A radially dependent photopeak efficiency model for Si(Li) detectors

Nuclear Instruments and Methods 178 (1980) 4 8 1 - 4 9 0 © North-Holland Publishing Company

A RADIALLY DEPENDENT PHOTOPEAK EFFICIENCY MODEL FOR Si(Li) DETECTORS David Damien COHEN The Australian Institute o f Nuclear Science and Engineering, Private Bag P.O. Sutherland 2232, Australia Received 3 July 1980

A simple five parameter model for the efficiency o f a Si(Li) detector has been developed. It was found necessary to include a radially dependent efficiency even for small detectors. The model is an extension o f the pioneering work of Hansen et al. [ 1 ] but correction factors include more up to date data and explicit equations for the mass attenuation coefficients over a wide range of photons energies. Four o f the five parameters needed are generally supplied by most commercial manufacturers o f Si(Li) detectors. S4Mn and 241Am sources have been used to calibrate a Si(Li) to ~±3% over the energy range 3 - 6 0 keV.

1. Introduction

of the sensitive volume and several correction factors. Hansen et al. [I ] express the efficiency in the form,

Recently, the necessity of absolute measurements of X-rays by Si(Li) detectors has increased in the fields of particle induced X-ray emission analysis, electron microprobe analysis and inner shell ionization in ion-atom collisions. These applications have created a demand for systematic efficiency calibration techniques in the low energy X-ray region (<100 keV). A comprehensive study of the properties of low energy photon detectors has been carried out by Hansen et al. [1]. This has been used as a starting point, expanded in a more explicit form, and more recent data and a radially dependent efficiency term have been added. The physical basis for a five parameter model to cover the energy range 1 keV to 100 keV is discussed in section 2. The variable parameters are, the detector's sensitive radius and depth, the thickness of the gold contact and silicon deadlayer, and a single parameter to describe the radial efficiency. All other variables are assumed to be known or measured. The validity of the model has been checked over the energy range 3 keV to 60 keV and measurements of the efficiency of a Si(Li) detector using two calibrated radioactive sources, S4Mn and 241Am are presented in section 3. The test detector had a measured full width half-maximum resolution of 142 eV at 5.9 keV and a 8/~m thick beryllium window.

e = e I " f g " fBe" f A u " f d " f R ,

(1)

where fg(E) is a photon energy dependent geometric factor, fBe, fAu, fd are transmission factors through the beryllium window, gold and frontal dead layers respectively and fR is the correction for radial dependent efficiency. We shall discuss each term in expression (1) for a Si(Li) detector and expand somewhat on the conclusions of Hansen. 2.1. Geometric f a c t o r

Photons of different energies will penetrate to different depths in the detector. For a detector of radius r, sensitive thickness D and front face distance d from a point source, the effective solid angle ~2o can lie between ~21 and 122 depending on the energy of the incoming photon, where 121 = nr2/d 2 ,

122 = rrr2/(d + D) 2 .

(2)

Hansen gives a mean interaction depth Z ( E ) for photons of energy E as, Z(E) =

1 - exp(-p.D)(1 +/~9) , /a[1 - exp(-/~D)]

(3)

where /a is the mass attenuation coefficient for silicon. Ref. 2 gives, /a(E) = 4.83 X 104E-2"79 cm -1 ,

2. Theory for the model

E > 1.838 keV,

(4)

The detection efficiency e of a semiconductor detector is the product of the intrinsic efficiency ei

for E in keV. 481

D.D. Cohen / Photopeak efficiency model

482

absorption in the source is given by,

Hence

gZo = nrZ/(d + Z) 2 ,

(5)

and the geometric correction factor for a point source ;g(E) is S~g(/:3 = fZl/~2o = (1 + Z / d ) z .

(6)

Table 1 shows that fg can become a very significant correction for high energy sources close to the detector front face. A further geometric correction is necessary when considering non-point sources for which the distance d to the front face of the detector is comparable with the detector diameter. A simple double integration over the source area gives the correction to the solid angle g2 relative to gZl as fg(E) = ~

=

1]

1

(1 + x ) v2

1

,

+Z]

where x = r2s/(d + Z) 2 and Z ( E ) is given by eq. (3).

