy C o m m l s s m n u n d e r contract with the U n i o n C a r b i d e C o r p o r a t i o n + C a s c a d e calculations similar to those reported here have also been carried o u t by Messel et al 4,5) a n d by NageD) F o r the case o f 50-MeV electrons incident on lead, the calculations o f Zerby a n d Messel have been c o m p a r e d a n d f o u n d to be in very g o o d a g r e e m e n t 7) a n d for the case of 1-GeV electrons incident on lead all three c a l c u l a t m n s h a v e been c o m p a r e d a n d f o u n d to be In g o o d agreementg)

r=025 TO 050 crn

i 10 0

2. E l e c t r o n - p h o t o n

"

:~

r=07 i TO I cm

7~ to_ ~ >_ ~D ~"

r =DISTANCEALONGRADIUS TARGETRADIUS=t cm

z

t0-2

-

I 0

I

2

3

4

5

DEPTHIN TARGET(RAD LENGTHS) (t RAD LENGTH= 03825 cm) Fig 1 Energy d e p o s m o n vs depth for 100-MeV electrons incident on a 1 912-era-thick Ta target

109

0

ENERGY

0 01 TO 1 0 MeV rOTO2OMeV 2OTO60MeV 6 0 TO 10 0 MeV

PHOTON

?,

POLAR

20

40

2 RAD

4 cm

LEldGTHS

60

+

x

= 0 7646

Y

80

cm

x

* x

+ .

ANGLE WITH RESPECT TO INCIDENT ELECTROrJ IIIRECTION IN UNITS OF C 9 DEGREES

THICKNESS=

TARGET

=

RADIUS

TA9’;ET

x

Fig 2 Energy-angle dlstrlbutlon of end-escape photons for 100-MeV electrons mcldent on a 0 7646-cm-thick Ta target

_

-I----

0 * + X

-

-

+

100

+

7,

POLAR

20

g;i,z;

ENERC

DIRECTION

ANGLE

PHOTON

RAMUS

x

40 IN

WITH

UNITS

RESPECT OF

TARGE .T THI’_KtIESS

TARGET

I .___

L

/

+ +

60 0 9

TO

i,

i

LEN’;rH:

DEGREES

INCIDENT

= 5 -AD

= 1 cm

80

r-

ELECTRON

A

I’ *-

!

~* -4

+

A

= , 912 cm

Fig 3 Energy-angle dlstrlbutlon of end-escape photons for _^^ __ _. IOU-MeV electrons mcldent on a 1 912-cm-thxk Ta target

r

(

100

i

ELECTRON-PHOTON TARGET

THICKNESS

-ENERGY -*-

ENERGY

TARGET

= 2

RAD

LENGTHS

INTERVAL=OOt-1 INTERVAL

THICKNESS=

RAO

-ENERGY

lNTERVAL=O01-4

---

INTERVAL=

ENERGY

cm

MeV

= 1-Z 5

= 0 7646

CASCADE

MeV LENGTHS

=

! 912

cm

MeV 1-2

MeV

TARGET

RADIUS=

4 cm

111

CALCULATIONS

results, particularly for photons, must be consldered to be more approximate than the high-energy results The geometry considered throughout this paper IS that of an Infinitely thm beam of electrons of energy E, lncldent normally along the axis of a cylmdrlcal target of radms t and thickness z 3. Energy deposition by the cascade and the energyangle distribution of escaping photons Table 1 gives the overall energy balance for lOO-MeV electrons Incident on two tantalum targets havmg l-cm radu and thicknesses of 2 and 5 radlatlon lengths For these relatively thin targets, an appreciable part of the lncldent energy escapes from the target and a large fracilon of the escaping energy IS m the form of photons For the thinner target, most of the energy escapes through the end of the target, while for the thicker target, the photons escapmg through the sides account for a goodly portlon of the escaping energy Smce the calculations are carried out by Monte Carlo methods, there 1s a statlstlcal error associated with each of the numbers m the table and, m general, the statIstIca error associated with the smaller numbers 1s

