Coefficients and of the four-order Padé Approximation
In all nuclear reactors some neutrons can be absorbed in the resonance region and, in the design of these reactors, an accurate treatment of the resonant absorptions is essential. Apart from that, the resonant absorption varies with fuel temperature, due to the Doppler broadening of the resonances (Stacey, 2001). The thermal agitation movement of the reactor core is adequately represented in microscopic cross-section of the neutron-core interaction through the Doppler Broadening function. This function is calculated numerically in modern systems for the calculation of macro-group constants, necessary to determine the power distribution in a nuclear reactor. This function has also been used for the approximate calculations of the resonance integrals in heterogeneous fuel cells (Campos and Martinez, 1989). It can also be applied to the calculation of self-shielding factors to correct the measurements of the microscopic cross-sections through the activation technique (Shcherbakov and Harada, 2002). In these types of application we can point out the need to develop precise analytical approximations for the Doppler broadening function to be used in the codes that calculates the values of this function. Tables generated from such codes are not convenient for some applications and experimental data processing.
This chapter will present a brief retrospective look at the calculation methodologies for the Doppler broadening function as well as the recent advances in the development of simple and precise analytical expressions based on the approximations of Beth-Plackzec according to the formalism of Briet-Wigner.
2. The Doppler broadening function
Let us consider a medium with a temperature where the target nuclei are in thermal movement. In a state of thermal equilibrium for a temperature , the velocities are distributed according to Maxwell-Boltzmann distribution (Duderstadt and Hamilton, 1976),
where is the total number of nucleus, is the mass of the nucleus and is Boltzmann’s constant.
Considering the neutrons as an ideal gas in thermal equilibrium, it is possible to write the average cross-section for neutron-nucleus interaction taking into consideration the movement of the neutrons and of the nucleus as:
where is the distribution function of Maxwell-Boltzmann as given by equation (1) and is the velocity of the target nuclei. Denoting the relative velocity between the movement of the neutron and the movement of the target nucleus and considering the isotropic case, that is, with no privileged direction, it is possible to separate the integration contained in equation (2) in the double integral:
It is possible to see clearly in equation (3) that the cross-section depends of the relative velocity between the neutrons and the target nuclei. As the nuclei are in thermal movement, the relative velocity can increase or decrease. This difference between relative velocities rises to the Doppler deviation effect in cross-section behaviour. After integrating equation (3) in relation to the azimuthal angle the average cross-section for neutron-nucleus interaction can be written thus:
Denoting so that , equation (4) takes the form of:
From the definition of the relative velocity one has the relation,
and, as a result,
With the aid of a simple substitution, using relations (6) and (27), equation (5) is thus written as:
In equation (8), the limits of integration are always positive due to the presence of the module. As a result, one should separate the integral found in equation (8) into two separate integrals, as follows,
It is possible to modify the limits of integration for equation (9) taking into account that the mass of the target nucleus is much larger than the mass of the incident neutron. In terms of relative velocity, equation (9) can be written as:
where it was defined . Introducing the variables for reduced velocities and , equation (11) is written by:
Integrating equation (12) in relation to V one gets to the expression:
For resonances (that is, for the energy levels of the composed nucleus) it is possible to describe the energy dependence of the absorption cross-section by a simple formula, valid for , known as Breit-Wigner formula for resonant capture, expressed in function of the energy of the centre-of-mass by,
where is the energy where the resonance occurs and is the energy of the centre-of-mass of the neutron–nucleus system. Apart from that, we find in equation (14) the term , that is the value of the total cross-section in resonance energy that can be written in terms of the reduced wavelength by:
where the statistical spin factor is given by the expression:
where is the nuclear spin and is the total spin (Bell and Glasstone, 1970).
In replacing the expression (14) in equation (13) one finds an exact expression for the average cross-section, valid for any temperature:
In a system with two bodies it is possible to write the kinetic energy in the centre-of-mass system, by
where is the reduced mass of the system.
For the problem at hand, of a neutron that is incident in a thermal equilibrium system with a temperature T, it is a good approximation to assume that . Thus, the ratio between the kinetic energy of the incident neutron and the kinetic energy of the centre-of-mass system is thus written
where A is the atomic mass of the target core. Resulting:
and denoting one finally obtains the expression for the cross-section of radioactive capture near any isolated resonance with an energy peak , as written by:
where is the module for relative neutron-nucleus velocity, is the module for neutron velocity, and
The Doppler width for resonance is expressed by:
All the other nuclear parameters listed below are well established in the literature,
= mass number;
= absolute temperature;
= energy of incident neutron;
= energy of centre-of-mass;
= energy where the resonance occurs;
= total width of the resonance as measured in the lab coordinates;
= Doppler width of resonance;
= neutron velocity module;
= module of the relative velocity between neutron movement and nucleus movement;
= module of the velocity for each target nucleus.
