Modified method of a determination in the 1/E 1+a epithermal neutron spectrum of reactor

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1 J Rdionl Nucl Chem (2010) 285: DOI /s x Modified method of determintion in the 1/E 1+ epitherml neutron spectrum of rector Trn Vn Hung Received: 19 Februry 2010 / Published online: 7 April 2010 Ó The Author(s) This rticle is published with open ccess t Springerlink.com Abstrct A modified method hs been developed for the determintion in the 1/E 1? epitherml neutron spectrum of rector. It is bsed on the Cd-covered nd without Cdcovered irrditions of two monitors. This method ws pplied to determine the vlue in the chnnels of Dlt rector nd the results were compred with those obtined by the other methods. It ppered tht the results of the modified method were in quite good greement with those of other methods. It lso showed tht the modified method ws simple in prcticl uses nd good ppliction in the experiment of determintion in the rector irrdition chnnels. Keywords Introduction Vlue Dlt rector Modified method As reported by Schumnn nd Albert [1] nd by Ryves [2], the epitherml neutron flux in rector irrdition chnnels is not proportionl to 1/E, but rther 1/E 1?, where is smll positive or negtive constnt nd mesure of the epitherml neutron flux devition from the idel distribution 1/E, nd E is the neutron energy. The vlues re smller thn unity in bsolute vlue, nd vry between the irrdition chnnels in the sme rector. T. Vn Hung (&) Reserch nd Development Center for Rdition Technology, 202A, Street 11, Linh Xun Wrd, Thu Duc District, HoChiMinh City, Vietnm e-mil: trnhungkeiko@yhoo.com In the idel cse, the resonnce integrl for 1/E epitherml neutron spectrum is written s: I 0 ¼ Z 1 E Cd rðeþ E de ð1þ with: E Cd is the effective Cd cut-off energy (=0.55 ev). The resonnce integrls, defined ccording to Eq. 1 nd tbled in litertures, re not vlid in non-idel cse. In the non-idel cse, the resonnce integrl for 1/E 1? epitherml neutron spectrum is defined s: I 0 ðþ ¼ Z 1 E Cd rðeþ:1 ev E 1þ de: ð2þ It indictes tht the resonnce integrls for prcticl uses re function of nd thus of the irrdition position. Thus, in the (n, c)-ctivtion nlysis with rector neutrons (NAA) using comprtor method, should be known to preserve the ccurcy of the nlysis results. The vlue should be determined either by experiment or by clcultion. In experiment, severl techniques hve been developed by De Corte et l. [3 5], nmely the Cd-covered multi-monitor method, the Cd-rtio for multi-monitor method, the bre multi-monitor method. However, in these methods, the vlues re found from implicit functions by the itertive method on computer. Consequently, they re merely pproximte methods. In this work, modified method for the determintion of prmeters in rector irrdition chnnels is to be presented. In tht, the prmeter is written s n explicit formul. The results of the determintion in irrdition chnnels of Dlt rector using the modified method re lso being reported.

