s
Nuclear instruments and Methods in Physics Research A 388 ( 1997) 254-259
INSTIIUMENTS
6METHoDS
IN PHYSBCS
__
__
@
REz%F
EISEVIER
Two-parametric
method for silicon detector calibration
and fission fragment spectrometry
S.I. Mulgin*, V.N. Okolovich,
Institute
of Nuclear
Received
Physics. National
Nuclear
Centre,
2 May 1995: revised form received
in heavy ion
S.V. Zhdanov
48lW82 Almaty,
I I November
Kazakhstan
1996
Abstract
A new procedure has been proposed for the calibration of silicon semiconductor detectors in heavy ion spectrometry. This
method exploits a two-parametric empirical equation for the value of pulse-height defect (PHD) in detectors versus masses
and energies of recorded ions. A convenient practical guide using the mean channel values of light- and heavy-fragment
groups in spontaneous fission of 2X’Cf has been developed for determining the parameters of the PHD description for a given
detector. The method being applied to heavy ion spectrometry with .%-surface-barrier detectors and p-i-n diodes has
demonstrated good results in a large range of masses and energies.
1. Introduction
One of the specific problems appearing in heavy ion and
fission fragment spectrometry
with Si-detectors
is the
necessity to take into account the so-called pulse-height
defect (PHD). PHD is commonly defined as a difference
between the energies of a heavy ion and of an alphaparticle yielding the same pulse height. Investigations of
PHD showed that its total value R consists of three
components, and each of them has a complicated dependence on the mass M and the kinetic energy E of recorded
ion, i.e.
R(M, E) = R,(M. E) + R,(M
E) + R&k’. E)
(1)
The quantity R, is due to energy losses in the entrance
window of a detector. If the metallic contact width is
known, the value is determined from the tables in Ref. [ 11.
The component R, results from energy losses at elastic
collisions with atoms of crystal lattice in a detector and
could be calculated with a good accuracy as outlined in
Refs. [2,3].
The third quantity R, is connected with the electronhole recombination in plasma produced along an ion track.
According to Ref. [4], the behaviour of R,(M, E) essentially depends on many factors (the characteristics of silicon,
the detector production technology,
the electric field
strength in the detector, etc.), which makes it difficult to
achieve a good quantitative description of this value.
* Corresponding author.
0168-9002/97/$17.00
PII
SO1 68.9002(96
Copyright
)O I2 I l-9
The difficulties connected with evaluation of recombination losses and the uncertainties in the widths of detector
entrance windows make it necessary to determine the value
R(M, E) for each given detector experimentally. The most
precise solution of the problem is to measure the dependence R(E) for every studied mass M with a heavy ion
accelerator. But, this technique is rather expensive. So, the
detector calibration methods are usually based on the
application of different empirical expressions for R&f, E),
whose parameters are determined either from measurements only for several selected masses and energies or
from the parameters of “‘Cf spontaneous fission fragment
spectra.
At present the PHD for Si-detectors is usually described
within the framework of the procedures proposed by
Schmidt et al. [5], by Kaufman et al. [6], and by Ogihara et
al. [7]. These techniques give satisfactory results for the
detectors whose parameters are close to those investigated
in these original works and in the range of heavy ion
masses and energies studied by these authors. In other
cases, as the analysis of the experimental information on
the PHD [3,4,6,8] shows, the application of the techniques
may lead to visible errors.
Under the circumstances, we defined the objectives of
this work as follows: (1) on the base of the published data
on PHD to develop a new method of PHD description for
Si-surface-barrier
detectors and p-i-n diodes in a large
range of masses and energies of recorded ions; (2) to
obtain a reliable and convenient practical guide for the
determination of energy calibration parameters for a given
detector.
0 1997 Elsevier Science B.V. All rights reserved
S.I. Malgin et al. f Nucl. instr. and Meth. in Phys. Res. A 388 (1997) 254-259
255
2. Empirical PHD description for surface-barrier
detectors
At present sufficient experimental information has been
accumulated
for the dependence
of total energy loss
R(M, E) of ions in the range of masses M = 40-200 amu
and of energies E = I O-150 MeV measured for the Sisurface-barrier
detectors
of the resistivity
p = 2002000a cm at the bias voltage U = lo-200V
[3-81. So,
we suppose that the aims of this work would be achieved
(without any additional measurements) only by obtaining
empirical expressions which make it possible to describe
all totality of the data with good accuracy.
