A method for determination of the 152Eu activity

A method for determination of the 152Eu activity

Nuclear Instruments and Methcnis 203 (1982) North-Holland Publishing Company 273-280 273 A METHOD FOR DETERMINATION OF THE ts2Eu ACTIVITY S. BABA, ...

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Nuclear Instruments and Methcnis 203 (1982) North-Holland Publishing Company

273-280

273

A METHOD FOR DETERMINATION OF THE ts2Eu ACTIVITY S. BABA, S. ICHIKAWA . T. SEKINE, 1 . ISHIKAWA and H. BABA Japan Aronde Energy Revearclr Irrsrirare, Tnkai, Nnkrr, Iharaki. Japan

Received 29 IXcember

1981

and in revised form

19 April 1982

A newmethod for the absolute measurement of the 152 Eu activity was established based on the 4s;ß-y spectroscopic coincidence method . It consists of a 4aß counter and a Ge(Li) detector. i n which the effective counting efficiencies of the 4-fr v,unter f.,r ß-rconversion electrons . andAugerelectrons were obtained by taking the intensity ratios for certain y-rays between the single spectrum and the spectrum coincident with the pulses from the 4aß counter. First. in order to verify the method. three different methods of the absolute measurement were performed with aprepared M'o'o source to find excellent agreement amongthe results deduced by them . Next. the 4nß-yspectroscopic coincidence measurement was applied to 352Eu sources prepared b, irradiating an enriched "' Eu target in a reactor. The result was compared with that obtained by the 1-ray spectrometry using a -Eu standard source supplied by LMRI . They agreed with each other within an error of IS .

1, Introduction Europium-152 is an important nuclide as a standard source for energy calibration and efficiency correction of the solid-slate y-ray detector. Its decay scheme or photon emission rates have . therefore. been measured very carefully [1-5]. On the other hand, the absolute determination of the 112 Eu activity is attended by great difficulty because of the complex decay scheme . One cannot apply ordinary coincidence measurement techniques there . The sole example of an t 52 Eu absolute measurement is 4my counting [6,71 taking advantage of the almost 100% emission of y- or X-rays perdisintegration. In the 4aycounting method,a large Nal(TI) scintillator or a pair of such detectors is used for determining the counting efficiency at each -y-ray energy. The total efficiency for ts2Eu is then calculated with the so obtained counting efficiency and the known decayscheme . It follows that this is an indirect absolute measurement requiring other primary standard sources to determine the y-counting efficiency of the detector. Furthermorc, the accuracy of the result is directly affected by the reliability of the decay scheme, though its effect decreases as the detection efficiency increases . It is reported [61 that an accuracy of I% at the confidence level of 3o has been attained using a 4" Cp x 5"" well-type Nal(TI) detector. In the present work, a new method of the absolute measurement of ts2Eu was established by developing the 4aß--t spectroscopic anti-coincidence method proposed by Kawada et al . [8] . This is a method consisting of a 4aß counter and a Ge(Li) detector to determine the 0167-5087/82/0000-0000/$02.75 b, 1982 North-Holland

intensity of the relevant y-ray in the anti-coincidence or coincidence spectrum gated by the output pulses from the ,ß counter. In order to investigate the reliability of this metluxL various kinds of absolute measurements were applied to a prepared "'Co source to see that the resulting value. of the disintegration rate agree well with one amuher. Then, the present 4nß-y spectniscopic coitrciderxr method was applied to prepared ts2Eu sources and the obtained result was compared with the source intensity deduced with the `Eu standard source supplied by LMRI (Laboatoire Primaire de Métrollgic des Itnements lonisants . Saclav) . 2. Pmpamtion of "'Eu stturces aM detumi»Miua y-ray sp«Irwtretry

be.