2 332 T. Vn Hung Bse of modified method From Eq. 2, the resonnce integrl of isotope i in the nonidel cse should be written s:! I 0i ðþ ¼ I 0i 0:426r 0i 0:426r 0i þ E ri ð2 þ 1ÞðE Cd Þ 1eV ð3þ where r 0i 2,200 m s -1 cross-section of nuclide i, E ri effective resonnce energy (ev) of nuclide i. Note tht Eq. 3 is only vlid when E Cd = 0.55 ev, since = 2(E 0 /E Cd ) 1/2 with E 0 = ev nd E Cd = 0.55 ev. Accordingly, Q 0 () = I 0 ()/r 0 cn be written (in ev unit) s below:! Q 0i ðþ ¼ Q 0i 0:426 0:426 þ ð2 þ 1ÞðE Cd Þ : ð4þ E ri We know tht, vlue is much smller thn unity in bsolute vlue. In prctice, in rector irrdition chnnels, the bsolute vlue is less thn 0.2 (in most cses, \ 0.1 nd this condition is stisfctory in rector core). We suggest substitution Q 0i () from Eq. 4 by the following pproximte formul: Q 0i ðþ ¼Q 0i expð i ðln E ri ÞÞ ð5þ where i is constnt for ech nuclide nd determined by fitting the vlues of Q 0i (), which re clculted from Eq. 4 in the rnge of B 0.2, ccording to the fitting function (5). Note tht, i for ech nuclide is dependent on the sign of. In this work, i vlues re determined by fitting Q 0i () using Kleigrph progrm. The vlues of i, correltion nd coefficient r of fitting function, nd the relevnt nucler dt for nuclides chosen s -monitors re given in Tble 1. As the comprison of Q 0i () between formul (4) nd (5), the results of clculting Q 0i () for 197 Au(n, c) 198 Au, 64 Zn(n, c) 65 Zn nd 94 Zr(n, c) 95 Zr for negtive nd positive re respectively shown in Tbles 2 nd 3. It indictes tht, when vlues re negtive nd less thn 0.2 in bsolute vlue, the differences of Q 0i () clculted from these formul re less thn 0.08% for 197 Au(n, c) 198 Au, 0.8% for 94 Zr(n, c) 95 Zr, nd bout 2% for 64 Zn(n, c) 65 Zn. In the cse of positive vlues, Q 0i () vlues for 197 Au(n, c) 198 Au mtch ech other very well with the differences less thn 0.04%, while Q 0i () vlues for 94 Zr(n, c) 95 Zr nd 64 Zn(n, c) 65 Zn re in good greement with \ When vlues re more thn 0.16, the differences of Q 0i () vlues clculted from these formul cn be more thn 2% for 94 Zr(n, c) 95 Zr nd 64 Zn(n, c) 65 Zn. However, in most cses, jj\ 0:1 nd this condition is stisfctory in rector Tble 1 Nucler dt nd i fctor for the nuclides chosen s -monitors, correltion coefficient r of fitting function Nuclide E ri ðevþ Q 0i i r \ 0 [ Au(n, c) 198 Au ± ± Zr(n, c) 95 Zr ± ± Zn(n, c) 65 Zn ± ± Dt for Au nd Zn were tken from De Corte [6] nd Zr dt from Simonits [7] Tble 2 The vlues Q 0i () for monitors clculted from formul (4) nd (5) with in intervl [0,-0.2] 197 Au(n, c) 198 Au 64 Zn(n, c) 65 Zn 94 Zr(n, c) 95 Zr Q 0i () from (4) Q 0i () from (5) Q 0i () from (4) Q 0i () from (5) Q 0i () from (4) Q 0i () from (5)