0

25

75

50 POLAR

ANGLE ELECTRON

100 WITH

RESPECT DIRECTION

125

475 TO INCIDENT

i deg i

Fig 4 Side-escape photon angular dlstrlbutlons for 100-MeV electrons mcldent on Ta targets (photon energies of 001 to 2 MeV)

photons In order to make the calculations as valid as possible at the lower energies, the complete energy dependence of the cross section 1s used In all cases Compton scattering 1s the only mechanism retained m the calculation by which a photon changes direction and contrlbutes to the lateral spread of the cascade Charged particles change dlrectlon by scattering from nuclei and from bound atomic electrons It 1s assumed that bremsstrahlung 1s produced m the dlrectlon of the electron or positron and that the electron or positron direction 1s not changed by a bremsstrahlung-producing colhslon Both members of a produced pair and electrons produced by photolomzatlon are assumed to be emitted m the direction of the lncldent photon Photons are followed m the calculation until their energy falls below 0 01 MeV and electrons and posltrons are followed until their energy falls below 2 MeV Photons or charged particles with energy less than these cutoff values are assumed to deposit all of their energy at their point of orlgln The approxlmatlons mentioned above are not equally valid over the entire energy range considered In general, the low-energy

TARGET

THICKNESS

-ENERGY -*-

ENERGY

TARGET

= 2

FAD

INTERVAL

LENGTHS=

=2-6

INTERVAL

= 6 - (0

THICKNESS=5

RAD

-ENERGY

INTERVAL=Z-6

---ENERGY

INTERVAL=6-

0 7646

cm

MeV MeV

LENGTHS=

4 912

cm

MeV 10

TARGET

MeV

RADIUS

;’

4 cm

,

t

25

50 POLAR

75

100

ANGLE WITH RESPECT ELECTRON DIRECTION

425

150

(75

TO INCIDENT tdeg)

Fig 5 Side-escape photon angular dlstrlbutlons for lOO-MeV electrons mcldent on Ta targets (photon energies of 2 to 10 MeV)_

112

R

G. ALSM1LLER,

JR

TABLE 1

Energy balance for 100-MeV electrons incident on tantalum targets (radius = 1 cm) MeV/modent electron Photons

Electrons

Target thickness = 2 rad lengths = 0 7646 cm Absorbed 2 67 x 10-2 3 55 x l0 s Reflected 3 81 x 10-1 2 32 x 10-2 Side escape 9 50 × 10-I 1 30 x 10.2 End escape 5 01 x 101 1 25 x 10l Target thickness = 5 rad lengths = I 912 cm Absorbed 6 08 x 10.2 6 84 x l0 s Reflected 5 54x 10-1 1 59x 10-2 Side escape 6 41 x 100 2 12 x 10-1 End escape 2 18 × 101 2 00 x 100

larger t h a n that associated with the larger n u m b e r s I n particular, the fact that the energy of reflected electrons is larger from the thin target t h a n from the thick target Is p r o b a b l y due to p o o r statistics I n fig 1 the energy deposited per u n i t v o l u m e by the cascade in the thicker t a n t a l u m target is shown as a f u n c t m n of depth in the target and of the distance (r) along its radius I n s o f a r as energy deposition Is a t0-1

AND

H

S. M O R A N

criterion, the cascade is reasonably well collimated at 5 radiation lengths, but the spreading of the cascade with thickness is quite evident Since the energy that escapes from the target Is primarily in the form of photons, the energy-angle d i s t r i b u t i o n of the escaping p h o t o n s Is of interest The distributions for the p h o t o n s escaping from the ends of the two t a n t a l u m targets dtscussed above are shown in figs 2 a n d 3 a n d the distributions for the p h o t o n s escaping from the sides of these targets are given in figs 4 a n d 5 I n figs 2 a n d 3 the points which are plotted are values of the distributions at the centers of equally spaced histogram intervals I n considering the figures, it should be carefully noted that the distributions are integrated over the indicated energy intervals a n d that these energy intervals are not of u n i f o r m width I n figs 4 a n d 5 it is s o m e t h i n g of an a n o m a l y that the a n g u l a r d l s t n b u t m n from the thick target is more sharply peaked in a forward direction t h a n that from the thin target This is because only side-escape p h o t o n s are shown If one adds the end escapes a n d s~de escapes, then indeed the a n g u l a r distribution from the thin target is more forwardly peaked t h a n that f r o m the thick target I n fig 6 the energy-angle d l s m b u t m n of the sideescape p h o t o n s from 150-MeV electrons incident on a m u c h thicker (20 radiation lengths) t a n t a l u m target is shown