Europium-152 was prepared by irradiating an err riched europium target (tS'Eu 925 and t"Ett :t'i) with reactor neutrons. The irradiated sample wits purified by separating other rare earth eknterus ov maincd irx the target by the cation exchange method 191, The purifW europium fraction was stored in the form of europium[ chloride as adiluted hydrochloric acid svùutioxa, Activity-measurement sources were prepared by taking an aliquot of a known amount of thestink solution by the gravimetric method. The spume for abwAuec measurement were made by drying up in a de-vatthe droplet mounted on a VYNS thin film 0Spg."cm~p both surfaces of which were coated with gold. The sources for y-ray spectrometry were made by drying rep the droplet mounted on a Mylar film utukrr an ink-d

274

S. Raha er nl. / Derermlamion nf -Eu

lamp and then covering it with Scotch tape. The standard source for they-ray spectrometry was a 1s=Eu source supplied by LMRI, the intensity of which was 10.24=0.18 pCi (99.7% confidence level) at October 18th, 1979. Use was made of a coaxial Gc(Li) detector with a relative detection efficiency of 15% and a resolution of 2.1 keV for the spectrometry. Thegeometry in thespectrometry was adjusted at an appropriate source distance between 20 and40 cm so as to maintain thetotal count rate below 300 cps. Quantitative determination of '5ZEu was carried out by comparing directly the peak areas of the same y-ray between the sample source and the standard. Use was made of the following five peaks: 344. 779, 964, 1112, and 1408 keV, for the determination. They are all of relatively large intensity and none of them is affected by y-rays emitted from t54 Eu contained in the 's 2 Eu source by a small amount. The decay correction for 152Eu activity was made using the half-life value of 13.1 =0.1 y [10]. 3. Principle of theabsolute measurement Suppose the ß-count rate N4 is measured for the radioactive nuclide A with the decay scheme illustrated in fig. 1, and the disintegration rate N is given by No = NplIfEl+, "

(1)

where f, is the branching ratio of theith ,ß-ray and cp, thecounting efficiency. Thesummation should be taken over all the ß-rays. Now, let us costruct a coincidence circuit consisting of a ß counter and a solid-state y-ray spectrometer, whose block diagram is shown in fig. 2. The peak area AJj) of the jth -y-my of interest was obtained in the coincidence y spectrum gatedby the output pulses from

-to(r

Fig. 2. A block diagram of the counting set-up for the spectrouopic absolute measurement.

theß counter. Then,A,(j) is expressed as

1

Ground State Fig . 1 . An illustrative decay scheme for the scopic absoute measurement .

4aß-y

spectro-

(2)

Ao(j) =No f c P,4,c w .

where ¢, is the relative branching ratio of thejth y-ray and c yr the photopeak efficiency. The fl-counting effi :iency cp, is then given by the ratio of AJj) to the peak area Ajj) of thecorresponding y-ray in the single spectrum : namely, ~.(I)

A,(I ) - NOffp,~lEYI A j I) NOf4jcy,

.

(3)

Hence, eq. f 1, is rewritten as No = Npl2fAtJj) .

If there exist cascade y transitions feeding to the relevant y-ray, as will be shown later, cp is notgiven by such a simple formula as eq . (3) [cf. eq. (5)]. Kawada et al. have used the anti-coincidence spectrum instead of the coincidence one [8] and, therefore. given the fl-efficiency Ea, = 1 - Aa(j) ,

A

4aß-y

(3")

where 0 .(j) = A,(j)1A.,(j) with the peak area A,(j)of yl in the anti-coincidence spectrum. The 41rß counter used for theß-ray detection assures an cp close to unity . so that the anti-coincidence method is favorable compared to the coincidence method because of its better statistics. However, this is hardly applicable for nuclides with complex decay schemes like ts: Eu since the correction becomes too complicated . In order to test the above-mentioned absolute measurement method utilizing the spectroscopic coincidence technique, the intensity of a 'Co source was de. termined by thespectroscopic coincidence as well as the anti-coincidence method,on the onehand. Theresult of either method wasthen compared with that obtained by the ordinary 41rß-y coincidence measurement consisting of a 4aßcounter and a Nal(TI) detector, on the other hand.

