34

Annual Report of the Radiaticn Centerof Osaka Prefecture, Vol. 10 (1969)

Sakai, Osaka

Range Distribution of 4- to 24-MeY Electrons

Tatsuo Teeare, Rinsuke lro, Shigeru Oxese and Yoshiaki FuJrrex

(Received l)ecemter 25, 1q69)

Experimental data are presented of the range distribution of electrons in the absorbers ofBe, A1, Cu, Ag, and Au. The measurement has been made with a thin charge collector insertedin the absorber, and the energies of the incident eLectrons investigated are 4.09, 7.79,77.5,74.9and 23.5 MeV. The distributions ohserved have been fitted rvith an empirical equation in rnhichan expression for the transmission curve given bv XAar is utiiized in a slight11, modified form.The most probable ranEe -R,,,, the practical ranqe Rr, the tnean projected range n, tUe t-,oz

range R,, and the ra.nqe straegling oa,iameter :/R have been determined frcm the distributions.Ttvo t1,pes of empirical equation for erpressing the ratios of the range parameters to thetheoretical mean range have been compared, and the one originally proposed b1- Harder andPoschet has been found to gire a better fit for ai1 the range parameters considered.

4-24MeV ETorftfE44 r*fi#)EJ., iFd+rfi4, mH e, +=€E*Be, Cu, Al, As, rb.lL\-Au tFt--i;ijZai-alftirijfi''-'.)'..:*-*-*,i*E?-=?E-ia. /nii)t1rj.€l:, 91!1i*tr

1., Zil.llEDihr-tri'at\7..-ffiL.EliilEqXtEaffjJ,l,, :,i.r.:!&!r1n1]*=rt-ll:iiiil;?,Elli7=:-L,',:i-a1T:i:" i'ErJ

"€1-t:jt7rr+- ll, 4 09, 7 79, 11.5, 14.9?j&Lr23 5X{eV t-a. f,EiErj ,l:ifiz{E,Fftfra*a-ir-L:c-;*.41:..

.,)Itli, 'il=Tor5.661;;4i1-r..ll_,f l,,larotirEL,+:t48ft=tir!,'Ll,E|t:t,-ai,dfr,r-/:i.:orirZi\, f!!,;&,tof,{prog.iZ,.Lit1l::':. ii:, i.Irj.iEr-:t:/-,tiiijt6, ftI,E7ftfER-, +H-7ftfliR2, *ji4J.E;fffER, 1,,/Tff Rr.;;.1r.-\7ft8,;Jlfittr lR rr:i..:ir":. 61'l{:irir}.{:6i!r-i-jt-r[f€iai,li-Z,LLlr{tiri-,lESAr\i:-:rLr-Ce-:&;JL/:ff#R, Harder & Poschet

ijJEnEL, lr,iilrrl:l,iielir:.:iitz-i:iiD,fr;ir, f iT -.b;1.ro*/i1- (;-d,rrf:',i-:.i ll tr, f"\ro.IfifEi.-rr+L{, :l[Af*+ L ]i:l -t-_r..-f"ti:;r--J- : & lll.]i,: /:"

I. INTRODUCTION

In a previous reoortl) 1pr'41ts r',-ere describecl cfthe rneasuremert of differential rense distri-

'.bution of electror-.0 in Ee and Al fcr incidentenerqies from 4"09 to 14.9 MeV. Additicnal da,ta

of the rairse distrihlrtion ha..,e been obtrined fcrthe abrorhers of Cu, Ag, a.nC Ar in the enelgl-regicn fr:cm .,i.09 to 23.5 l\{eV, nnd for A1 a-t 2,?.5

I4eV. T1:: l-.,rrnose of this paoer is to presenta nu-merical '-ztb't.,.: of these da-ta as well as thatof variou:r rair?e Darameiers determined fromthe distrihuticns cbserved. Comparisons of em-pir:ica). r'e!.ations are aiso described. A completeaccount of tiris experiment -ril1 be publishedel.servhere.!)