4. Photon track length and neutron yield w~

X to~z

Cd Q:: bJ Wl-O_ Z

a: co 10-3

tt3 ta3 ~12 a_ I--- t 0 - 4 o

So ~ sc 5 N

t0. 6 0

25

50

75

t00

t25

t50

175

200

The n e u t r o n yields from thick targets may be obtained from the calculated p h o t o n track lengths a n d the measured p h o t o n e u t r o n p r o d u c t i o n cross sections In calculating these yields, three a p p r o x l m a t m n s , all of which are t h o u g h t to be quite good here, are introduced First, it is assumed that p h o t o n e u t r o n p r o d u c t i o n may be neglected m calculating the p h o t o n track length Second, it is assumed that p h o t o n " b a c k - s c a t t e r i n g " may be neglected, at least for those p h o t o n energies above the p h o t o n e u t r o n p r o d u c t i o n threshold; that is, it is assumed that, for energies greater t h a n the p h o t o n e u t r o n threshold, the p h o t o n track length in a cylindrical v o l u m e of radius r o a n d thickness z o in a target of radms r ( > ro) a n d thickness : ( > zo) is the same as the p h o t o n track length in a cylindrical target of radms t o a n d thickness z o Third, no correction is m a d e for n e u t r o n a b s o r p t i o n In the target With these a s s u m p t i o n s , one has

POLAR ANGLE WITH RESPECT TO INCIDENT ELECTRON DIRECTION (deg)

Fsg 6 Side-escape photon angular distribution for 150-MeV electrons incident on a 7 646-cm-thlck Ta target

Y(Eo) = where

leo et.

T(E ~,Eo,r,z)mr(E,)dE,,,

ELECTRON-PHOTON

t0

-ql

z 0

,

113

CASCADE CALCULATIONS

;TT Tj#T %7£Z

'

b LLJ

U L=J W

1~ t

Z

£

t0-2

z w

~0-3

o i]i

,

t0 4

0

~0

20

30

40

50 60 ~-;< (MeV)

70

80

90

100

Fig 7 P h o t o n track length vs p h o t o n energy for 100-MeV electrons mctdent on Ta targets

Y(Eo) = Eth = T(E7,E,r,z) = n = cr(E~) =

neutron yield, photoneutron producnon threshold, photon track length, number density of target nuclei, photoneutron producnon cross section

In fig 7 photon track lengths for the case of 100-MeV electrons incident on tantalum are given * The histograms are the results of the Monte Carlo calculanons and the smooth curves drawn (by eye) through the histograms were used m calculating the yields from eq (1) The stansncal fluctuanons shown m fig 7 are reasonably representanve of the fluctuanons obtained in all of the track-length calculations Large energy intervals were used at the higher energies because the h~gh-energy photons do not contribute apprecmbly to the neutron yield The track lengths decrease very rapMly with increasing photon energy The shape of the track-length curves changes very slowly w~th target thickness, becoming shghtly more peaked toward the lower energies as the thickness increases Fig 8 shows the photoneutron yields from 34-MeV electrons mcMent on copper as a funcnon of target thickness for target rad~ of 1 and 5 cm The photoneutron cross secnon in copper has been measured both by Fultz et al 9) and by Bacm et al ~0), so calculanons • Numerical values o f all of the track lengths used m th~s paper are available u p o n request

using each of these slightly different cross secnons are presented For th~s rather low electron energy, there is appreciable spreading of the cascade as evidenced by the difference in yield from the two targets Also shown m the figure are the expemmental results of Barber and George ~ ) The target of 5-cm radius corresponds roughly to the experimental target The calculated values are thus lower than the experimental values by apprommately 20 ° o In figs 9 and 10, the yields from 34- and 100-MeV electrons incident on a lead target of 5-cm radms are gwen At the lower energ,es (less than approximately (×40 a) 50

- -

FULTZ e t o/ CROSS SECTION

------

BACIU

•

o 4-

ef

0/

CROSS

SECTION

I

BARBER-GEORGE (EXPERIMENTAL)

i f :Bern

25 F:~cm

v

• = TARGET

>-

RADIUS

I 2 TARGET

4 THICKNESS

(~ R A D

LENGTH

(RAD - t 43t8

6 LENGTHS) crn)

Fig 8 N e u t r o n yield vs target thickness for 34-MeV electrons mctdent on a Cu target

114

R G. A L S M I L L E R , JR. A N D H. S. M O R A N

I--FULLER

150 ..... •

HA~WARC

M I L L E R e/ ~/

I

CROSS

still increasing at a target thickness o f l0 r a d i a t i o n lengths The yield values for a target having a thickness o f 20 r a d i a t i o n lengths are

SECTIDN

- R O e~ CECTtON

HARVEY el o~ c r ~ c s q SECTION {EYPERIMENTAL]

BARBER-GEORGE

./J

£

-\ m o

.........-----

Y = 5 09 x 10 -2 neutrons/electron F u l l e r - H a y w a r d cross section,

I

"

Y = 4 54 x 10- 2 neutrons/electron M d l e r et al cross section.