2.2. Beryllium w i n d o w correction The correction due to photon absorption between the source and detector front face including self

Table 1 Geometric correction factors. D = 5 mm, point source. F

.

z

jg

jg

(keV)

(cm -1 )

(mm)

d = 10 mm

d = 64 mm

5 10 20 40 60

542 78 11.3 1.64 0.528

0.018 0.13 0.87 2.2 2.4

i .004 1.026 1.182 1.488 1.538

1.001 1.004 1.027 1.070 1.076

(10)

fBe = exp[--/'IBeXBe] • Ref. 2 gives, /.IBe(E) = 1098E -2.92 cln -1

for E > 0.2 keY ; (11)

ifxue is measured in gm and E in keV then

(7)

2 '

(9)

where #i is the total attenuation coefficient of the ith element and xi is the thickness of the ith absorber place between the source and the detector front face. Generally for both source and detector in vacuum the main absorber is the beryllium window and eq. (9) becomes

fBe = exp[--0.1098XBe E-2"92] , '

where x = (rs/d) 2 and r s is the source radius. Expansion of eq. (7) in powers of x shows that the solid angle correction is of order ( - 3 x / 4 ) and for r s ~ d is usually negligible. In most modern Si(Li) detectors the front face lies ~ 1 0 mm behind the beryllium window, making it impossible to place a source closer than d = 10 mm. For an 8 mm diameter source hard up against the b e r y l l i u m window this correction would be ~11%. The corrections of eqs. (6) and (7) tend to balance out and the final geometric correction is __

fa=exp[~tJixi]

for E > 0.2 keV . (12)

For most modern detectors XBe G 10 /.lm and so for 3 keV X-rays the absorption in the window is ~<4%.

2.3. GoM layer correction For Si(Li) detector efficiency measurements below 5 keV corrections must be made [eq. (9)] for the thickness of the evaporated gold layer on the front face of the detector. The M I - M 5 gold absorption edges produce sharp discontinuities in the X-ray transmission through this layer. Transmission can differ by 1 0 - 1 2 % above and below the M4 and M5 edge although for thick beryllium windows this effect can be washed out somewhat. Gold contact layer thickness is usually measured by fluorescing the layer with photons of energy greater than 11.92 keV, the gold L3 absorption edge. The yield from the gold L X-rays give an estimate of the layer thickness when the appropriate fluorescent yields and Koster Cronig transition probabilities of Krause [3] are used. Ref. 2 gives //Au(E) = 8.085 X 10SE -2"ss cm -1 , for 3.4 < E < 11.9 k e V ,

(13a)

= 2.8 X 104 cm -1 ,

for 2.2 < E < 3.4 k e V ,

(13b)

= 1.120 X l0 s E -2A6 cm -1 , for 0.76 < E < 2.2 keV ;

(13c)

D.D. Cohen/Photopeak efficiency model ifxAu is measured in/am and E in keV then fAu = exp[--80.85XAu E-2"ss] , for 3.4 < E C 11.9 k e V ,

(14a)

= exp [--2.8XAu ] , for 2.2 < E C 3.4 k e V ,

(14b)

= exp[-11.20XAu E-2"16 ] , for 0.76 C E <

2.2 k e V .

(14c)

The nature of Si(Li) detectors requires that XAu be as small as possible and is generally ~<0.03/am. Shima et al. [4] have measured Si(Li) detector gold layers for several detectors and found 0.014 < XAu < 0.026/am. The manufacturer's specification for our detector was XAu = 0.02 /am. The L3 upper energy edge limit of i 1.9 keV in eq. (14a) is of no consequence because of the extremely thin nature of the gold layer and f a u 1000 for all photon energies greater than this. For photon energies between the gold M1 and M5 edges (2.2 < E < 3.4 keV) an average value of the mass attenuation coefficient has been used. A more complete study of the effects of the gold absorption edges in the spectrum of energy dispersive Si(Li) detectors is given by Bromback [5]. The thickness of an evaporated 0.02 /am gold layer would be extremely non uniform.and Bromback suggests large.open area in the film may make up as much as 25% of the irradiated detector area. For this reason any measurement is generally only an average value over the detector sensitive area. 2.4. Dead layer correction Incomplete charge collection is attributed to a dead layer at the entrance face of the detector. The reason for this dead layer is that some of the electrons, which are formed at the photon interaction point, diffuse into the surface (where they are lost) before their motion in the collecting electric fields removes them from the region of the surface. Table 1 shows that for low energy photons (<5 keV) the interaction point is extremely close to the front face of the detector. Goulding [6] assumes that electrons are lost if their diffusion distance exceeds their drift distance and produces the following expression for the dead layer thickness, R Tm XSi -

,

(15)

eu s

where R is the gas constant, T is the temperature (K),

483

e the electronic charge, m the electron mobility and Vs the terminal velocity of electrons in the applied electric field. For silicon at 7 7 K , m ~ 4 X 104 cm 2 s -~ and Vs ~ 107 cm s -~ giving Xsi ~ 0.28 /am. This rather thick dead layer is produced by basic physical processes and represents a fundamental limit rather than being a consequence of manufacturing processes. The manufacturer's specification for our detector was xsi = 0.1/am. Eq. (9) is again used to determine fd. Ref. 2 gives, /asi(E) =

4,83 ×

1 0 4 E -2"79

cm -1 ,

for E > 1.838 k e V ,

(16a)

= 3631E-3.o3 cm-1 , for 0.2 < E <

1.838 keV ;

(16b)

ifxsi is measured in/am and E is in keV, fd = exp[--4.830xsi E-2"79] , for E > 1.838 k e V ,

(17a)

= exp[-0.3631Xsi E-3.o3 ] , for 0.2 < E <

1.838 k e V .