'i Research ileactor Institate, I(-r-oto ,nruu."ity, Kuma-tori, Osaka

II. EXPERIX{ENTAL },iETHOD -{\DDEFINI'I-ION OF RANGtr PARAMtrTtrRS

The electrcn bra.n used in t1.re preselt experi-ment r';as oro-.-iCed fcr energies up to 14.9MeVb..' the iireo,r e.:celerator cf the Radiation Ceirterof Ot:i<a Plefectr-rre rRCOr. and for 23.5 l\{e\r bythat cf 1.1.e Kvctc Unir-erslt-t Research ReactorInstitute (KURRI). The e:spefifflental arrange-;:rert at ti:e RCO has been ilescribed in theorevi.ons paner.r) The experiment at the KIIRRIn,a.s oerformeC rvith a- -oimiJ.ar arra-nqement ex-cept'ior the foiJowinqs: No shielding *-all existed}:etween the anai)rzrng magnet and the experi-r:rental area, the thickness of the a,iuminumrnindox, at the beam erit rvas 0.05,19,icm2 (a.A27 g/c:n: at the RCO), ancl the distance betrveen thelvindorv and the absorb:" s-,'sterfl was 7.6 cm(i.9.n at the RCO).

A thin charge collector, u'hich consisted of thesame material as the absorber and rvas put inan insulating sheath, was moved progressivelytlrrough the absorber thickness (see Ref" 1 or )for details of the absorber system). Currentsfrom the coliector and the absorber assembll'were respectiveiy amplified with picoammetersand then fed to current integrators.

The charge distribution 1(;r) is detenninedfrom

:t(x't -1Q8t i-1QG), Q(xi|,tx, (1)

where ,JQ(r) is the charge measured at the clepthr by the collector of thickness ,Jx, and Q(;r) isthe charge collected by the absorber assembll'.The charge distribution otrserved can be inter-preted as representing the range distribution ofthe incident electrons except the initial part(regir:n of small .r), nhere the secondarl- emission shott,s an appreciable contribution becauseof boundary effect.

The foilorving parameters have been deter-mined from the observed distribution 1'(-r):

The most probable range R,,,: the deptir atrvhich y r reaches the maxirnum

The range straggling /R: FWI{M of y(r)

The mean projected range -R :R- i* ylxlxdx,0

The practical or extrapoiated rartge Rp : R2

.-R,,, ; T(R,,,') \''' R,,,'), u'here 1' .v - I l' t' .r'' 'rr-r'''"JriThe I-'+, range R1 : the <lepth at u'hich -r' -r.

has fallen to 1t., of its maximum.

III. RESULTS

The distribution y(r) has been measured forBe, Al, Cu, Ag, and Au at incident energies:4.09,7.79, 11.5, 1.1.9, and 213.5 MeV. The measure-ment for Be rvas exciuded al ):.1.5 MeV because.the diameter and the total thickness of theabsorber disks used rvere insufficient at thisenergy. Part of the results has been presented

** Though 7(.r) ma1, approximately be equal to thetransmission function 4(:r) of electrons, tlte t$'o func'tions do not allvays coincide with each othei'becauseof differences in clefinitions involved. 7(r) is defined

for an effectively semi-infi.nite absorber as the numberof electrons absorbed in the region of depth from rto m divjded b-v the total number of obsorbed electrons,

whereas ,(r) is defined lot a plonc ficrallel absorber

of thickness * as the number of electrons transmittedthrough it divided by the number of incident electrons.

35

in figures of the previous report.l) All the setsof data zrre numerically given in Table i. Someof the distribution curves are shown in Figs" Iand 2 for demonstrating the trends of the distri-butions obserr.ed: the ordinate and the abscissaare in dimensionless pararneters of y(r)ry(R,,,)and x'L, respectir,ely, rvhere L is the mean ra1lge(or the average of the path length) of electronscalcuiated by Berger and Seltzersr from

(2)

Here E, is the incident energy, p is the densityof the absorber, and dI) ds is the mean energ-\'loss of electrons per unit path length.

As illustrated in Figs. 1 atrd l, the r,alue ,R,,,'lof the peak position increases gradua111 to-nvard

Fig. 1

0 0.2 0.4 0.6 0. 8 r. 0 1.2

xLRange distributions of electrons in Cu absorberat incident energiesof 4.09, 7.79, 11.5, 14.9, and

23.5 MeV. Experimental points have been sup-pressed for simplicity of representation.

! r.! i.4 r.! 0.8 .! l.Z,l

2 Range distributions of 11.5-N4eV electrons inabsorbers of Be, A1, Cu, Ag, and Au. Experi-mental points have been suppressed for simpli-city of representation.

