A model-based efficiency calibration of a Si(Li) detector in the energy region from 3 to 140 keV

NUCLEAR

INSTRUMENTS

AND

METHODS

I22

(t974)

4o5-414;

©

NORTH-HOLLAND

PUBLISHING

CO.

A M O D E L - B A S E D EFFICIENCY CALIBRATION OF A Si(Li) D E T E C T O R IN THE ENERGY REGION F R O M 3 TO 140 keV* W I L L I A M J. GALLAGHERT and S A M J. C I P O L L A

Department of Physics, Creighton University, Omaha, Nebraska 68178, U.S.A. Received 14 June 1974 and in revised form 25 September 1974 T h e full-energy peak efficiency in a particular geometry o f a collimated 3 m i n x 30 m m 2 Si(Li) detector with 0,5 rail thick beryllium and Mylar windows was determined over the energy range 3.3-136 keV by m e a s u r i n g the absolute intensities o f Xa n d low-energy ;v-rays from the decays o f intensity-calibrated sources o f 241Am, '~7Co, 1 3 7 C s a n d 65Zn. The measured efficiencies were fitted to a simple analytic function o f p h o t o n energy, with the fit s h o w i n g good agreement with the d a t a over the entire range o f energies. The physical basis o f the model as well as the m e a n i n g f u l n e s s o f its parameters is discussed. A discrepancy between the fit and the m e a s u r e d efficiency o f the

17.75 keV L , + L n X-rays from 241Am decay required a re-evaluation o f the intensity per decay for these transitions. Some peculiarities o f the low-energy efficiency response are discussed and are attributed to source serf-absorption effects. The extension o f the calibration to lower energies would be possible if more low-energy p h o t o n sources with k n o w n X-ray intensities were available. The use o f L X-rays and L / K intensity ratios calculated from published data for m e d i u m - Z elements is discussed and is considered to be feasible for this extension purpose. Some radionuclides that are likely to be useful in the low-energy region are suggested.

1. Introduction

measurements must be accompanied by direct efficiency measurements s-8) using intensity-calibrated photon sources, and are only useful for interpolation purposes or for pointing out anomalies in the efficiency data.

High-resolution Si(Li) and Ge(Li) photon detectors are being increasingly used in pure and applied science. An accurate knowledge of the detection efficiency of these detectors is often required for quantitative work. In contrast to other types of photon detectors such as scintillation detectors and proportional counters, each particular semiconductor detector is unique in its detailed efficiency response due to the impossibility of precise control on the fabrication of these devices. The manufacturer's specifications of the detector's dimensions along with tabulations of photon absorption coefficients 1) can be used to calculate the efficiency. For the sake of comparison, this has been done for the 3 m m x 30 mm 2 Si(Li) detector/) that was investigated in this work, and the result is shown in fig. I. Nominal manufacturer specifications of the gold-contact thickness, silicon dead-layer thickness, beryllium window thickness and depletion depth of the active region, along with the measured thickness of an additional Mylar window, were used in calculating this curve. Such calculated curves are unreliable since the detector specifications may be in error by 100%, or more3). It is generally agreed that the efficiency of these detectors must be determined empirically and should be redetermined periodically. Techniques for measuring the detector dimensions have been described in the literature4). Generally these * W o r k supported by the Research Corporation, U.S.A. Present address: Physics D e p a r t m e n t , M a s s a c h u s e t t s Institute o f Technology, Cambridge, Massachusetts.

4O5

Si dead layer (0.5/1) ! A u contact (40p, g/¢m z) ! Be window (0.5 mid 1.0:

== 0.1 ~_~

O.O]

T T F Y - - ~ r ~ - r

10 Energy (key)

III

lO0

Fig. 1. Calculated full-energy peak efficiency as a function o f p h o t o n energy for the 3 m m x 30 mm'> Si(Li) detector, based on m a n u f a c t u r e r ' s n o m i n a l specifications and using the equation, tel. eft. = exp ( - ~'~Jeab~tab) [1 --exp (--.Usipsi'3 ram)l, with the "7 mass absorption coefficients taken from ref. 1.

406

WILLIAM

J. G A L L A G H E R

The procedure followed here uses a mathematical model fitted to the measured efficiency vs photon energy data, to give the detection efficiency as an analytic function of the photon energy from which the efficiency can be reliably calculated instead of being read off a graph or interpolated from the data. For a well-collimated detector like the one used in this work, this approach gives quite satisfactory results in a more expedient manner than has been done previously. The first sections of this report describe the experimental procedure that was followed to measure the efficiency of the Si(ki) detector at 16 different photon energies ranging from 3.3keV to 136keV, using intensity-calibrated sources of 65Zn, 57C0, 137Cs and 24'Am. Particular attention is paid to the low-energy region ( E < 5 . 0 keV), where the use of L and M X-rays from the medium and heavy elements would be most helpful. The L X-rays from the decay of 1 3 7 C s w e r e used here for the first time for this purpose. It has been found in the course of this work that calculated values of L X-ray intensities may be reliable enough to employ in extending the efficiency calibration to lower energies by using the L X-rays from the standard calibration sources in addition to the K X-rays.