(17b)

For a silicon dead layer of Xsi = 0.3 /am, fd = 0.77 just above the silicon K edge and fd = 0.98 just below the silicon edge and at 1 keV fd = 0.90 which is still not as large a correction as for 0.02 tam gold layer at 1 keV where fAu = 0.80. Again the silicon dead layer is thus of little consequence for photon energies above 5 keV. The silicon dead layer may not be of uniform thickness, Richard et al. [7] found their detector had a central active area with a very thin dead layer, surrounded by an area with dead layer whose thickness increased rapidly with radial distance from the centre of the front face. Campbell and McNelles [8] observed a similar effect in both Si(Li) and Ge(Li) detectors. The effect of a non-uniform dead layer of this nature is to produce a localised maximum in the efficiency versus energy curve, for an uncoliimated detector, around photon energies of 20 keV. This will be discussed further in section 3. 2.5. Absorption in the sensitive volume When escape through the sides and front face of the detector are ignored the fractional absorption in the sensitive volume is, e I = 1 - exp[--/~t D] ,

(18)

D.D. Cohen/Photopeak efficiency model

484

where D is the depth of the sensitive volume and/.t t is the total mass absorption coefficient. In most experiments the counts in the photopeak only are considered and losses due to Compton scattering are ignored. This implies/at(E) is replaced by the photopeak absorption coefficient ~tphoto(E ). Ref. 9 gives for silicon //photo(E) = 1.502 × 10SE -3"22 cm -1 , for E > 20 k e V ,

Table 2 Ratios of escape peak to photopeaks. Element

Energy (keV)

Ratio (%)

Ca V Mn Fe Ni Zn

3.69 4.95 5.90 6.40 7.48 8.64

0.77 0.57 0.35 0.24 0.20 0.14

-+0.04 -+0.016 +-0.01 -+0.01 -+0.01 +-0.006

(19)

hence for a Si(Li) detector with D in /am and E in keV, ei = 1 - e x p [ - 1 5 . 0 2 D E -3"22] for E > 20 k e V .

(20)

For most Si(Li) detectors D ~ 5 mm and ei is not significantly different from unity for E N 20 keV, however at 100 keV, el = 0.027 and the major loss of detector efficiency for high photon energies is due to this effect. There is a finite probaIiility that an X-ray of energy E in the sensitive volume will interact with a silicon atom and produce a silicon K X-ray which will then completely escape from the sensitive volume. The enegy deposited in the detector is then (E - 1.74) keV. These events are lost from the full energy peak and reappear in the spectrum as a separate peak of energy ( E - 1 . 7 4 ) keV. This escape probability increases with decreasing incident X-ray energy down to the K absorption edge of silicon. Table 2 shows the ratio of the escape peak to photopeak for our Si(Li) detector. The ratio is <1% for most elements of interest. These events were not added into the photopeak for efficiency calculation.

The full energy peak consists of a Gaussian peak and a fairly flat low energy tail extending to zero. The tail arises from incomplete charge collection from certain regions of the detector. For a given detector the tail/peak ratio is a function of both detector bias voltage and amplifier time constant, decreasing as both parameters increase until a minim u m saturation values is reached. This minimum is different for each detector and is also a function of photon energy. The tail to peak event ratio usually lies between 1 and 15% for most modern detectors. In order to obtain the true counts in the photopeak all events in this low energy tail must also be included. Table 3 shows the effects of each of these correction factors for a typical Si(Li) detector as a function of the ingoing X-ray energy.

2.6. Radial efficiency Kushelevski and Alfassi [10] measured the off centre gamma ray detection efficiencies of cylindrical single open ended Ge(Li) detectors and found that

Table 3 Correction factors for a Si(Li) detector. r s = 4 mm, d = 50 mm, D = 4.26 mm, r = 2 mm, XBe = 8 ~tm,xsi = 0.1 t~m,XAu = 0.02 tzm,fR = 1.00. Energy (keV)

fg

fBe

fAu

fd

eI

•

1 2 3 4 5 10 20 30 40 60

0.995 0.995 0.995 0.996 0.996 1.000 1.029 1.060 1.072 1.079

0.415 0,890 0,965 0,985 0,992 0.999 1,000 1.000 1.000 1.000

0.799 0.951 0.946 0.954 0.974 0.995 0.999 1.000 1.000 1.000

0.964 0.933 0.978 0.990 0.995 0.999 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 0.984 0.674 0.359 0.113