~

5U

,

Y = 2 56 × 10 -2 n e u t r o n s / e l e c t r o n H a r v e y et al cross section I

I ~ARGET r H I C P N E S 3

{F,/4D L L N ~IH~}

R~D LENSTH = 0 5 1 3 ~ u m )

Fig 9 N e u t r o n yield vs target thickness for 34-MeV electrons incident on a Pb target

22 MeV) three q m t e different p h o t o - n e u t r o n - p r o d u c tion cross sectnons have been r e p o r t e d in the hteralure ~1-14) A t higher energies, a cross sectnon t h a t is r a t h e r slowly v a r y i n g has been m e a s u r e d by Jones a n d Terwzlliger 15) Because of the dlscrepancms m the m e a s u r e d values at low energies, it is not clear h o w the cross section should be e x t r a p o l a t e d to the higher energnes Therefore, for the p u r p o s e o f the c a l c u l a n o n s r e p o r t e d here, t h e simple expedient o f a s s u m i n g the cross section to be c o n s t a n t a b o v e a p p r o x i m a t e l y 22 M e V was followed C a l c u l a t i o n s using each o f the cross sections have been carried out a n d are shown m t h e figures Also shown m fig 9 are the m e a s u r e d values o f B a r b e r a n d G e o r g e ~ ) T h e calculations using the cross section m e a s u r e d by M d l e r et al 13) are m g o o d a g r e e m e n t with the experimental m e a s u r e m e n t s In fig 10 It s h o u l d be n o t e d t h a t the ymld curves are

In fig 11 the yield f r o m 100-MeV electrons incident on t a n t a l u m is shown as a function o f target thickness for various target radii The p r o d u c t i o n cross section was t a k e n f r o m the m e a s u r e m e n t s of Fuller and Weiss 16) a n d Jones a n d Terwdliger ~s) At this energy t:.~ cascade is r a t h e r well collimated a n d the change m ylc;d in going f r o m a target o f l - c m radius to a target o f 5-cm r a d i u s is n o t large {,to 4) r=~cm

300

-,~200 r=OZ5cm

LU -

100

•

TARGET

RA[,IUS

I

! J

2

4

TARGET

THICKNESS

6

8

tO

CRAD L E N G T H S )

( t RAD L E N G T H = O 3 8 2 3 c m )

Fng l 1 N e u t r o n yield vs target thickness for 100-MeV electrons incident on a Ta target

--MI'_'_ER 4E]C

.....

e' 2

c ~, -

HAR E'r el o

rp

E ~1 /N

,-~ SEt q ,ON

/

In fig 12 the ynelds f r o m electrons of several energies incident on a 1-cm-radlus t a n t a l u m target is shown as a f u n c n o n o f target thickness for a thickness o f less t h a n 10 r a d i a t i o n lengths T h e yields for a target thickness o f 20 r a d i a t i o n lengths are

/

/

Y(100 MeV) = 3 24 x 10 - 2 neutrons/electron Y(150 MeV) = 5 26 × 10 - 2 n e u t r o n s / e l e c t r o n

I

Y(200 MeV) = 6 96 × 10 - z neutrons/electron

t

~©o

"~

TARe ET R~[,IIIS:5, m

.'J

F

i

~

P

0

4

U

TZIRc,ET

t{b

THIc KNESS (F, AD LENCTHC, I

, t PAP LENGTH = ') 5 t z = m J

Fig 10 N e u t r o n yield vs target thickness for 100-MeV electrons lncxdent on a Pb target

A l s o shown in the figure is a single e x p e r i m e n t a l p o i n t f r o m the w o r k o f B a r b e r a n d G e o r g e s 1) T h e n e u t r o n yield as a function o f incident electron energy IS, o f course, o f interest T h e ratios of the yield to the incident electron energy for s o m e o f the thicker targets are shown in table 2 B o t h the t a n t a l u m a n d the lead results indicate t h a t for a 10-radiation-length

115

E L E C T R O N - P H O T O N CASCADE C A L C U L A T I O N S

]

(~i0-4)

600 • BARBER-GEORGE (EXPERIMENTAL} TARGETR

A

D

~

I

~

TABLE 3 Ratio of neutron yield from < 30-MeV photons to neutron yield from photons of all energies

Z

,

(rad lengths)

