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A deformed Doppler Broadening Function considering the Tsallis speed distribution
Annals of Nuclear Energy 128 (2019) 414–421
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
A deformed Doppler Broadening Function considering the Tsallis speed distribution q Guilherme Guedes a,b,⇑, Daniel A.P. Palma c, Alessandro C. Gonçalves b a
Federal Center for Technological Education Celso Suckow da Fonseca – CEFET/RJ, Nova Friburgo, Brazil Department of Nuclear Engineering, Federal University of Rio de Janeiro – PEN/COPPE/UFRJ, Rio de Janeiro, Brazil c Brazilian Nuclear Energy Commission – CNEN, Rio de Janeiro, Brazil b
a r t i c l e
i n f o
Article history: Received 1 August 2018 Received in revised form 14 December 2018 Accepted 14 January 2019
Keywords: Deformed Doppler Broadening Function Quasi-Maxwellian distribution Tsallis distribution
a b s t r a c t The Doppler Broadening Phenomenon provides a good description for the effects of thermal agitation on the cross-section of the neutron-nucleus interaction and has its theoretical basis on Quantum Mechanics through the Single Level Breit-Wigner Formalism, as well as on Statistical Mechanics, with the Maxwell Boltzmann distribution. In this paper, the consequences of replacing the Maxwell Boltzmann speed distribution by the non-extensive Tsallis distribution in the evaluation of the absorption cross section in three-dimension is discussed. The Tsallis speed distribution introduces a deformation parameter q that measures the deviation from the Maxwellian speed distribution and, by taking the limit q ! 1, the deformation is removed. In this context, and considering the Tsallis distribution, an integral form is obtained for a deformed Doppler Broadening Function in the scope of the single-level Breit-Wigner formalism and Bethe-Placzek approximations. This deformed Doppler Broadening Function reproduces the wellestablished conventional Doppler Broadening Function at the limit when q ! 1 in which the deformation is removed. In order to study the range of values where the effect is significant, numerical tests were carried out considering several values for the q parameter. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction In nuclear reactors, some neutrons can be absorbed in the resonance region, and in designing certain types of reactors, an accurate treatment of resonant absorptions is essential. The Doppler Broadening Phenomenon is adequately represented mathematically in the microscopic cross section of the neutron-nucleus interaction through the Doppler Broadening Function wðx; nÞ. Physically, this function describes the broadening phenomenon of the isolated resonances with the temperature, which causes an increase in the range of energies in which a neutron can be absorbed. Mathematically, wðx; nÞ can be interpreted as a convolution integral between a Gaussian and a Lorentzian function (Gonçalves et al., 2008). In the context of reactor physics, wðx; nÞ is well established, its properties being known, its analytical formulation being found in the literature (Palma et al., 2006). Function wðx; nÞ is obtained from the confluence of two large areas of knowledge, namely, Quantum Mechanics and Statistical Mechanics. In the scope of Quantum Mechanics, it is based on the 1936 result of Breit and Wigner
q
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⇑ Corresponding author.
https://doi.org/10.1016/j.anucene.2019.01.023 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
(1936), that describes the resonant part of the cross section for a state of natural width C:
rðEÞ ¼
C
h i; 2p ðE Er Þ2 þ ðC=2Þ2
ð1Þ
where E is the incident neutron energy and Er is the energy where the resonance occurs. Eq. (1) is a Lorentzian-type function and is still widely used. We can highlight some reasons for this, such as the fact that it is easy to implement, it is possible to use resonance parameters already published and abundant in the literature, and that it allows an analytical approximation of wðx; nÞ (Mamedov, 2009), and can be used analytically in reactor physics applications. Given the extensive set of advantages for its use, the BreitWigner formalism will be considered here, leaving the Quantum Mechanics formalism unchanged. The main focus of this paper will be on the change of the Statistical Mechanics theory. The content related to Statistical Mechanics consists of considering the target nuclei distribution as in a gas whose molecules vibrate with different velocities, these being distributed according to the MaxwellBoltzmann statistics (Pathria and Beale, 2011; Casquilho and Teixeira, 2012), which can be written in the following way:
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G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
f ðV; TÞ ¼
M 2pkB T
3=2
2
e
MV 2k T B
;
ð2Þ
where T is the absolute temperature, M is the target nucleus mass, ~ V is its velocity and kB is the so-called Boltzmann constant. Eq. (2) has a Gaussian behaviour and indicates that particles with high energies as well as those with very low energies are unlikely. In applications related to Pressurized Water Reactors (PWR), for example, during their normal operation, the target nuclei are in a thermal equilibrium situation such that the Maxwell-Boltzmann statistics can be satisfactorily considered, being used in different codes, such as NJOY (MacFarlane et al., 2012). However, it has been well known that for long-range interacting systems, as well as long-term temporal correlations systems (Tsallis, 1995; Tsallis, 1995), such statistics may be inadequate. Therefore, efforts to generalize the Maxwell-Gibs entropy concept, that generates the MaxwellBoltzmann speed distribution, have been made over the last decades, generating deformed statistics or non-Gaussian statistics, which are generalisations of the Maxwell-Boltzmann statistics meant to describe systems that have some restrictions to the applicability of usual Statistical Mechanics. In this context, in addition to Tsallis statistics, several others have been developed, such as those by Lévy (1925), Lévy (1937) and Kaniadakis (2001). The scheme shown in Fig. 1 shows how changes in quantum or statistical formalisms generate consequences in the broadening of the resonances. Non-extensive statistical mechanics and thermodynamics were introduced in 1988 (Tsallis, 1988), and further developed in 1991 (Curado and Tsallis, 1991) and 1998 (Tsallis and Mendes, 1998), to extend the range of applicability of statistical mechanical procedures to systems where the Boltzmann-Gibbs thermostatistics and standard thermodynamics present serious mathematical difficulties or simply fail. The Tsallis non-extensive distribution is based on a generalisation of the usual Boltzmann-Gibbs thermostatistics, with explicit functional dependency of a q parameter that measures the deviation in relation to the Gaussian behaviour of the system. Formally the q-distribution of velocities is written as Silva et al. (1998):
f q ðV; TÞ ¼ AðqÞexpq
! MV 2 ; 2kB T
ð3Þ
where:
AðqÞ ¼
M 2pkB T
3=2 ðq 1Þ
C 1=2 ð3q 1Þ ð1 þ qÞ 2
2
1 þ q1 ;
1 2
C
1 q1
and the q-exponential function is defined by: 1
expq ðzÞ ¼ ½1 þ ðq 1Þzq1 :
The q-parameter is such that q > 1 and if the limit q ! 1 in Eq. (3) is taken, it reproduces the standard Maxwell-Boltzmann distribution. In the same limit, Eq. (5) is reduced to the usual exponential function. It is possible to see that the positive nature of the power argument means that Eq. (3) has a thermal cut-off at the maximal allowed speeds. The components of the velocities lie within the ½L; L interval, where:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kB T : L¼ Mðq 1Þ
ð6Þ
This paper is organized as follows: Section 2 presents a brief review of the Doppler Broadening Effect considering a Maxwell-Boltzmann (MB) distribution of target nuclei velocities, Section 3 is dedicated to the development of a Doppler Broadening Function based on the Tsallis distribution, Section 4 focuses on the results obtained for the phenomena of the Doppler Broadening associated with the Tsallis distribution, Section 5 presents the conclusions. 2. The conventional Doppler Broadening Function In nuclear reactor cores, the target nuclei are usually considered to be in thermal equilibrium at a temperature T, their random movement ruled by the Maxwell-Boltzmann speed distribution (Pathria and Beale, 2011; Casquilho and Teixeira, 2012). The random nature of this motion generates relative speeds between neutron and nucleus that may be greater or smaller than the neutron speeds. This difference in relative speeds gives rise to the so-called Doppler Broadening Phenomenon. This way, the expressions for the cross-section of radioactive capture near any isolated resonance with an energy peak as given by the Single Level BreitWigner (SLBW) formalism is written as:
r c ðE; TÞ ¼ r0
Cc C
1=2 E0 Wðx; nÞ; E
where
ð4Þ
ð5Þ
Wðx; nÞ ¼ 2pn ffiffipffi
R þ1 2E
C 0
dy 1þy2
ð7Þ
v r ðyÞÞ2 þ exp ðv ðxÞ 2 2v th ð8Þ v r ðyÞÞ2 : exp ðv ðxÞþ 2v 2 th
The function defined by Eq. (8) is the so-called Doppler Broadening Function. In this expression, ~ v is the neutron velocity, ~ vr ¼ ~ v ~ V is the relative velocity between the neutron and nucleus movement, with v ðxÞ and v r ðyÞ being respectively their modules. The x and y variables are defined by the following expressions:
x¼ y¼
Fig. 1. Schematic representation for obtaining and applying the Doppler Broadening Function in the context of nuclear reactors physics.