Our selection of an analytic function type for R(M, E)
was based on the following considerations:
I. According to the experimental data: (a) the function
value monotonously increases with the growth of M or E;
(b) the condition a2Rl tIE2 c 0 should be satisfied at any M
or E.
2. The analytic function R(M, E) should be simple
enough and convenient for practical use.
We have investigated a large group of functions meeting
the requirements and have established that the best results
are obtained with the relation:
NM, E) =
l
+
;f,,<
+ aMdE
+ PM/E,
where M is expressed in amu; E and R in MeV, a. b, c, d
and f are constants which are common for all surfacebarrier detectors studied here; cy and p are description
parameters which take into consideration individual features of a given detector.
In order to determine the constants a, b, c, d, f we have
analysed all experimental
data on R(M, E) from Refs.
[3-81 within the least squares’ method (the X2-method) in
code MINUIT. These constants were common for all types
and all operation conditions of detectors, and description
parameters (Y and p were common only for the data on
R(M, E) measured with a given detector of certain resistivity at fixed bias voltage. The analysis results in the
following empirical expression:
0.55E
R(M, E) = , + ,3,9E,M
+ aME
0
I2
IO
6
6
4
2
n
Fig. 1. Pulse-height
defect R versus the mass and the energy E of
ions. (A) 0 The experimental data from Ref. [3] for
‘% thermal neutron-induced fission fragments measured at bias
voltage 41 V. (B) The experimental data from Ref. [6], (--) the
description results from Ref. 161, (-. -) the description according
to Refs. [X6]. In both parts (-)
the results of the description
proposed in this work. Numbers are the masses of heavy ions
fission fragments.
recording
Solid
curves perform our results of calculations
on
dashed curves the description from the original
works, and dashed-dotted
curves the description according
R(M, E).
+ PE.
We should point out that the minimum value of ,$ is
achieved at the power coefficients c, d, f slightly differing
from the integer numbers performed in Eq. (3), where
c = 1, d = 1, and f = 0. However, we decided to use these
integer values, since, in this case, the significant simplification of the description function leads only to unessential
increase of the x1.
The quality of the experimental data description with
Eq. (3) is demonstrated in Figs. l-3. In Figs. I and 2 the
data from Refs. [3,6,7,8] measured for silicon detectors of
resistivity p = 200-2000 R cm are shown with circles.
0
20
40
60
60
100
120
f40
E Nell)
Fig. 2. Pulse-height defect R versus the mass and the energy E of
recording ions. (A) (0) The experimental data from Ref. [7], (--)
the description results from Ref. [7]. (B) (0) The experimental
data from Ref. [Sl. In both parts (-)
the results of the
description proposed in this work. Numbers are the masses of
ions.
256
S.I. Mulgin et al. I Nucl. lnstr. and Meth. in Phys. Res. A 388 (1997) 254-259
of a heavy ion and the channel number P in which the ion
is registered. It is convenient to perform such relation as:
P(M, E) = B[E - R(M, E)] + C ,
or taking into consideration
Eq. (3):
0.55E
1 + 13.9E,M+aME+/?E
0
0
0
20
40
60
80
100
E (MeV)
Fig. 3. Pulse-height defect R(M, E) versus the bias voltage in the
detector with p = 342 f2 cm: circles, triangles, and rectangles experimental
data from Ref. [3]; curves-the
description
of
present work. Numbers are the masses of fragments.
to Schmidt et al. 15.61. These data reflect the variation of
the individual properties of all totality of the detectors
studied in the present work. In these figures we see that the
description with Eq. (3) performed by solid curves reproduces experimental dependencies R(M, E) with good
accuracy in the range of masses between 40-200 amu and
of energies up to 150 MeV. Besides, one can note that the
agreement between the experimental data and this description is not worse and, in some cases. is even better than in
the above-mentioned
original works.