0.318 0.786 0.888 0.927 0.958 0.993 1.012 0.714 0.385 0.122

D.D. Cohen / Photopeak efiiciency model the dependence of the efficiency on the radial distance can be approximated by a bell shape function of tile form

e(r)

eo exp[ -o~r2] ,

the detector plane through this pinhole as,

c°s'L

(21) -

where eo is the efficiency at the centre of the detector and ~ is a constant characteristic of the detector and the gamma ray energy. Alfassi and Northman [11] also found that a similar relationship held for Si(Li) detectors where the sensitive area is usually nmch smaller, c~ was found to be an approximately linear function of the inverse of the X-ray energy. If fR is the ratio of the integrated efficiency of a detector of radius r with radial dependence, to the efficiency of the same detector without radial dependence (c~ = 0), then ]

r

./"R = r " f_ exp[ o~t2 ] dt , o

(22)

and c~ the radial dependence term in eq. ( 2 l ) can be found by iteratively solving the following

x-l'Fe,

1 =: P(x)

485

(23)

where x = rx/(2c0 and P(x) = [1/x/(2rr)] ;f, exp[-c~/2] dt is the standard probability function tabulated in nrany mathematical tables [12]. For a 4 mm diameter detector with f ~ = 0.72 the solution is a = 0.285 which when substituted into eq. (21) gives an edge efficiency (r = 2 mm) of 32% of the central value. The radial efficiency is best checked experimentally by moving a finely collimated point source over the surface of the detector and monitoring the variation of the counts into the detector. In practice this is very difficult because most Si(Li) detectors are only a few millimeters in diameter and generally lie 10 mm behind a beryllium window. Point sources and highly collimated beams also have extremely low count rates. To overcome these problems we have devised a technique using a strong extended source of radius r s and a long narrow fixed slit width aperture of variable slit length x ' placed directly against the beryllium window. Tile counts into a source peak are monitored as the slit length scans across the detector face from +_x' mm, the origin being the line joining the source centre and tire detector centre. For a narrow enough slit each element of tile slit can be considered as a pinhole. Simple algebraic manipulation gives the area of intersection of the image of the source, radius ri, at

g{[(ri + x ) 2

r2][r 2

(ri--X)2])

1/2

(24)

where x = x'd/(d - a), r i = ars/(d a), d is the s o u r c e - d e t e c t o r distance and a is the d e t e c t o r - s l i t distance and x is the distance of the centre of the image of the source in tile detector plane from the detector centre, hA is only defined for Ir ril < x < Ir+rii, s o f o r O < x < l r r i l w e d e f i n e 6 A =rrri2and for x > Ir + ri[ hA = 0. Integration of eq. (24) and normalisation to Irr~ enables us to predict the number of counts being detected as a function of the slit length exposed, normalised to the slit fully open count. Comparison of eq. (24) with experiment will enable us to not only determine the detector sensitive radius (which may be quite different to the geometric radius) but also plots of the count versus slit length will give us estimates of the radial efficiency dependence of the detector. The advantage o f this system is that strong extended sources can be used which are placed externally to the detector housing. A comparison with experiment is discussed in section 3. The integrated hA is not a very sensitive function of r for ri > r and so for the best results the source diameter and the d e t e c t o r - s l i t distance should be kept as small as possible. This can also be achieved by increasing d, the d e t e c t o r - s o u r c e distance, but this reduces the count rate into the detector and a compromise must be struck. Radial dependence is included by putting r = ~x in eq. (21) and multiplying it by eq. ( 2 4 ) b e f o r e integration to find the integrated efficiency with slit length x ' .

3. The experiment Two calibrated X-ray sources were prepared for the absolute efficiency nmasurements. Particular care was taken to make them as thin as possible to keep self absorption losses to a mininmm. Standard solutions were evapouratcd onto 8 mm diameter stainless steel backings, producing a layer less than 5/ag cm -2 thick, and covered with VYNS foil less than 10 /ag cm -z thick. The total source thickness (~15/~g cm -2) corresponded to less than 1%. absorption for X-rays of energy greater than 2 keV. An 241Anl and S4Mn source of strengths (385.2 -+ 0.8) kBq and (390.3 -+

486

D.D. Cohen/Photopeak efl~ciency model Table 4 Si(Li) detector specifications.