2
C 2
C
ðE E0 Þ; ðECM E0 Þ;
ð10Þ
where C is the total width of the resonance as measured in lab coordinates, ECM is the two-body system energy in centre-of-mass coordinates, E0 is the energy where the resonance occurs and E is the incident neutron energy. The following also needs to be defined:
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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4E0 kB T ; CD ¼ A n¼
C ; CD
v th
rffiffiffiffiffiffiffiffi kB T ; ¼ M
ð11Þ
sffiffiffiffiffiffiffi 2E ðE þ ECM Þ E ECM pffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffi : v ðxÞ v r ðyÞ mn 2mn E 2mn E
ð12Þ
Thus, one can write the exponential function argument that appears in Eq. (17) as follows:
ð13Þ
where A is the target nucleus mass number, and v th is the thermal speed and also the scale parameter of the distribution. Due to the fact that the Doppler Broadening Function with a functional form such as given by Eq. (8) does not have an analytical solution and also has a complicated form, the possibility of making some approximations can be very useful. To deal with this problem, Bethe and Placzek, studied the resonance effects in nuclear processes and especially the Doppler Broadening Function, to suggest in their 1937 paper (Bethe and Placzek, 1937) some approximations for energies near the resonant peak, which can be presented as: 1. The second exponential in Eq. (8) is dismissed, that is, it is considered that:
½v ðxÞ þ v r ðyÞ2 ½v ðxÞ v r ðyÞ2 ;
ð14Þ
2. Based on the fact that the ratio between the energy of neutron incidence and the practical width is large, it is possible to extend the lower integration limit of the remaining integral to 1:
2
C
E0 ! 1;
ð15Þ
3. Using the fact that we are dealing with massive target nuclei, it is possible to write that:
1=2 pffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi ECM E ECM E : E 1þ ECM ¼ E 1 þ E 2E
ð16Þ
With the first two approximations, given by Eqs. (14) and (15), Eq. (8) takes the following shape:
n Wðx; nÞ pffiffiffiffi 2 p
Z
þ1 1
dy e 1 þ y2
ðv ðxÞv r ðyÞÞ2 2v 2 th
:
ð17Þ
To implement the third approximation, it is necessary to work a little on Eq. (16):
pffiffiffiffiffiffiffiffi pffiffiffiE þ ECM ðE þ ECM Þ pffiffiffi : ¼ ECM E 2E 2 E
Based on the fact that for M mn we have that leads us to:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lv 2r ðyÞ ðE þ ECM Þ ðE þ ECM Þ pffiffiffi ) v r ðyÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2mn E 2 E
ð18Þ
l mn , this
where it was considered that the two-body system energy in the centre-of-mass coordinates is given by:
ECM ¼
1 2 lv ðyÞ: 2 r
1 mn v 2 ðxÞ; 2
we may write:
ð21Þ
ð23Þ
where Eqs. (9) and (10) were used. Using Eqs. (11) and (12), we have:
½v ðxÞ v r ðyÞ2 n2 ðx yÞ2 : 4 2v 2th
ð24Þ
Finally, the following result for the Doppler Broadening Function with the Bethe and Placzek approximations, well established in the literature, is obtained:
n
Wðx; nÞ wðx; nÞ ¼ pffiffiffiffi 2
p
Z
þ1
1
dy n2 ðxyÞ2 e 4 : 1 þ y2
ð25Þ
Eq. (25) will be used as a comparison parameter for the Doppler Broadening Function based on the Tsallis speed distribution, which will be obtained in the next section. 3. The Tsallis Doppler Broadening Function In the 2017 paper (Guedes et al., 2017), G. Guedes et al., in their study of the Doppler Broadening Function in the scope of the Kaniadakis speed distribution (Kaniadakis, 2001), obtained the following expression for the Doppler Broadening Function for any even speed distribution:
rffiffiffiffiffiffiffiffiffiffiffi Z Z v ðxÞþv r ðyÞ þ1 2kB T dy n dVVfgen ðV; TÞ: 2 M C2 E0 1 þ y v ðxÞv r ðyÞ
Wgen ðx; nÞ ¼ p
ð26Þ
where subscript gen stands for generalized speed distribution. Given that the Tsallis distribution, as defined by Eqs. (3) and (4), preserves such parity, the cross-section of the radioactive capture near any isolated resonance in the scope of the Tsallis speed distribution (Silva et al., 1998) can be written thus:
r cq ðE; TÞ ¼ r0
Cc C
1=2 E0 Wq ðx; nÞ; E
ð27Þ
where Wq ðx; nÞ is the Tsallis Doppler Broadening Function, which can be obtained from Eq. (26) with f gen ðV; TÞ being replaced by f q ðV; TÞ, given Eq. (3). In order to obtain an expression for Wq ðx; nÞ, we have to take into account the cut-off speed defined by Eq. (6), as otherwise the interval of integration would contain some intervals where the Tsallis speed distribution function would not be well defined. So, we first perform the integration on the V variable:
v ðxÞv r ðyÞ
dVVfq ðV; TÞ ¼
Z v ðxÞþv r ðyÞ v ðxÞv r ðyÞ
dVVAðqÞexpq
! MV 2 : 2kB T
ð28Þ
To work with Eq. (28), one needs to obtain the indefinite integral of the expq function:
Z
Z
expq ðzÞdz ¼
ð20Þ
Then, using the fact that the neutron energy in lab coordinates is given by:
E¼
½v ðxÞ v r ðyÞ2 AC 2 ð x y Þ 2 ; 2 4EkB T 4 2v th
Z v ðxÞþv r ðyÞ ð19Þ
ð22Þ
1
½1 þ ðq 1Þzq1 dz:
ð29Þ
To perform such an integration, a simple substitution of the type u ¼ 1 þ ðq 1Þz is used, leading to:
Z
expq ðzÞdz ¼
1 ½1 þ ðq 1Þzexpq ðzÞ þ C: q
Defining the iexpq function through the expression:
ð30Þ
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G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421 Table 1 Legendre points xi and their respective weights xi .
Table 3 Non-zero percentage deviations between wBG ðn; xÞ and wðn; xÞ.
i
xi
xi
n
x
wBG ðn; xÞ
wðn; xÞ
%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.9879925 0.9372734 0.8482066 0.7244177 0.5709722 0.3941513 0.2011941 0.0000000 0.2011941 0.3941513 0.5709722 0.7244177 0.8482066 0.9372734 0.9879925
0.0307532 0.0703660 0.1071592 0.1395707 0.1662692 0.1861610 0.1984315 0.2025782 0.1984315 0.1861610 0.1662692 0.1395707 0.1071592 0.0703660 0.0307532
0.25 0.30 0.30 0.35 0.35 0.40 0.40 0.40
2 8 10 8 10 4 8 10
0.18324 0.07042 0.03880 0.05724 0.02815 0.17359 0.04566 0.02109
0.18325 0.07043 0.03881 0.05726 0.02816 0.17360 0.04569 0.02110
0.005 0.014 0.026 0.035 0.036 0.006 0.066 0.047
iexpq ðzÞ ¼
2 xq ¼ pffiffiffiffiffiffiffiffiffiffiffiffi : n q1 By defining BðqÞ as:
3=2 BðqÞ ¼ AðqÞ 2pMkB T ¼
1 ½1 þ ðq 1Þzexpq ðzÞ; q
1 1 þ 2 q1 ¼ 14 ðq 1Þ1=2 ð3q 1Þð1 þ qÞ 1 ðq1 Þ
ð31Þ
expq ðzÞdz ¼ iexpq ðzÞ þ C:
ð32Þ
After a simple substitution of the type u ¼ V 2 =2v 2th on Eq. (28), and the substitution of Eq. (32), the following is obtained:
R v ðxÞþv r ð yÞ ðv ðxÞv r ð yÞÞ2 þ dVVf ð V; T Þ ¼ A ð q Þiexp q q v ðxÞv r ð yÞ 2v 2th v r ð yÞÞ2 : AðqÞiexpq ðv ðxÞþ 2 2v
ð33Þ
th
By imposing the results which come from the Bethe and Placzek approximations, as given by Eqs. (14) and (24), Eq. (33) then becomes:
v ðxÞv r ðyÞ
"
n2 dVVfq ðV; TÞ AðqÞiexpq ðx yÞ2 4
#
ð34Þ
ð38Þ
we get to:
Z
Z v ðxÞþv r ðyÞ
ð37Þ
we can finally write the following expression for the Tsallis Doppler Broadening Function:
Wq ðx; nÞ wq ðx; nÞ ¼
h 2 i R xþx dy 2 n : ¼ 2pn ffiffipffi BðqÞ xxqq 1þy 2 iexpq 4 ð x yÞ
ð39Þ
Taking into account that function expq at the limit q ! 1 reproduces the usual exponential function, it is easy to realize, with the help of Eq. (31), that:
lim iexpq ðxÞ ¼ e x :
ð40Þ
q!1
By using the asymptotic limit of Gamma Function (Spiegel, 2013):
CðxÞ
pffiffiffiffiffiffiffi 2pxx1=2 ex ; x ! þ1;
ð41Þ
it can be seen directly that:
The iexpq function which appears in Eq. (34) is defined by Eq. (31) in terms of the expq function. Then, this argument must also satisfy the positive nature of the power argument wich gave rise to the result presented in Eq. (6). In that way, the definition given by Eq. (5) leads to the following:
1 4 : 1 þ ðq 1Þ ðx yÞ2 P 0 ) ðx yÞ2 6 2 4 n ðq 1Þ
ð35Þ
lim BðqÞ ¼ 1:
ð42Þ
q!1
Finally, Eq. (37) leads to:
lim xq ¼ þ1:
ð43Þ
q!1
With the results shown in Eqs. (40), (42) and (43) it is easy to see that:
lim wq ðx; nÞ ¼ wðx; nÞ;
This condition can be written as:
ð44Þ
q!1
x xq 6 y 6 x þ xq ;
ð36Þ
where
that is, if the limit q ! 1 is taken, the usual Doppler Broadening Function is recovered as expected.
Table 2 Doppler Broadening Function values using Eq. (25). The wðn; xÞ function n
x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.04309 0.08384 0.12239 0.15889 0.19347 0.22624 0.25731 0.28679 0.31477 0.34135
0.04308 0.08379 0.12223 0.15854 0.19281 0.22516 0.25569 0.28450 0.31168 0.33733
0.04306 0.08364 0.12176 0.15748 0.19086 0.22197 0.25091 0.27776 0.30261 0.32557
0.04298 0.08305 0.11989 0.15331 0.18325 0.20968 0.23271 0.25245 0.26909 0.28286
0.04267 0.08073 0.11268 0.13777 0.15584 0.16729 0.17288 0.17360 0.17052 0.16469
0.04216 0.07700 0.10165 0.11540 0.11934 0.11571 0.10713 0.09604 0.08439 0.07346
0.04145 0.07208 0.08805 0.09027 0.08277 0.07043 0.05726 0.04569 0.03670 0.03025
0.04055 0.06623 0.07328 0.06614 0.05253 0.03881 0.02816 0.02110 0.01687 0.01446
0.03380 0.03291 0.01695 0.00713 0.00394 0.00314 0.00289 0.00277 0.00270 0.00266
0.01639 0.00262 0.00080 0.00070 0.00067 0.00065 0.00064 0.00064 0.00064 0.00063
418
G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
to obtain the values for both the Doppler Broadening Function wðx; nÞ and the Interference function vðx; nÞ for several values of x and n parameters. These tables are reported in several textbooks (Lamarsh, 1966; Duderstadt and Hamilton, 1976; Stacey, 2007). It is widely known that the nuclear reactors in operation nowadays has their designs based on the Maxwell-Boltzmann distribution. Thus, for some operating condition beyond the design bases, or in new reactor designs that may be developed, it would be relevant to consider another kind of speed distribution, although not expecting a large deviation from the Maxwellian distribution. Thus, in order to study the influence of some deviation from the Maxwellian behaviour, the interval from the non-deformed case q ¼ 1:0 to q ¼ 1:5, with an increment of 0:1 will be considered. In this range of values it is possible to see the influence of considering the Tsallis distribution in the Doppler Broadening Phenomenon. Table 2 shows the calculated values for the Doppler Broadening Function, Eq. (25), using the 15th order Gaussian Quadrature method implemented in C language. From the values obtained, only eight of them differ from those presented in the table for the values available in Beynon and Grant (1963) and these, as well as the percentage deviation, are shown in Table 3, where the column for wBG ðn; xÞ represents the values for the Doppler Broadening Function obtained by Beynon and Grant in 1963 and the column for wðn; xÞ has the values calculated using the Gauss-Legendre quadrature method. Table 2 will be taken as a reference in this paper. By analyzing Table 3, one can see that the greatest deviation between wBG ðn; xÞ and wðn; xÞ corresponds to the value of 0:066% obtained for n ¼ 0:40 and x ¼ 8, that is, the deviations are negligible, and can be attributed to the uncertainties introduced by the numerical methods used.