275

S. Baba et al. / Determiaatiaa of 152E. artfeity

4. 4%pry spectroscopic coincidence measurement of ts"Eu Europium-152 decays via p --ray emission by 27% and via electron capture (with a negligible amount of ,6' emission) by 73% [I1). Though there are twelve ß-rays emitted in the f3- decay, only four of them possess branching ratios of more than 1%: namely, 170 keV (1.9%), 370 keV (2.5%), 690 keV (13%), and 1492 keV (8%). For the first three out of the above four /3- -rays, the ß-counting efficiencies can be obtained from eq. (3) by applying the spectroscopic coincidence method with the 1299, 1090, and 779 keVy-rays, respectively . Regardingthe 1492 keV,6-ray, theassociated 344 keV y-ray to be used contains feeds from the other /3 transitions, so that eq. (3) cannot be aprlied directly in this case. Instead . the fl-counting efficiency must be calculated by eq, (5) : (1492) = (25.290(344)-13 .0,,(690)



-2.43,,(370)-1 .86,,(170))/8,

(5)

with thepeak area ratio A, (344) and the knowncounting efficiencies, E, (170), E, (370), and E, (690) . Fig. 3 shows an effective decay scheme of 152 E u simplified from theaboveviewpoint. The EC decayscheme is no less complex than the ß" decayscheme . However, oneneeds to consider only the contribution of internal conversion andAuger electrons in the absolute measurement, since the ß' emission is of negligible amount. In the traditional absolute measurement-by means of p-ray detection, there existed a limitation for its reliability in the case of i52Eu due to the inaccuracies in the counting efficiency for the conversion electrons and in the values of the conversion coefficient . Furthermore, application of the efficiencytracer method [t2) was not promising either because of the existence of both hard K-Auger and soft L-Auger electrons. "Eu

On the contrary, the present spectroscopic coincidence method solves such difficulties. That is. the amount of the contribution of conversion electrons emitted in the 122 keV y transition (cf. fig- 3) is determined by the use of a y-ray possessing a cascade relation with the 122 keV transition. Here, theresulting value is not merely the counting efficiency for the conversion electrons but the product of the efficiency and the rate of the internal conversion, as shown in eq(6) : A~(Y,)

0~(Y,) =

q.(Y,) Ne%,E rr

+.-)),J122)

Nef,E .,

=

a .z2 ce(122) = e122, 1 + a,z2

where a, 22 and,,(122) are . respectively, theconveaioa coefficient and thecounting efficiency for theelectrons emitted in the 122 keV transition. Furthermore, Y, denotes any of the y transitions in the cascade relation with the 122 keV transition. One sees that the error involved in the value of the conversion coefficient is ineffective in eq . (6) because the contribution is measured as a whole. As is clearly seen, there are four y-rays available for determining the amount of the contribution of themust sizable conversion electrons emitted in the 122 keV transition . It follows that onecanobtain four independent observations for it. assuring quite high accurcy Ay transition cascading to the 122keV transition is in thecascade relation with theAuger electrons emitted in the preceding EC decay, too. Hence, strictly speaking, eq. (6) should he replaced by eq. (6') including the contribution of Auger electrons : k(y,)=e .22'( 1- nlzz)SA'

(6-)

where SA is the contribution of Auger electrons which is expressed as SA =

Fig. 3. A amplified dmay scheme of "Ea effective to the 4,70 -y spatroscopic :oincidence measurement .

{ a az/ (1

O - WK)EK+ ( 1- WL)EL

with the X-ray fluorescence yield W and the countiag efficiency EA for Auger electrons. Here, the subscripts K and L denote K X-rays or K-Auger and L X-ray on L-Auger, respectively. The spectroscopic coincidence counting with the 12_-1 keV y-ray supplies information for the contribution cA the conversion and Auger electrons emitted prior to the transition as a whole: 0.(122) = (e)+ (1- (e))SA,

(8)

S. Bnha e1 nl. / Defernrinnlinn of `7F. IIW«r

276

5. Results and discussiott

and

~Ilo~~l 1 +(r~)} E el

(c)= '

