A Monte Carlo study of the response of a germanium detector to electrons and positrons

Nuclear Instruments and Methods 211 (1983) 391-402 North-Holland Publishing Company

A MONTE CARLO STUDY OF THE RESPONSE ELECTRONS AND POSITRONS *

391

OF A GERMANIUM

DETECTOR

TO

H . N O M A ** a n d F . T . A V I G N O N E , III Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA D . M . M O L T Z a n d K.S. T O T H Oak Ridge National Laboratory ***,Oak Ridge, Tennessee 37830, USA Received 7 June 1982

A Monte Carlo code has been developed which predicts electron and positron response functions of a planar intrinsic germanium (Ge) detector. The operational flow chart for the main program is given and the function of each subroutine is discussed. The characteristics of the response functions are studied by considering effects due to bremsstrahlung, annihilation radiation and the total electron transit distance. It is found that electron spectra are distorted primarily near the full energy peak by bremsstrahlung while positron spectra are seriously distorted by both annihilation radiation and bremsstrahlung. Total transit distances calculated here are found to be in good agreement with those calculated using the Bethe-Bloch formula. The effects of bremsstrahlung, annihilation radiation and total stopping distance, calculated independently in the complete code, are then used to construct an abbreviated version which reduces the computer time by a factor of 16. Sample computations are presented for a 16 mm diameter by 7 mm thick, planar detector. The calculated response functions reproduce experimental spectra accurately; they are also used to analyze positron spectra and obtain endpoint energies.

1. Introduction Nuclear spectroscopy studies involve the experimental d e t e r m i n a t i o n of the energies, spins a n d parities of nuclear levels as well as transition probabilities a n d decay energies. In beta decay (negatron or positron) a n extremely f u n d a m e n t a l q u a n t i t y of interest is the Q-value which is determined from the beta e n d p o i n t energy a n d knowledge of the d a u g h t e r level scheme. In studies of nuclides far from /3 stability, the half-lives of interest are short a n d the well proven techniques of magnetic spectrometry can not be applied in most cases. For e n d p o i n t energies not m u c h larger t h a n 1 M e V the Si(Li) detector is ideal because it has exellent energy resolution, collects events corresponding to all energies at once, and has a low atomic number, which minimizes the shape distortion effects of electron scattering. F o r m e a s u r e m e n t s involving /3 spectra with e n d p o i n t energies of between 2 a n d 10 MeV, single Si(Li) detectors * Supported by the US Dept. of Energy under contracts DE-AS09-79ERI0434 and DE-AC05-760R00033. ** Permanent address: Department of Physics, Faculty of Science, Hiroshima University, Hiroshima (730), Japan. *** The Oak Ridge national Laboratory is operated by the Union Carbide Corporation for the US Department of Energy under contract W-7405-eng-26. 0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d

c a n n o t be utilized because it is difficult to drift lithium into silicon to the required depletion depths. In 1977 Aleklett [1] introduced a system in which the electrons enter the Si(Li) detector on a flat cut edge of one of its sides. Electrons which scatter out of the faces are detected in two other Si(Li) detectors used in anticoincidence. This technique, though complicated, was successful. In early a t t e m p t s to use intrinsic G e detectors for fl e n d p o i n t studies, M o o r e et al. [2] used a magnetic filter to eliminate the interference of y rays. It was later s h o w n by G i r a r d a n d A v i g n o n e [3] that even in cases involving complex decay schemes with m a n y y rays, accurate e n d p o i n t energies can be determined from linear F e r m i - K u r i e plots if the overall volume of the detector is chosen small enough to avoid f l - y summing, with the d e p t h being n o thicker t h a n necessary to stop fl-particles at the e n d p o i n t energy. This technique has b e c o m e a s t a n d a r d tool in fl e n d p o i n t energy determinations [4-6] with on-line mass separators. Recently, A v i g n o n e et al. [7] have d e m o n s t r a t e d that the major difficulty associated with using this technique for p o s i t r o n m e a s u r e m e n t s is a n effective shift in the F e r m i - K u r i e plot and a non-linearity near the endpoint, b o t h associated with the interference from annihilation radiation. The techniques used to calculate the corrections due to this effect were tested using positron spectra from the decays of 82Sr, 27Si a n d 58Cu a n d

H.Noma et al. / Response of Ge detector

392

subsequently used to determine the Q-values of the positron decays of 76Rb, 77Rb and 77Kr [8]. The present paper describes an extension of these Monte Carlo calculations to the determination of the /3- and /9 + response functions, including the effects of multiple electron scattering, bremsstrahlung and annihilation radiation interference. In earlier work of this type, Rehfield and Moore [9] examined semiempirical response functions for beta particles. They approximated the effects of the backscatter and bremsstrahlung with a flat tail but later [10] improved their approximation to account for backscattering by using a flat tail and for bremsstrahlung with a triangular component. A more complete study, both theoretical and experimental, has recently appeared in print by Varley et al. [11]. In their earlier work, Decker et al. [4,5] experimentally studied the electron response of Ge using monergetic electrons from 1.6 to 8 MeV and a magnetic spectrometer. They divided the spectra into three components: a back- and side-escape tail, a bremsstrahlung escape tail and the primary or full energy peak. In this paper we describe the details of two Monte Carlo codes; Code I is a detailed and complete code while Code II uses parameterized functions from Code I to calculate /3- and fl+ response functions with less computer time. The Ge detector analyzed in these calculations is 16 mm in diameter and 7 mm thick. These dimensions had been determined [3] earlier to be optim u m for a practical balance between efficiency and freedom from 7 ray interference. Calculational techniques used here are similar to those used by Berger et al. [12] in their analyses of the response functions for Si(Li) detectors.

When the electron is stopped in the detector, ETOT is equal to the initial energy El. At each step (i), there is an associated energy lost from the detector DE,rein , due to escaping bremsstrahlung. The total energy which escapes from the detector due to this effect is called ESUM and is given by, ESUMb . . . .

=

E DE~ . . . .