3. The Bethe and Placzek approximations
The expression proposed by Bethe and Placzek for the Doppler broadening function is obtained from some approximations, as follows:
it is a good approximation to extend the lower limit for integration down to in equation (22), given that the ratio between the energy of neutron incidence and the practical width is big.
being the energy of the system in the centre-of-mass system and the energy of the incident neutron, the following relation is always met:
where it was denoted that . Equation (25) can be expanded in a Taylor series and, to the first order, is written by
In terms of the masses and velocities, equation (26) is written as follows:
where is the reduced mass of the system. For heavy nucleus and equation (27) can be written as:
The approximations made in this section apply in almost all the practical cases, and are not applicable only in situations of low resonance energies () and very high temperatures.
4. Properties of the Doppler broadening function
The function as proposed by the approximation of Bethe and Placzek has an even parity, is strictly positive and undergoes a broadening as that variable diminishes, that is, varies inversely with the absolute temperature of the medium. For low temperatures, that is, when temperature in the medium tend to zero, the Doppler broadening function can be represented as shown below:
Equation (31) is known as an asymptotic approximation of the Doppler broadening function. For high temperatures, that is, when the temperature of the medium tends to infinite, the Doppler broadening function can be represented through the Gaussian Function, given that:
The area over the curve of the Doppler Broadening function is written as below and, as it consists of separable and known integers it is possible to write:
From equation (33) once concludes that the area over the curve of the Doppler Broadening function is constant for the intervals of temperature and energy of interest in thermal reactors. This property is valid even for broadened resonances as shown in Figure 1, considering the different values of variable .
5. Analytical approximations for the Doppler broadening function
This section describes the main approximation methods for the Doppler broadening function, according to the approximation of Bethe and Placzek, equation (30).
5.1. Asymptotic expansion
A practical choice to calculate the Doppler broadening function is its asymptotic expression resulting from the expansion of the term in equation (30) in a Taylor series around .
Despite equation (35) being valid only for , it is quite useful to determine the behaviour of the Doppler Broadening function in specific conditions. For high values of , it is possible to observe that function presents the following asymptotic form:
5.2. Method of Beynon and Grant
Beynon and Grant (Beynon and Grant, 1963) proposed a calculation method for the Doppler broadening function that consists of expanding the exponential part of the integrand of the Doppler broadening function in the Chebyshev polynomials and integrate, term by term, the resulting expression, which allows writing:
where and and still,
For values where the condition was met (Beynon and Grant, 1963) is recommended the use of the asymptotic expression of the function , equation (36). It should be pointed that the results obtained by this method have become a reference in several works on the Doppler broadening function.
5.3. Method of Campos and Martinez
The core idea of the method proposed by (Campos and Martinez, 1987) is to transform the Doppler broadening function from its integral form into a differential partial equation subjected to the initial conditions. Differentiating equation (30) in relation to one obtains:
Acknowledging in equation (40) the very Doppler broadening function and the term of interference as defined by the integral:
it is possible to write:
Deriving equation (42) again in relation to , after expliciting function in the same equation, one has:
The right side of equation (43) can be written in another way, given that ,
In replacing the result obtained in (44) in equation (43) one obtains the differential equation where Campos and Martinez based themselves to obtain an analytical approximation for the broadening function :
subjected to the initial conditions:
Admitting that function may be expanded in series,
and all the other terms are calculated from the following relation of recurrence:
The representation in series for the Doppler broadening function, as given by equation (47), is valid only for . For the cases where , (Campos and Martinez, 1987) used the asymptotic form as given by equation (35), as well as proposed by Beynon and Grant.
5.4. Four order method of Padé
The Padé approximation is one of the most frequently used approximations for the calculation of the Doppler broadening function and its applications and can efficiently represent functions, through a rational approximation, that is, a ratio between polynomials. For the four-order Padé approximation (Keshavamurthy& Harish, 1993) they proposed the following polynomial ratio:
where and are the following polynomials:
5.5. Frobenius method
In this method the homogeneous part of the differential equation that rules the Doppler broadening function, equation (45), is solved using the Frobenius Method (Palma et. al., 2005) that consists fundamentally of seeking a solution of the differential equation in the form of series around the point , with a free parameter, that is, as follows:
with and where s is the parameter that grants the method flexibility.