.400

Ta target ( 10 (l-cm-radms)

200

2O

EO-30Me~

I

o 2

4

~

target the yield rises shghtly faster than the incident electron energy as one goes from electron energies of the order of 30 MeV to electron energies of the order of 100 MeV In the tantalum case this may m part be due to the small target radius and the appreciable spread of the low-energy cascade The tantalum results for both 10- and 20-radiation-length targets lndtcate that for electron energies greater than 100 MeV the yield is very nearly proportional to the incident electrons

TABLE2 Ratio of neutron yield to incident electron energy for tantalum and lead targets

z

E0

Y( Eo) / Eo

(rad lengths)

(MeV)

(10 -4 n/MeV) -

[

30 100 150

200 20

100 150 200 34

Pb target [ 1 0 100 (5-cm-radms) [20 100

a Using Fuller-Haywardl2) cross sections b Using Miller et a113) cross sections c Using Harvey et a114) cross sectmns

100 150 200 100 150 200

0 92 0 88 0 85 091 0 88 0 86

100

0 0 0 0 0 0

(5-cm-radlus) 2O

Fig 12 Neutron yield vs target thickness for 30- to 200-MeV electrons incident on a Ta target

10

YdY

Pb target ( 10

TARGET THICKNESS (RAD LENGTHS) I RAD LENGTH= 0 3823 cm)

Ta target (1-cm-radms)

Eo

(MeV)

2 45 3 06 3 29 314 3 24 351 3 48 3 86 a 3 15b 2 08 c 491 a 4 39b 2 46 c 5 09 a 4 54 b 2 56 c

100

90 a 84 b 95 e 90 a 84b 95 c

a Using Fuller and Hayward 12) cross sect,ons b Using Miller et al 13) cross sectmns c Using Harvey et al 14) cross sections

energy It is not possible to ascribe meaningful errors to the numbers given in the table It must be understood, however, that variations of the order of 5% are surely not significant, because this much variation can be introduced by the way the smooth curve is drawn through the track-length histogram (fig 7) In particular, the decrease in Y/Eo as one goes from 150- to 200-MeV electrons m tantalum is not significant At the higher incident electron energies, the contribution to the yield from htgh-energy photons is not negligible This is shown m table 3, where the raUo of the partial yield, Yp, calculated from photons with energy less than 30 MeV, to the total yield is gwen for several cases It msa pleasure to thank Dr C D Zerby of the Union Carbide Research Institute, Tarrytown, New York, for m a n y helpful discussions concerning the operation of the code and the electron-photon cascade m general. We would also like to thank J Bansh of the Central Data Processing Faclhty, Oak Ridge Gaseous Diffusion Plant, for his help in carrying out some of the numerical computations References

1) C D Zerby and H S Moran, Studies of the longltudmal development of H~gh-Energy electron-photon cascade showers m copper, ORNL-3329 (1962) 2) C D Zerby and H S Moran, A Monte Carlo calculation of the three-dlmenslonal development of High-Energy electronphoton cascade showers, ORNL-TM-422 0962)

116

R G. ALSMILLER, JR

3) C D Zerby and H S Moran, Neutron Phys DIV Ann Progr Rept 2, ORNL-3499 (1963) 3 4) H Messel et a l , Nucl Phys 39 (1962) I 5) D F Crawford and H Messel, Nucl Phys 61 (1965) 145 0) H H Nagel, Z Physlk 186 (1965) 319 7) R G Alsmlller, J r , Shielding calculations for High-Energy accelerators, ORNL-TM-1298 (1965), wdl also be pubhshed in the Proc first Syrup Accelerator Radiation Doslmetry and Experience (Brookhaven National Laboratory, Nov 1965) 8) W R Nelson et a l , Electron mduced cascade showers in

AND H S MORAN copper and lead at 1 GeV, Stanford Linear Accelerator Center, SLAC-PUB-163 (1966) 9) S C F u l t z e t a l , P h y s Rev 133(1964) Bl149 10) G Bacm et a l , Rev Roumalne Phys 9 (1964) 977 ll) W C Barbex a n d W D George, Phys Rev 11611959) 1551 12) E G Fuller and E Hayward, Nucl Ph~s 33 (1902) 431 13) j Mdler et a l , Nucl Phys 32 (1962) 236 14) R R Harvey et al , Phys Rev 136 (1964) B126 15) L W Jones and K M Terwllhger, Phys Rev 91 (1953) 699 16) E G Fuller and M S Weiss, Phys Rev 112(1958)560