Fig. 3 illustrates the description quality for data on
R(A4.E) measured for the detector of p = 342 0 cm at
three bias voltage values. It is easy to see that the proposed
description
R(M, E) (solid, dashed and dashed-dotted
curves) enables to reproduce also the dependence of PHDvalue on the bias voltage in the detector.
In Figs. l-3 only a part of the experimental information
analysed in this work is presented. The description quality
of the remaining totality of the data on R(M, E) is
analogous to the one shown here. The standard deviation
of the calculation results through all the data totality
(above two hundred experimental points) AR = 0.17 MeV,
that is less than typical errors of a single experimental
point measurement AR = 0.2-0.4 MeV.
3. Energy calibration
method
An energy calibration of a heavy ion spectrometer
requires to reveal a relation between the mass and energy
I>
+C
In this expression the scale coefficient B and the zero shift
value C are easily defined within a standard procedure if a
precision pulse generator and several alpha-sources were
used. The PHD for alpha-particles makes zero according to
its definition. So, our task is only to determine the
parameters (Y and p.
The problem could be solved by measuring the R(M, E)
at two or more different combinations of ion masses and
energies. In the first case, parameters LYand p are found
from the solution of a system of two linear equations for
P, (M, , E, ) and P?(M,. E,). If the number of measurements
II > 2. the system is solved n(n - I)/2 times, then the
values LYand p obtained at all independent combinations
are averaged.
Another method to make the energy calibration is based
on the analysis of pulse height spectra of “‘Cf spontaneous fission fragments. This method is similar to that
outlined in Refs. [5,6]. But, in our case, the mean channel
numbers of the light- and heavy-fragment groups (P), and
(P), have been involved in the calibration routine instead
of the channel positions of the corresponding peaks. The
advantages of this version are:
(a) The values (P), and (P), could be determined with
a higher accuracy than peak positions.
(b) The quantities (P),, and (P), are calculated through
all masses and energies of fragments released in *?f
spontaneous fission. Consequently, the R(M, E) description
parameters a and p extracted from these values are
averaged over a large range of masses and energies. This
circumstance increases the parametrization reliability at its
application to masses and energies of fragments which are
far from the most probable M and E of light and heavy
fission fragments of “‘Cf.
(c) Besides, the quantities (P), and (P), depend only
on the parameters cr and p of a given detector and are not
sensitive to measured spectrum shape variations.
The method used in this work for determining the
experimental values (P), and (P), is illustrated in Fig. 4,
where one of measured *“Cf spontaneous fission fragment
spectra is shown. One can easily note that the distributions
of the light- and heavy-fragment groups visibly overlap in
the central part of the spectrum, thus. it is very difficult to
decompose the spectrum onto groups correctly. So, we
have to establish the following classification of counts: all
counts to the left from the mean channel position (P), of
the total distribution belong to the heavy group of frag-
S.I. Mulgin
et al.
/ Nucl. Instr.
and Meth.
ments, and all events to the right from (P), to the light
group’. Besides, in order to exclude the influence of
background events on the spectrum tails we assume that:
(a) from the lower side of the spectrum is limited by the
channel P, defined by the condition N(P,) = O.OSN,,
where NH is the number of counts per channel at a
heavy-fragment
group peak; (b) from the upper side the
spectrum is limited by the channel P1 defined from the
relation N(Pz) = O.O_5N,,,where N, is the number of counts
per a channel at a peak of a light-fragment group. So,
in Phys. Res. A -388 (1997)
(P),,= 2
(p), =
B{6% - [ (,+p;s95;E,M),,
++fE),
+
03, =
PW,
+ C.
B{(E),,
- [(j+;&j,
(F(M, E)) = c
c
FM
M Ek
(P)q
P=P,
P=P,
In order to reveal the quantitative
relations between the
(P),_,
(P),and (Y, p a special program has been designed
for computing the “‘Cf spontaneous
fission fragment
spectra at fixed values B, C. LYand p. In the program
relative numbers of counts per channel Y(P)were calculated with the equation:
Y(P)= i
M
(F(M, E)),in Eq. (7) was
( 1 - MIAE,lY,(E,)
3
(r-1,
c N1PV’I c NV’)
V’),, =
+a(ME).