0.8) kBq respectively were prepared using this technique. The sources were mounted in a specially designed Perspex holder attached to the Si(Li) detector and the distance between the front face of the source and the beryllium window was known to -+0.3 mm and could be varied between 5 and 100 ram. Great care was taken in the design of the holder to have the source and detector centres correspond to better than +0.05 mm over the full source travel distance. The detector was an Ortec 7900T-449 Si(Li)with a measured resolution of 142 eV at 5.9 keV. The manufacturer's geometric spe.cifications are given in table 4. The 241Am and S4Mn sources were chosen to completely cover the X-ray energy range 3 - 6 0 keV. A typical 24~Am spectrum over the energy range 0 - 3 4 keV is shown in fig. 1. The Np M(c~137), Np L(cq3~,) and the 26 keV, 33 keV and 59 keV photons give a complete cover of the energy range. The Np M lines are not completely resolved and the eleven Np L series lines that are resolved were fitted by Gaussian peaks with exponential tails to extract the peak areas. The tailing due to inefficient charge collection

1 D 1

.

.

.

.

.

.

•

,

•

,

•

,

•

,

•

Detector active diameter Detector sensitive depth Beryllium window thickness Gold layer thickness Silicon dead layer thickness Window to detector distance

4.0 mrn 4.26 mm 8 pm 0.02 #m 0.1 pm 7.0 mm

discussed in section 2.4 is clearly seen for the Np I_~ (~7%) and the 26 keV gamma ray (~15%) peaks. The small peak at 9.7 keV corresponds to the Au Ixt peak and is produced by fluorescence of the gold contact layer by all photons from the source above 11.9 keV, the gold L3 absorption edge. An upper estimate [1] of 0.05 /Jm was obtained for the average thickness of the gold contact layer. The background under the Au L~ peak suggests that the thinnest detectable layer for these run conditions was 0.015 /Jm in good agreement with the results of Shima et al. [4] who measured layers in the range 0 . 0 1 4 - 0 . 0 2 6 #m by the fluorescence technique.

,

..a

,

,

.

,

,

,

.

,

.

,

,

,

,

,

.

,

,

~. ..a

T = 5000 sees

DIST. = 64 mm DIAM., 4 mm 10 3

5

".¢

cJ ~.

tt)

-a

r-,

I0 z

8

10=

10'

.0

2.0

4-0

6.0

8.0

I0.0

12.0

lfl.O

1610

18.0

2010

221.0

2410

26.0

28.0

30.0

32.0

3,'.,.0

ENERGY E x (keY)

Fig. 1. Typical spectrum for an 241Am source of strength (385.2 +- 0.8) kBq distance 64 mm from detector front face. Run time of 5000 s.

D.D. Cohen /Photopeak efficiency model The Cr, Fe, and Ni K series lines are due to fluorescence of the source stainless steel backing.

0.~

3.1. Semiconductor location

0.6

It is important to determine the position of the front face o f the detector with respect to the beryllium window. This was done by the conventional application of the (1/d 2) law. The 241Am source was placed at various known distances d ' from the detector's beryllium window, in a vacuum, and the inverse square root of the total counts N in the detector plotted as a function o f d ' . The intercept of the leastsquares fitted straight line to these points with the d' axis gave the d e t e c t o r - w i n d o w distance. A measured value of (7.1 -+ 0.2) mm was obtained which compares well with the manufacturers value of 7 mm. Corrections o f the type discussed in eq. (8) were < 5 % and a difference of (0.7-+ 0.4) mm was measured between the d ' axis intercepts for the Np I ~ and the 26 keV gamma ray, in reasonable agreement with the value obtained from eq. (3) o f ~1 ram.

3.2. Detector efficiency The photopeak efficiency o f our detector at a photon energy E is given by eexpt = 47rNE/(~'2oTRPE),

(25)

w h e r e NE is the number o f counts in the full energy peak in time T from a source o f disintegration rate R

..........

I .........

487 I .........

I .........

I .........

i ....

0.7

0.5 w H

uJ

0.4 0.3 a.2 0.1

0.0

........

' .........

0.0

' .........

~

.........

20.0

' ......... 40.0

X-RAY

ENERGY E

' .........

' ...... 60.0

[KEV)

Fig. 2. Absolute detector efficiency vs photon energy. The solid curve is a fit to the experimental points using eq. (1) and the following parameters, r s = 4 mm, d = 64 mm, XAu = 0.02 ttm, xsi = 0.1 ,um, XBe = 8 #m, D = 4.11 mm, fR = 0.720.

and PE is the number of photons o f energy E emitted per decay, f2 o is obtained from eq. (5). Both the 2glAre and S4Mn sources were counted for 5000 s at a d e t e c t o r - s o u r c e distance d = (64.0 -+ 0.3) mm in a vacuum. The experimental efficiency results are shown in table 5 together with values of branching ratios P used in this work and the theoretical efficiencies predicted using eq. (1). The choice o f branching ratios used is discussed in section 3.3. Fig. 2 is a plot o f the absolute detector efficiency

Table 5 Comparison of experimental and theoretical detector efficiencies. Line

Energy (keY)