2. Experimental methods 2.1. THE E X P E R I M E N T A L SYSTEM Fig. 2 shows a block diagram of the experimental configuration used in this work. The Si(Li) detector is contained in a standard liquid-nitrogen cooled cryostat that is directly coupled to the aluminum research chamber so that the only material directly between the photon source and the detector is a 0.5 rail thick beryllium window on the cryostat and an additional sheet of 0.5 rail thick Mylar. Between the windows is a graded C-A1-W collimator that is A" thick (W) with a 3 m m diameter opening. The distance between the source and the front end of the collimator is 1.05", which is the same separation that will be used in Radioactive Source Beryllium Window

A N D SAM J. C I P O L L A

particle-excited X-ray studies with this system. The X-ray sources were secured onto the target holder of the research chamber at an inclination of 45 ° to the Si(ki) detector. A rotary oil pump constantly evacuated the chamber during the measurements. The signal processing electronics consisted of a Kevex Model 2000 resistive feedback preamplifier with a cooled FET, a Canberra Model 1413 Amplifier, and a 1024-channel Nuclear Data Model 2200 multichannel analyzer. The analyzer was modified to provide dccoupling to the - 5 V unipolar output from the amplifier and to enable it to accept the long pulses from the amplifier. Data from the analyzer were read out by a paper-tape punch, a point plotter, and a typewriter. 2.2. CALIBRATION SOURCES The radioactive calibration sources used in the efficiency determination were 57C0, 65Zn, ~37Cs and Z4~Am. The first three sources were prepared by drop evaporation onto a 0.49 mg/cm / thick Mylar sheet 9) with a similar sheet covering the dried deposit. They were intensity calibrated using a NaI(TI) detector and IAEA intensity standards in a fixed geometry. The 241Am source was purchased from the Monsanto Research Corporation in the form of AmO2 uniformly deposited on platinum with no cover, and with a manufacturer-specified alpha activity of 0.082pCi + 2 % . Table 1 presents the calibration data for these sources. The actual efficiency measurements were carried out about 9 months after the time of the source calibrations. Intensity corrections had to be made on the 57Co, 65Zn and 137Cs source calibrations to account for photon absorption within the covers, since the highenergy 7-rays employed in the intensity calibration of these sources are essentially unaffected by the Mylar covers. The correction factors f had the form: f = exp ( - l a , f 2 ) ,

where # is the mass absorption coefficient of Mylar at the particular photon energy, and t is the Mylar thickness. The 45 ° geometry necessitated the introduction of ,."2 factor. The mass absorption coefficients of Mylar (C1o Ha 04) were calculated using I 0)

[ -1000 V Bias I Supply )

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r] ]' Detector ~L ~ J

i

~, Preamp t J :

(1)

mp tier

> Channel Analyzer j

I Collimator Mylar Window

Fig. 2. Block d i a g r a m of the experimental configuration used in m e a s u r i n g the efficiency response of the system. T h e region between the source and the detector was evacuated.

= y~ u , , ~ , ,

(2)

i

where the #~ are the energy-dependent mass absorption coefficients ~) of the constituent elements of Mylar, and the w, are the mass fractions of the constituent elements. Above 20 keV the correction factor f is essentially unity.

MODEL-BASED EFFICIENCY

407

CALIBRATION

TABLE 1 Data for activity calibration of sources.

Isotope

Half-life T~

65Zn 57Co ]37Cs 24ZAm

246 270 30 454

Emission energy used (MeV)

Emission type

1.115 0.122+0.136 0.661 5.44 + 5 . 4 8

7 V V c~

d d y y

Although source thickness effects may be important at low energies, no corrections were attempted to account for the finite thickness of any of the sources. 2.3. EXPERIMENTAL PROCEDURE The duration of a measurement on any source depended on the source strength and the intensities of the transitions of interest, with a compromise made in time so that electronic drifts in the system did not cause severe problems. Measurement times ranged from 1 d to 2k d. A control measurement on one of the sources was made at the beginning and at the end of the series of measurements in order to check the long-term stability of the system. The amplifier shaping time constant that resulted in the best energy resolution (173 eV at 6.4 keV) Was 6/~s, 5xlO~

i 7x10~ I

][

~.122

I

s~

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3 J

_= d

Activity

Ao

(/~Ci)