"

(2)

When a positron is stopped, it annihilates with an electron and produces annihilation photons. Since twophoton annihilation is strongly dominant, we assume that two photons of hu = me c2 are produced back-toback. We call the total energy deposited in the detector by these two photons Eannih~.The total energy losses, which are deposited in the detector are given by EB-= E T O T - ESUM b. . . . .

(3)

and Eft+= ETOT - ESUM b . . . .

+ gannihi.

(4)

Throughout this article we refer to the multistep, scattering energy E(multistep) as ETOT.

2.2. Multiple scattering The direction of the momentum vector of beta particles in matter is mainly changed by nuclear multiple elastic scattering. We assume that at each distance W,, the particle's momentum vector is rotated through angles (0, ~) which are selected by using random variables with weightings in accordance with the Moli6re multiple scattering formula [13]. The angular distribution associated with this scattering is given by,

W( O )2crOdO = F( O2)dO 2,

(5)

where 0 is a parameter defined by O --- 0/Xc~/B. Also,

2. Theoretical considerations

Xc -= (~/1 - f12/f12 ) ( 4 4 . 8 Z ) ( o / , , t ) I/2, The theoretical bases for the formulae used in our Monte Carlo codes are presented only briefly. The effects of electron and positron scattering are given equivalent treatment theoretically, while the treatment for the distortion due to annhilation radiation is treated in the usual way appropriate to 7 ray scattering and absorption.

and o-= o W i, where p is the density of the detector material and B -= v/c.The values of the function F(O 2) are tabulated in ref. 13, while B is a slowly varying parameter dependent on the number of collisions n and has values of 3.36, 6.29 and 8.93 for logl0 n = 1, 2 and 3 respectively. Given the quantity lOgl0n, the step distance Wi, in cm, is calculated by

2.1. Energy losses W, = The passage of a beta particle (fl+ or /3-) through the detector is divided into many steps. We assume that the particle loses an energy of DE i in a step distance, W,. This incremental loss DE i is mainly due to inelastic collisions with atomic electrons and to the emission of bremsstrahlung in the Coulomb field of nuclei. The total energy loss, called ETOT, is given by ETOT ~- Y~ DEi. i

(1)

A Z2/3(1.13 + 3.767 2 ) Oy 2

l0 s,

(6)

where s --- lOgl0 n - 8,215 and y -= Z/137fl. For a given step distance, the angle O is chosen using a random variable technique that reproduces the Moli6re angular distribution while q5 is chosen uniformly. We essentially replace a very large number of interactions by a smaller number, one at each distance W~. For each value of the index i, the scattering angles 0 and q~ are selected and

H.Norna et al. / Responseof Ge detector the electron moves in the new direction, dictated by these angles, for a distance W, with a reduced energy.

2.3. The energy loss distribution The distribution of energy losses DE i at each step is given by the function q~(k) derived by Landau [14] which for large values of X, is given as 1

exp{,&,2 lel?,,

1},

(7)

where X is a dimensionless parameter, X ~ ( D E / - D E M ) / a W i.

(8)

In eq. (8), DE M is the most probable energy loss and

393

d o ( k , El) over k, E o t ( E i ) = f ' do, Emin

(12)

where the m i n i m u m photon energy E m i n w a s arbitrarily adopted to be Ei/100 for these calculations. To simplify the Monte Carlo code, a relative photon energy distribution OR(EK ) was calculated in terms of the relative energy E K = k / E i , by using eq. (10) and (11). An average distribution was used for beta energies in the interval 0.1 < E i ~ 5 MeV introducing an error of about 10%, but leading to a significant simplification of the code.

2.5. The photoelectron angular distribution

a ~ O.153(pZ/Afl 2) M e V / c m . An expression for DE M, which includes a correction Dpo l, a density effect due to the polarizability of the material, was derived by Sternheimer [15] as follows: DEM = a W i { In[ aWimc2fl2/12(1 -/32)] -/32 + K - Dpo,}

(9)

In eq. (9), I is the mean excitation energy and K = 1.12 as given in ref. 15. After choosing a value for X, in accordance with the distribution q,(X), the energy loss for the ith step is computed by use of eq. (8),

2.4. Bremsstrahlung cross sections The differential cross section d o ( k , Ei) for the production of a bremsstrahlung photon of energy k by a beta particle of initial energy E i, can be calculated using formulae given by Koch and Motz [16]. For beta particle energies below 0.2 MeV, a nonrelativistic Born approximation without screening corrections called the Heitler approximation was used as follows:

dok

(Zr°)2(16]dk

1-~ \ 3 - l k -

l~ln(P°+P]

p02

p~Z-~_pj,

(10)

whereP0 and p are the initial and final electron momenta, respectively and ro=e2/mc2=2.82 x 10 13 cm. For beta particle energies above 0.2 MeV, a relativistic Born approximation without screening was used. This formula, [16] is the Bethe-Heitler approximation and can be written as

d o,

4Z2r02d[ 137k

1+

E00

The energy losses due to escaping annihilation radiation bremsstrahlung are accounted for in a code similar to those used earlier [7,17]. In the present calculations we also account for energy lost by photoelectrons escaping from the crystal. For this section of the code, we use the techniques discussed above to determine when the electron escapes and how much energy it has deposited prior to escape. The only new feature concerns the angular distribution of the electrons. The direction of the photoelectron is chosen, relative to the direction of the absorbed photon, in accordance with an angular distribution W(8). The nonrelativistic approximation of Sauter is used [18] in cases where photon energies are less than 100 keV.