P (photon/decay)

eexpt

et~)eory [eq. (1)]

Np M total Np M(o433,) L1L3 Cr Kc~ Cr K/3 Cr K(c~t3) LI La Ln I43 L'r 26 33 59

3.30 3.35 4.82 5.41 5.95 5.47 11.89 13.94 15.87 17.8 20.8 26.35 33.2 59.54

0.0635

0.48 + 0.04

0.250 0.0086 0.132 0.0039 0.1925 0.0485 0.024 0.0014 0.359

0.70 0.73 0.72 0.72 0.74 0.73 0.59 0.40 0.085

0.650 0.653 0.687 0.696 0.700 0.696 0.720 0.723 0.726 0.729 0.716 0.604 0.410 0.0856

a) Parameters used same as fig. 2.

I .........

± 0.02 ± 0.03 -+0.02 ± 0.3 -+ 0.02 -+ 0.02 -+ 0.04 -+ 0.03 ± 0.009

D.D. Cohen / Photopeak efficiency model

488

versus the p h o t o n energy. The solid curve is a fit to the data using the radial d e p e n d e n t factor f k = 0.72 and the parameters o f table 2. This value o f fl~ has been obtained by forcing the curve to pass through the points for the Cr K(a/3) and the Np l_xv lines for which the branching ratios are well k n o w n and the total errors are only ~+2%. There are four main points o f interest about fig. 2. (a) The absolute efficiency is never 100%. Errors in the experimentally measured efficiency (+3%) will not account for such a low absolute efficiency. There are two obvious explanations either the effective detector radius is 1.7 m m and not 2 mm or the detector has some radially d e p e n d e n t efficiency such that the integrated efficiency over the whole sensitive volume is only ~ 7 2 % . Liebert et al. [13] reported a similar effect with an f a = 0.88 for their d e t e c t o r , no satisfactory explanation was given by t h e m for having a value o f f R < 1.00. Substitution o f f R = 0.72 and r = 2.0 mm into eq. (23) and solving for c~ gives,

e(r) = eo exp [-0.285r=1 ,

(26)

for the radial dependence o f our detector. Fig. 3 shows the radlal efficiency and the integrated radial efficiency, normalised to unity, obtained from eq. (26) for our 4 m m diameter detector. The total integrated efficiency for the full detector is 72% and for a 2 m m diameter aperture is 91%. This is a fairly severe drop off with radius for such a small detector and an independent check was p e r f o r m e d . Copper apertures ranging in diameter from 0.5 m m to 3 mm were placed in front o f the d e t e c t o r and the e x p e r i m e n t repeated with 241Anl source. The experi-

0.8 -'--'''''..

z 0.6 C... o.~

,_~

I ¸

02 i

05

J

1.0 RADIUS ~mm)

i

i

1.5

2.0

Fig. 3. Radial efficiency of Si(Li) detector for c~= 0.285. The solid curve is the integrated efficiency normalised to unity. The dashed curve is the radial efficiency. (+) are the experimental points for the apertures shown and should follow the solid curve.

1.2 ........,.........,.........,.........,.........,.........,.........,.........,.........,........., 1.0

~>x~C~v~

co 0.8 F-

r~

~ o ,

g

/

O

o

o

0.4

0.0

'/

A

/x

\\

oL

................................................ ~"..... ............' ....... -4.0

-2.0 0.0 2.0 DISTANCE ( r a m )

4.0

l:ig. 4. Normalised counts in the detector vs the distance from the centre of the detector, r = 2 m m , d =44 ram, a = 13 ram. The dashed curve is obtained by integration of eq. (23) with radial dependence included ~ = 0.285. ((:) horizontal scan, (X) vertical scan. The solid curve is the differentim of the dashed curve.

mentally obtained points are shown in fig. 3 and show a significant d r o p - o f f in efficiency with increasing aperture size. No significant variation with p h o t o n energy over the energy range 13 keV to 26 keV was observed. A f m t h e r check was made using a 500/2m wide slit o f variable length placed just in front o f tire beryllium window. Both horizontal and vertical scans o f the detector were made using tire S4Mn source. The results are shown in fig. 4. The agreement between eq. (23) with radial d e p e n d e n c e and tile experimental horizontal and vertical scans is good. Confirming a sensitive area o f radius 2 m m and the radial dependence determined by a = 0.285. The solid curve o f fig. 4 shows the n u m b e r of counts in the d e t e c t o r as a function of distance from the detector centre. The radially d e p e n d e n t ternr rounds the flat top and steepens the sides o f the curve. The fit o f the dashed curve to tire data is not very sensitive to variations in d e t e c t o r radius o f less than 10%. However, for no radial dependence (c~ = 0) and r = 1.7 mm poorer fits in the r > +1 mm region were obtained. (b) The difference in the experimental and theoretical efficiency at the Np M energy can not be explained by errors in the peak counts. The main Np M(c~/33,) peak between 3 keV and 4 keV can be fitted very easily by three Gaussians at the appropriate energies and the total Np M count is ~ 1 0 % higher than the Np M(c~/33,) count. Two explanations are possible, the branching ratio P = 0.0635 is wrong and