'- f'a - ) 14,,4u \)'ou

Eou-a

o.+

(

- a.4

BeAICuAqAu

Fig.

T l.SMeV

3ri

Table l. Range distribution -rr(.t) of electrons in the absorbers of Be, Al, Cu, Ag, and Au. y(r) is expressed in

units of 10 1 cmz/g and absorber thickness r is in g,/cmz.

Eo:4.09 MeV

Be (Z:4) At (z:13) Ct (.2=29) As (Z:47) Ar:. (Z:79)

! (r) i (i) y (x) y (x) i (i)

0.07 -1.590. 17 -0. 60

0.27 -0.290.45 0.050.64 0.520.83 0.940. 97 1.441.17 2.501. 35 3.681. 5,1 5. 411 no I aa

1..82 8.151.94 5.23

2.04 9.462.14 9.252.24 8.662.32 7.86

2.42 5. 89

2.45 5. 15

2.51 3.90

2.55 3. 10

2.61 1.95

2.65 1.33

2.69 1.13

2.73 0.90

2.75 0.282.83 0. 16

2.84 0. 13

2.87 0 06

294 001

0 07 -0.720.12 0.180.21 0.790.34 1.570.48 2.360.61 3.020. 75 4.190.89 5.081.02 6.051.16 6.76L.29 7.381. 45 7. 68

1.59 7.511.72 6.831.86 5.461. 99 4. 11

2.0s 3, 31

2.1,3 2.362. 18 1.87, o7 1 1'oacn70

2.40 0. 38

2.45 0.232.54 0.03

0. 07 -0. 71

0. 08 0.470.16 2.050.34 3.870.51 5. 31

0.62 6.120.78 7.100.95 7.551.13 7.421.40 6.021.57 4.761.84 2.552.01 t.21.2.11 0.76

2.28 0.26o c1 n Jo

2.46 0.072. 55 0. 03

2.;2 0,02

0. 07

0. 09

0. 18

0. 28

0. 38

0. 48

0. 61

0. i20. 82

0. 93

1.021.251..45

1. 68

1. 89

2. 09

2.312.512. t-1

0.241..67

3, 81

5.176.41/. JO

7.968. 32

8. 39

7. 98

7.545. 70

3. 80

2.070. 95

0. 33

0. 09

0. 03

0. 01

0.07 0.480. 17 5.04u.tl /.D1

0. 37 8. 8,1

0. ,15 9. 69

0. s4 i0.00.65 10. 1

0.74 9. 31

0.83 8.5s1.0s 6.311.25 4.13

t.42 2.38t.62 1.231.80 0.52

2.01 0.20

2.21 0.05

2. 39 0. 03

2. 59 0. 03

2.;i 0. 01

Eo :7. 79 MeV

Be (Z:4) A1 (.2=\3)

j (x) tG)

An (Z:79)'":=Yx y(x)

0. 09 -1. 48

0. 19 -0. 73

0.47 -0.331. 19 0.071.96 0.582.70 1.56aoo ,oe

3.60 4.14J.6Z 4. OL

4.02 5.344.20 5.534. 39 5.434.57 4.804.72 3.92

0. 10 -1. 11

0. 23 -0. 30

0. 37 -0. 03

0.64 0. 33

0.91 0. 69

1. 18 1.051.48 1.611.75 2.192.A2 2.792.29 3.402.56 3.902. B0 4.243.07 4.353.34 4.17

0. 10 -0. 91

0.14 -0. 19

0.28 0.570.55 1. 33

0.99 2.261.43 3.211.87 3.862.05 3.95

2.23 3.912.50 3. BB

2.94 3.263. 11 2.803.38 2.093.82 i.07

0. 1i -0.310.22 0.730.32 1.40

0.52 2.09c.77 2.76

0.97 3.321.18 3. 70

1.39 3.961.59 4.17

1. 8,1 4. 08

2.a4 3.902.24 3.652.56 3.012.90 2.31

0. 10 -0.240.14 0.880.20 1.560.30 2.340.48 3.290.68 4.040.86 4.52

1.08 4.82L.28 4. Bs

1.45 4.69

1.65 4.302.04 3.392.34 2.60

2.62 1.82

Ae e:4i)x tQ) x t@)