W(O)2~rsinOdO-

C sin30 d 0

(1 + E z , / 2 m c 2 - / 3

cos

0)4'

(13)

where C is a normalization constant. A relativistic formula [18] is utilized when the photon energies exceed 100 keV: W(0)2~r sin 0 dO -

C sin30[1 + F ( 1 - / 3 c o s 0)] dO

(1 -/3 cos 0)4 (14)

where 1 - 3j1 - B 2 + 2(1 - / 3 2)

F=-

2(1 - / 3 2 ) 3/2

In both eq. (13) and (14), the photon energy k is contained within y in the usual way. -3Eoo

2.6. Selection of random variables (11)

In eq. (ll), E 0 and E are the initial and final electron energies in units of mo c2, respectively. The total bremsstrahlung cross section or(El) is obtained by integrating

There are several techniques employed in Monte Carlo calculations which are used to select a value of a variable for a given trial, so that the known distribution of values for that variable is reproduced. These techniques can be shown to be equivalent to utilizing a

H.Noma et a L / Response of Ge detector

394

response-distribution function X ( R ) , where R is a uniformly distributed random number ( 0 < R < 1). The function X ( R ) is related to a given distribution q~(x) such that when a statistically significant set of random numbers ( R ) is selected and related to a distribution ~ ( x ) as follows: R=

x

)d~/f~m ' a"O (x)dx,

CODE I

I INPUT: E,, PI, SI I

t

I En - El, Pn - PI, Sn - SI, ESUMb. . . . = 0.0 999"~'~

CALL ESCAT (En, Wn+l, THET, FI) [ CALL LANDAU (En, Wn+l, DEn+l} ]

t

(15)

then the distribution in x is determined and is in fact proportional to @(x). The variable x could be 0 in an angular distribution or ?t in the Landau distribution, for example. Each part of a Monte Carlo code, involving a given distribution @(x) and variable x, can be tested independently by performing the operations indicated in eq. (15) and determining if the distribution of selected values of x actually reproduces @(x).

I

t

En+l - En

DEn+I' Pn+l

Pn + Wn+l X (Sn + Sn+1)/2 I

CALL BREMS (En, Wn+l, DEbrem~) ESUMbfem ~ = ESUMbrems + DEbrem s

t'

Pn'l IS IN DETECTOR

I

~ES l E,,+I > 0003 YES 999 ~

I NO

RETURN

STOP: ETOT = El

En = En+ I P,, = Pn+l

t

3. Operation of the Monte Carlo codes

:

ETOT = En

CALL IENS1 (Pn' Eannihi)

SO=Sn+1

Schematic diagrams of the functional steps of the detailed Code I and the abbreviated Code II are shown

ESCAPE

Eann,hi = O.O

i

Epos,tfon - ETOT ESUMbrems + Eanmh i Eelectlon - ETOT ESUMbrems Emult,ste{; - ETOT

t

CALL SMEAR

1

t

OUTPUT CHANNELS OF SPECTRUM ] CODEI

BREMS. ESCAPE

/

~

SIDEESCAPE

~

PHOTO-ELECTRON . ESCAPE

SOURCE

",

c'

Ib

~<%= ~

I ~ , FORWARDESCAPE

.z"

I

Ge DETECTOR CODE 17. I

BREMS.

//I

//"

//

SOURCE+C .....

"',,

..,q

"-,

~.,r" TAIL

4 511 keg GAMMA RAY

/~ ,--4i " I REMS STEP I

J I

;'0. JGAMMARA¥ i

j-T.. L_

Fig. 1. Schematic diagram of the processes calculated with Codes I and II.

Fig. 2. Flow chart of Code I.

in fig. 1. The corresponding operational flowchart for Code I is shown in fig. 2. The distribution of beta particle trajectories is consistent with that for an isotropic source. The history of each beta particle is followed until it is stopped in the detector or escapes. Bremsstrahlung, annihilation photons and photoelectrons are followed until they either escape or are absorbed. Finally, the total energy for that event is computed, Below we discuss in detail the logic by which the code accomplishes this. Let us assume that a beta particle with energy El arrives at the face of the detector from the source. Let us further assume that the entrance point is defined by the vector PI which has a unit vector SI in the direction of its momentum on the surface of the detector. Further, let us assume that the beta particle suffers many collisions and that after n steps, it has an energy E,, is located by vector P, and has a direction unit vector S,. After the (n + 1)th step, the total distance traveled is IV,+ 1 while the scattering angles (0,~)n+l are calculated using subroutine ESCAT [based upon eq. (5) and (6)]. In subroutine ESCAT, logl0n is chosen for the three different energy intervals corresponding to values of 1, 2, and 3. In table 1 we give corresponding values of Wi, The next direction vector S, + i is then calculated from the scattering angles (0, +1, @, + i)-

395

H.Noma et al. / Response of Ge detector

Table 1 Results of the subroutine ESCAT (E, W, THET, FI) and the most probable energy-loss DE M. logmn

Energy E(MeV)

W(I0 -3) (cm)

DEM(keV)

1 2 3

0 . 0 0 3 < E < 0.2 0.2 < E < 3.00 3.0 < E < 1 0 . 0

0.033- 0.115 1.15 - 2.00 20.0 -20.3

0.59- 5.68 6.5 - 7.9 100 -110

Subroutine L A N D A U is used to calculate the energy loss D E , + 1 of the (n + 1)th step by use of eq. (8), and by using the L A N D A U distribution q~(X) as discussed in section 2-6. At this point, the most probable energy loss is calculated by eq. (9). We note from table 1 that D E M is 5.68 keV at E = 3 keV and lOgl0n = 1. The beta particle is considered to be stopped. Generally when the particle reaches a point at n + 1, it has just lost energy D E , + 1 and has a direction vector Sn+ 1. The energy E,+ 1 is given by, E , + , = E, - D E , + , ,

(16)

while the location vector Pn+ 1 is given by P , + , = P. + W , + I ( S , + S , + 1 ) / 2

(17)