D.D. Cohen / Photopeak efficiency model

489

0.8 0,7 0.6

E

o.s o.L,

uJ

0.3 0.2 0.! 0.0

0.0

e0.0 40.0 X-RAYENERGYE fKEV)

60.0

Fig. 5. Same as for fig. 2 with the following parameters changed: xsi = 1.56/~m,D = 3.89 mm, fR = 0.736.

should be P = 0.047 or the silicon dead layer thickness Xsi = 0.1 /am is too low and should be more like 1.6/am. Fig. 5 shows a fit to the same data as fig. 2 with Xsi = 1.56 /am and fR = 0.736. The Np M, L~ and L3' points are better fitted, however the well established Cr K(a/3) point lies well above the curve. Most other workers report dead layers Xsi ~ 0.3/am, so 1.56 /am does seem excessive especially as the manufacterer quotes Xsi = 0.1 /am. However, similar effects have been reported b y Liebert et al. [13] in Si(Li) detectors and they give Xsi = 1.8/am. Self-absorption in the source at 3.3 keV is 41%.

We prefer to believe that Xsi =0.1 /am and the branching ratio is wrong. (c) The fit to the data in fig. 2 has been made to pass through the well established Cr K(a/~) and Np La points, this leaves the Np I.~ and L3' points slightly high, although not significantly so. Richard et al. [7] and Campbell et al. [8] have observed a bump of magnitude 5 - 1 0 % , in the efficiency curve, in the 20 keV region. It was caused by a dead layer whose thickness increases rapidly with radial distance from the centre, and effect that can only be accounted for in the present model by an energy dependent a = a(E). No such dependence has been observed in the energy range 1 4 - 2 6 k e V , a n d the 1 - 2 % " b u m p " in the present work is attributed to inaccuracies in the branching ratios. (d) The effect of the silicon dead layer thickness on the size of the silicon K edge can be clearly seen in figs. 2 and 5. For Xsi = 0.1/am the edge j u m p is ~5% while for Xsi = 1.56 /am the jump is ~40%. This would be a very sensitive technique for silicon deadlayer thickness measurement, however, reliable sources in this photon energy region are not available. More cumbersome and less accurate fluorescence techniques would have to be used.

3.3. Branching ratios Table 6 compares low energy intensities per disintegration for S4Mn and 241Am sources obtained by various workers. A variety of data is thus available,

Table 6 Decay data for S4Mn and 241Am" Line

Energy (keV)

IAEA (1971)

Hansen (1973)

Np M Np M(t~#7) LI L3 Cr Ka Cr K# Cr K(o43) L1 La L~ I4~ L'r 26 33 59

3.3 3.35 4.82 5.41 5.95 5.47 11.89 13.94 15.87 17.8 20.8 26.35 33.2 59.54

0.0635 ± 0.006

0.635 ± 0.006

0.250 ± 0.002 0.008 ± 0.001 0.135 ± 0.003

0.250 ± 0.002

a) Ref. 16.

Gehrke a) (1971)

Campbell (1975)

0.135 ± 0.003

0.0081 ± 0.0007 0.126 ± 0.009

0.2514 -+0.0008 0.0086 ± 0.0003 0.132 ± 0.003

0.184 ± 0.004 0.050 ± 0.001 0.025 ± 0.002

0.210 ± 0.004 0.050 ± 0.001 0.025 ± 0.002

0.191 ± 0.014 0.0475 ± 0.0035 0.022 ± 0.002

0.1925 ± 0.006 0.0485 ± 0.002 0.024 ± 0.001

0.359 ± 0.007

0.359 ± 0.006

0.359 ± 0.006

0.359 ± 0.006

This work (1980) 0.047 0.043 0.0017 0.220 0.029 0.251 0.0087 0.132 0.0038 0.194 0.0496 0.0236 0.0014 0.355

± 0.003 ± 0.002 ± 0.005 ± 0.002 ± 0.002 ± 0.007 ± 0.0003 ± 0.003 ± 0.002 ± 0.004 ± 0.002 ± 0.0001 ± 0.0001 ± 0.009