1.497 × 104 1.098 x 105 9.627 x 104 3.03 x 103

0.825 2.974 3.092 0.082

and this value was used throughout the measurements. Proper pole-zero cancellation in the amplifier was accomplished with the aid of a pulser connected to the preamplifier test input, so that the pulser-generated output signals perfectly matched the detector signals. Any dc-level shift was cancelled by using the " h i g h " setting of amplifier's base-line restorer. The dc level of the amplifier was adjusted to place the zero energy level at approximately channel zero in the dc-coupled multichannel analyzer. The live timer of the analyzer was used to set the duration of each measurement, and dead times throughout these measurements were always less than 1%. Different amplifier gain settings were used depending on the range of energies being measured from a particular source and on the accuracy of definition of the peaks that was desired. 3. Analysis

of data

3.1. SPECTRUM ANALYSIS

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Fig. 3. A typical spectrum m e a s u r e m e n t on the 57Co source, s h o w i n g the K , (6.40 keV) and Ke (7.06 keV) X-rays and the low-energy 14.4 keV v-ray from the d a u g h t e r .57Fe. The insert shows the high-energy 122 keV and 136 keV v-rays from 57Fe. The low-energy and high-energy spectra were recorded at different amplifier gain settings. The escape peaks, which are about 0.5% the intensity o f the m a i n peaks, do n o t s h o w up on this scale and were n o t included in the analyses described in section 3.1.

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I 370

380

400

420 Channel

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Fig. 4. A typical spectrum m e a s u r e m e n t on the 65Zn source s h o w i n g only the region o f interest containing the K s (8.04 keV) and K~ (8.91 keV) X-rays from the daughter 65Cu.

408

WILLIAM

J. G A L L A G H E R

calibration sources. The information contained in these plots served as input data for an interactive computer program, called X S P E C ~ ) , developed to accurately determine the energies and intensities of the spectral peaks. Briefly, XSPEC searches the spectrum for distinct clusters, or closely situated groups of peaks, based on a modified smoothed second difference inspection of the spectrum. Once such a cluster is located, the search is temporarily halted while a suitable polynomial function is fitted to the background in the vicinity of the cluster, after which it is subtracted from the cluster. A Gaussian function is then fitted to the most resolved peak in the cluster as a quadratic polynomial fitted to the natural logarithm of the count. The fitted peak is subtracted from the cluster, and the peak-stripping process is continued until all the peaks in the cluster have been fitted. For poorly resolved multiplets in a cluster, the stripping process is successively repeated until there are no significant changes in the parameters of each member of the multiplet. Certain parameters related to the quality of the fit indicate whether a fitted peak is actually a single photopeak or is composed of several photopeaks; if it appears to be more than one peak, then one or more

20X103;

4xlO t x-rays

A N D SAM J. C I P O L L A

additional Gaussians can be fitted to it. As an illustration of the XSPEC analysis, fig. 7 is presented to show the final result of a fit to the K~,-K~ doublet in the ~3VBa X-ray spectrum from the decay of ~37Cs. 10! r

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Fig. 6. A typical s p e c t r u m m e a s u r e m e n t o n the S4lAm source, clearly showing the M (3.30 keV), L c ( 11.88 keV), L ~ ( 13.95 keV), L,2 (13.76 keV), L, (15.87 keV), La~ (17.74 keV), Laz (16.84 keV), a n d L v (20.08 keV) X-rays from the daughter 2S7Np, as well as the 26.4 keV and 59.5 keV y-rays. A l t h o u g h the 33.1 keV ;e-ray is hardly discernible on this scale, it was analyzed by the c o m p u t e r p r o g r a m XSPEC. T h e peaks between the M X-rays and the Lt peak are L X-rays f r o m the fluorescence o f the platinum backing of the source. F- . ,

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Fig. 5. A typical s p e c t r u m m e a s u r e m e n t on the 1 3 7 C s source s h o w i n g the L X-rays (weighted-average E = 4 . 6 0 keV), the partially resolved K ~ (32.19 k e V ) - K ~ 2 (31.82 keV) doublet, a n d the Kpt (36.38 keV) a n d Kpz (37.26 keV) X-rays from the daughter lS7Ba. T h e insert in the upper left-hand corner shows the L X-ray cluster in m o r e detail, a n d taken at a higher amplifier gain setting. The three L X-ray peaks were identified as the L~ (4.45 keV), La~ (4.83 keV) and the La~ (5.16 keV) transitions in taTBa. The broad peak just below the energy o f the K , peaks is due to backscatter.

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Fig. 7. Typical s p e c t r u m analysis o f the c o m p u t e r p r o g r a m X S P E C on the K ~ - K % doublet in the s p e c t r u m from zaTCs decay. The dashed lines indicate the final G a u s s i a n fit to the two peaks, after a linear b a c k g r o u n d (also indicated by a dashed line below the peaks) was first subtracted from the doublet.

MODEL-BASED

EFFICIENCY

A simpler method of analysis to determine the total peak counts could have been used, but XSPEC is the main calculational tool that will be used in all further work with this system. It is important that the analysis procedure used during the efficiency calibration be the same as that used in later experiments which make use of the measured efficiency response of the system.