The results obtained with this deformed Doppler Broadening Function as given by Eq. (39) will be reported in the next section. 4. Results In this section the results obtained for the Tsallis Doppler Broadening Function wq ðx; nÞ proposed in this paper will be reported. In order to obtain the results for the integral in Eq. (39), we used the Gauss-Legendre quadrature method (Burden and Faires, 2011). The N th order method consists of performing an approximation for a defined integral according to the expression:
Rb a
f ðxÞdx ¼ ba 2 ba 2
R 1 ba f 2 x þ bþa dx 1 2 N X
xi f
ba 2
ð45Þ
xi þ bþa ; 2
i¼1
where xi is the point of quadrature, and xi is its corresponding weight. The Gauss-Legendre quadrature points are the roots of the Legendre polynomial in the interval ½1; 1, as generated from the Rodrigues’ formula:
n i 1 d h 2 x 1 : n 2 n! dx n
Pn ðxÞ ¼
ð46Þ
n
In this paper a 15th order method was implemented, whose Legendre points xi and their respective weights are shown in Table 1. A traditional reference in the area of nuclear reactor physics lies in the tables obtained in 1963 by Beynon and Grant (1963), who used the Chebyshev Polinomial expansion (Arfken et al., 2013), and the Gauss-Hermite Method (Abramowitz and Stegun, 1972)
Table 4 Tsallis Doppler Broadening Function values for q ¼ 1:1 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.04663 0.09058 0.13202 0.17111 0.20802 0.24287 0.27580 0.30694 0.33639 0.36426
8.2 8.0 7.9 7.7 7.5 7.4 7.2 7.0 6.9 6.7
0.04662 0.09052 0.13183 0.17069 0.20724 0.24160 0.27389 0.30423 0.33274 0.35952
8.2 8.0 7.9 7.7 7.5 7.3 7.1 6.9 6.8 6.6
0.04660 0.09034 0.13127 0.16943 0.20492 0.23781 0.26822 0.29626 0.32203 0.34566
8.2 8.0 7.8 7.6 7.4 7.1 6.9 6.7 6.4 6.2
0.04651 0.08964 0.12904 0.16449 0.19589 0.22326 0.24670 0.26639 0.28256 0.29548
8.2 7.9 7.6 7.3 6.9 6.5 6.0 5.5 5.0 4.5
0.04614 0.08688 0.12048 0.14604 0.16346 0.17328 0.17649 0.17439 0.16834 0.15967
8.1 7.6 6.9 6.0 4.9 3.6 2.1 0.5 1.3 3.0
0.04552 0.08245 0.10739 0.11962 0.12068 0.11349 0.10146 0.08766 0.07439 0.06297
8.0 7.1 5.6 3.7 1.1 1.9 5.3 8.7 11.8 14.3
0.04467 0.07660 0.09131 0.09025 0.07879 0.06322 0.04844 0.03697 0.02925 0.02452
7.8 6.3 3.7 0.0 4.8 10.2 15.4 19.1 20.3 18.9
0.04361 0.06965 0.07398 0.06266 0.04577 0.03106 0.02148 0.01635 0.01388 0.01266
7.5 5.2 1.0 5.3 12.9 20.0 23.7 22.5 17.7 12.4
0.03558 0.03084 0.01242 0.00462 0.00322 0.00292 0.00278 0.00270 0.00266 0.00262
5.3 6.3 26.7 35.2 18.3 7.0 3.8 2.5 1.5 1.5
0.01526 0.00135 0.00073 0.00068 0.00066 0.00065 0.00064 0.00064 0.00063 0.00063
6.9 48.5 8.8 2.9 1.5 0.0 0.0 0.0 1.6 0.0
Table 5 Tsallis Doppler Broadening Function values for q ¼ 1:2 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.04992 0.09680 0.14087 0.18230 0.22129 0.25798 0.29254 0.32510 0.35580 0.38476
15.9 15.5 15.1 14.7 14.4 14.0 13.7 13.4 13.0 12.7
0.04991 0.09673 0.14065 0.18181 0.22038 0.25650 0.29032 0.32196 0.35158 0.37927
15.9 15.4 15.1 14.7 14.3 13.9 13.5 13.2 12.8 12.4
0.04988 0.09653 0.13999 0.18035 0.21768 0.25210 0.28374 0.31271 0.33917 0.36325
15.8 15.4 15.0 14.5 14.1 13.6 13.1 12.6 12.1 11.6
0.04977 0.09571 0.13739 0.17458 0.20717 0.23519 0.25878 0.27815 0.29360 0.30546
15.8 15.2 14.6 13.9 13.1 12.2 11.2 10.2 9.1 8.0
0.04934 0.09248 0.12741 0.15312 0.16955 0.17742 0.17803 0.17300 0.16404 0.15279
15.6 14.6 13.1 11.1 8.8 6.1 3.0 0.3 3.8 7.2
0.04862 0.08731 0.11218 0.12253 0.12041 0.10962 0.09443 0.07862 0.06472 0.05392
15.3 13.4 10.4 6.2 0.9 5.3 11.9 18.1 23.3 26.6
0.04763 0.08049 0.09354 0.08891 0.07356 0.05542 0.04022 0.03011 0.02444 0.02148
14.9 11.7 6.2 1.5 11.1 21.3 29.8 34.1 33.4 29.0
0.04638 0.07240 0.07362 0.05812 0.03872 0.02443 0.01705 0.01400 0.01267 0.01195
14.4 9.3 0.5 12.1 26.3 37.1 39.5 33.6 24.9 17.4
0.03702 0.02816 0.00858 0.00360 0.00302 0.00283 0.00272 0.00266 0.00263 0.00260
9.5 14.4 49.4 49.5 23.4 9.9 5.9 4.0 2.6 2.3
wq ðn; xÞ
%
0.01381 15.7 0.00091 65.3 0.00071 11.3 0.00067 4.3 0.00065 3.0 0.00064 1.5 0.00064 0.0 0.00063 1.6 0.00063 1.6 0.00063 0.0
419
G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421 Table 6 Tsallis Doppler Broadening Function values for q ¼ 1:3 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.05298 0.10260 0.14908 0.19265 0.23351 0.27185 0.30785 0.34166 0.37343 0.40331
23.0 22.4 21.8 21.2 20.7 20.2 19.6 19.1 18.6 18.2
0.05297 0.10252 0.14883 0.19209 0.23247 0.27016 0.30530 0.33807 0.36861 0.39706
23.0 22.4 21.8 21.2 20.6 20.0 19.4 18.8 18.3 17.7
0.05294 0.10228 0.14808 0.19040 0.22938 0.26512 0.29778 0.32751 0.35447 0.37882
22.9 22.3 21.6 20.9 20.2 19.4 18.7 17.9 17.1 16.4
0.05282 0.10134 0.14509 0.18379 0.21734 0.24578 0.26929 0.28813 0.30268 0.31332
22.9 22.0 21.0 19.9 18.6 17.2 15.7 14.1 12.5 10.8
0.05232 0.09763 0.13363 0.15921 0.17437 0.18006 0.17792 0.16995 0.15827 0.14482
22.6 20.9 18.6 15.6 11.9 7.6 2.9 2.1 7.2 12.1
0.05149 0.09169 0.11617 0.12433 0.11886 0.10453 0.08666 0.06970 0.05626 0.04700
22.1 19.1 14.3 7.7 0.4 9.7 19.1 27.4 33.3 36.0
0.05035 0.08386 0.09491 0.08651 0.06751 0.04770 0.03338 0.02555 0.02186 0.02000
21.5 16.3 7.8 4.2 18.4 32.3 41.7 44.1 40.4 33.9
0.04892 0.07460 0.07240 0.05286 0.03199 0.01953 0.01472 0.01295 0.01208 0.01155
20.6 12.6 1.2 20.1 39.1 49.7 47.7 38.6 28.4 20.1
0.03816 0.02506 0.00584 0.00329 0.00292 0.00277 0.00269 0.00264 0.00261 0.00258
12.9 23.9 65.5 53.9 25.9 11.8 6.9 4.7 3.3 3.0
wq ðn; xÞ
%
0.01212 26.1 0.00083 68.3 0.00069 13.8 0.00066 5.7 0.00065 3.0 0.00064 1.5 0.00064 0.0 0.00063 1.6 0.00063 1.6 0.00063 0.0
Table 7 Tsallis Doppler Broadening Function values for q ¼ 1:4 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.05587 0.10804 0.15676 0.20230 0.24487 0.28469 0.32197 0.35687 0.38958 0.42024
29.7 28.9 28.1 27.3 26.6 25.8 25.1 24.4 23.8 23.1
0.05586 0.10795 0.15648 0.20166 0.24369 0.28278 0.31910 0.35283 0.38415 0.41320
29.7 28.8 28.0 27.2 26.4 25.6 24.8 24.0 23.3 22.5
0.05583 0.10768 0.15562 0.19975 0.24019 0.27708 0.31060 0.34092 0.36823 0.39271
29.7 28.7 27.8 26.8 25.8 24.8 23.8 22.7 21.7 20.6
0.05568 0.10661 0.15223 0.19226 0.22657 0.25524 0.27848 0.29663 0.31012 0.31943
29.5 28.4 27.0 25.4 23.6 21.7 19.7 17.5 15.2 12.9
0.05511 0.10240 0.13924 0.16445 0.17811 0.18145 0.17647 0.16564 0.15151 0.13636
29.2 26.8 23.6 19.4 14.3 8.5 2.1 4.6 11.1 17.2
0.05418 0.09566 0.11949 0.12521 0.11627 0.09860 0.07868 0.06156 0.04956 0.04228
28.5 24.2 17.6 8.5 2.6 14.8 26.6 35.9 41.3 42.4
0.05288 0.08678 0.09555 0.08327 0.06102 0.04065 0.02838 0.02293 0.02048 0.01914
27.6 20.4 8.5 7.8 26.3 42.3 50.4 49.8 44.2 36.7
0.05125 0.07630 0.07046 0.04720 0.02614 0.01654 0.01359 0.01237 0.01171 0.01129
26.4 15.2 3.8 28.6 50.2 57.4 51.7 41.4 30.6 21.9
0.03905 0.02170 0.00440 0.00314 0.00285 0.00273 0.00266 0.00262 0.00259 0.00257
15.5 34.1 74.0 56.0 27.7 13.1 8.0 5.4 4.1 3.4
wq ðn; xÞ
%
0.01029 37.2 0.00079 69.8 0.00068 15.0 0.00066 5.7 0.00064 4.5 0.00064 1.5 0.00063 1.6 0.00063 1.6 0.00063 1.6 0.00063 0.0
Table 8 Tsallis Doppler Broadening Function values for q ¼ 1:5 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.05861 0.11317 0.16399 0.21134 0.25548 0.29666 0.33508 0.37096 0.40448 0.43581
36.0 35.0 34.0 33.0 32.1 31.1 30.2 29.3 28.5 27.7
0.05860 0.11307 0.16367 0.21062 0.25417 0.29451 0.33187 0.36645 0.39842 0.42798
36.0 34.9 33.9 32.8 31.8 30.8 29.8 28.8 27.8 26.9
0.05856 0.11277 0.16271 0.20848 0.25024 0.28814 0.32239 0.35316 0.38069 0.40519
36.0 34.8 33.6 32.4 31.1 29.8 28.5 27.1 25.8 24.5
0.05840 0.11156 0.15891 0.20008 0.23499 0.26373 0.28656 0.30388 0.31619 0.32408
35.9 34.3 32.5 30.5 28.2 25.8 23.1 20.4 17.5 14.6
0.05776 0.10683 0.14434 0.16896 0.18093 0.18177 0.17395 0.16042 0.14422 0.12793
35.4 32.3 28.1 22.6 16.1 8.7 0.6 7.6 15.4 22.3
0.05670 0.09927 0.12223 0.12530 0.11283 0.09214 0.07094 0.05469 0.04473 0.03926
34.5 28.9 20.2 8.6 5.5 20.4 33.8 43.1 47.0 46.6
0.05525 0.08932 0.09556 0.07936 0.05444 0.03472 0.02518 0.02143 0.01962 0.01857
33.3 23.9 8.5 12.1 34.2 50.7 56.0 53.1 46.5 38.6
0.05341 0.07758 0.06792 0.04144 0.02158 0.01495 0.01293 0.01199 0.01145 0.01111
31.7 17.1 7.3 37.3 58.9 61.5 54.1 43.2 32.1 23.2
0.03972 0.01824 0.00384 0.00304 0.00280 0.00270 0.00264 0.00260 0.00258 0.00256
17.5 44.6 77.3 57.4 28.9 14.0 8.7 6.1 4.4 3.8