a1

4.92 2.615.10 1.40

5.29 0.545.39 0.235. 47 0. 15

5.57 0.0s5.67 0.02

3. 61 3.60

3.88 2.694.18 1.574. 3t 1 .1.4

4.45 0. 67

4.58 0.414 72 0.224. 85 0.094.99 0.03

5.13 0.01

4.09 0 55

4.27 0 35

4.50 0 14

4. 68 0.07486 0035. 13 0.01

3. 3i 1. 35

3.72 0,624.17 0.184.37 0.094.58 0.044.78 0.01

2.89 1.21

3.22 0.663.77 0.173.98 0. 10

4.18 0.054.36 0.034.56 0.024.74 0.01

Eo:11.5 I\{eV

Be (.2:4) At (z-r3) Ct (Z-25) As Q:47) Au (Z:79)

x v (.x) I y(x) x tQ) t tQ) rc ! (rc)

0. 14 -1.250. 33 -0. 53

1.04 -0. 11

2.01 0.082.57 0.2+I O( n ?t

3.51 0. 63

3.85 0.864.44 l. 43

4.97 2.12

5.38 2.76

5.72 3.255. 90 3.476.10 3.77

6.28 3.896.47 3. 87

6.61 3.716.79 3. 3,1

6.94 3. 06

7.13 2.547. 36 1.887. 41 1 .547.51 1.267 .61 0. 90

7.68 0. 75

7.79 0. 49

7.91 0.328.01 0. i88.08 0. 12

B. i8 0.068.28 0.038.38 0.02

0. 13 -1. 03

0. 18 -0.640.32 -0.350. 54 -0. 14

0.94 0.091. 35 0. 30

1.78 0. 60

2.1S 0. 92

2.59 1.322.96 1.70

oooJ. J/ '.

LL

3.78 2.614.21 3.014.62 3.135.02 3.045.43 2.585.76 1.976.17 1.226. 36 0.876. 57 0. 55

6. 76 0. 31

7. 00 0. 1.1

7.19 0.067.41 0.02

0.t2 -0.830.57 0.371.00 0. 86

1.46 r.261. 89 1. 75

2.34 2.189 :- 9 ;1

2.95 2.63aol ,-,

3.40 2.743.66 2.654.1t 2.484.33 2.294.78 1.90

5.39 1.18

5.67 0.85

5. 84 0. 66

6.10 0.436.28 0.316. 55 0. 15

6.73 0. 10

6. 98 0.04

7 .16 0. 02

0. 14 -0. ,18

0. 19 0.06

0.34 0. 53

0.68 1. 11

1.21 1.83

1.75 2.,1,5

2.27 2. i'6

2.52 2,87

2.81 2.79

3.34 2.61

3.88 2.084.40 1. 48

4.94 0. 85

5.36 0.48

5.90 0.19

6. 43 0.066.68 0.036.97 0. 01

7.23 0.01

0. 13 -0. 35

a.22 0.72

0.32 1.12

0.50 1.72

0.80 2. 30

1.10 2.79

1.47 3. 18

t.67 3.24

1.85 3.23

2.07 3.22

2.54 2.893.04 2.31

3. 51 i.694.01 1. 11

1. 30 0.79

1. 58 0. s7

1.98 0. 31

r2i 0.17

r;; 0.11

;.83 0 0s

6. 1,1 0. 03

6. c2 0.01

E6:14.9 MeV

Be (Z:4) At (z--13) Cr (Z:29) Ae Q-47) Ar (Z:79)

x v(x) x t@) x t@) x t@) x y (.1)