TABLE 2 Transition energies and intensities for the calibration sources. Isotope

Emission energy

N~/t = Ao l f e x p ( - 0 . 6 9 3 TITs),

(3)

where I is the relative intensity, or photons per decay, of the particular photon emission of energy E, f is calculated from eq. (1), T is the time interval between the calibration of the source and the measurement, and T½ is the half-life of the source. The detection efficiency e is then defined as ]Vm/t

N~/t e = (Nm/A o Ift) exp (0.693 TITs).

8.15 1115.0

Relative emissions per decay, /

Ref. a

K~+Ka

0.7633 4-0.0069 1.00

4, 5, 6

y-ray

4, 4, 4, 4,

12

57Co

6.46 14.36 121,97 136.33

K~+Ka y-ray ),-ray ;v-ray

0.5668 0.0979 0.8879 0.1186

137Cs

4.60 b 32.10 36.50 661.0

L~ + L a K, +K% K~I+K~2 y-ray

0.00896 4- 0.00047 0.0658 4-0.0011 0.0158 4-0.0003 1.00

e 4, 5, 6 4, 5 , 6 12

M X-rays Lg L~X-rays L~+L,~ Lr X-rays ;v-ray y-ray y-ray c~-particles

0.0635 4-0.0060 0.00808 4- 0.00073 0.1341 4-0.0028 0.2085 +0.0038 0.04971 4-0.00096 0.0237 4-0.0014 0.00104:50.00011 0.3590 4-0.0060 1.00

4 5 4, 5 4, 5 4, 5 4, 5 5 4, 5 12

241Am

3.30 l 1.88 13.90 17.75 20.08 26.35 33.20 59.54 5477 + 5435

4-0.0114 4-0.0017 4-0.0049 4-0.0030

5,7 5, 7 5, 7 5, 7

a The values o f I are weighted averages from values in the references, unless otherwise noted. b Intensity-weighted average energy according to m e t h o d o f ref. 4. See sect. 3.3. e Calculated in sect. 3.3. T h e uncertainty is calculated from the s t a n d a r d deviation o f the m e a s u r e m e n t o f ref. 16.

(4)

Table 2 gives the relative intensities I for the calibrations sources. The values of I listed in this table for which more than one reference is cited are weighted average values from the various references. The uncertainties in the references, unless otherwise stated, were treated as standard deviations and are reported here as such.

Emission type

E (keV) '~5Zn

3.2. EFFICIENCY CALCULATIONS The area under an integrated photopeak gives the number of X-rays Nm of a particular energy measured by the system during the live time t of the measurement. The registered number of X-rays per unit time, Nm/t, is compared with the total number of X-rays/s, Ne/t, of that energy which were emitted by the radioactive source. If A0 denotes the calibrated activity of the source in emission particles per unit time (see table 1), as determined at the time the source was calibrated, then the number of photons emitted per second by the source at the time of the measurement is given by

409

CALIBRATION

TABLE 3 Data for calculating IL/IK in the 137Cs decay. Parameter

Value

Reference

~L/~K nrm ~L a)K

0.178 4- 0.007 0.887 0.093 4-0.012 0.901 :t:0.026

14 15 13 13

3.3. L X - R A Y INTENSITIES FROM 1 3 7 C s DECAY In order to increase the number of low-energy data points to use in the overall efficiency determination, full advantage was made of the photon energies available from the sources on hand. Besides the K X-rays from 137Cs decay, use was made of the L X-ray spectrum from this source, which is shown in the insert of fig. 5. The computer program XSPEC found three peaks in this cluster, which were identified as the I_~, Lp, and La2 transitions in 137Ba. The total number of

counts in each peak was calculated in the manner described in section 3.1, and the total number of all L X-rays was calculated from the sum. The intensityweighted average energy of these L X-rays is 4.60 keV where the method of Hansen et al. 4) was employed. The value of I, giving the relative number of photons per decay (see table 2), was calculated in the following manner. The relative intensity of L X-rays to K X-rays

410

WILLIAM J. GALLAGHER AND SAM J. CIPOLLA

f r o m an internal conversion source such as ~37Cs is given by t3)

IL/IK = (0~L/0~K"}-~KL) ~L/O-)K ,

(5)

where c¢ is the internal conversion coefficient, n~.e is the average total number of primary L-shell vacancies produced in the decay of a K-shell vacancy, ~L is the average L-shell fluorescence yield and oK is the K-shell fluorescence yield. The values of these parameters used in this analysis are found in table 3. Using these values in eq. (5), the relative intensity is

IL/I K = 0.1 10+0.015.

(6)

This result is in excellent agreement with the measured value of Nix et al.~6), who found ILl1 K = 0.110_+0.006. F r o m table 2 the total K X-ray intensity for ~37Ba is I~=0.08153_+0.00118. So, the total L X-ray intensity is I~ = (IL/I~) IK, which is IL = 0.00896-+0.00119.

(7)

This is the relative intensity per decay for X37Ba L Xrays which was entered in table 2 and used in this work.