In Tables 4–8, the values for the Tsallis Doppler Broadening Function calculated using Eqs. (39) and (45) and their respective deviations from the values of the usual Doppler Broadening Function shown in Table 2 can be found for several values of the q parameter. Figs. 2–6 show, the Tsallis Doppler Broadening Function defined by Eq. (39) for several values of the q parameter as well as the Maxwellian Doppler Broadening Function given by Eq. (25) considering some values for the n parameter. Analyzing the graphs above, it can be seen that as the deformation parameter q increases, the peak of the Doppler Broadening Function gets higher, that is, closer to the peak value 1 given by the Single Level Breit-Wigner formalism, for the non-broadened Function. In other words, the introduction of the deformation parameter q leads to an attenuation of the Doppler Broadening Effect.
wq ðn; xÞ
%
0.00840 48.7 0.00076 71.0 0.00068 15.0 0.00065 7.1 0.00064 4.5 0.00064 1.5 0.00063 1.6 0.00063 1.6 0.00063 1.6 0.00063 0.0
5. Conclusions This paper tackles the consequences of considering the targetnuclei random movement as given by the non-extensive Tsallis distribution in the evaluation of the absorption cross section in three-dimension. The Tsallis distribution can be mathematically formulated as a deformation of the usual Maxwell-Boltzmann distribution, which is dependent on a q > 1 parameter and, if the q ! 1 limit is taken, the deformation is removed and the usual Maxwell-Boltzmann distribution is recovered. In this context, a new deformed Doppler Broadening Function called wq ðx; nÞ as defined by Eq. (39), dependent on a deformation parameter q, was obtained, and when the q ! 1 limit is taken, the usual Doppler Broadening Function is retrieved. This new expression, which presents in its form an integral that, as in the Maxwellian case, does
420
G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
Fig. 2. Tsallis Doppler Broadening Function for n ¼ 0:05.
Fig. 4. Tsallis Doppler Broadening Function for n ¼ 0:25.
Fig. 3. Tsallis Doppler Broadening Function for n ¼ 0:15.
not have an analytical solution, was evaluated using the 15th order Gauss-Legendre quadrature method, a numerical method widely used and well established in the literature. With the said numerical method, Table 2 was constructed, reproducing the results obtained by Beynon and Grant (1963) which were taken as a comparison parameter for this paper. Tables 4–8, varying the parameter q with a 0:1 increment between the q ¼ 1:1 and q ¼ 1:5 values, where the percentage deviation from the usual Doppler Broadening Function was also obtained. Still using the Gaussian quadrature method, the graphs shown in Figs. 2–6 were generated where, for each of them, by making the parameter n dependent on the temperature constant, it is possible to visualise the behaviour of the Doppler Broadening Function as it varies the deformation parameter q. In analysing the graphs shown herein, it is possible to see that, for a given resonance, if parameter n is kept constant, the Doppler Broadening Effect is deformed and depends on parameter q, and it is also possible to see that, as q rises, the Doppler Broadening Effect is softened, that is, the deformation parameter q makes the Doppler Broadening Effect smoother. The profile change for the
Fig. 5. Tsallis Doppler Broadening Function for n ¼ 0:35.
Doppler Broadening Function can affect both the resonant absorption and the resonant scattering, in which case the interference term also needs to be evaluated. In PWR reactors, the theory based on the distribution of Maxwell Boltzmann is well established, and has been applied through decades producing good agreement with experimental data. However, it is also widely known that precise and accurate cross section data is of fundamental importance for the operation of nuclear reactors, and it is possible that in new reactor designs as well as
G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
421
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.anucene.2019.01.023. References
Fig. 6. Tsallis Doppler Broadening Function for n ¼ 0:45.
in specific situations for the reactors currently in operation, the use of non-Maxwellian statistics will become important. The present paper, together with the previous one by Guedes et al. (2017), considering the Kaniadakis statistics, form a theoretical basis to study any quasi-Maxwellian statistics that preserves an even parity in variable V. Thus, if in the study of the influence of quasi-Maxwellian distributions on the Doppler Broadening Phenomenon, the importance of using a distribution other than Tsallis or Kaniadakis becomes evident, it is reasonable to assume that the theoretical framework for the treatment with any other distributions has been paved. Acknowledgments This research project is supported by the following Brazilian institutions: Brazilian Council for Scientific and Technological Development (CNPq), Brazilian Nuclear Energy Commission (CNEN), and Research Support Foundation of the State of Rio de Janeiro (FAPERJ).
Gonçalves, A.C., Martinez, A.S., Silva, F.C., 2008. Solution of the doppler broadening function based on the fourier cosine transform. Ann. Nucl. Energy 35, 1878– 1881. https://doi.org/10.1016/j.anucene.2008.04.003. Palma, D.A., Martinez, A.S., Silva, F.C., 2006. The derivation of the doppler broadening function using frobenius method. J. Nucl. Sci. Technol. 43 (6), 617–622. https://doi.org/10.1080/18811248.2006.9711141. Breit, G., Wigner, E., 1936. Capture of slow neutrons. Phys. Rev. (Ser. I) 49, 519–531. https://doi.org/10.1103/PhysRev. 49.519. Mamedov, B.A., 2009. Analytical evaluation of doppler functions arising from resonance effects in nuclear processes. Nucl. Instrum. Methods Phys. Res. Sect. A 608, 336–338. https://doi.org/10.1016/j.nima.2009.07.008. Pathria, R.K., Beale, P.D., 2011. Statistical Mechanics. Elsevier, USA. Casquilho, J.A.P., Teixeira, I.C., 2012. Introdução á Física Estatística. Livraria da Física, São Paulo, Brasil. MacFarlane, R.E., Muir, R.M., Boicourt, D.W., Kahler, A.C., 2012. The NJOY Nuclear Data Processing System. Version 2012. Tsallis, C., 1995. Non-extensive thermostatistics: brief review and comments. Physica A 221, 277–290. https://doi.org/10.1016/0378-4371(95)00236-Z. Tsallis, C., 1995. Stock exchange: a statistical model. Chaos Solitons Fractals 6, 561– 567. https://doi.org/10.1016/0960-0779(95)80063-M. Lévy, P.P., 1925. Calcul des Probabilites. Gauthier Villars, Paris. Lévy, P.P., 1937. Thóeorie del’addition des Variables Aléatoires. Gauthier Villars, Paris. Kaniadakis, G., 2001. Non-linear kinetics underlying generalized statistics. Physica A 296, 405–425. https://doi.org/10.1016/S0378-4371(01)00184-4. Tsallis, C., 1988. Possible gereralization of boltzmann-gibbs statistics. J. Stat. Phys. 52, 479–487. https://doi.org/10.1007/BF01016429. Curado, E.M.F., Tsallis, C., 1991. Generalized statistical mechanics: connection with thermodynamics. J. Phys. A: Math. General 24 (2), L69–L72. https://doi.org/ 10.1088/0305-4470/24/2/004. Tsallis, C., Mendes, R.S.P.A.R., 1998. The role of constraints within generalized nonextensive statistics. Physica A: Stat. Mech. Appl. 261, 534–554. https://doi. org/10.1016/S0378-4371(98)00437-3. Silva Jr., R., Plastino, A.R., Lima, J., 1998. A maxwellian path to the q-nonextensive velocity distribution function. Phys. Lett. A 249, 401–408. https://doi.org/ 10.1016/S0375-9601(98)00710-5. Bethe, H.A., Placzek, G., 1937. Resonance effects in nuclear processes. Phys. Rev. (Ser. I) 51, 450–484. https://doi.org/10.1103/PhysRev. 51.450. Guedes, G., Gonçalves, A.C., Palma, D.A.P., 2017. The doppler broadening function using the kaniadakis distribution. Ann. Nucl. Energy 110, 453–458. https://doi. org/10.1016/j.anucene.2017.06.057. Spiegel, M.R., 2013. Variáveis Complexas. McGraw-Hill, São Paulo, Brasil. Burden, R.L., Faires, J.D., 2011. Numerical Analysis. Brooks Cole, Boston, USA. Beynon, T.D., Grant, I.S., 1963. Evaluation of the doppler-broadened single-level and interference functions. Nucl. Sci. Eng. 17 (4), 547–550. https://doi.org/ 10.13182/NSE63-A18446. Arfken, G., Weber, H., Harris, F., 2013. Mathematical Methods for Physicists: A Comprehensive Guide. Elsevier, New York, USA. Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions. Dover, New York, USA. Lamarsh, J.R., 1966. Introduction to Nuclear Reactor Theory. Addison-Wesley, New York, USA. Duderstadt, J.J., Hamilton, L.J., 1976. Nuclear Reactor Analisys. John Wiley & Sons, New York, USA. Stacey, W.M., 2007. Nuclear Reactor Physics. John Wiley & Sons, Weinheim, Deutschland.