(9)

The summation in eq. (9) should he taken over all the y-rays cascading to the 122keV transition. On the other hand, no information is available for the contribution of conversion electrons emitted in y transitions decaying directly to the ground state. However, the contribution of conversion electrons is small enough to be neglected in this case. A typical example of such transitions is the 1086 keV transition, for which -have the relation : ~~(1086)=dA+(1-~ .x) ~'Ik(~kI(I +(rk))Eck x k (~' =S(10) (+Ik k Here, we conversely take advantage of the contribution of conversion electrons being negligibly small in the 1086 keV transition for obtaining SAin eqs . (6) and (8). The net contribution of conversion electrons is now deduced as -e .22-

1Jy,)-SA 1-SA

and (e)=

',

(122)-SA 1-SA

(12)

with the known value of SA.

TableI summarizes the results of three kinds of absolute measurements applied to a prepared 'Co source . The 47rß-y spectroscopic method can be applied to either of the two y-rays . 1173 and 1332 keV, so that tv o independent measurements are available for the ,I.stintegration rate. They are both listed in the table. The second column in table I gives the dead time Tt) of each equipment which was determined by the twosource method 1131 . The values in the third column represent the disintegration rates deduced only with the correction against the counting loss in NR. The 4aß-y spectroscopic anti-coincidence method gives a disintegration rate relatively close to that obtained by the ordinary 4aß-y coincidence method . whereas the spectroscopic coincidence method results in obvious underestimation of the disintegration rate as compared with the other two. Moreover. this deviation in the disintegration rate increases as the width of the coincidence gate-pulse TR (the fourth column) increases. Since the two values of the disintegration rate obtained in each spectroscopic measurement are very close to each other. the results are considered to be of high quality. The source of the above-mentioned deviation in the disintegration rate is found in (1) elongation of the clock time in the coincidence or anti-coincidence mode counting, (2) the effect of accidental coincidences. and (3) the effect of the detection sensitivity of theßcounter against y-rays. The first effect is effective only in the 4aß-y spectroscopic measurement while the others are equally valid for the ordinary 4aß-y coincidence and thespectroscopic method. The ß-ray counting efficiency with corrections for such effects is given by A,- 2NP" ,R- p, 13 R 1-TR (2NP+Ny)-EPY

Table I Comparison of the results of various methods of the absolute measurement for "Co. Notations are explained in the text . Method

Tn( Ps)

No (dps)

4aß-y coincidence

3.0

6979=""-15

4aß-y spectroscopic coincidence

1.6

6742=27 6730=27

1 .6 1 .6 1 .6

4aß-yspociroscopic anti-coincidence

TR(Ps)

No(dps)

1.0 6.0

6997=15

6850=27 6884=15

3.0

6984=27 7019=' 15

7016=12 7018=11

6.0

7011=12 7012=11

7034=12 7029=11

3.0

7027=12 7024=11

7010=28 6996=28

S. Baba er al.

1131.

In the case of the 47r,6--y spectroscopic coincidence method . oneobtains clr = Or~l(Y) _ Aa(y) - No:T, :- cß,(I+ N,;T,) (I

-

;vR) EA,)(l+Nr

NG'rE

N,,( I

0jy) (I-,,)(I+N,;Tn)-N,Tr.

-NßTto)

(15)

(16)

(1 -NATO)

and T,, represents the gate width effective to accidental coincidences of the y pulses in the pulse height analyzer with ßsignals. It was found that T,: was smaller than Ta by 0.5 Ws irrespective of the pulse height. as shown in fig. 4, Thedisintegration rate N) is nowequally given by No Na

(17)

cp+2etf,(1-cn)

Nuclide

Disintegration rate per mg solution (dp,) ut I September, 1980

1"'Eu

4537 < 43 33 " - 3

-Eu

solu-

(14)

Here, N o; is the number of gate pulses per unit time given by N ;=

277

irin-Table 2 Result of the -pray spectrometry for the prepared 1`- Eu tion .