The initial equation of the problem, obtained when , remembering that is
From equation (55), as , one obtains that or . Using first and one obtains the following relations of recurrence:
Considering the case where , one obtains the other series linearly independent with the first term, not null, denoted by :
With this the homogeneous solution assumes the following form:
where the coefficients are all known from the relations of recurrence, equations (56) to (59). In writing function as:
it is possible to determine all the coefficients equalling, term by term, equations (60) and (61) so to write:
Acknowledging the expansion of the cosine and sine functions, one obtains an analytical form to solve the homogeneous part of the differential equations that rule the Doppler broadening function:
In order to obtain the particular solutions of equation (45), and consequently its general solution, it is possible to apply the method of parameter variation from the linearly independent solutions:
Supposing a solution thus,
where functions and are determined after the imposition of the initial conditions expressed by equations (46a) and (46b) and of the imposition of the nullity of the expression:
That, along with the condition,
Which results from the very equation (45), form a linear system whose solution is given by the equations:
Integrating equations (69) and (70),
it is possible to write the solution particular of equation (45) as follows:
As the general solution of differential equation (45) is the sum of the solution of the homogeneous and particular equations, the initial conditions are imposed, as expressed by equations (46a) and (46b), to determine the constants:
Finally, according to the Frobenius Method, the Doppler Broadening function can be written thus:
5.6. Fourier transform method
In doing the transformation of variables in the full representation of the Doppler broadening function, equation (30), one obtains the expression
that can be mathematically interpreted as the convolution of the Lorentzian function with a gaussian function, as exemplified by the equation below:
where is the lorentzian function and the gaussian function. Function admits a full representation through the Fourier cosine transform (Polyanin and Manzhirov, 1998), as being
where and .
The integrals on the right side of equations (3.25) and (3.26) are known as complementary error functions, in which case one can conclude that:
With some algebraic manipulation it is easy to prove that the Fourier transform method and the Frobenius method, equations (85) and (76) respectively, provide identical results.
5.7. Fourier series method
From the representation of the Doppler broadening function in a Fourier cosine transform, equation (82), it is possible to write
where function is even and can be expanded into a Fourier series in cosines:
5.8. Representation of function using Salzer expansions
Although the formulations obtained for function from the Frobenius method, Fourier transform and Fourier series methods only contain functions that are well-known in literature, it can be inconvenient to work with error functions that contain an imaginary argument. One of the ways to overcome this situation is to calculate the real and imaginary parts of function using the expansions proposed by Salzer (Palma and Martinez, 2009)
where auxiliary functions and are written by:
5.9. The Mamedov method
Mamedov (Mamedov, 2009) put forward an analytical formulation to calculate function , based on its representation in the form of a Fourier transform, equation (82). Using the expansions in series of the exponential and cosine functions,
and the well-known binomial expansion
Mamedov proposed the following expressions for the Doppler broadening function:
where , and are the well-know incomplete Gamma functions and are binomials coefficients defined by:
6. Numerical calculation of function
The numerical calculation of the Doppler broadening function consists of calculating a defined integral. There are many methods in the literature for this calculation, but in this chapter we will describe a numerical reference method based on the Gauss-Legendre quadrature. In basic terms, the Gauss-Legendre quadrature method consists of approximating a defined integer through the following expression:
where is the order of the quadrature, is the point of the quadrature and the weight corresponding to the point of quadrature. The points of the Gauss-Legendre quadrature are the roots of the polynomials of Legendre (Arfken, 1985) in the interval , as generated from the Rodrigues‘ formula,
for an isotope at a given temperature, that is, for a fixed value for variable , the function decreases rapidly and a very high value is not necessary for what we will consider our numerical infinite. This fact can be evidenced at Figure 1.
For that an adequate numerical infinite was considered, as well as a high-order quadrature , whose points of Legendre and respective weights are found in Table 3. The results obtained with this method, whose handicap is the high computing cost, can be seen in Table 4.
A brief retrospective look at the calculation methodologies for the Doppler broadening function considering the approximations of Beth-Plackzec according to the formalism ofBriet-Wigner was presented in this chapter.
This research project is supported by Instituto Alberto Luiz Coimbra de Pós-graduação e PesquisaemEngenharia/Universidade Federal do Rio de Janeiro (COPPE/UFRJ), ComissãoNacional de Energia Nuclear (CNEN), Centro de Desenvolvimento de Tecnologia Nuclear (CDTN). It was financially supported by through National Institute of Science and Technology of Innovative Nuclear Reactors,Brazilian Council for Scientific and Technological Development (CNPq,) and Research Support Foundation of the State of Minas Gerais (FAPEMIG).