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A deformed Doppler Broadening Function considering the Tsallis speed distribution q Guilherme Guedes a,b,⇑, Daniel A.P. Palma c, Alessandro C. Gonçalves b a
Federal Center for Technological Education Celso Suckow da Fonseca – CEFET/RJ, Nova Friburgo, Brazil Department of Nuclear Engineering, Federal University of Rio de Janeiro – PEN/COPPE/UFRJ, Rio de Janeiro, Brazil c Brazilian Nuclear Energy Commission – CNEN, Rio de Janeiro, Brazil b
a r t i c l e
i n f o
Article history: Received 1 August 2018 Received in revised form 14 December 2018 Accepted 14 January 2019
Keywords: Deformed Doppler Broadening Function Quasi-Maxwellian distribution Tsallis distribution
a b s t r a c t The Doppler Broadening Phenomenon provides a good description for the effects of thermal agitation on the cross-section of the neutron-nucleus interaction and has its theoretical basis on Quantum Mechanics through the Single Level Breit-Wigner Formalism, as well as on Statistical Mechanics, with the Maxwell Boltzmann distribution. In this paper, the consequences of replacing the Maxwell Boltzmann speed distribution by the non-extensive Tsallis distribution in the evaluation of the absorption cross section in three-dimension is discussed. The Tsallis speed distribution introduces a deformation parameter q that measures the deviation from the Maxwellian speed distribution and, by taking the limit q ! 1, the deformation is removed. In this context, and considering the Tsallis distribution, an integral form is obtained for a deformed Doppler Broadening Function in the scope of the single-level Breit-Wigner formalism and Bethe-Placzek approximations. This deformed Doppler Broadening Function reproduces the wellestablished conventional Doppler Broadening Function at the limit when q ! 1 in which the deformation is removed. In order to study the range of values where the effect is significant, numerical tests were carried out considering several values for the q parameter. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction In nuclear reactors, some neutrons can be absorbed in the resonance region, and in designing certain types of reactors, an accurate treatment of resonant absorptions is essential. The Doppler Broadening Phenomenon is adequately represented mathematically in the microscopic cross section of the neutron-nucleus interaction through the Doppler Broadening Function wðx; nÞ. Physically, this function describes the broadening phenomenon of the isolated resonances with the temperature, which causes an increase in the range of energies in which a neutron can be absorbed. Mathematically, wðx; nÞ can be interpreted as a convolution integral between a Gaussian and a Lorentzian function (Gonçalves et al., 2008). In the context of reactor physics, wðx; nÞ is well established, its properties being known, its analytical formulation being found in the literature (Palma et al., 2006). Function wðx; nÞ is obtained from the confluence of two large areas of knowledge, namely, Quantum Mechanics and Statistical Mechanics. In the scope of Quantum Mechanics, it is based on the 1936 result of Breit and Wigner
q
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⇑ Corresponding author.
https://doi.org/10.1016/j.anucene.2019.01.023 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
(1936), that describes the resonant part of the cross section for a state of natural width C:
rðEÞ ¼
C
h i; 2p ðE Er Þ2 þ ðC=2Þ2
ð1Þ
where E is the incident neutron energy and Er is the energy where the resonance occurs. Eq. (1) is a Lorentzian-type function and is still widely used. We can highlight some reasons for this, such as the fact that it is easy to implement, it is possible to use resonance parameters already published and abundant in the literature, and that it allows an analytical approximation of wðx; nÞ (Mamedov, 2009), and can be used analytically in reactor physics applications. Given the extensive set of advantages for its use, the BreitWigner formalism will be considered here, leaving the Quantum Mechanics formalism unchanged. The main focus of this paper will be on the change of the Statistical Mechanics theory. The content related to Statistical Mechanics consists of considering the target nuclei distribution as in a gas whose molecules vibrate with different velocities, these being distributed according to the MaxwellBoltzmann statistics (Pathria and Beale, 2011; Casquilho and Teixeira, 2012), which can be written in the following way:
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G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
f ðV; TÞ ¼
M 2pkB T
3=2
2
e
MV 2k T B
;
ð2Þ
where T is the absolute temperature, M is the target nucleus mass, ~ V is its velocity and kB is the so-called Boltzmann constant. Eq. (2) has a Gaussian behaviour and indicates that particles with high energies as well as those with very low energies are unlikely. In applications related to Pressurized Water Reactors (PWR), for example, during their normal operation, the target nuclei are in a thermal equilibrium situation such that the Maxwell-Boltzmann statistics can be satisfactorily considered, being used in different codes, such as NJOY (MacFarlane et al., 2012). However, it has been well known that for long-range interacting systems, as well as long-term temporal correlations systems (Tsallis, 1995; Tsallis, 1995), such statistics may be inadequate. Therefore, efforts to generalize the Maxwell-Gibs entropy concept, that generates the MaxwellBoltzmann speed distribution, have been made over the last decades, generating deformed statistics or non-Gaussian statistics, which are generalisations of the Maxwell-Boltzmann statistics meant to describe systems that have some restrictions to the applicability of usual Statistical Mechanics. In this context, in addition to Tsallis statistics, several others have been developed, such as those by Lévy (1925), Lévy (1937) and Kaniadakis (2001). The scheme shown in Fig. 1 shows how changes in quantum or statistical formalisms generate consequences in the broadening of the resonances. Non-extensive statistical mechanics and thermodynamics were introduced in 1988 (Tsallis, 1988), and further developed in 1991 (Curado and Tsallis, 1991) and 1998 (Tsallis and Mendes, 1998), to extend the range of applicability of statistical mechanical procedures to systems where the Boltzmann-Gibbs thermostatistics and standard thermodynamics present serious mathematical difficulties or simply fail. The Tsallis non-extensive distribution is based on a generalisation of the usual Boltzmann-Gibbs thermostatistics, with explicit functional dependency of a q parameter that measures the deviation in relation to the Gaussian behaviour of the system. Formally the q-distribution of velocities is written as Silva et al. (1998):
f q ðV; TÞ ¼ AðqÞexpq
! MV 2 ; 2kB T
ð3Þ
where:
AðqÞ ¼
M 2pkB T
3=2 ðq 1Þ
C 1=2 ð3q 1Þ ð1 þ qÞ 2
2
1 þ q1 ;
1 2
C
1 q1
and the q-exponential function is defined by: 1
expq ðzÞ ¼ ½1 þ ðq 1Þzq1 :
The q-parameter is such that q > 1 and if the limit q ! 1 in Eq. (3) is taken, it reproduces the standard Maxwell-Boltzmann distribution. In the same limit, Eq. (5) is reduced to the usual exponential function. It is possible to see that the positive nature of the power argument means that Eq. (3) has a thermal cut-off at the maximal allowed speeds. The components of the velocities lie within the ½L; L interval, where:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kB T : L¼ Mðq 1Þ
ð6Þ
This paper is organized as follows: Section 2 presents a brief review of the Doppler Broadening Effect considering a Maxwell-Boltzmann (MB) distribution of target nuclei velocities, Section 3 is dedicated to the development of a Doppler Broadening Function based on the Tsallis distribution, Section 4 focuses on the results obtained for the phenomena of the Doppler Broadening associated with the Tsallis distribution, Section 5 presents the conclusions. 2. The conventional Doppler Broadening Function In nuclear reactor cores, the target nuclei are usually considered to be in thermal equilibrium at a temperature T, their random movement ruled by the Maxwell-Boltzmann speed distribution (Pathria and Beale, 2011; Casquilho and Teixeira, 2012). The random nature of this motion generates relative speeds between neutron and nucleus that may be greater or smaller than the neutron speeds. This difference in relative speeds gives rise to the so-called Doppler Broadening Phenomenon. This way, the expressions for the cross-section of radioactive capture near any isolated resonance with an energy peak as given by the Single Level BreitWigner (SLBW) formalism is written as:
r c ðE; TÞ ¼ r0
Cc C
1=2 E0 Wðx; nÞ; E
where
ð4Þ
ð5Þ
Wðx; nÞ ¼ 2pn ffiffipffi
R þ1 2E
C 0
dy 1þy2
ð7Þ
v r ðyÞÞ2 þ exp ðv ðxÞ 2 2v th ð8Þ v r ðyÞÞ2 : exp ðv ðxÞþ 2v 2 th
The function defined by Eq. (8) is the so-called Doppler Broadening Function. In this expression, ~ v is the neutron velocity, ~ vr ¼ ~ v ~ V is the relative velocity between the neutron and nucleus movement, with v ðxÞ and v r ðyÞ being respectively their modules. The x and y variables are defined by the following expressions:
x¼ y¼
Fig. 1. Schematic representation for obtaining and applying the Doppler Broadening Function in the context of nuclear reactors physics.