3 Modified method of determintion 333 Tble 3 The vlues Q 0i () for monitors clculted from formul (4) nd (5) with in intervl [0,0.2] 197 Au(n, c) 198 Au 64 Zn(n, c) 65 Zn 94 Zr(n, c) 95 Zr Q 0i () from (4) Q 0i () from (5) Q 0i () from (4) Q 0i () from (5) Q 0i () from (4) Q 0i () from (5) core; s for chnnel R4V4 of the DR-3 rector (Ri/, Denmrk), = ± [8], Eq. 5 cn be good use to replce Eq. 4 in -prctice. It cn be seen lter in the discussion bout the error estimtion of the method. Thus, in the two-detector method of Ryves using Cd-rtio modified by De Corte et l. [4], cn be found s the root of the eqution: I 01 ðr Cd2 1Þ ðr Cd1 1Þ ¼ r 01 0:426 E r1 þ 0:426 ð2þ1 I 02 r 02 0:426 E r2 þ 0:426 Note tht: Q 0i ðþ ¼ Q 0i 0:426 0:426 E ri þ ð2 þ 1Þ:0:55 : Eqution 6 cn be written s: ÞE Cd ð2þ1þe Cd : ð6þ ðr Cd2 1Þ ðr Cd1 1Þ ¼ Q 01ðÞ ð7þ Q 02 ðþ Where i is denotes the ith monitor nd R Cdi is Cd rtio of ith monitor. Substituting Eq. 5 into Eq. 7, it cn be written: ðr Cd2 1Þ ðr Cd1 1Þ ¼ Q 01ðÞ Q 02 ðþ ¼ Q 01 expð 1 ðln E r1 Þ ð8þ Q 02 expð 2 ðln E r2 Þ nd thus, prmeter cn be written: 1 ¼ ð 2 ln E r2 1 ln E r1 Þ ln ðr Cd2 1ÞQ 02 : ð9þ ðr Cd1 1ÞQ 01 In the cse of Cd-covered co-irrdition of two-detector, from bse ctivtion eqution, is esily written: 1 ¼ ð 2 ln E r2 1 ln E r1 Þ ln A sp1k 0Auð1Þ e 2 Q 02 ð10þ A sp2 k 0Auð2Þ e 2 Q 01 A spi ¼ A pi wsdc where A pi mesured verge ctivity of the full-energy pek, A pi = N pi /t m with N pi net number of counts under the full-energy pek collected during mesuring time t m ; w weight of the irrdited element; S ¼ 1 e kt irr ; k = decy constnt, t irr = irrdition time; D ¼ e kt d ; t d = decy time; C ¼ 1 e ktm kt m k 0Au(i) is k 0 -fctor of ith isotope to gold (see Ref. [3]) nd e i is full-energy pek detection efficiency of energy E i. Thus, in experiment using pirs of 197 Au 94 Zr nd 197 Au 64 Zn, we only need the determintion of R Cdi rtios (Cdrtio method) or A spi (Cd-covered irrdition only) of monitors. Then will be clculted using Eqs. 9 or 10, respectively. Error estimtion of the method Errors of the method should be estimted in two kinds: systemtic nd sttisticl errors. In this report, the errors due to pproximtion of Eq. 5 re considered s systemtic errors, while the errors of the vribles in Eqs. 9 nd 10 using clcultion of re sttisticl errors. The 197 Au 94 Zr nd 197 Au 64 Zn pirs were pplied. The choice of 197 Au 94 Zr nd 197 Au 64 Zn monitor pirs is very suitble for the experiment of the -determintion. The reson is the E ri effective resonnt energies of these isotopes re in wide region (E 197 r ð AuÞ ¼ 5:65 ev; E 64 r Zn ¼ 2; 560 ev; E 94 r ð ZrÞ¼6; 260 evþ nd nucler prmeters re suitble for rector irrdition. Moreover, the smple preprtion is prticulrly esy, the product nucleus hve simple decy scheme nd their cross-section hve been determined in detil nd with high ccurcy. Error estimtion due to pproximtion of Eq. 5 Reviewing Tbles 2 nd 3, we see tht the Q 0i ()-vlues of 197 Au(n, c) 198 Au clculted from Eqs. 4 nd 5 re in very