4. Physical model of the efficiency response 4.1.

FUNCTIONAL FORM OF THE EFFICIENCY-ENERGY RELATIONSHIP

For ease of handling the efficiency information, it is desirable to analytically reduce the efficiency ~ to a simple function of the energy E. W o o d et al. 8) are the only group known to the authors who have addressed themselves to this problem. They fit the efficiency data with a function of the form e = O exp (--//btb--//~tds) [1 -- exp (--//sptas)],

(8)

where the fitted parameters were ~2, the fractional solid angle subtended by the detector; tb, the beryllium window thickness ; td~, the detector's silicon dead layer thickness; and t~s, the detector's active silicon depth. The energy dependence was carried by Yb, It,, and ~M, the total mass absorption coefficients of beryllium and silicon and the photoelectric mass absorption coefficient of silicon, respectively. W o o d et al. expressed the mass absorption coefficients, away from absorption edges, as tt = exp [A + BIn E + C(ln E)Z],

X-rays, was negligible. The method assumes that • is energy-independent, which they verified experimentally below 15 keV and which should be true for the wellcollimated detector used in the present study. For the geometry in fig. 1, additional terms would have to be added to eq. (8) to account for the Mylar window and the gold contact on the detector. Applying W o o d ' s method would therefore require five preliminary fits to the various absorption coefficients and the determination of four thickness parameters describing the low-energy behavior from essentially four or five data points which define this part of the efficiency curve in our work. A simpler method of fitting the experimental poinis is required. Various empirical functions describing the efficiency as a function of energy were investigated. These included various polynomials of ~ and In e in terms of E, E and In E. Curve fitting by splines~S), i.e. joining fits o f the above types over portions of the energy range such that the resulting curve was continuous and had a continuous logarithraic derivative, was also tried. None of these proved adequate for describing the shape of the efficiency curve over the entire energy range without reproducing the point-to-point r a n d o m data fluctuations. This failure led to a search for a simple analytical function which closely approximates the physical behavior of the detector. Use of the Born approximation for photons with energies at least several times greater than the binding energies of atomic electrons and much less than the electron's rest energy shows ~9) that the photoelectric cross section is proportional to E -3'5. Less restrictive assumptions ~ ' 2 ° ) show that the same trend in energy dependence is continued to lower energies as long as the p h o t o n energy is at least twice the electron's binding energy. This suggests that eq. (9) can be simplified by dropping the (In E) z term and setting B = - 3 . 5 . Inspection of total cross sections') base3 on experimental data shows that, to a good approximation, Ft oc E B, with B ~ - 2.9 right up to the absorption edges. If it is assumed that the B's are the same for all the absorbers*, the energy dependence in eq. (8) can lye explicitly represe:ated as = O exp (~E p) [1 - exp (?E~)].

(10)

(9)

where A, B, and C were determined from fits to tabulations of # given in ref. 17. Their experimental configuration did not have a Mylar window and their detector's gold-contact thickness, as determined by a technique 6) of p h o t o n excitation of gold K and L

* Inspection of the total cross sections in ref. 1 shows that for the low-Z absorbers, Be, Si. and the H, C, and O in Mylar, B ~ -2.9; so eq. (10) can be used with confidence above the Si absorption edge at 1.84 keV. This is not true for gold as the L and M absorption edges occur in the region from 2.2 to 14.3 keV. The gold thickness should however be small enough to ignore. (See sect. 5.1.)

MODEL-BASED EFFICIENCY

The preliminary fits to the mass absorption coefficients need no longer be made, as eq. (10) can be fit directly. In the course of the calibration measurements it became obvious that small variations in the source position relative to the detector were having a significant effect on the count rates and hence the measured efficiencies. Campbell and McNelles 6) have investigated the significance of this effect. We have added additional terms to eq. (10) to account for this. These additional terms have the effect of modifying the geometrical factor in eq. (10) each time a source is repositioned. The equation fit to the data then takes on the form: e~ = Q exp (ccE~)(1 - exp (?E~)) x × (1

+aXIi+bXzi+cX3i)+ri,

(l 1)

at each discrete energy E~, where [1 if data point number i is from ~3VCs decay; Xll if not;

The normalization procedure embodied in eq. ( l l ) has proved to be most useful in determining the overall efficiency response of the system when there exists an unavoidable uncertainty in the source-detector separation among the various sources or between measurements on the same source. Although it would seemingly be a simple matter to ensure reproducibility in the placement of the sources in the research chamber, the asymmetry of the sources as regards the location and shape of the active deposit makes such a sourcenormalization procedure valuable. 4.2. FITTING PROCEDURE AND RESULTS Eq. (1 1) is nonlinear in its parameters (~2, ~, fl, y, 6, a, b, and c). A weighted version of Marquardt's nonlinear least squares algorithm z~'22) was used to fit this model. The algorithm finds a minimum in the weighted sum of the squares of the errors,

l0

X2i

=

10 if data point number i is from the X-ray or low-energy 7 from STCo decay; if not;