,

while for thespectroscopic anti-coincidence method the following equation is derived : 1-cp = S.'( Y) _

r`,Eu -

/ Deterrnirrution af

in the case of the ordinary 4aß-y coincidence method. Here. Tais the resolving time of themixer unit, assumed to be equal to To . ip, is they-ray detection efficiency of the# counter for a single 'Co y-ray taken to be 0.0044

for the ordinary 4aß-y coincidence and the 4r.ß-y spectroscopic method, and the final results are summarized in the fifth column of table l . Here. Np give the ß count rate corrected for the count loss dueto the dead time T, ) and background 8: Niï =

NP/( 1

(i11)

- N'TO)-a .

They are in good agreementwith one another for different conditions of various counting methods. This implies that the present counting method assures quite high accuracy for determination of the'Co activity and theabove-mentioned corrections are in theright course. Table2 gives the result of the -y-ray spectrometry with the prepared ts`Eu sample solution, calibrated against the standard source supplied by LMRI. Table3 summarizes theresults of the 4aß-y spectroscopic coincidence measurement performed with four rs` Eu sources prepared from the above sample solutivvi. The top four 9.'s in the table are the peak-area ratio, with respect to the y-rays associated with four fl-s. 10

--

122 keV

---- 344 keV

0-5

779 keV 10

i=

1

T

N

1

C

1 11

Q

2

â 3

0-5 a

10

i

-6

-4

-2 0 2 4 ß- Delay (psec)

G

Fig . 4. l'hotopeak intensity in the coincidence y-ray spectrum displayed as a function of the delay time fed to the gate-pulse . The width of the gate-pulse wasset to 6 ps. Three y-ray peaks are chosen in order to demonstrate that the effective width is equally 5.5 ;Ls for all y-rays though the coincidence timing shifts forward as the y-ray energy decreases.

05 4 0

0

05

10 E. (MeY)

15

20

Fig. 5. Beta-ray efficiency curves of he 4,*--- O-W by the 4mß- y,pectnucopic -incidence ttte1h.L

278

S. Baba e1 ni. / De-ft-ion nf "Eu aetiolly

Table 3 Result of the 4aß-y spectroscopic coincidence absolute measurement of Source no.

1 7501

NP 9,044)

'=

6

1s2 Eu.

Detailed explanation is given in the text. 3

2 6595

=

7

4 .=

7132

7

7952

-

8

0.9694 .e 0.9262'=

0.0027 0.0027

0.9803=0.9421 .2

0.0080 0.0030

0.9690=0.9278 t

0.0027 0.0027

0.9557--^ 0.9102 `-

0.0027 0.0027

0.9740= 0.9308=

0.0062 0.0062

0.9779= 0.9397=

0.0066 0.0066

0 .9833= 0 .9421,

0.0065 0.0065

0 .9760= 0 .9304=

0.0060 0 .0060

9,(10901

, .941 '= 0.898 s

0.022 0.022

0 .988 = 0 .950 =

0.024 0 .024

0 .943 =0.902 "_

0.024 0 .024

0.903 s 0.858 =

0.020 0.020

.1,(1299)

0.828 "= 0.785 =-

0 .024 0 .024

0.906 = 0.868 -=-

0 .025 0 .025

0.811 0.771

0.022 0,022

0.801 0.756

s

0.020 0M0

1,(1492) .1,(1086) SA 9,(122) (e)

0.9601s 0.18520.l447= 0.2081= 0.0202=

0 .0081 0.0028 0.0028 0.0010 0.0030

0.9608= 0.2244= 0.l884s 0.2395= 0.0107-

0 .0089 0.0034 0.0034 0.0010 0.0036

0.9490 = 0.1592 -= 0.1206-"0.1750 "= 0.0130 "

0.0083 0.0027 0.0027 0.0008 0.0028

0.9291= 0,1563 = 0.ll36= 0.1772= 0 .0185s

0.0074 0.0025 0.0025 0.0008 0.0027

9,(245)