4 334 T. Vn Hung Reltive error (%) Monitor Zn-64 Monitor Zr-94 α Fig. 1 Survey of f -vlues on when \ 0 Reltive error (%) α 0.25 Fig. 2 Survey of f -vlues on when [ 0 Monitor Zn-64 Monitor Zr-94 good greement with the difference of less thn 0.08% in the rnge of B 0.2. We cn think tht they re quite coincident. The uncertinty of the -vlues depends only on the differences of the Q 0i ()-vlues of 94 Zr(n, c) 95 Zr nd 64 Zn(n, c) 65 Zn clculted from Eqs. 4 nd 5, respectively. From Eq. 5, it cn be written s: ¼ðlnQ 0 ðþ lnq 0 Þ=ln E r : ð11þ From the error propgtion eqution, the percentile error or uncertinty due to pproximtion of Eq. 5 cn be written s: f ¼ r ¼ 1 : 1 DQ 0 ðþ ð12þ ln E r Q 0 ðþ where r is bsolute uncertinty of ; DQ 0 ()is the difference of the Q 0i ()-vlues clculted from Eqs. 4 nd 5. From Eq. 5, f is dependent upon ech isotope chosen s monitor nd inverse proportion to -vlue. The survey of f -vlues on, in which \ 0.2, for 94 Zr nd 65 Zn were crried out in Figs. 1 nd 2. From Figs. 1 nd 2, the systemtic uncertinty (f )of the 197 Au 94 Zr pir is lower thn tht of 197 Au 64 Zn one. It is obvious tht Q 0 of 94 Zr (Q 0 = 5.306) is bigger thn one of 64 Zn (Q 0 = 1.908). Furthermore, DQ 0 ()of 64 Zn clculted from Eqs. 4 nd 5 is bigger thn one of 94 Zr nd the effective resonnce energy of 94 Zr E ri ¼ 6; 260 ev is lso bigger thn the one of 64 Zn E ri ¼ 2; 560 ev : Sttisticl errors This error cn be estimted from the errors of the vribles in Eqs. 9 or 10. The bsolute uncertinty in cn be clculted from the uncertinties of the vribles (denoted x j ) which determine in Eqs. 9 or 10: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X u r ¼ r 2 o 2 t x j ox j j ð13þ where q/qx j nd rx j re the corresponding prtil derivtives nd the uncertinties of x j -vribles, respectively. According to the customry error propgtion theory, the error propgtion functions cn be written s: Z ðx j Þ¼ o. oxj x ¼ o x j j ox j : ð14þ And reltive error: s ðx j Þ¼Z ðx j Þ Dx j : x j ð15þ Applying the bove formul Eqs. 14 9, we get: 1 Z ðe r1 Þ¼ 2 ln E r2 1 ln E r1 ð16þ 2 Z ðe r2 Þ¼ 2 ln E r2 1 ln E r1 ð17þ 1 ln E r1 Z ð 1 Þ¼ 2 ln E r2 1 ln E r1 ð18þ 2 Z ð 2 Þ¼ 2 ln E r2 1 ln E r1 ð19þ Z ðq 01 Þ¼Z ðq 02 Þ¼ ln E r2 1 ln E r1 ð20þ Z ðr Cd1 Þ¼ 1 R Cd1 ðr Cd1 1Þð 2 ln E r2 1 ln E r1 Þ ð21þ Z ðr Cd2 Þ¼ 1 R Cd2 ðr Cd2 1Þð 2 ln E r2 1 ln E r1 Þ : ð22þ In the cse of Cd-covered co-irrdition of two-detector, the vlue is clculted using Eq. 10, nd we obtin Eqs for the error propgtion functions of E ri nd i. The other vribles cn be clculted s following: Z ða sp1 Þ¼Z ða sp2 Þ¼Z ðk 0Auð1Þ Þ¼Z ðk 0Auð2Þ Þ ¼ Z ðe 1 Þ¼Z ðe 2 Þ¼ 1 1 ð 2 ln E r2 1 ln E r1 Þ : ð23þ

5 Modified method of determintion 335 Fig. 3 Ground pln of Dlt rector Experimentl As the first check, in this work, the 197 Au 94 Zr pir ws pplied. The E ri effective resonnt energies of these isotopes re in wide region (E 197 r ð AuÞ ¼ 5:65 ev; E 96 r Zr ¼ 248 ev; 94 Er ð ZrÞ ¼ 6; 260 evþ nd nucler prmeters re suitble for rector irrdition. Moreover, the irrdition of these monitor foils with bre nd Cdcover, we cn simultneously determine the vlue by bre multi-monitor method s presented in [3], s comprison. The experiment of the -determintion ws crried out in the chnnels 7-1, 1-4 nd neutron trp of Dlt rector. These irrdition positions re situted in rector core (Fig. 3). As stndrd mterils, % Au Al wire (dim. 1 mm) nd high-purity Zr foils of mm thickness were used. In this cse, the monitor foils of Au nd Zr with bre nd Cd-covered