490

D.D. Cohen/Photopeak efficiency model

m a k i n g efficiency calibration o f Si(Li) detectors to better than -+5% a little difficult. The Np M photon intensities have all been taken from the work of Karttunen et al. [14] using Ge(Li) detectors. Significantly larger errors occur for this measurement since the Ka and I ~ escape in germanium from the 13.9 keV Np La falls within the M X-ray region. We feel our present value of 0.047 -+ 0.003 is a more realistic result. The evidence is that the 241Am I g intensity does indeed exceed the IAEA value, but the 14% excess quoted by Hansen differs markedly from the ~5% of other workers and the present work. Campbell and McNelles [15] have defined a consistent set for the Np L lines o f Z41Am. This set most closely approaches the results o f the present work when normalised to the Np La value of 0.132 and has been used to calculate our experimental efficiencies. Once a reliable efficiency curve of the type shown in fig. 2 has been obtained, the reverse process can be performed and branching ratios for our S4Mn and 24~Am sources obtained. Column seven of table 6 gives the branching ratios obtain in the present work using the theoreticaI efficiencies of table 5.

The author thanks Dr. G.C. Lowenthal for the preparation of the S4Mn and 241Am sources, Dr. D. Lang for helpful discussion leading to the derivation of eq. (24), Mr. H. Broe for preparation o f the source holders and variable slit apertures and the Australian Institute of Nuclear Science and Engineering for its financial support throughout this work.

4. Conclusion

References

We have shown that eq. (1) with all the correction factors described can adequately model the absolute efficiency for a Si(Li) detector to -+3% for X-ray energies between 3 keV and 60 keV. Two easily prepared thin sources were used to cover tiffs energy range. With more precise work on the 241Am Np M branching ratio the theories described here can be used to give good estimates of the silicon dead layer thickness x s i , the crystal thickness D, the gold layer contact thickness XAu, the radius of the sensitive volume r, and the radial dependent efficiency of a Si(Li) detector a. All the model parameters have been presented in explicit form making it easy to code up on a programmable desk top calculator. More sophisticated least-squares fitting routines can be used by larger computers to fit the model parameters to the experimental points using the manufacturers specifications as a starting point. The X-ray energy range 3 - 6 0 keV covers all elements from Argon up by using either K or L series lines. As a means of calibrating detector efficiencies for PIXE analysis or a t o m - i o n collision experiments the technique described here is fast and simple. There is however, no reason to limit the model to energies above 3 keV and in all the equations describing the correction factors energies well below this have been

[1] J.S. Hansen, J.C. McGeorge, D. Nix, W.D. Schmidt-Ott, I. Unus and R.W. Fink, Nucl. Instr. and Meth. 106 (1973) 365. [2] J.W. Mayer and E. Rimini, Ion beam handbook for material analysis (Academic Press, N.Y., 1977) p. 454. [3] M.O. Krause, J. Phys. Chem. Ref. Data 8 (1979) 307. [4] K. Shima, K. Umetani and T. Mikumo, J. Appl. Phys. 51 (1980) 846. [5] J.D. Bromback, X-ray Spectrom. 7 (1978) 81. [6] F.S. Goulding, Nucl. Instr. and Meth. 142 (1977) 213. [7] P. Richard, T.I. Bonner, F. Furuta, I.L. Morgan and J.P. Rhodes, Phys. Rev. A1 (1970) 1044. [8] J.L. Campbell and L.A. McNelles, Nucl. Instr. and Meth. 98 (1972) 433. [9] E.F. Plechaty and J.R. TerraU, Lawrence Radiation Laboratory Report UCRL - 50178 (1966). [10] A.P. Kushelevski and Z.B. Affassi, Nucl. Instr. and Meth. 131 (1975) 93. [11] Z.B. AU'assiand R. Nothman, Nucl Instr. and Meth. 143 (1977) 57. [12] M. Abramowitz and LA. Segun, ltandbook of Mathematical functions (Dover Publ., N.Y., 1968). [13] R.B. Liebe'rt~,.,iT. Zabel, D. Miljamic, H. Larson, V. Valkovic and G.C. Phillips, Phys. Rev. A8 (1973) 2336. [14] E.I. Karttunext, H.U. Freud and R.W. Fink, Phys. Rev. A4 (1971) 1695. [15] J.L. CampbeU and L.A. McNeUes, Nucl. Instr. and Meth. 125 (1975) 205. [16] R.J. Gehrke and R.A. Lokken, Nucl. Instr. and Meth. 97 (1971) 219.

considered. However tile availability of calibrated X-ray sources at such low energies makes the model difficult to test and errors in the efficiency would be larger than -+3%, the error aimed at throughout this work. The absolute efficiency of our detector with no apertures never reached 100% even in the most sensitive 1 0 - 2 0 keV energy region. Various apertures (both circular and slit) showed that the central region was 100% efficient and a radially dependent efficiency model was needed to explain this loss. Previous workers have only considered radial dependence for much larger detectors, we feel this is a strong case for measuring the radially dependent efficiency even for the smallest o f detector sensitive areas.