0.6318= 0.5172=

0.0044 0.0052

0.6501= 0.5192=

0.0049 0.0060

0.6199 = 0 .5198=

0 .0044 0 .0052

0 .6256= 0 .5251-

0.0040 0.0047

9,(964)

0.62000.5033=

0.0045 0.0053

0.6496=' 0.5186-

0.0049 0.0060

0 .6141 s 0.5132=

0.0046 0.0054

0.6165 s 0.5148=

0.0043 0.0050

:1,(1112)

0 .6456= 0 .5334=

0.0050 0 .0058

0.65981, 0.5312=

0 .0055 0 .0065

0.6293= 0.5305=

0.0049 0.0056

0.6171= 0.5155=

0.0046 0.0053

0 .0032 0.0047

0.6268= 0.5277 "

0.0039 0.0048

0.6247= 0.5241=

0.0037 0.0045

0.0070 0.5210s 0.9339s 0.0133 0.4106= 0.0054 0.6629- 0.0075 9946 = 113 1 .011 = 0 .012

0.5228=0 .9269= 0.3881 `

9,(779)

1

A,(1408)

0.6232~ 0.5071=

0.0029 0.0041

0.5153s 0.0134 0.9260= 0.0113 0.3928s 0.0094 0 .6430= 0.0099 s 180 11678 1 .014s 0.016


N N ;N. 1

0.6468= 0.5151=-

emitted with branching ratios greater than 1%. For each of them, the upper numerical value represents the obrecved peak-area ratio and the lower one gives the corrected value according to eq. (14) in which c ps is neglected. 17re observed values of 9(122) and (e) are found below (em )_ in table 3. By the use of the counting efficiencies found for four - -rays . one can construct a fl-counting efficiency curve as a function of the ß-ray energy, Ep , as shown in fig. 5. 'I1ten the counting efficiencies for eight ß-rays possessing branching ratios greater than 0.1% (cf. fig . 3) were

read from the thus constructed efficiency curve to calculate the average ß-ray counting efficiency as -

i f1p,l i f .

0.0079 0.0115 0 .0058 0.6385s 0 .0074 11174 =132 1 .011 ~ 0.013

0.5199s 0.0055 0. l06= 0.0101 0.3826= 0.0042 0.6286= 0.0060 12672 ' 122 1 .014 -"_ 0.010

1.013=0.006

(Nô/N.-I )_

<1p)

==

(19)

The overall ß-counting efficiency, Jp , including contributions of conversion and Auger electrons, is approxi-

mately expressed as jp=0.27{(E,)+(I-(Ep))EE,7)+(E .),

(20)

where (c .) is the contribution from the EC-decay side: (E~)=0.63[SA+(1 -SA)(1Epr+e122 +(e)(1-e122))]

+0 .10{SA+(1-8A)11ß7) " (21)

Furthermore, It ., is the overall detection efficiency of the ß counter for y-rays, which is estimated to be 0.0068 1141. The above-mentioned quantities are also listed in table 3. The y-ray spectrometry for the ts2Eu sample solution revealed that it contained the "Eu activity by 0.73%. The fl-counting efficiency for 15° Eu was approximated by
279

S. Rahn er al. / Docrrninatimt of 1 ' 2En aetici(r

purpose of correction for such a small amount of impurity. Hence, the disintegration rate, Nô52, of 15"Eu is given by Nôs'

Np .0073f114<£ß> ' SIl + 0 !154 is the total fl-ray

_c

(22)