0. 17 -1.720. 37 -0. 54

0.74 -0.291. 11 -0. 19

0. 17 -0. 90

0. 30 -0. 45

0. 43 -0. 30

0.71 -0. 15

0.17 -0.860.62 0.131.05 0.421.50 0.71

0. 19 -0. 51

0.40 0.23

0.73 0.631.26 1.09

0. 18 -0. 35

0. 39 0.660.56 1.02

0.76 1. 30

38

2.06 -0.03298 0093 90 0.254.80 0. s2

s.73 0.976 4,1 1. 59

6.96 2.147.34 2.51i.77 3. 06o 10 c ac

8.31 3.218.49 3. 19

8.74 2.Slon, ,q)9 21. 2.109. 57 1.269. 95 0. 51

10.14 0.2510.32 0. t310. 52 0. 0.1

10. 70 0.0210.89 0.0r

Eo-.23. 5 IVIeV

1.251812.36,a-

3. 41

3. 95

4. 51

5. 06

6. 07

6. 61

7.177. ,15

7.727.99o oa

8. 49

8.779. 04

9. :11

9. 58

9. 87

0. 03i )l

0.420. 68

0. 97

1.341.7 4

2.172. ?.6

c 10

2391971.681. 31

1030 7,1

0. 16

0. 25

0.120. 05

0. 01

0. 01

1.912.392.82

3. 71

.1. i64. 38

4. 83

5.265. T7

6.156. 60

7. 03

7. 48

7.92B. 37

8. 80

9. 21

9.64

1. 00

1.301. 64

1. 88

z. l.)

2.222.222.1s2. 06

1. 80

1..52

1.1s0. 85

0.520. 30

0. i40. 06

0. 02

0. 01

1. 80oa9

2.863. 39

3. 93

4. 4r4. 99

5. 41

5. 95

6. 1B

7.548. 08

8. 61

9.159. 67

1.561. 93

2202292.222.021.671. 3,1

0. 94

0. 61

0. 34

0.170.070. 03

0. 0i0. 01

Ag Q:47)

1. 16 i.80i.53 2.201.73 2.392.72 2.602.3?r 2.642.50 2.602.70 2.562. 88 2.523. 10 2.44a an a ac

! la a on

3.67 2.094.06 1. 75.t.44 t. 42

5.01 0. 97

5.41 0. 68

6.00 0.376.38 0.246.98 0.11

7. 35 0.057. 9.1 0. 02

8. 32 0.01

Au (Z-79)

t Qc)

0. 32 -4. 2t069 0511.29 0.911 76 1.192.26 1.432 73 1,64oooln-,.). L J

3.70 1.81

4.20 1.795. 1; 1. 57

6.1i 1.18;.1i 0.768 08 0.459.0J 0.21

10.02 0.0910.99 0.03

At (Z:L3) Cl1 (Z:29)

x t@) r t@)

0.29 0.76

0. 83 -0. 12

2.17 0.113. s5 0. 30

s. 58 0. 70

7, 19 1. 13

8. 30 1. 37

9. 37 1. 57

10.87 1. 34

11.41 i. 14

12.48 0. 6,1

13. 02 0.4213. s9 0. 18

11.13 0.0911.6; 0.02

0.29 -0.;8f. i7 0. 1i2.91 0.624. 50 1.065.38 1.27A')7 1 /)

7. 15 7.428.04 1. 33

8.88 1. 17

9.76 0.9010.6s 0.6011. 53 0.371.2. 12 0. 15

13. r-.9 0. 07

13. 9; 0. 01

, (r)

0.33 0.310.8; a.2l1. 40 0.492.46 0.893.53 1.284.59 1.505. 13 1. 55

5.55 1.556.62 1.387.68 1.088.75 A.72

9.81 0.4110.83 0. 18

11.90 0.0612.96 0.01

uility u,ith increasing E6 or decreasing atomicnlrmber Z. This is due to the decrease of theeffect of multiple scattering detours.

'lhe range parameters fl,,,, /?, Rp, R1, and therange straggling ;lR obtained in this experimentiire listed in T:lble ,.+**

*+* Some o{ the vaiues reported in Ref. I of R*, tri,

and ,R, for Be and A1 have been revised accordingto additional measurements and re.examination ofcorrections.