were simultneously irrdited in the mentioned rector chnnels. The irrdition durtions were 20 min nd 1 h, respectively. In both cses, decying nd mesuring time were h nd 30 min. The counting ws performed with 70 cm 3 coxil GeHP detector pired to 4096 chnnels nlyzer. The results which were summrized in Tble 4, were compred with those obtined by the three-detector method without Cd [3] nd by -determintion method using neutron spectrum clculted by MCNP code [9]. As n exmple for the error estimtion of, we crried out the results of the error estimtion of in 7-1 chnnel of Dlt rector using R Cd method (Eq. 9). Indeed, with = 0.044, the uncertinty of R Cd in the experiment bout 1%, the uncertinties Q oi nd E ri from report [6, 7], we

6 336 T. Vn Hung Tble 4 -Vlues in irrdition chnnels of Dlt rector Chnnels 197 Au 94 Zr (Eq. 9), (10-2 ) 197 Au 94 Zr (Eq. 10), (10-2 ) Three-detector method with Cd, (10-2 ) Clculted (10-2 )[9] Neutron trp -3.1 ± ± ± ± Chnnel -3.6 ± ± ± ± Chnnel -4.4 ± ± ± ± 0.4 used Eqs nd obtined: s ðe rzr Þ1:4%; s ðe rau Þ 1:4%; s ð Au Þ0:03%; s ð Zr Þ0:13%; s ðq 0Au Þ 6:5%; s ðq 0Zr Þ7%; s ðr CdAu Þ5:2%; s ðr CdZr Þ 7:6%: The uncertinty of in totl ws bout 13% (with f & 2%). Similrly, in the cse using the method of Cdcovered co-irrdition of two-detectors ( = 0.045), the uncertinty of estimted from Eqs nd Eq. 23 ws bout 15%, wheres using of the three-detector method without Cd, the uncertinty of ws bout 30%. The comprison of the vlues in Tble 4 shows tht the vlues clculted from Eqs. 9 or 10 re in good greement with ech other. Moreover, they gree well with -determintion method using neutron spectrum clculted by MCNP code [9] nd the three-detector method without Cd. Conclusion From Tble 4, it is obvious tht the modified method presented in this report is suitble for rpid -determintion in experiment. This is in complete greement with the results of the other methods. Moreover, the results from studying on the error due to pproximtion of Eq. 5 suggest tht in cse bsolute vlue is less thn 0.25 (this condition is stisfctory in irrdition chnnels locted in the rector core), the using of this method is quite possible with resonble errors. Open Access This rticle is distributed under the terms of the Cretive Commons Attribution Noncommercil License which permits ny noncommercil use, distribution, nd reproduction in ny medium, provided the originl uthor(s) nd source re credited. References 1. Schumnn P, Albert D (1965) Kernenergie 2:88 2. Ryvers TB (1969) Metrologi 5: de Corte F, Moens L, Simonits A, de Wispelere A, Hoste J (1979) J Rdionl Chem 52: de Corte F, Moens L, Sordo-El Hmmmi K, Simonits A, Hoste J (1979) J Rdionl Chem 52: de Corte F, Sordo-El Hmmmi K, Moens L, Simonits A, de Wispelere A, Hoste J (1981) J Rdionl Chem 62: de Corte F, Simonits A (1989) J Rdionl Nucl Chem 133:43 7. Simonits A, de Corte F, Vn Lierde S, Pomme S, Robouch P, Eguskiz M (2000) J Rdionl Nucl Chem 245: de Corte F, Moens L, Jovnovic S, Simonits A, de Wispelere A (1986) J Rdionl Nucl Chem 102:37 9. Hung TV (2010) J Rdionl Nucl Chem 283:719