where emission rate of Is4Eu, which is 0.9998 1111. The result of the absolute measurement of "Eu is given in 1010, too. For comparison, the activity concentration of the sample solution was then compared with that obtained by the y-ray spectrometry, and the ratio is presented at the bottom of table 3. The ratio being close to unity assures not only the reliability of both the 4rry method adopted by LMRI and the present method but also the accuracy of the y-branching ratios of "Eu available. It is quite unlikely that the agreementis merely accidental. Theerror attached to theLMRI 15'Eu source is 1 .5% on the 99 .7% confidence level (3o), while that for the present method amounts to 1.8% on the same confidencelevel. Therefore. they arein agreementon the95% confidence level (2a) but are slightly different from each other on the 68%confidence level (to). Thedispersion amongfour values of theintensity ratio in table 3 is quite small, which indicates that the major part of the errors is of systematic character. The accuracy of the4aß-y spectroscopic deterntination dependson theevaluation of the contribution from theEC decay expressed by eq. (21) since little room is left for incorrectness of the magnitude of (ca) because of the total ß'-branching ratio being as small as 0.27 . In theevaluation of (c~), the contribution of Auger electrons, SA , shows a strong influence and 8A is in turn largely affected by the content of detection, c i,Y, of y-rays cascading to the 1086 keV y-ray. Furthermore. an increase in SA causes a decrease in both e, 22 and (e), and the latter is far more effective to the magnitude of (c c ). In conclusion, an c ar of 0.01% is not negligible in the evaluation of (ce) . With consideration of each factor an error of 0.6% would be a reasonable amount for the4aß-y spectroscopic coincidence determination of 152Eu, It follows that one must discern the possible existence of significant differences between the two methods. The authors are grateful to Prof . T. Watanabe and Dr . H. Miyahara of Nagoya University for their valuable discussions.

a

11

D

n; n,

p.channpt

a

n

ltl

:v

b

t_charp~él

coinadence channel

r,me -r

Fig. 6. Classification of accidental coincidences : (I) accidental coincidence of a )i- and ay-signal, (11) accidental coincidenof aß-signal and acoincident signal. (I11) accidental coincidence of ay-signal and acoincident signal. and (IV) accidental coincidence between twocoincident signa),. first-order term only with respect to the resolving time TR. Here . two cases, 11-a and III-a, are included in the true coincidence, and therefore the remaining four are thecases to be considered . There are two viewpoints valid in considering accidental coincidences. If one regards the signal overlapping the preceding pulse as virtual, oneshould consider cases 11-b and 111-b as accidental coincidences as well as case 1 . This :esulis in the identical consequence for tile' number of accidental coincidences N,,1, with that usually used (IS-191; namely, +1 R

N,(N;-N!')

=rtt ( 2NßNY-N'(Nß+NY)) .

(=`)

where the primeletters give observed countrates including background . The number of true t»oinciderccs. is then derived as N, - 2TR N;N, N"= NN (24) 1 _TR(

P+N7) .

On theother hand, the above-tttentioned merlappiog pulses can be regarded as real signals instead of virtual ones by taking the dead time correction int, account at the same time. In this point of view, the cases 11-b and Ill-b are true coincidences andonly thecase I is left a< theaccidental coincidence ; hence N_ =2 TR( NB-N")(N;-N°),

(2sÀ

The number of true coincidences is now given by

Appendix Correction for

1

accidental coincidem-

Accidental coincidences are divided into six cases as indicated in fig. 6, when one restricts oneself to the

where the third term on theright-hand side of eq, (-%) is the carrtxtion for thedead time loss in the ,A%wkensignals, the case IV in fig. 6. Hence, -obtains

S. fhaer a). / Derer)ni-Nou aJ r "k'a a"rici(v A.. _,Tx N,,NY I-2Tx (N,+NY)

(27)

The difference in the N"'s between eqs . (27) and (24) comes from the consequences with and without the dead time correction. Using eq . (27), the fl-counting efficiency c,{ is given by (28) whereas A~' in eq. (24) is related to E,t as A') "A'' = P"C ".

(29)

Here. AY' = N,',i( I - TR AY) - By is the y-count rate corrected for dead time losses while N, is the observed y-count rate minus background By . and p is the probability of detection due to dead time losses (IS] given by - Tx N ') . pr, = I 

(30)

Both eqs. (28) and (29) result with trivial approximalions in the same expression for c,, : N./N,-2T x N1

References III S. Baba and T. Suzuki. J. Radioan .A . Chem. 29 (1976) 301 .

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