IV, ANALYSIS AND DISCUSSION

A. Range Distribution

The differelltial range distributions obsen'edcan indirectly be compared with results of theMonte Carlo calculation or the experiment thatis concerned with transmission curves by usingthe relation

y(x): - dT(.x) ldx

:39

Table 2. Values of R,,, R, Rp, Rr, and ,lR obtained in this experiment. Values of Rp given by the empiricalequation of Ebert ei al. (Ref.6) are alsoshown for comparison. Parentheses are attached to thevaluesin the cases where the application of the empirical equation goes outside the scope of their experi-ment.

AbsorberR,,

(s cmj)

2 0,1 i0.024.231:0.03

6.36-0.048.20 - 0. 05

1.45+0.023.04+0.04.1.65-0.06

6. 10 r0.079.8010. 17

1. 01::0. 02

2. 23t 0. 04

3. 40:l:0. 03

4.36+0.056.80:0. 12

0.785:0.0131.69+0.032.66r.0.043. 42!4. A7

5. 32t 0. 07

0.585 r 0.013

1.20r0.021.81 :0.032.33-0.033.80.0.07

R(e.'"*z;

1. 87 :r 0. 0:t

3.88rl:0.08

5.85+0. 13

7 .75 t:0.77

1.30+0.032.7710.064.27+0.095.64 :0. 12

8.71+0.19

1.02:0.032. 16'-0.063. 32 0. 10

1.40 0. 13

6. i2 -0. 19

0.855+0.03i1.80'r0.072.77+4.103. 54+0. 13

5.39+0. 19

0.7121:0.039

l. 45+0.07

2. 14 .0. 10

2.81'.0.13-1. 19r -0. 19

Eo(NIeV)

2. 4"i - 0. 02

4. 8l -0. 03

7.3010.049.43+0.05

2.01+0.024.0210.046.00+0.067. 81 -i 0. 0B

12. 18r0. 18

1. .15 0 02

3.0i '0.034.49+0.045.77 -r 0. 07

8.69+0.08

1.194+0 014

2.45+0.033.67+0.034.70+0.036.95-0.08

Constants

2.8610.01) ,r/ f!l- \ll

8.26+0.0410.60+0.05

2. 50+0.01

4. 97 +0.02

7.32+0.039.:18:0.05

14.;8-0. 12

2.34 t.0.02

4. 57+ 0. 02

6.72 f0.03

8.7110.0512. 90 _0. 10

2. 12 - r-. L1

I .ft I a.

a :'t l:; I ,. '-:'

:a) ,a

tR(g,1cm?)

0.99t0.031. 71 i0. 09

2. ,14a0. 10

2. 9,1+ 0. 11

1.29::0.022.28 r0.07

3. 12'r 0. 10

3.96 f0.106.11 r 0.34

7.22+0.042.45+0.083. 50+0. 14

4.32+0. t46.40+0.27

0.98t0. 04

2.07+0.063.09+0. 06

3. 82+0. i35.45+A.20

0. 971+0. 004

0. 447: 0. 014

1.51+0.0911.0+0.3

... b

(4. 30)

(6. 56)

(8. 63)

Cu

(z--2e)

Ag(Z - 41-)

Au(z-ie)

4. 09

i .79

11. s

14.9

1. 89

3. 98

6. 08

(8. 00)

(12. 86)

1..2totrn

3. 95

(5.20)(8.38)

4. 09

7 .79

11.5

14. I

,1. 90

T. i911.5

14. I

1. 16

3.11

4.75(.6.26)

(10. 08)

4. 09

7. is11.5

14. Ioaf,

- drii)/dx, 3

where l(r) is the transmission coefficient ofelectrons for the absorber of thickness -r, and

the sign of approximate equality is used in con-sideration of differences in the definition of. T(x)and r(r).

Mara) expressed his Monte Carlo result fortransmission curves obtained in the energy regionfrom 0.4 to 10 MeV with an equation of thefollowing form originally developed by Makhovs)

T@) : exp 1-1*',1,f ctsEoloo('z-ou1-au), ( 4 )

where the a;'s O-1,2,..., 6) are constants. Valuesof the a,'s determined by Mar are listed in column2 of Table 3. Ebert et al.6) have reported thatMar's expression agrees fairly well with their

Table il. \-a1ues of constants in the empirical relationsfor the transmission curve. Ertors attachedto the present results are those of least-squares fit, and € is the weighted rms devi-ation of the calculated values of the rangedistribution from the experimental data.