2
C 2
C
ðE E0 Þ; ðECM E0 Þ;
ð10Þ
where C is the total width of the resonance as measured in lab coordinates, ECM is the two-body system energy in centre-of-mass coordinates, E0 is the energy where the resonance occurs and E is the incident neutron energy. The following also needs to be defined:
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G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4E0 kB T ; CD ¼ A n¼
C ; CD
v th
rffiffiffiffiffiffiffiffi kB T ; ¼ M
ð11Þ
sffiffiffiffiffiffiffi 2E ðE þ ECM Þ E ECM pffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffi : v ðxÞ v r ðyÞ mn 2mn E 2mn E
ð12Þ
Thus, one can write the exponential function argument that appears in Eq. (17) as follows:
ð13Þ
where A is the target nucleus mass number, and v th is the thermal speed and also the scale parameter of the distribution. Due to the fact that the Doppler Broadening Function with a functional form such as given by Eq. (8) does not have an analytical solution and also has a complicated form, the possibility of making some approximations can be very useful. To deal with this problem, Bethe and Placzek, studied the resonance effects in nuclear processes and especially the Doppler Broadening Function, to suggest in their 1937 paper (Bethe and Placzek, 1937) some approximations for energies near the resonant peak, which can be presented as: 1. The second exponential in Eq. (8) is dismissed, that is, it is considered that:
½v ðxÞ þ v r ðyÞ2 ½v ðxÞ v r ðyÞ2 ;
ð14Þ
2. Based on the fact that the ratio between the energy of neutron incidence and the practical width is large, it is possible to extend the lower integration limit of the remaining integral to 1:
2
C
E0 ! 1;
ð15Þ
3. Using the fact that we are dealing with massive target nuclei, it is possible to write that:
1=2 pffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi ECM E ECM E : E 1þ ECM ¼ E 1 þ E 2E
ð16Þ
With the first two approximations, given by Eqs. (14) and (15), Eq. (8) takes the following shape:
n Wðx; nÞ pffiffiffiffi 2 p
Z
þ1 1
dy e 1 þ y2
ðv ðxÞv r ðyÞÞ2 2v 2 th
:
ð17Þ
To implement the third approximation, it is necessary to work a little on Eq. (16):
pffiffiffiffiffiffiffiffi pffiffiffiE þ ECM ðE þ ECM Þ pffiffiffi : ¼ ECM E 2E 2 E
Based on the fact that for M mn we have that leads us to:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lv 2r ðyÞ ðE þ ECM Þ ðE þ ECM Þ pffiffiffi ) v r ðyÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2mn E 2 E
ð18Þ
l mn , this
where it was considered that the two-body system energy in the centre-of-mass coordinates is given by:
ECM ¼
1 2 lv ðyÞ: 2 r
1 mn v 2 ðxÞ; 2
we may write:
ð21Þ
ð23Þ
where Eqs. (9) and (10) were used. Using Eqs. (11) and (12), we have:
½v ðxÞ v r ðyÞ2 n2 ðx yÞ2 : 4 2v 2th
ð24Þ
Finally, the following result for the Doppler Broadening Function with the Bethe and Placzek approximations, well established in the literature, is obtained:
n
Wðx; nÞ wðx; nÞ ¼ pffiffiffiffi 2
p
Z
þ1
1
dy n2 ðxyÞ2 e 4 : 1 þ y2
ð25Þ
Eq. (25) will be used as a comparison parameter for the Doppler Broadening Function based on the Tsallis speed distribution, which will be obtained in the next section. 3. The Tsallis Doppler Broadening Function In the 2017 paper (Guedes et al., 2017), G. Guedes et al., in their study of the Doppler Broadening Function in the scope of the Kaniadakis speed distribution (Kaniadakis, 2001), obtained the following expression for the Doppler Broadening Function for any even speed distribution:
rffiffiffiffiffiffiffiffiffiffiffi Z Z v ðxÞþv r ðyÞ þ1 2kB T dy n dVVfgen ðV; TÞ: 2 M C2 E0 1 þ y v ðxÞv r ðyÞ
Wgen ðx; nÞ ¼ p
ð26Þ
where subscript gen stands for generalized speed distribution. Given that the Tsallis distribution, as defined by Eqs. (3) and (4), preserves such parity, the cross-section of the radioactive capture near any isolated resonance in the scope of the Tsallis speed distribution (Silva et al., 1998) can be written thus:
r cq ðE; TÞ ¼ r0
Cc C
1=2 E0 Wq ðx; nÞ; E
ð27Þ
where Wq ðx; nÞ is the Tsallis Doppler Broadening Function, which can be obtained from Eq. (26) with f gen ðV; TÞ being replaced by f q ðV; TÞ, given Eq. (3). In order to obtain an expression for Wq ðx; nÞ, we have to take into account the cut-off speed defined by Eq. (6), as otherwise the interval of integration would contain some intervals where the Tsallis speed distribution function would not be well defined. So, we first perform the integration on the V variable:
v ðxÞv r ðyÞ
dVVfq ðV; TÞ ¼
Z v ðxÞþv r ðyÞ v ðxÞv r ðyÞ
dVVAðqÞexpq
! MV 2 : 2kB T
ð28Þ
To work with Eq. (28), one needs to obtain the indefinite integral of the expq function:
Z
Z
expq ðzÞdz ¼
ð20Þ
Then, using the fact that the neutron energy in lab coordinates is given by:
E¼
½v ðxÞ v r ðyÞ2 AC 2 ð x y Þ 2 ; 2 4EkB T 4 2v th
Z v ðxÞþv r ðyÞ ð19Þ
ð22Þ
1
½1 þ ðq 1Þzq1 dz:
ð29Þ
To perform such an integration, a simple substitution of the type u ¼ 1 þ ðq 1Þz is used, leading to:
Z
expq ðzÞdz ¼
1 ½1 þ ðq 1Þzexpq ðzÞ þ C: q
Defining the iexpq function through the expression:
ð30Þ
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G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421 Table 1 Legendre points xi and their respective weights xi .
Table 3 Non-zero percentage deviations between wBG ðn; xÞ and wðn; xÞ.
i
xi
xi
n
x
wBG ðn; xÞ
wðn; xÞ
%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.9879925 0.9372734 0.8482066 0.7244177 0.5709722 0.3941513 0.2011941 0.0000000 0.2011941 0.3941513 0.5709722 0.7244177 0.8482066 0.9372734 0.9879925
0.0307532 0.0703660 0.1071592 0.1395707 0.1662692 0.1861610 0.1984315 0.2025782 0.1984315 0.1861610 0.1662692 0.1395707 0.1071592 0.0703660 0.0307532
0.25 0.30 0.30 0.35 0.35 0.40 0.40 0.40
2 8 10 8 10 4 8 10
0.18324 0.07042 0.03880 0.05724 0.02815 0.17359 0.04566 0.02109
0.18325 0.07043 0.03881 0.05726 0.02816 0.17360 0.04569 0.02110
0.005 0.014 0.026 0.035 0.036 0.006 0.066 0.047
iexpq ðzÞ ¼
2 xq ¼ pffiffiffiffiffiffiffiffiffiffiffiffi : n q1 By defining BðqÞ as:
3=2 BðqÞ ¼ AðqÞ 2pMkB T ¼
1 ½1 þ ðq 1Þzexpq ðzÞ; q
1 1 þ 2 q1 ¼ 14 ðq 1Þ1=2 ð3q 1Þð1 þ qÞ 1 ðq1 Þ
ð31Þ
expq ðzÞdz ¼ iexpq ðzÞ þ C:
ð32Þ
After a simple substitution of the type u ¼ V 2 =2v 2th on Eq. (28), and the substitution of Eq. (32), the following is obtained:
R v ðxÞþv r ð yÞ ðv ðxÞv r ð yÞÞ2 þ dVVf ð V; T Þ ¼ A ð q Þiexp q q v ðxÞv r ð yÞ 2v 2th v r ð yÞÞ2 : AðqÞiexpq ðv ðxÞþ 2 2v
ð33Þ
th
By imposing the results which come from the Bethe and Placzek approximations, as given by Eqs. (14) and (24), Eq. (33) then becomes:
v ðxÞv r ðyÞ
"
n2 dVVfq ðV; TÞ AðqÞiexpq ðx yÞ2 4
#
ð34Þ
ð38Þ
we get to:
Z
Z v ðxÞþv r ðyÞ
ð37Þ
we can finally write the following expression for the Tsallis Doppler Broadening Function:
Wq ðx; nÞ wq ðx; nÞ ¼
h 2 i R xþx dy 2 n : ¼ 2pn ffiffipffi BðqÞ xxqq 1þy 2 iexpq 4 ð x yÞ
ð39Þ
Taking into account that function expq at the limit q ! 1 reproduces the usual exponential function, it is easy to realize, with the help of Eq. (31), that:
lim iexpq ðxÞ ¼ e x :
ð40Þ
q!1
By using the asymptotic limit of Gamma Function (Spiegel, 2013):
CðxÞ
pffiffiffiffiffiffiffi 2pxx1=2 ex ; x ! þ1;
ð41Þ
it can be seen directly that:
The iexpq function which appears in Eq. (34) is defined by Eq. (31) in terms of the expq function. Then, this argument must also satisfy the positive nature of the power argument wich gave rise to the result presented in Eq. (6). In that way, the definition given by Eq. (5) leads to the following:
1 4 : 1 þ ðq 1Þ ðx yÞ2 P 0 ) ðx yÞ2 6 2 4 n ðq 1Þ
ð35Þ
lim BðqÞ ¼ 1:
ð42Þ
q!1
Finally, Eq. (37) leads to:
lim xq ¼ þ1:
ð43Þ
q!1
With the results shown in Eqs. (40), (42) and (43) it is easy to see that:
lim wq ðx; nÞ ¼ wðx; nÞ;
This condition can be written as:
ð44Þ
q!1
x xq 6 y 6 x þ xq ;
ð36Þ
where
that is, if the limit q ! 1 is taken, the usual Doppler Broadening Function is recovered as expected.