(1)
l tC.
P(E),I>
The mean of any quantity
computed with the equation:
N(P)P/ ij N(P),
157
relations which were easily obtained within the theorem of
the mean and Eq. (4):
+
(P),= 2 N(P,P/ il N(P),
P-P,
P=P2
2.54-259
Y,[E,(M,P)]a~~(M,P)/~P,
(6)
-_(I
where A = 252 is the mass number of nucleus “‘Cf;
E,(M, P) is the total kinetic energy E, of two coincident
fragments at which a fragment with the mass M falls into
the channel P; Y,(E,)is the relative yield of fragments
with mass M and energy E,.
The values E,(M,P) at fixed B, C, a and p are found by
solving Eq. (4) where the energy E of the fragment with
the mass M at fixed E, is defined according to the
conservation of energy and of linear momentum as E =
(1- M/A)E,.The data on Y,,,(E,)
for the spontaneous
fission of “‘Cf were borrowed from our work [9], which
demonstrated that our results and those obtained earlier by
Schmidt et al. [5,10] are in agreement. The agreement, in
particular. means that the energy standard for “*Cf
generally accepted in fission investigations
is conserved
at
the transition
to the detector
energy calibration method
used in the present work.
On the second step, the values (P),,P, and P, were
found from the analysis of the distribution Y(P),
then (P),
and (P),were determined with Eq. (6). The same quantities (P), and (P), were calculated on the bases of
’ Of course. this classification
is not strict, but the definition
freedom. as it will be shown below, does not influence on the
extraction accuracy of a and p.
where the summation through M and E, was organised so
that: if the value P, found with Eq. (4) for fixed M and E
falls into the interval [P,,
(P),],
the addendum in this
equation corresponds to the heavy-fragment group (i = H):
if P, belongs to the interval [(P),,
P,].then i = L. This
summation
routine provides
for the absolute correspondence of the experimental
values (P), and (P),.
which were determined on the bases of the above mentioned count classification, to those calculated with Eq. (7).
The test calculations made with the program showed
that the variations of parameters (Y and p cause visible
alterations of the Y(P)distribution shape. However, at any
reasonable parameters (Y and /3 the values (F(M, E)),are
constant and make:
(E),= 103.76;
(ME),~
= 11436;
0.55E
I + 13.9EIM L = 4'07'
(E),
=
78.68: (ME), = 11198;
(8)
0.55E
I + 13.9EIM
>fi= 5'oo
So, the experimental values (P),,
(P),and Eqs. (7) and
(8) allow to determine the parameters LYand p:
@=2.997X
x lo-J
IO-“7 (p), - c _ 3,953
(p)H - c - 7.593 x 1o-J
B
(9)
p=4.355x W'v-4.266
x lo-* (p)H -c
B
+ 1.0445.
An example of the calibration method application to the
description
of experimental
*?f
spontaneous
fission
fragment spectra is demonstrated
in Fig. 4, where the
calculated dependence N(P)is shown with the solid curve.
S.I. Mulgin et al. I Nucl. Instr. und Meth. it1 Phw.
258
0
0
40
80
IL0
Channel
Fig. 4. Pulse height spectrum for “5’Cf spontaneous
fission
fragments. (0) The experimental data. (---)
the results of the
description with Eq. (4) and (6). In the insertion: pulse-height
defect dependencies corresponding to this spectrum; (-) the results
of the description with Eq. (3). (--)
the description defined
according to Refs. [5,6]. Numbers are the masses of fission
fragments.
Res.
(1997)
254-259
Though having been developed with a completely different
application in mind, the performance of p-i-n diodes for
detecting fission fragments is successful. in this connection
we tried to spread our approach onto the PHD description
for p-i-n diodes. Unfortunately, we are short of detailed
experimental information on the behaviour of the PHD in
detectors of this type. Actually. the only systematic
investigation of the detectors has been reported in Ref.
[I I], where the R(M, E) dependencies in four p-i-n diodes
were measured with a LOHENGRIN spectrometer. The
results of the experiment are demonstrated with circles in
Fig. 5. The attempt to describe the data with Eq. (3) using
the parameters obtained for surface-barrier detectors has
shown that the PHD dependencies
for these types of
detectors are different. This result could have been quite
expected, since the structure and the production technology
of p-i-n diodes significantly differ from those of surfacebarrier detectors.