MaraEq. (4)

0. 848

0.23

0. 634

49tr

0.24

0. 971 - 0, 001

0.290-0.0030. 775 - 0. 007

6.06+0.023.54+0.09

Eq. (a) Eq. (5)

A7

A2

AA

A+

Ai

A6

A7

a Ref. 4.b Parameter not used.

G/c:^\ 1?r

Present data Eberl pt al.

1. 6e +0. 02 | r. oo I 2. 45 r-0. 01 | t. asr o. o0

3.43 r0.04 | :. st | 4.7e r'0.03 | 2.571.0.11

5.23 r-0.04 | s.:z I z. i Lr o. oa I a.6ato. ts

6.71 +0. 05 | 12. oe; I o. zt to. os 4. 55' 0. I 1

10.34:::0. 13 (i1.38) I ta. oe+0. io 6.98+0' 31

7.5+0.7

Be

(z:4)

4. 09

i.7911.5

14. I

AI(z -13)

2426 9tO-

40

experimental transmission curves at 4 MeV butthat the agreement is poor at higher energies.A similar trend is also seen in the comparisonof the present results for the range distributionwith -drt/dx in which Mar's expression is used.A rather good agreement can be seen for thelimited cases of Z{29 and Eo{7.79MeY. Thepeak of -drt/dx is generally at larger depththan the present values of R., and the relativedeviation increases with increasing Eo and Z.This discrepancy, however, cannot necessarilybe considered as showing the defect of Mar'sMonte Carlo method because of the indirect andapproximating procedure used in the comparison.

In order to obtain a better fit, the method ofleast squares was applied to the present datawith -dr/dx in which Eq. (4) is used f.or lt(x).

The calculation was carried out with a computorby using a Fortran program "ALSQM" developedby the present authors. The feature of thisprogram consists in iterative procedure includ-ing adjustment of approximate values for speci-fied unknowns subject to large variations. Theresult of the fit are shown in column 3 of Table 3.

The maximum deviation of R* is 16fu, and thefit is not satisfactory enough.

For the purpose of further improvement, aslight modification of Eq. (4) was tried:

q,c) :exp \- (x.,(Z+a1)o"/ arEo)onz "u), ( 5 )

where a7 is a new constant, and the constant a5

used in Eq. (4) has been dropped instead. Theresult of least-squares fit by the use of thisequation is presented in column 4 of Table 3.

While the weighted rms deviation e is reducedonly a little as shown in the lowest row of thetable, the maximum deviation of R* is 7.8fu,being considerably smaller than before. Some

t.0

0.4 !.6 0. I,,iL

Fig. 3 (a)

r'LFig. 3 (b)

Fig. 3 Comparison of the experimental range distribu-tions with those calculated from the originalMar's expression IEq. (4) with Mar's values ofconstants] and the modified expression IEq.(5)] for the transmission curve. Solid circles:experimental, dashed lines: calculated from1\{ar's expression, and solid lines : calculatedfrom the modified expression.

of the range distribution curves obtained fromEq. tSl are compared in Fig. 3 ri'ith the experi-mental data and also rvith the curr.es calculatedfrom original NIar s expression.

A more refined empirical equation for therange distribution is under investigation on thebasis of a modification proposed by Ebert et a1.,6)

and the result will be reported later.!]

B. Range Parameters

1. Comparison with Other Restr.ltsThe values of the practical range R, for AI

determined in this experiment show a goodagreement with the empirical equation of l(atzand Penfold?) as already described.l) In Table2, r,alues of R, obtained from the empiricalrelation recently reported by Ebert et al.6) arepresented for comparison. Their relation wasdetermined from transmission curves experiment-ally obtained for Eo:4.0-12.0MeV and Z-G-92.Therefore, the values for Be and En)14.9 MeV(shown in parentheses) are outside the scope oftheir experiment. While the values in paren-theses generally show an appreciable deviationfrom the present results, no significant differencecan be seen within the scope of their experi-ment. The present results for R, and ,F arealso in good agreement with the results obtainedby Harder and Poschets) from the measurementof transmission curves and those determined by

E 0.6E

- 0.4

0.8

: 0.6.c-

-:a o.t

N.2

t. 23. 5 M"v

0

0

liIt

a 4.2 4.4 0.6 0.8 1.0 1.2 1.4

\i

4l

Table 4. Values of constants in the empirical relations [Eqs. (6) and (7)] for R,/L, where R represents one of the

experimental range parameters and I is the mean range calculated under the continuous slowing-down

approximation. Errors attached are those of least-squares flt, and e is the weighted rms deviation.