10 if data point number i is from a high-energy ? from 57Co decay; if not; a, b, and c are the normalization parameters, and r~ is the random error associated with the measurement of e~. The 2 4 1 A m s o u r c e was chosen as the source with no correction factor, since the construction of this source was such that it could be very reliably positioned. Two correction factors had to be introduced for SVCo as the source position was changed between the low- and high-energy calibration measurements. No correction term could be introduced for °SZn as this source yielded only one calibration point*. An alternative way to look at the added terms in eq. (11) is to view them as nomalization factors and to define the normalized efficiency as X3~ =

e; =

gi

411

CALIBRATION

z = 2

(13)

where ai is the standard deviation of the measured efficiency of the ith data point, by taking initial parameter estimates and successively incrementing them by an amount determined by means of an interpolation between increments found by the method of linear

,if T /

,7

/ /

J

-1

l

'4

(12)

1 + aXll + bX21 + cX3i" It is convenient to plot the efficiency data in this normalized form for comparison with the prediction equation, eq. (10), as has been done in fig. 8.

I

I

* The positioning of the 65Zn source turned out to be highly reproducible in the source holder. I f this were not the case and the introduction o f a separate geometrical factor had been necessary, this point would have yielded no useful information. This is true for all sources which yield only one calibration point and it constitutes a problem for all onepoint calibration sources.

:i

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Energy (key) Fig. 8. The experimentally determined full-energy peak efficiency of the 3 m m x 30 m m 2 Si(Li) detector as a function of p h o t o n energy. The efficiency, eq. (10), is plotted along with the data in normalized form, eq. (12). Table 4 gives the parameters and statistics of the fit.

412

WILLIAM

J. G A L L A G H E R

TABLE 4 P a r a m e t e r s o f the fit to the efficiency d a t a . Parameter

O c~ fl )' a b c reduced Z2

Value

(

3.95 :t:0.16) x 10 4 - 4 . 6 0 4-1.33 - 1 . 3 2 4-0.20 ( - 3 . 3 8 4-0.58) x 1 0 -5 - 3.62 4-0.04 -0.3364-0.018 -0.2204-0.012 0a 1.57

a T h i s p a r a m e t e r w a s n o t i n c l u d e d in the fit. See sect. 4.2.

least squares applied to a first order Taylor series expansion of the model in terms of the parameter increments and increments of parameters in a direction opposite to the gradient o f z 2. The incrementing of the parameters continues until a minimum of Z2 is located. Initial parameters were easily chosen from a rough graph of the data, and no relative minima, other than the absolute minima, were encountered. The parameter c was insignificantly non-zero, and it was dropped from eq. (11), which was then refitted. The resulting values of the model parameters and X 2 a r e tabulated in table 4. The fitted curve, plotted according to eq. (10), is given in fig. 8. 5. Discussion 5.]. INTERPRETATIONOF THE FITTED CURVE With the possible exception of the 17.75 keV point (considered further in sect. 5.2) the fitted curve and the measured efficiency data in fig. 8 agree quite well. The fitted value of the high-energy-dependence parameter 6 = -3.60_+0.04 is in reasonable agreement with the theory discussed in sect. 4.1 ( 6 - = 3 . 5 ) and with tabulations ~) of experimental photoelectric cross sections for silicon (6 ~ -3.1). The fitted value of the low-energy-dependence parameter /3 = - 1.32 +_0.20 is significantly lower than expectations (theory:/3 = - 3.5; experimental total cross section:/3 = - 2.9). If one is to believe in the physical model of eq. (10), then the disparity in the low-energy region must be due to some unaccounted effect on the low-energy data that is suppressing the measured efficiency. There could be several things that would affect the intensity of the low-energy photons and not appreciably affect the high-energy data. One possible explanation is that the absorption in

AND

S A M J. C I P O L L A

the gold contact layer is not negligible and that gold is in fact the dominant absorber in the low-energy region. Averaging through the L and M absorption edges in the 3.3-14.4 keV region for gold gives a total cross section roughly proportional to E -~5, which is closer to the fitted result of E - ~.32 ±o.20. The average thickness of gold required to make this explanation plausible ( ~ 350 pg/cm 2) is almost an order of magnitude greater than the manufacturer's specifications. The fact that others 4.8) have measured gold thicknesses in accordance with specifications makes this explanation unlikely. This conclusion was verified by using the ?-rays from a strong 57Co source to excite the L and K X-rays from the gold layer ~' 8), with the tungsten collimator removed so that X-rays fluoresced from tungsten did not interfere with the intensity measurement of the gold X-rays. The gold X-rays were hardly discernible in the measurement as they were below the limit of sensitivity of the system. Thus, our results conclusively show that the gold layer is insignificant in modifying the lowenergy efficiency response. A thicker than normal silicon dead layer on the detector could also diminish the low-energy response. Wood et al. 8) investigated two similar Si(Li) detectors under the same conditions and they found that not all detectors have the nominal silicon dead-layer of about 0.5 #m, but that it can be an order of magnitude thicker. However, it is difficult, if not impossible, to reconcile the measured low-energy-dependence parameter / 3 = - 1 . 3 2 _ + 0 . 2 0 with the expected value of / 3 = - 2 . 9 in attempting to attribute the low-energy efficiency attenuation to photon absorption in the silicon dead layer. The same argument holds true if an attempt is made to attribute the low-energy response to a thicker than specified beryllium window. The most plausible explanation for the unexpected low-energy behavior is that the emitted intensity of the low-energy photons from the source is being reduced more severely than that of the higher-energy photons. Self-absorption in the drop-evaporated 57Co, 65Zn and ~a7cs sources could cause this type of behaviour. (The 24~Am source, prepared by another technique, should have a negligible thickness. This is supported by the lack of fluoresced americium X-rays in the 2 4 1 A m spectrum of fig. 6.) Peterman 23) calculated the selfabsorption of photons in uniform disk-shaped sources and found that, even at energies as high as 14.4 keV, significant self-absorption occurs in reasonably thin sources (0.01 mm). Correcting for such self-absorption in drop-evaporated sources would be difficult due to the significant non-uniformity of the deposit, but such a correction could be expected to raise the data points