In this way, the location and energy constantly evolve in finite steps until the electron either escapes or is absorbed. Subroutine B R E M S gives the bremsstrahlung escape energy, DEbrems, as a function of the current beta energy E n and the step distance W~+l. When the beta energy E , is larger than 3 MeV, the step distance is divided into ten intervals. In each interval, DEbrem s is calculated with BREMS. The bremsstrahlung emission probability is obtained from W,+I and the total bremsstrahlung cross section or(E,, ), given by eq. (11) and (12). The photon energy k = E K - En is then selected by use of the relative photon energy distribution function OR(EK ), discussed in section 2-4. The angular distributions are assumed to be isotropic because low energy photons are primarily emitted. The energy loss DEbrCm s is then calculated using the photon response code used in our earlier work [17]. If after the (n + 1)th step P,+~ still describes a point inside of the detector, and E,+ 1 > 3 keV, the code returns to statement 999 where another representative interval is calculated. This process is repeated until either P, is outside of the detector, in which case the electron escapes, and E,.nihi = 0, or P~ is inside of the detector and Es < 3 keV indicating a stopped particle. At this point, the energy deposited by the annihilation radiations is calculated using subroutine 1EN511 which is also fashioned after the ~, ray code described in ref. 17. One major difference is that in subroutine 1EN511, the escape of both Compton electrons and photoelec-

trons is accounted for in a manner similar to that of the main code. It is found that approximately 10% of the 511 keV photopeak is lost due to electron escape from the small detector analyzed in this work (16 mm diam. by 7 mm deep). The shape of the Compton tail is not noticeably changed by electron escape. The deposited energies are smeared according to a Gaussian smearing routine S M E A R discussed in ref. 17. The smeared energies are converted into channel number and the distribution of events calculated in this way represents the calculated spectrum. An operational flowchart describing the abbreviated Code II is shown in fig. 3. Code II is a simplified version of Code I, and depends on parameters and two data sets calculated by Code I. The parameters are the stopping rate, RSTOP, the average stopping depth or distance from the detector face, Z, and the annihilation radiation absorption rate, R511, defined by R511 =

N u m b e r of events Eannihi > 0.1 keV N u m b e r of stopped particles

(18)

CODE I]

i ,NPOT: E, I t

[ CALL PARAMT (El, RSTOP,~-,RANG, R511) ]

I

t

? NO ~

I

STOP ,~ DETECTOR

~ YES I pEAK: ETOT = El' ESUM. . . .

~

=0,J =0 ]

/

i

ESUMbrems : O I EanmhI : 0 ]

| ~ YES

CALL BREMS IE, Wi+l, DEbrems) ESUMbrems= ESUMbrems + DEbtems J=J+l

i I , <,STEP

I

DATA: 511SHAPE,R511 I m

Eoos~t,on = ETOT - ESUMbrems + Eannihi ] Eelectron = ETOT ESUMbrems Em~mstep = ETOT

t f

CALL SMEAR ] ] OUTPUT: CHANNELS OF SPECTRUM ]

Fig. 3. Flow chart of Code II. The parameters (RSTOP, Z and R511) and the data (TAIL and 511 SHAPE) were calculated using Code I.

396

H.Noma et aL / Response of Ge detector

These parameters are calculated as functions of the initial beta energy El. In Code II, we calculate the range, R A N G , of electrons of energy E1 with the Bet h e - B i t c h formula [19] which includes density and bremsstrahhng effects. When RSTOP is selected as a weighted random variable, it is then determined whether or not a given beta particle stops. If a particle is determined to have only partially deposited its energy, then this energy is determined by the distribution calculated in detail by Code I, the results of which are shown in fig. 4. The calculation assumes a source-to-detector distance of 8 mm which is a standard distance one would use with this size detector (results do vary with this distance). In Code II, when the particle itself escapes from the detector, we neglect the bremsstrahlung and annihilation radiation effects, hence, ESUMb .... = Eannihi = 0. When the particle is stopped, the energy loss due to bremsstrahlung is taken into account. Photons are emitted when the beta particle traverses the range, R A N G . This distance is divided into JSTEP steps and a photon is emitted at each step. This parameter was fixed at 100 in Code II by comparing spectra generated by both codes. The bremsstrahlung energy loss at each step distance is chosen to be the same in Code II. The depth at which a bremsstrahlung photon is emitted from the detector surface is given by 1

Zi

= (Z/RANG) E Wj,

(19)

j=l

where Z is the average stopping depth. The X - Y coordinates of the stopped particle are chosen randomly for simplioity. The escape energy for bremsstrahlung is calculated using subroutine BREMS, which when summed over all steps yields the total bremsstrahlung escape energy E S U M b . . . . " It is also assumed that the annihilation radiation is emitted from a depth Z. The probability of absorption

of a given energy by annihilation photons is given by the parameter R511 for Eann~hi > 0.1 keV, while the energy distribution of E a n n i h i is given by subroutines D A T A and 511 SHAPE. These routines are based on the normalized shape of the annihilation energy losses and are independent of the depth Z, a fact verified by Code I. Using the results from Code I, we assume that the 511 keV total energy rate is given by R511 ×0.011. The energy losses Emultistep, E B and EB+ are obtained by using the equations in section 2 above. These energies are converted into channel numbers with subroutine S M E A R to obtain the calculated spectra. Code II then, is a significantly simplified code which depends critically on results obtained using the detailed Monte Carlo Code I. Its results were also directly checked with Code I.

4. Results and discussion

The positron spectrum at a full kinetic energy EI = 4.0 MeV, calculated with the full version Code I is shown in fig. 5. The general features are in agreement with those shown in fig. 6 of ref. 11. Three main features are coded individually, namely, the electron escape tail, the bremsstrahhng escape, and the annihilation radiation contribution. The chief sources of spectral distortion in the region of the full energy peak are due to bremsstrahlung and annihilation radiation associated with the stopped positrons. With the approximation that the only difference between positrons and electrons is the distortion due to annihilation radiation, a positron spectrum is obtained by summing all three effects while an electron spectrum is obtained by eliminating annihilation radiation effects. As one test of our methods, we

POSITRON SPECTRUM AT SL = 8 mm SOURCE I

LEGEND POSITRON ;,;~ ;~, ANNIHILATION PART BREMSSTRAHLUNG PART TAIL OF ESCAPE PART x

104 PEAK

tso

x

×

x V5

~_ lo3 |

8

t0o 50

0

I

2 3 4

5 6 7 8 9 IO fl t2 f5 14 15 ~6 17 IS ~9 20 CHANNEL

Fig. 4. The data called TAIL and parameter RSTOP in Code II deduced from this multistep spectrum obtained by Code I at S L = 8 mm for 10 3 events at each initial beta energy El. Interval~ of 0.2 MeV have been used and each tail is divided into 20 channels. The primary peak is due to stopped beta particles in the detector.