Table 2 Doppler Broadening Function values using Eq. (25). The wðn; xÞ function n
x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.04309 0.08384 0.12239 0.15889 0.19347 0.22624 0.25731 0.28679 0.31477 0.34135
0.04308 0.08379 0.12223 0.15854 0.19281 0.22516 0.25569 0.28450 0.31168 0.33733
0.04306 0.08364 0.12176 0.15748 0.19086 0.22197 0.25091 0.27776 0.30261 0.32557
0.04298 0.08305 0.11989 0.15331 0.18325 0.20968 0.23271 0.25245 0.26909 0.28286
0.04267 0.08073 0.11268 0.13777 0.15584 0.16729 0.17288 0.17360 0.17052 0.16469
0.04216 0.07700 0.10165 0.11540 0.11934 0.11571 0.10713 0.09604 0.08439 0.07346
0.04145 0.07208 0.08805 0.09027 0.08277 0.07043 0.05726 0.04569 0.03670 0.03025
0.04055 0.06623 0.07328 0.06614 0.05253 0.03881 0.02816 0.02110 0.01687 0.01446
0.03380 0.03291 0.01695 0.00713 0.00394 0.00314 0.00289 0.00277 0.00270 0.00266
0.01639 0.00262 0.00080 0.00070 0.00067 0.00065 0.00064 0.00064 0.00064 0.00063
418
G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421
to obtain the values for both the Doppler Broadening Function wðx; nÞ and the Interference function vðx; nÞ for several values of x and n parameters. These tables are reported in several textbooks (Lamarsh, 1966; Duderstadt and Hamilton, 1976; Stacey, 2007). It is widely known that the nuclear reactors in operation nowadays has their designs based on the Maxwell-Boltzmann distribution. Thus, for some operating condition beyond the design bases, or in new reactor designs that may be developed, it would be relevant to consider another kind of speed distribution, although not expecting a large deviation from the Maxwellian distribution. Thus, in order to study the influence of some deviation from the Maxwellian behaviour, the interval from the non-deformed case q ¼ 1:0 to q ¼ 1:5, with an increment of 0:1 will be considered. In this range of values it is possible to see the influence of considering the Tsallis distribution in the Doppler Broadening Phenomenon. Table 2 shows the calculated values for the Doppler Broadening Function, Eq. (25), using the 15th order Gaussian Quadrature method implemented in C language. From the values obtained, only eight of them differ from those presented in the table for the values available in Beynon and Grant (1963) and these, as well as the percentage deviation, are shown in Table 3, where the column for wBG ðn; xÞ represents the values for the Doppler Broadening Function obtained by Beynon and Grant in 1963 and the column for wðn; xÞ has the values calculated using the Gauss-Legendre quadrature method. Table 2 will be taken as a reference in this paper. By analyzing Table 3, one can see that the greatest deviation between wBG ðn; xÞ and wðn; xÞ corresponds to the value of 0:066% obtained for n ¼ 0:40 and x ¼ 8, that is, the deviations are negligible, and can be attributed to the uncertainties introduced by the numerical methods used.
The results obtained with this deformed Doppler Broadening Function as given by Eq. (39) will be reported in the next section. 4. Results In this section the results obtained for the Tsallis Doppler Broadening Function wq ðx; nÞ proposed in this paper will be reported. In order to obtain the results for the integral in Eq. (39), we used the Gauss-Legendre quadrature method (Burden and Faires, 2011). The N th order method consists of performing an approximation for a defined integral according to the expression:
Rb a
f ðxÞdx ¼ ba 2 ba 2
R 1 ba f 2 x þ bþa dx 1 2 N X
xi f
ba 2
ð45Þ
xi þ bþa ; 2
i¼1
where xi is the point of quadrature, and xi is its corresponding weight. The Gauss-Legendre quadrature points are the roots of the Legendre polynomial in the interval ½1; 1, as generated from the Rodrigues’ formula:
n i 1 d h 2 x 1 : n 2 n! dx n
Pn ðxÞ ¼
ð46Þ
n
In this paper a 15th order method was implemented, whose Legendre points xi and their respective weights are shown in Table 1. A traditional reference in the area of nuclear reactor physics lies in the tables obtained in 1963 by Beynon and Grant (1963), who used the Chebyshev Polinomial expansion (Arfken et al., 2013), and the Gauss-Hermite Method (Abramowitz and Stegun, 1972)
Table 4 Tsallis Doppler Broadening Function values for q ¼ 1:1 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.04663 0.09058 0.13202 0.17111 0.20802 0.24287 0.27580 0.30694 0.33639 0.36426
8.2 8.0 7.9 7.7 7.5 7.4 7.2 7.0 6.9 6.7
0.04662 0.09052 0.13183 0.17069 0.20724 0.24160 0.27389 0.30423 0.33274 0.35952
8.2 8.0 7.9 7.7 7.5 7.3 7.1 6.9 6.8 6.6
0.04660 0.09034 0.13127 0.16943 0.20492 0.23781 0.26822 0.29626 0.32203 0.34566
8.2 8.0 7.8 7.6 7.4 7.1 6.9 6.7 6.4 6.2
0.04651 0.08964 0.12904 0.16449 0.19589 0.22326 0.24670 0.26639 0.28256 0.29548
8.2 7.9 7.6 7.3 6.9 6.5 6.0 5.5 5.0 4.5
0.04614 0.08688 0.12048 0.14604 0.16346 0.17328 0.17649 0.17439 0.16834 0.15967
8.1 7.6 6.9 6.0 4.9 3.6 2.1 0.5 1.3 3.0
0.04552 0.08245 0.10739 0.11962 0.12068 0.11349 0.10146 0.08766 0.07439 0.06297
8.0 7.1 5.6 3.7 1.1 1.9 5.3 8.7 11.8 14.3
0.04467 0.07660 0.09131 0.09025 0.07879 0.06322 0.04844 0.03697 0.02925 0.02452
7.8 6.3 3.7 0.0 4.8 10.2 15.4 19.1 20.3 18.9
0.04361 0.06965 0.07398 0.06266 0.04577 0.03106 0.02148 0.01635 0.01388 0.01266
7.5 5.2 1.0 5.3 12.9 20.0 23.7 22.5 17.7 12.4
0.03558 0.03084 0.01242 0.00462 0.00322 0.00292 0.00278 0.00270 0.00266 0.00262
5.3 6.3 26.7 35.2 18.3 7.0 3.8 2.5 1.5 1.5
0.01526 0.00135 0.00073 0.00068 0.00066 0.00065 0.00064 0.00064 0.00063 0.00063
6.9 48.5 8.8 2.9 1.5 0.0 0.0 0.0 1.6 0.0
Table 5 Tsallis Doppler Broadening Function values for q ¼ 1:2 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.04992 0.09680 0.14087 0.18230 0.22129 0.25798 0.29254 0.32510 0.35580 0.38476
15.9 15.5 15.1 14.7 14.4 14.0 13.7 13.4 13.0 12.7
0.04991 0.09673 0.14065 0.18181 0.22038 0.25650 0.29032 0.32196 0.35158 0.37927
15.9 15.4 15.1 14.7 14.3 13.9 13.5 13.2 12.8 12.4
0.04988 0.09653 0.13999 0.18035 0.21768 0.25210 0.28374 0.31271 0.33917 0.36325
15.8 15.4 15.0 14.5 14.1 13.6 13.1 12.6 12.1 11.6
0.04977 0.09571 0.13739 0.17458 0.20717 0.23519 0.25878 0.27815 0.29360 0.30546
15.8 15.2 14.6 13.9 13.1 12.2 11.2 10.2 9.1 8.0
0.04934 0.09248 0.12741 0.15312 0.16955 0.17742 0.17803 0.17300 0.16404 0.15279
15.6 14.6 13.1 11.1 8.8 6.1 3.0 0.3 3.8 7.2
0.04862 0.08731 0.11218 0.12253 0.12041 0.10962 0.09443 0.07862 0.06472 0.05392
15.3 13.4 10.4 6.2 0.9 5.3 11.9 18.1 23.3 26.6
0.04763 0.08049 0.09354 0.08891 0.07356 0.05542 0.04022 0.03011 0.02444 0.02148
14.9 11.7 6.2 1.5 11.1 21.3 29.8 34.1 33.4 29.0
0.04638 0.07240 0.07362 0.05812 0.03872 0.02443 0.01705 0.01400 0.01267 0.01195
14.4 9.3 0.5 12.1 26.3 37.1 39.5 33.6 24.9 17.4
0.03702 0.02816 0.00858 0.00360 0.00302 0.00283 0.00272 0.00266 0.00263 0.00260
9.5 14.4 49.4 49.5 23.4 9.9 5.9 4.0 2.6 2.3
wq ðn; xÞ
%
0.01381 15.7 0.00091 65.3 0.00071 11.3 0.00067 4.3 0.00065 3.0 0.00064 1.5 0.00064 0.0 0.00063 1.6 0.00063 1.6 0.00063 0.0
419
G. Guedes et al. / Annals of Nuclear Energy 128 (2019) 414–421 Table 6 Tsallis Doppler Broadening Function values for q ¼ 1:3 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.05298 0.10260 0.14908 0.19265 0.23351 0.27185 0.30785 0.34166 0.37343 0.40331
23.0 22.4 21.8 21.2 20.7 20.2 19.6 19.1 18.6 18.2
0.05297 0.10252 0.14883 0.19209 0.23247 0.27016 0.30530 0.33807 0.36861 0.39706
23.0 22.4 21.8 21.2 20.6 20.0 19.4 18.8 18.3 17.7
0.05294 0.10228 0.14808 0.19040 0.22938 0.26512 0.29778 0.32751 0.35447 0.37882
22.9 22.3 21.6 20.9 20.2 19.4 18.7 17.9 17.1 16.4
0.05282 0.10134 0.14509 0.18379 0.21734 0.24578 0.26929 0.28813 0.30268 0.31332
22.9 22.0 21.0 19.9 18.6 17.2 15.7 14.1 12.5 10.8
0.05232 0.09763 0.13363 0.15921 0.17437 0.18006 0.17792 0.16995 0.15827 0.14482
22.6 20.9 18.6 15.6 11.9 7.6 2.9 2.1 7.2 12.1
0.05149 0.09169 0.11617 0.12433 0.11886 0.10453 0.08666 0.06970 0.05626 0.04700
22.1 19.1 14.3 7.7 0.4 9.7 19.1 27.4 33.3 36.0
0.05035 0.08386 0.09491 0.08651 0.06751 0.04770 0.03338 0.02555 0.02186 0.02000
21.5 16.3 7.8 4.2 18.4 32.3 41.7 44.1 40.4 33.9
0.04892 0.07460 0.07240 0.05286 0.03199 0.01953 0.01472 0.01295 0.01208 0.01155
20.6 12.6 1.2 20.1 39.1 49.7 47.7 38.6 28.4 20.1
0.03816 0.02506 0.00584 0.00329 0.00292 0.00277 0.00269 0.00264 0.00261 0.00258