Eq. (6)

ba

bz

b.t

b+

e

d1

u!

ds

d.+

6

0.514+0.0210.649:r:0.013

0. 460+ 0. 010

0.907 r 0.015

0.8%

1.49 +0.08

0.750+0.056

0.31610.017

0.223+0.0121.2?l

0. 176+0.018

0. 811 :l0.0270.633+0.020

0.903+0.010

1.2?,

1. 17 .0.020.235 +A.027

0. 557--0.030

0. ,124 r 0. 019

1' 42'

0.215+0.020

0.937+'0.025

0. 521 + 0. 019

0.973-r 0.015

r.t g,6

1. 23 ,+:0.05

0.368+0.041

0.508+0.027

0.270+0.015

2.lcl

0.307+0.076

0. s95i0.0830.93410. 118

0.813+0.020

Z C'.)

1- 25 + 0. 0,1

0.340+0.076

0.477+0.078

0.75310.111

2.6?/;

Wittigsl through a Monte Carlo calculation (see

Ref. 2 for a detailed comparison).2. Empirical Equatiort

Harder and Poschets) presented empirical rela-tion of the following form to express Rr/L andRiL as a function of Zmoc2,l Eo'.

RiL- (b(Zmocz,/Eo)t/t i 611 t', (6)

where R stands for R2 or R, b and b' are con-stants for each of the range parameters, andmocz is the rest energy of the electron. Since aplot of the present values of R L us Znt6c! Eo

shows that the results for different values of Zlie on the curves slightll' shifted from each

other, the following modification is consideredto give a better fit:

R/L-(b1Zb"(moc2f Es)'.*DrJ 1, (7 )

where R now stands for any of the four rangeparameters determined in this work, and thebis (i-1,2,3,4) are constants for each of them.An inspection of the aforementioned plot onlogarithmic scales shows a possibility of fittingwith another equation

Ri L: d&x?(-- dzZd'(moc2 ,/ Eo)d ") , ( 8 )

where the d;'s (i:1,2,3, 4) are constants for each

range parameter. The method of least squareswas applied to the present data with these equ-ations. The results are given in Table 4. Bycomparing the rms deviations shown in thistable, it can be concluded that Eq. (7) gives a

slightly better fit for all the range parametersconsidered.

Some other empirical or semiempirical equ-

ations being developed rvill be reported.:)

ACKNOWLEDGMENTS

The authors are indebted to the members ofthe linear accelerator groups of the RCO andthe KURRI for their kind cooperation. Theyalso wish to thank Mr. M. Hayashi of the KURRIfor his valuable aid in carrying out the com-putor calculations of least-squares fit.

REFERENCES

1r T. Tabata, R. Ito and S. Okabe, Ann. Rep.

Rad. Ctr. Osaka 9. 3l .1968.r.

2, T. Tabata, R. Ito, S. Okabe and Y. Fujita,to be published.

3) M.J. Berger and S.M. Seltzer, National Aero-nautics and Space Administration Report No.

NASA SP-3012, 1964 (unPublished).

4) B.W. Mar, Nuclear Sci. Eng. 24, 193 (1966).

5) A.F. Makhov, Fizika Tverdogo Tela 2, 216l(1960) lEnglish transl.: Soviet Phys.-Solid State2, t934 (196r).1.

6) P.J. Ebert, A.F. Lauzon and E.M. Lent, Phys.

Rev. 183, 422 (1969).

T L. Katz and A.S. Penfold, Rer'. Mod. Phys.

24,28 (1967).

8) D. Harder and G. Poschet, Phys. Letters 248,519 (1967).

9) S. Wittic, Johann-Wolfgang-Goethe UniversityInstitute for Nuclear Physics Report No' IKF-20, 1968 (unpublished).

R Rb R* Rr

Eq. (7)