0.0

0.5

1.0

1.5

2.0 2.5 3.0 ENERGY (MeV)

3.5

4.0

4.5

5.0

Fig. 5. Positron spectrum given by Code ] at E l = 4 M e V and SL = 8 mm for 105 events. The solid line shows the brems-

strahlung effects due to stopped particles in the detector. The dashed line shows the escape tail due to the multistep losses and the shaded area shows the annihilation radiation effects.

H.Noma et aL / Response of Ge detector 105

397

q

I

I

t 4000

t 4020

~POSITRON ---'-ELECTRON - ~ MULTISTEP 104

I'

~

1°2I,

~

El = 4 Mek/"

103

L~?

:

r~" ~

<

z D ©

El = 1 MeV

>'< a: 10 2

i

=!

0

50

100

150

200

250

300

350

BREMSSTRAHLUNG ENERGY (keV)

Fig. 6. Bremsstrahlung spectra of ESUMbrem s due to particles stopped in the detector with El =1, 4 and 6 MeV. Each spectrum contains 10 3 events using Code I and SL = 8 mm.

10 0 I t 3960

t 3980

ENERGY (keV)

Fig. 7. Peak spectra showing the net positron and electron

have calculated an electron spectrum at kinetic energy EI = 4.0 MeV and found it to be in good agreement with the experimental spectrum given in ref. 4. The bremsstrahlung contribution consists of a low energy part for ESUMbrem s < 20 keV and a triangular part. Fig. 6 shows the bremsstrahlung contribution to the spectrum for EI = 1, 4 and 6 MeV, calculated with Code I. The triangular portion for ESUMbrem s > 20 keV shows a strong dependence on EI. We also have noted that the relative intensity of the bremsstrahlung spectra are not sensitive to the source-to-detector distance, if it is greater than about 8 ram. The low energy portion of the bremsstrahlung escape spectrum consists mainly of K X-ray escape due to bremsstrahlung. Fig. 7 shows both electron and positron spectra in the region of the full energy peaks. The pure " M U L T I S T E P " peak neglects photon effects. The low-energy tail was identified as a contribution due to the low energy portion of the bremmstrahlung contribution. The peak intensity difference between the multistep and electron spectra is due to the bremsstrahlung contribution while the difference between the electron and positron peaks comes from the summing of positron and annihilation radiation energies. The results of a detailed calculation of the total transit distance W = EiW, for stopped beta particles is shown in fig. 8. It is evident that for values of EI up to 8 MeV, the predicted total transit distance W agrees well with the range, R A N G , given by the Bethe-Bloch formula, including density, bremsstrahlung and polarization effects [19]. The stopping distance, Z; form the entrance face of the detector also shows essentially a linear dependence on EI.

peaks and the pure multistep peak, calculated using Code I at EI = 4 MeV and SL = 8 mm for 10 5 events. The high energy events beyond the positron peak are due to the annihilation radiation.

I

I

I

I

a TOTAL T R A N S I T DISTANCE o STOPPING DISTANCE: 2 - - B E T A RAN

0.9

I

I

all

W

~"-.I

a.zv/

I

0.8

0.7

0.6 E

~

0.5

.

0.I

-

o o

oooooo oooooo ooooo __

o °°°

o o o o

0

I 2

[

I

i

I

I

3

4

5

6

7

8

E.[ (MeV)

Fig. 8. The total transit distance W = Y~W, for stopped beta particles calculated with Code I. Also shown is the average stopping distance ( Z ) from the detector surface. The beta

range (RANG) was calculated with the Bethe-Bloch formula.

398

H.Noma et al. / Response of Ge detector I

~.o f

I

I

I

I

I &&&z z~z~ A &z~z~

0.5 f o

z~& z~z~

0.2

& 0.t

--

~ ~.o

0.05

-

o

o BACK ESCAPE a. SIDE ESCAPE • FORWARD ESCAPE

~" '~" ''Q ~. o@ •

ELECTRON BACKESCAPE (THEORY) ----- POSITRON BACKESCAPE(THEORY)

0.02

@

I

I

I

I

I

I

2

3

4

5 EI (MeV)

6

7

8

Fig. 9. Back, side and forward escape rates calculated wi~h Code I at SL = 8 mm for 103 events a't each energy. The solid and dashed lines are the backscattering coefficients of electrons and positrons respectively given in ref. 9.

o.8

L - - .~. •~ ~ ~

06

I ~ ~

(2)RSTOP SLI 8 m m

I

I

I

R ( l ] ' ) ~-

]

(~) RSTOP Sk~ PARALLEL

~

-

az

(3)

R5tt

1

RATE O~

o

-

-

Np ( l y ) N t ( E > 0.1 ) '

(20)

where Np(1y) is the total n u m b e r of events u n d e r the o n e - a n n i h i l a t i o n - p h o t o n escape peak and, N t ( E > 0.1) is the total n u m b e r of annihilation p h o t o n s which deposit more than 0.1 keV in the detector. The quantity R (l y) for a variety of source-detector distances is tabulated in table 2. Each value given was calculated for a total of 105 beta events using Code I. One notes that R ( I y ) has essentially a constant value of 0.011 for

xX\\\

0.5

< oi4

The back, side, a n d forward escape rates were also calculated as functions of the initial particle energy E1 using Code I and are shown in fig. 9. As expected, backscattering escape decreases while side escape increases with increasing El. This clearly demonstrates the value of good collimation. Forward escape does not become i m p o r t a n t until the energy corresponds to a range that exceeds the thickness of the detector. The theoretical backscattering curves are those given by Rehfield et al. [9]. The consistency of these results to the theoretical curves shows that this effect is not strongly d e p e n d e n t on the size of the detector• The stopping rates R S T O P for SL, the source-detector distances of 8 a n d 100 mm, are shown in fig. 10. These rates differ but they b o t h gradually decrease with increasing initial energy. The difference between these two curves is as expected a n d again shows the merits of using " g o o d " geometry a n d the importance of having equivalent geometries for comparing calculation and experiments. The annihilation radiation absorption rate R511, defined by eq. (18), is also shown in fig. 10. It was observed that the curve R511 was not sensitive to source detector distances SL > 8 mm. The curve R511 increases gradually for EI = 1 to E1 --- 4 MeV, where it essentially becomes c o n s t a n t with a value of a b o u t 0.35 up to 8 MeV. The shape of the spectral c o n t r i b u t i o n due to the annihilation radiation is itself found to be almost i n d e p e n d e n t of initial particle energy. The single escape peak rate R ( 1 y ) for annihilation radiations is defined as follows:

1

t

1

1

1

1

1

2

3

4

5 E[ (MeV)

6

7

8

Fig. 10. The stopping rates RSTOP: (1) for a parallel beta particle beam, (2) at SL = 8 mm, (3) the annihilation absorption rate R511 and (4) the bremsstrahlung rate RB. These results were calculated with Code I.

Table 2 Escape peak rate R(17) of a single 51 l-keV gamma ray for the annihilation shape. El (MeV)

SL

R(1y)

1.0

8 mm parallel 8 mm parallel 8 mm parallel

0.01232 _+0.00078 0.01124+0.00073 0.01112 + 0.00071 0.01082 + 0.00065 0.01268 _+0.00082 0.01166 4- 0.00069

1.0 4.0 4.0 6.0 6.0

H.Noma et aL / Response of Ge detector EI = 1.0 to 6.0 MeV and for SL = 8 m m to infinity. A quantity which we call the bremsstrahlung rate RB defined in eq. (21), is also shown as a function of EI in fig. 10. We define

RB-

N(B) U•(stop) '

(21)

where N(B) is the total n u m b e r of bremsstrahlung photons in the bremsstrahlung spectrum, for which E S U M b . . . . > 0.1 keV, and Na(stop) is the total n u m b e r of beta particles that stop in the detector. The quantity RB is not sensitive to the source distance SL when SL > 8 mm, but increases with EI almost linearly. The detector efficiency for an electron event to appear in the full energy peak is then RSTOP(1-RB) while for a positron it is RSTOP( 1-RB- R511). The central processing unit ( C P U ) time with an IBM 370, or equivalent c o m p u t e r is 17.4 s, on the average, per 100 electrons analyzed with Code I at EI = 4.0 MeV. The average n u m b e r of steps NS, each of distance W,, has an approximately constant value - 2 0 0 for El = 2.2 to 5.0 MeV. We have noted, however, that NS increases from a value of 94 at EI = 0.2 and decreases to a value of 125 over the interval EI = 5.0-8.0 MeV. Code II was developed to significantly shorten the c o m p u t e r time by exploiting the constant or smoothly varying nature of some of these parameters. The bremsstrahlung contribution is c o m p u t e d in Code II by allowing the bremsstrahlung p h o t o n s to be emitted in a total n u m b e r of steps, called JSTEP (see fig. 3). The bremsstrahlung rate RB actually d e p e n d s on how JSTEP is chosen. This parameter was varied in Code II over a range from 10 to 1000. It was found that when JSTEP = 100, spectra calculated with Code II are very similar to those calculated with Code I. Fig. 11 shows a positron spectrum for EI = 4.0 MeV, calculated with Code II and with JSTEP = 100. This calculation required 1.1 s per 100 events, a factor of 16 less than the same calculation using Code I. This factor is somewhat d e p e n d e n t on EI.

10 3

tO 2

3.15

3.7

3.8

3.9

4.0 4.I 4.2 ENERGY (MeV)

4.3

4.4

4.5

4,6

Fig. 11. A positron spectrum with 104 events for initial particle energy El = 4 MeV and JSTEP = 100 using Code II.

399

It has clearly been demonstrated, then, that an elaborate and detailed M o n t e Carlo Code can be used to generate the behavior of certain parameters for use in a simplified code with essentially equivalent results. This abbreviated code can then be used to do very repetitious calculations, as exemplified by the extraction of the best e n d p o i n t energy from a raw beta spectrum with an iterative technique [20]. In the present work, the multiple scattering distributions as well as the energy losses are assumed to be

Table 3 Numbers of electrons which escape or are stopped or backscattered Energy (MeV)

Stop number

Back escape

Side escape

Forward escape

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60 6.80 7.00 7.20 7.40 7.60 7.80 8.00

654 679 646 689 678 678 687 719 719 699 689 710 692 675 680 678 669 677 624 629 628 618 628 587 605 562 568 575 533 548 551 524 511 482 490 454 437 404 388 360

342 311 328 285 272 285 245 210 204 194 187 157 156 140 126 140 113 120 139 124 125 112 97 107 71 77 86 58 71 78 57 57 61 60 44 30 33 49 29 29

4 10 26 26 50 37 68 71 77 107 124 133 152 185 194 182 218 203 237 247 247 270 275 306 324 361 346 367 395 372 390 412 419 ,149 447 488 496 496 516 532

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 7 9 9 19 28 34 51 67 79

1

18 18 25 18 21 9 14 16 10 13 22 16 20 20 27 17 19 21 20 29 19 25 31 36 44 45 37 30 38 39 34 39 37 45 40 34 57 48 57 55

Energy (MeV)/bin

0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000 3.200 3.400 3.600 3.800 4.000 4.200 4.400 4.600 4.800 5.000 5.200 5.400 5.600 5.800 6.000 6.200 6.400 6.600 6.800 7.000 7.200 7.400 7.600 7.800 8.000

47 29 29 24 30 16 18 17 22 18 26 15 28 21 20 31 26 22 41 26 39 37 30 29 24 28 44 37 45 57 41 37 36 34 37 54 54 36 57 62

2

44 25 23 31 28 31 23 18 19 19 19 19 18 24 25 25 28 21 28 34 35 27 22 34 34 35 40 37 25 31 43 40 43 49 53 48 46 47 45 53

3 48 35 32 36 31 30 23 13 23 20 22 24 17 25 21 20 22 25 25 29 33 27 21 40 32 31 34 23 37 29 35 49 44 39 43 41 44 52 38 38