12.9 23.9 65.5 53.9 25.9 11.8 6.9 4.7 3.3 3.0
wq ðn; xÞ
%
0.01212 26.1 0.00083 68.3 0.00069 13.8 0.00066 5.7 0.00065 3.0 0.00064 1.5 0.00064 0.0 0.00063 1.6 0.00063 1.6 0.00063 0.0
Table 7 Tsallis Doppler Broadening Function values for q ¼ 1:4 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.05587 0.10804 0.15676 0.20230 0.24487 0.28469 0.32197 0.35687 0.38958 0.42024
29.7 28.9 28.1 27.3 26.6 25.8 25.1 24.4 23.8 23.1
0.05586 0.10795 0.15648 0.20166 0.24369 0.28278 0.31910 0.35283 0.38415 0.41320
29.7 28.8 28.0 27.2 26.4 25.6 24.8 24.0 23.3 22.5
0.05583 0.10768 0.15562 0.19975 0.24019 0.27708 0.31060 0.34092 0.36823 0.39271
29.7 28.7 27.8 26.8 25.8 24.8 23.8 22.7 21.7 20.6
0.05568 0.10661 0.15223 0.19226 0.22657 0.25524 0.27848 0.29663 0.31012 0.31943
29.5 28.4 27.0 25.4 23.6 21.7 19.7 17.5 15.2 12.9
0.05511 0.10240 0.13924 0.16445 0.17811 0.18145 0.17647 0.16564 0.15151 0.13636
29.2 26.8 23.6 19.4 14.3 8.5 2.1 4.6 11.1 17.2
0.05418 0.09566 0.11949 0.12521 0.11627 0.09860 0.07868 0.06156 0.04956 0.04228
28.5 24.2 17.6 8.5 2.6 14.8 26.6 35.9 41.3 42.4
0.05288 0.08678 0.09555 0.08327 0.06102 0.04065 0.02838 0.02293 0.02048 0.01914
27.6 20.4 8.5 7.8 26.3 42.3 50.4 49.8 44.2 36.7
0.05125 0.07630 0.07046 0.04720 0.02614 0.01654 0.01359 0.01237 0.01171 0.01129
26.4 15.2 3.8 28.6 50.2 57.4 51.7 41.4 30.6 21.9
0.03905 0.02170 0.00440 0.00314 0.00285 0.00273 0.00266 0.00262 0.00259 0.00257
15.5 34.1 74.0 56.0 27.7 13.1 8.0 5.4 4.1 3.4
wq ðn; xÞ
%
0.01029 37.2 0.00079 69.8 0.00068 15.0 0.00066 5.7 0.00064 4.5 0.00064 1.5 0.00063 1.6 0.00063 1.6 0.00063 1.6 0.00063 0.0
Table 8 Tsallis Doppler Broadening Function values for q ¼ 1:5 calculated using Eq. (39). The wq ðn; xÞ function and this percentage deviation from wðn; xÞ. x¼0
x ¼ 0:5
x¼1
x¼2
x¼4
x¼6
x¼8
x ¼ 10
x ¼ 20
x ¼ 40
n
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
wq ðn; xÞ
%
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
0.05861 0.11317 0.16399 0.21134 0.25548 0.29666 0.33508 0.37096 0.40448 0.43581
36.0 35.0 34.0 33.0 32.1 31.1 30.2 29.3 28.5 27.7
0.05860 0.11307 0.16367 0.21062 0.25417 0.29451 0.33187 0.36645 0.39842 0.42798
36.0 34.9 33.9 32.8 31.8 30.8 29.8 28.8 27.8 26.9
0.05856 0.11277 0.16271 0.20848 0.25024 0.28814 0.32239 0.35316 0.38069 0.40519
36.0 34.8 33.6 32.4 31.1 29.8 28.5 27.1 25.8 24.5
0.05840 0.11156 0.15891 0.20008 0.23499 0.26373 0.28656 0.30388 0.31619 0.32408
35.9 34.3 32.5 30.5 28.2 25.8 23.1 20.4 17.5 14.6
0.05776 0.10683 0.14434 0.16896 0.18093 0.18177 0.17395 0.16042 0.14422 0.12793
35.4 32.3 28.1 22.6 16.1 8.7 0.6 7.6 15.4 22.3
0.05670 0.09927 0.12223 0.12530 0.11283 0.09214 0.07094 0.05469 0.04473 0.03926
34.5 28.9 20.2 8.6 5.5 20.4 33.8 43.1 47.0 46.6
0.05525 0.08932 0.09556 0.07936 0.05444 0.03472 0.02518 0.02143 0.01962 0.01857
33.3 23.9 8.5 12.1 34.2 50.7 56.0 53.1 46.5 38.6
0.05341 0.07758 0.06792 0.04144 0.02158 0.01495 0.01293 0.01199 0.01145 0.01111
31.7 17.1 7.3 37.3 58.9 61.5 54.1 43.2 32.1 23.2
0.03972 0.01824 0.00384 0.00304 0.00280 0.00270 0.00264 0.00260 0.00258 0.00256
17.5 44.6 77.3 57.4 28.9 14.0 8.7 6.1 4.4 3.8
In Tables 4–8, the values for the Tsallis Doppler Broadening Function calculated using Eqs. (39) and (45) and their respective deviations from the values of the usual Doppler Broadening Function shown in Table 2 can be found for several values of the q parameter. Figs. 2–6 show, the Tsallis Doppler Broadening Function defined by Eq. (39) for several values of the q parameter as well as the Maxwellian Doppler Broadening Function given by Eq. (25) considering some values for the n parameter. Analyzing the graphs above, it can be seen that as the deformation parameter q increases, the peak of the Doppler Broadening Function gets higher, that is, closer to the peak value 1 given by the Single Level Breit-Wigner formalism, for the non-broadened Function. In other words, the introduction of the deformation parameter q leads to an attenuation of the Doppler Broadening Effect.
wq ðn; xÞ
%
0.00840 48.7 0.00076 71.0 0.00068 15.0 0.00065 7.1 0.00064 4.5 0.00064 1.5 0.00063 1.6 0.00063 1.6 0.00063 1.6 0.00063 0.0
5. Conclusions This paper tackles the consequences of considering the targetnuclei random movement as given by the non-extensive Tsallis distribution in the evaluation of the absorption cross section in three-dimension. The Tsallis distribution can be mathematically formulated as a deformation of the usual Maxwell-Boltzmann distribution, which is dependent on a q > 1 parameter and, if the q ! 1 limit is taken, the deformation is removed and the usual Maxwell-Boltzmann distribution is recovered. In this context, a new deformed Doppler Broadening Function called wq ðx; nÞ as defined by Eq. (39), dependent on a deformation parameter q, was obtained, and when the q ! 1 limit is taken, the usual Doppler Broadening Function is retrieved. This new expression, which presents in its form an integral that, as in the Maxwellian case, does
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Fig. 2. Tsallis Doppler Broadening Function for n ¼ 0:05.
Fig. 4. Tsallis Doppler Broadening Function for n ¼ 0:25.
Fig. 3. Tsallis Doppler Broadening Function for n ¼ 0:15.
not have an analytical solution, was evaluated using the 15th order Gauss-Legendre quadrature method, a numerical method widely used and well established in the literature. With the said numerical method, Table 2 was constructed, reproducing the results obtained by Beynon and Grant (1963) which were taken as a comparison parameter for this paper. Tables 4–8, varying the parameter q with a 0:1 increment between the q ¼ 1:1 and q ¼ 1:5 values, where the percentage deviation from the usual Doppler Broadening Function was also obtained. Still using the Gaussian quadrature method, the graphs shown in Figs. 2–6 were generated where, for each of them, by making the parameter n dependent on the temperature constant, it is possible to visualise the behaviour of the Doppler Broadening Function as it varies the deformation parameter q. In analysing the graphs shown herein, it is possible to see that, for a given resonance, if parameter n is kept constant, the Doppler Broadening Effect is deformed and depends on parameter q, and it is also possible to see that, as q rises, the Doppler Broadening Effect is softened, that is, the deformation parameter q makes the Doppler Broadening Effect smoother. The profile change for the
Fig. 5. Tsallis Doppler Broadening Function for n ¼ 0:35.
Doppler Broadening Function can affect both the resonant absorption and the resonant scattering, in which case the interference term also needs to be evaluated. In PWR reactors, the theory based on the distribution of Maxwell Boltzmann is well established, and has been applied through decades producing good agreement with experimental data. However, it is also widely known that precise and accurate cross section data is of fundamental importance for the operation of nuclear reactors, and it is possible that in new reactor designs as well as
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421
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.anucene.2019.01.023. References
Fig. 6. Tsallis Doppler Broadening Function for n ¼ 0:45.
in specific situations for the reactors currently in operation, the use of non-Maxwellian statistics will become important. The present paper, together with the previous one by Guedes et al. (2017), considering the Kaniadakis statistics, form a theoretical basis to study any quasi-Maxwellian statistics that preserves an even parity in variable V. Thus, if in the study of the influence of quasi-Maxwellian distributions on the Doppler Broadening Phenomenon, the importance of using a distribution other than Tsallis or Kaniadakis becomes evident, it is reasonable to assume that the theoretical framework for the treatment with any other distributions has been paved. Acknowledgments This research project is supported by the following Brazilian institutions: Brazilian Council for Scientific and Technological Development (CNPq), Brazilian Nuclear Energy Commission (CNEN), and Research Support Foundation of the State of Rio de Janeiro (FAPERJ).
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