In order to develop the description of PHD in p-i-n
diodes we have analysed the data from Ref. [I I] with
using Eq. (2) in order to determine the description function
and its parameters. The analysis has led to the following
final relations:
R(M.E)=
The calculation was made with Eq. (6) at the parameters a
and p found with Eq. (8). In our opinion. the reasonable
agreement of the calculation results with the experiment is
a good argument in favour of this calibration method. In
the insertion in Fig. 4 the solid curves show the energy
dependencies R(M, E) calculated for masses between 60180 amu at the same LYand p. The dashed lines, here,
perform dependencies
of the PHD obtained within the
calibration procedure from Refs. [S] and [6]. One can see
that these descriptions coincide at masses and energies of
fragments which are close to the mean masses and energies
of light- and heavy-fragment
groups in the spontaneous
fission of ‘?f.
But. if the fragment masses are far from
these mean values. the R(M, E) dependencies calculated
according to Refs. [5.6] considerably differ from those
performed by solid curves which are more close to the
experimental points. Also, the fact could be easily noted in
Fig. IB, where the dashed-dotted
curves show the results
of Schmidt et al. calibration with the parameters from Ref.
A 388
0.4876
I + 698SEIM’
+aME+/3E,
0.487 E
I + 698SEIM’
I4 ,
_M
I
2
a
[61.
4. Energy calibration of p-i-n diodes
In recent years a new class of solid state detectors. the
so-called p-i-n diodes, which usually serve as photo
diodes, switches, attenuators, limiters. and modulators of
ultra high frequency oscillations, is widely used in spectrometry of heavy ions and of fission fragments. The p-i-n
diodes are made from intrinsic silicon being doped on the
front and rear side of the wafer by ion implantation.
E (MeV)
Fig. S. Pulse-height defect R in p-i-n diodes versus the mass and
the energy E of fission fragments. Opened and closed circles - the
experimental data from Ref. [I I], curves - the results of the
description with Eq. (10). Numbers are the masses of ions.
S.I. Mulgin et al. I Nucl. Instr. and Meth. in Phys. Res. A 388 (1997) 254-259
a = 2.974
x
lo-4
x lo-4 70%-
PL - c
____
B
/?=4.312X
c-
3,919
1.892 x
IO-‘,
(12)
10m’v-4.236
x]O-‘(P)~-c+].1382.
B
The quality of the PHD description in the p-i-n diodes is
in Fig. 5, where the experimental
results from
Ref. [ 1 I ] and the calculations with Eq. (IO) are shown by
circles and by curves, respectively. One can see that the
proposed
description
reasonably
agrees with the experimental dependencies. However, it should be noted that
this PHD description should be applied to p-i-n diodes
with care, since it was tested only over comparatively
small totality of data.
So. one can see that the universal Eq. (2) proposed in
the present work for describing PHD in silicon semiconductor detectors used in heavy ion and fission fragment
spectrometry could be easily modified to energy calibration
of surface-barrier detectors and p-i-n diodes. We can hope
that this equation and genera1 points of this approach are
also applicable to the calibration of another types of
Si-detectors, but, unfortunately, now we have no reliable
experimental
data in order to confirm or reject this
supposition.
illustrated
resistivity p = 200-2000 fl cm at the bias voltage U = IO200 V in the range of masses M = 40-200 amu and of
energies E = IO- 150 MeV
The convenient practical guide using mean channel
values of light- and heavy-fragment groups in spontaneous
fission of *“Cf has been developed for determining the
energy calibration parameters of these spectrometers.
The calibration method has been modified to describe
the mass
and energy
dependencies
of PHD
in p-i-n
diodes.
References
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[21
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]41
]51
]61
r71
@I
]91
5. Summary
A two-parametric
empirical expression has been proposed for the description of PHD in silicon semiconductors
in the dependence
on the mass and the energy of recorded
heavy ion. The description is applicable to heavy ion
spectrometry by means of surface-barrier detectors of the
159
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