MODEL-BASED

EFFICIENCY

TABLE 6

TABLE 5 l n t e n s i U e s o f t h e Lo a n d L n X - r a y s p e r d e c a y o f 241Am. Value 0.1946 0.210 0.184 0.191

4- 0 . 0 0 4 6 4-0.006 4- 0 . 0 0 4 4-0.014

Reference This study 4 IAEA 5

between 4.6 and 11.9 keV in fig. 8. The efficiency would then be more in accordance with the expected behavior of fig. 1 and with the expected energy dependence. In light of these problems with the measurement of the low-energy efficiency response, it is recommended that calibration sources be prepared by a technique other than drop evaporation to insure uniformity and thinness of the source deposit. It is also worth mentioning that Hansen et al. 4) found discrepancies in their work in the same energy region. They surmised that the values of the average energies of the X-rays used for each of the low-energy points may be in error. Because the efficiency is rapidly changing with energy in this region, such errors could drastically affect the shape of the efficiency curve. Better results might be obtained if single X-rays, where they are clearly resolved, were used for low-energy calibration purposes. 5.2.

THE

Lo

AND

L,

X-RAY

INTENSITIES IN THE

D E C A Y OF 2 4 1 A m

The accuracies of the IAEA-recommended intensities for the average 17.75 keV energy of the Lp and L,t X-rays from the decay of 241Am have been questioned by Hansen et alff), who used their efficiency curves for Ge(Li) detectors to calculate a different value for this quantity. Their recommended value was heavily weighted in the weighted average value for these transitions listed in table 2, and when it was used in this work it gave the worst point on the graph in fig. 8. Calculating another value for this intensity, based on our efficiency fit without the 17.75 keV point, gives a value more in line with the IAEA recommendation and that calculated by Gehrke and Lokken s) in a similar manner. For comparison, these values are listed in table 5. 5.3.

413

CALIBRATION

E X T E N D I N G THE L O W - E N E R G Y E F F I C I E N C Y MEASUREMENT

There is a clear need in all efficiency determinations on Si(Li) detectors to have available more information at low energies, especially < 5 keV. In this work we

Some additional low-energy calibration sources.

Source

Transitions

Average energy (keV)

Ref.

75Se

( 1 2 0 d)

As - L

1.3

8

SaSr

(64 d)

Rb - L

1,75

6, 17

1°9Cd

(453 d)

Ag - L

3,1

8

11aSn

(118 d)

In - L

3,5

8

141Ce

(32.5 d)

Pr Pr Pr Pr

M L~ Lo Lv

0.93 5.0 5.6 6.3

16

159Dy

(144d)

Tb-M Tb - Lz T h - Lo Tb - L v

1.25 6.3 7.1 8.1

16

170Tin

(130d)

Yb-M Yb - L, Yb - Le Yb-L~

1.54 7.4 8.4 9.8

12, 17

2°aHg

(47 d)

TI T1 TI TI TI

- M -L e - L, - Lt~ - Lv

2.3 8.95 10.3 12.3 14.3

6, 17

'~l°Pb

(22 y)

Bi Bi Bi Bi Bi

-

2.4 9.4 10.8 13.0 15.2

8

-

M Le Lz Le L,2

have shown that L X-rays can be taken advantage of in determining the low-energy efficiency response. This realization naturally leads to a consideration of other radionuclides that can further enhance the efficiency determination in this region. Additional sources that could be used for this purpose are given in table 6, which only emphasizes the L and M X-ray transitions of interest here. (The K X-rays and 7-rays from these sources have been often used in this kind of work by other investigators.) It would be well if individual g and M X-ray intensities from medium- and high-Z elements were measured as in the case of the 241Am decay. Recent literature 24'2s'16) shows that experimental work is in progress toward obtaining relative intensities from such elements. In the absence of experimental data, however, calculations of L X-ray intensities such as was