4 37 32 25 22 32 27 30 26 25 20 18 22 22 27 21 20 22 26 26 25 23 35 24 26 28 22 18 31 40 32 28 32 29 36 35 35 33 41 34 39

5 28 40 32 36 23 23 28 31 18 28 22 12 22 24 26 19 30 23 32 34 20 23 30 29 22 21 29 34 34 28 28 23 30 35 36 47 33 39 36 37

6 23 24 32 16 26 36 29 21 31 30 22 19 31 19 19 23 25 18 26 28 27 26 23 28 23 28 28 33 36 27 25 37 29 27 31 36 40 33 49 31

7 22 20 33 25 23 32 20 20 24 32 28 24 17 24 18 19 26 20 28 24 33 26 23 28 31 32 23 22 33 25 29 24 34 34 35 39 32 25 34 31

8 13 24 27 27 22 19 25 17 11 13 19 27 19 25 24 13 25 17 30 36 30 23 31 16 33 29 27 36 29 20 31 29 29 24 36 21 24 27 23 24

9 15 16 20 15 25 20 17 28 17 18 21 18 17 18 19 17 16 24 18 15 18 18 26 28 20 25 25 34 23 33 21 33 21 34 18 18 23 23 26 29

10 12 18 20 12 7 13 16 15 15 19 20 14 19 19 22 22 21 20 22 17 20 25 15 23 18 20 21 13 24 16 24 23 27 25 25 25 24 42 27 29

11 13 11 10 13 15 12 17 12 16 25 15 18 22 24 23 20 13 16 15 17 15 18 23 20 14 19 16 20 25 22 27 29 29 26 23 24 20 30 34 34

12 6 12 11 8 10 17 14 14 14 9 18 15 15 11 13 21 13 15 17 ll 16 15 20 16 14 24 17 16 16 20 19 21 20 22 15 15 27 29 24 34

13 7 4 10 11 12 11 7 7 9 12 10 20 12 12 11 15 16 21 10 12 9 8 12 14 17 12 19 14 11 24 18 17 10 22 24 28 27 32 41 42

14 4 6 11 8 3 9 10 13 8 9 10 10 8 6 16 16 10 8 10 11 10 12 14 14 19 21 20 11 15 19 12 13 28 13 17 25 14 38 24 26

15 2 3 4 2 6 8 8 5 6 8 7 4 3 15 7 6 6 11 12 11 9 14 11 14 5 19 14 18 11 12 12 10 20 20 17 18 27 22 19 28

16 2 I 7 4 4 5 7 1 6 4 8 5 7 6 5 8 8 8 7 6 5 10 8 8 7 12 11 9 8 8 7 8 10 10 11 17 12 16 20 18

17

Numbers of electrons in the various " t a i l bins" shown in fig. 4. The corresponding values in the full energy peak are given in table 3 as "'stop number".

Table 4

5 1 2 1 2 3 5 6 4 4 2 5 8 2 3 3 4 3 5 3 7 7 4 5 5 11 5 4 9 8 8 9 6 13 8 8 15 11 14 20

18 0 1 0 1 2 1 1 1 2 0 1 3 3 3 0 7 1 2 4 1 4 6 4 5 3 1 4 3 2 2 5 2 7 10 5 11 9 11 9 7

19 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 2 0 2 0 0 0 0 2 3 0 0 6 0 1 1 0 0 1 2 2 4 1 3

20

H.Noma et a L / Response of Ge detector

the same for electrons and positrons. We can, however, estimate the differences which would be reflected in the response functions by considering the theoretical backscattering coefficients shown in fig. 9. The coefficient for positrons is several percent less than that for electrons for an incident energy between 0.1 and 10 MeV [9]. In addition, Berger et al. [21] have estimated the positron energy loss to be approximately 3% less than that for electrons over an energy range of 0.5-10.0 MeV. We then estimate that the escape tail of the positrons is between 10 and 20% less than that for

401

electrons. N e v e r t h e l e s s the spectra are strongly d o m i n a t e d by the full energy peaks primarily distorted by bremsstrahlung and annihilation radiation from s t o p p e d particles. We conclude from this that the calculation of positron e n d p o i n t energies for continuous positron spectra should not be significantly affected by neglecting the difference in the multiscattering processes between electrons and positrons. The response functions calculated with these codes are found to reproduce the experimental spectra very well. The primary goals of this study are to permit the

Table 5 Parameters for a multistep spectrum. Energy (MeV)

Avg. step no

Avg. stop distance

Avg. stop radius

Avg. transit distance

Range

0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60 6.80 7.00 7.20 7.40 7.60 7.80 8.00

94.2 93.8 101.0 113.5 124.6 137.0 150.4 165.9 175.3 177.1 190.8 211.1 212.0 218.0 231.2 219.3 217.2 223.7 197.3 198.4 196.8 197.3 204.0 186.2 197.3 184.3 181.6 184.3 172.8 180.9 177.1 171.7 171.0 162.7 162.4 150.7 148.4 142.1 133.8 125.3

0.00332 0.00905 0.0163 0.0238 0.0331 0.0401 0.0513 0.0598 0.0711 0.0803 0.0908 0.0990 0.113 0.119 0.127 0.141 0.154 0.169 0.181 0.186 0.200 0.219 0.225 0.236 0.244 0.264 0.272 0.291 0.295 0.307 0.319 0.339 0.353 0.357 0.373 0.376 0.401 0.397 0.406 0.417

0.469 0.479 0.471 0.463 0.458 0.464 0.466 0.469 0.459 0.464 0.460 0.469 0.461 0.473 0.468 0.471 0.472 0.471 0.463 0.468 0.472 0.471 0.482 0.475 0.489 0.484 0.476 0.481 0.484 0.485 0.471 0.484 0.481 0.480 0.483 0.485 0.490 0.485 0.484 0.494