414

W I L L I A M J. G A L L A G H E R

done in sect. 3.3 for 137Cs decay can be expected to give reliable results, although the uncertainties involved will probably be larger than they would be if experimental data were used. 6. Summary and conclusion A technique for expediently determining the efficiency of a well-collimated Si(Li) detector over energies ranging from 3 keV to 140 keV has been described. The technique involves the accurate measuring of the efficiency at several discrete energies using calibrated photon sources and the fitting of these data to a simple function of the energy. The agreement between the fit and the data was excellent and the method could easily be employed for routine efficiency calibrations and recalibrations. Using lower energy photons from the L and M X-ray transitions from medium and heavy elements it would be possible to extend the technique down to the 1.84 keV silicon K absorption edge. For energies below 1.84 keV, a separate fit would have to be made that is independent of the fit above the edge. Disadvantages of the technique are that it does not directly yield meaningful detector parameters and that the technique cannot, in its present form, be applied to energy regions where there are significant efficiency discontinuities caused by X-ray absorption edges. Finally, it must be emphasized that the detection efficiency of Si(Li) detectors should be recalibrated periodically. We have seen a change (decrease) in the efficiency of our detector over the course of these and preliminary measurements. The authors are grateful to J. L. Duggan and C. C. Sachtleben for preparing and calibrating the 65Zn, 57C0 and '37Cs sources at the Oak Ridge Associated Universities Special Training Division in Oak Ridge, Wennessee~

References 1) W. J. Veigele, Atomic Data 5 (1973) 51. 2) Super Mark A, Model 3010, Kevex, Inc., Burlingame, Calif,

A N D SAM J. C I P O L L A 3) p. Lure, Kevex Corp., private communication. 4) j. S. Hansen, J. C. McGeorge, D. Nix, W. D. Schmitt-Ott, I. Unus and R, W. Fink, Nuc!. Instr. and Meth. 106 (1973) 365. 5) R. J. Gehrke and R. A. Lokken, Nucl. Instr. and Meth. 97 (1971) 219. 6) j. L. Campbell and L. A. McNelles, Nucl. Instr. and Meth. 98 (1972) 433. 7) j. L. Campbell, P. O'Brien and L. A. McNelles, Nucl. Instr. and Meth. 92 (1971) 269. s) R. E. Wood, P. V. Rao, O. H. Puckett and J. M. Palms, Nucl. Instr. and Meth. 94 (1971) 245. 9) G. Gleason, Special Training Div., Oak Ridge Assoc. Univ., private communication. 10) B. L. Henke and R. L. Elgin, in Advances in X-ray amdysis vol. 13 (eds. B, L. Hencke, J. B. Newkirk and G. R. Mallett; Plenum Press, New York, 1970) p. 648. it) XSPEC is similar to the programs described by M. A. Mariscotti, Nucl. Instr. and Meth. 50 (1967) 309; and J. T. Routti and S. G. Prussin, Nucl. Instr. and Meth. 72 (1969) 125. A complete description o f XSPEC will be given in a forthcoming paper. 12) C. M. Lederer, J, M. Hollander and 1. Perlman, Table of isotopes, 6th ed. (J. Wiley, New York, 1967). 13) W. Bambynek, B. Crasemann, R. W. Fink, H.-U. Freund, H. Mark, C. D. Swift, R. E. Price and P. V. Rao, Rev. Mod. Phys. 44 (1972) 716. 14) y . y. Chu and M. L. Perlman, Phys. Rev. 135 (1964) B319. 15) p. V. Rao, M. H. Chert and B. Crasemann, Phys. Rev. A5 (1972) 997. 16) D. W. Nix, J. C. McGeorge and R. W. Fink, Proc. Intern. Conf, on Inner shell ionization phenomena and fitture applications, Atlanta, Ga., vol. 1 ( C O N F - 720404, Jan. 1973) p. 214. 17) E. Storm and H. I. Israel, Nucl. Data Tables A7 (1970) 565. is) j. G. Hayes, Numerical approximation to functions and data (The Anthlone Press, London, 1970) p. 12. 19) K. Gottfried, Quantum mechanics, vol. 1 (W. A. Benjamin, Inc., New York, 1966) p. 463. 20) R. H. Pratt, A. Ron and H. K. Tseng, Rev. Mod. Phys. 45 (1973) 273. 21) D. W. Marquardt, J. Soc. lndust. Appl. Math. 11 (1963) 431. 2-)) p. R. Bevington, Data reduction and error analysis for the physical sciences (McGraw-Hill Book Co., New York, 1969) p. 235. 2z) B. F. Peterrnan, Nucl. Instr. and Meth. 101 (1972) 611. 24) A. G. Pinho, L. T. Auler and A. G. da Silva, Phys. Rev. C9 (1974) 2056. 25) E. Karttunen, H. U. Freund and R. W. Fink, Phys. Rev. A4 (1971) 1695.