0.0115 0.0325 0.0575 0.0818 0.110 0.138 0.167 0.190 0.222 0.241 0.273 0.304 0.323 0.352 0.379 0.398 0.429 0.459 0.473 0.503 0.520 0.554 0.582 0.601 0.642 0.650 0.674 0.716 0.728 0.760 0.777 0.800 0.822 0.843 0.878 0.869 0.899 0.932 0.923 0.q57

0.0143 0.0368 0.0622 0.0914 0.120 0.149 0.177 0.205 0.233 0.260 0.287 0.314 0.340 0.366 0.392 0.418 0.443 0.468 0.492 0.517 0.541 0.564 0.588 0.611 0.634 0.656 0.678 0.700 0.722 0.743 0.766 0.787 0.808 0.828 0.849 0.869 0.899 0.909 0.929 0.949

402

H.Noma et al. / Response of Ge detector

m e a s u r e m e n t of beta endpoints, a n d perhaps beta b r a n c h i n g ratios, using p l a n a r intrinsic G e detectors. The Q-value measurements afforded by the technique developed in the work discussed in this paper constitute a major effort in our studies of nuclei far from stability performed with the U N I S O R [22] on-line isotope separator. N e u t r o n deficient nuclides (e.g. 77Rb, 76Rb and 7VKr in ref. 8) are produced in reactions induced by heavy ions from the Oak Ridge isochronous cyslotron a n d from the Holifield Heavy Ion Research Facility V a n de G r a a f f accelerator. To check the present calculations we measured the well k n o w n positron spectrum from the decay of 27Si, produced in the reaction 27Al(p, n)27Si. In the iterative technique, a series of allowed positron spectra are calculated a n d distorted by our response codes for m a n y e n d p o i n t energies. These endpoints are varied until the F e r m i - K u r i e functions of the calculated, distorted-spectrum has the best overlap with the measured spectrum. Using this procedure we extracted an e n d p o i n t energy of 3777 _+ 30 keV for this decay, in excellent agreement with the accepted value of 3787_+ 1.3 keV [23]. A typical scintillation detector m e a s u r e m e n t would exhibit an order of m a g n i t u d e larger error, a clear d e m o n s t r a t i o n of the power of this technique. We also note that for simple decay schemes, the a p p r o x i m a t i o n s used in ref. 7 suffice. For the/3 + transition between ground states in the decay of 27Si, our a p p r o x i m a t e corrections of ref. 7 yield a beta e n d p o i n t of 3779 + 23 keV, which when c o m p a r e d to the value given above clearly demonstrates that either technique applies. The response functions calculated in the present work can be used to o b t a i n beta b r a n c h i n g ratios and for this purpose they are also given numerically in tables 3 and 4. The first column in table 3 are the unsmeared, peak channel n u m b e r s which correspond to the continua given in table 4. Table 5 gives the other numerical values used in constructing a simplified code corresponding to our Code II. We wish to express our appreciation to the U N I S O R staff for their help during this work.

References [1] K. Aleklett, P h . D . Thesis, University of Gothenburg, Gothenburg, Sweden (1.977).

[2] R.B. Moore, S.I. Hayakawa and D.M. Rehfield, Nucl. Instr. and Meth. 133 (1976) 457. [3] T.A. Girard and F.T. Avignone, III, Nucl. Instr. and Meth. 154 (1978) 199. [4] R. Decker, K.D. Wtinsch, H. Wollnik, E. Koglin, G. Siegert and G. Jung, Z. Physik A294 (1980) 34. [5] R. Decker, K.D. Wtinsch, H. Wollnik, G. Jung, J. Munzel, G. Siegert And E. KOglin, Z. Physik A301 (1981) 165. [6] D.S. Brenner et al., Bull. Am. Phys. Soc. 26 (1981) 568. [7] F.T. Avignone, III, H. Noma, D.M. Moltz and K.S. Toth, Nucl. Instr. and Meth. 189 (1981) 453. [8] D.M. Moltz, K.S. Toth, F.T. Avignone, III, H. Noma, B.G. Ritchie, and B.D. Kern, Phys. Lett. l13B (1982) 16. [9] D.M. Rehfield and R.B. Moore, Nucl. Instr. and Meth. 157 (1978) 365. [10] D.M. Rehfield, R.B. Moore and D. Hetherington, Nucl. Instr. and Meth. 178 (1980) 565. [11] B.J. Varley, J.E. Kitching, W. Leo, J. Miskin, R.B. Moore, K.D. Wtinsch, R. Decker, H. Wollnik and G. Siegert, Nucl. Instr. and Meth. 190 (1981) 543. [12] M.J. Berger, S.M. Seltzer, S.E. Chappell, J.C. Humphreys and J.W. Motz, Nucl. Instr. and Meth. 69 (1969) 1981 [see also NBS Technical Note 489 (1969)]. [13] G. Moli6re, Z. Naturforsch. 3A (1948) 78. [14] L. Landau, J. Phys. 8 (1944) 201. [15] R.M. Sternheimer, Phys. Rev. 88 (1952) 851. [16] H.W. Koch and J.W. Motz, Rev. Mod. Phys. 31 (1959) 920. [17] F.T. Avignone, III, Nucl. Instr. and Meth. 174 (1980) 555. [18] F. Sauter, Ann. Physique. 5 Bll (1931) 454. [19] R.H. Pratt, H.K. Tseng, C.M. Lee, L. Kissel, C. Maccallum and M. Riley, At. Data Nucl. Data Tables 20 (1977) 175. [20] D.M. Rehfield, Nucl. Instr. and Meth. 157 (1978) 351. [21] M.J. Berger and S.M. Seltzer, Tables of energy losses and ranges of electrons and positrons, NASA SP-3012 (1964). [22] UNISOR is a consortium of the University of Alabama in Birmingham, Emory University, Furman University, Georgia Institute of Technology, University of Kentucky, Louisiana State University, Oak Ridge National Laboratory, Oak Ridge Associated Universities, The University of South Carolina, University of Tennessee, Tennessee Technological University, Vanderbilt University, and Virginia Polytechnic Institute and State University, and is supported by them and by the U.S. Department of Energy. [23] A.H. Wapstra and K. Bos, At. Data Nucl. Data Tables 19 (1977) 175.