Approximate Model of Neutron Resonant Scattering in a Crystal
aa r X i v : . [ c ond - m a t . o t h e r] S e p Approximate Model of Neutron ResonantScattering in a Crystal
Arnaud Courcelle , John Rowlands Commissariat `a l’Energie Atomique, CEA/Cadarache, Bat 151, 13108 Saint PaulLez Durance, France
81 South Court Avenue, Dorchester, Dorset DTI 2DA, United KingdomE-mail: [email protected], [email protected]
Abstract.
In the theory of resonant scattering, the double differential cross section involves thecomputation of a multifold integral of a 4-point correlation function, which generalizesthe traditional 2-point correlation function of Van-Hove for potential scattering. Inthe case of a neutron-crystal interaction, the numerical computation of these multifoldintegrals is cumbersome. In this paper, a new approximation is suggested. It isbased on a factorization of the differential cross section into one function describingthe exchange of kinetic energy between the neutron and the bound nucleus (phononsdynamic) and a function related to the nuclear scattering amplitude. This formalismis then applied to the modeling of resonant scattering of a neutron by U in a U O crystal lattice. pproximate Model of Neutron Resonant Scattering in a Crystal
1. Introduction
The computation of the double differential scattering cross section (DDCS) at lowincident neutron energies is required to solve neutron transport problems. Wigner andWilkins [1] used a two-body kinematic approach to study potential scattering in a freegas. Under the same classic assumptions, Blackshaw and Murray [2] studied the caseof an energy-dependent cross section and further generalizations were investigated byOuisloumen and Sanchez [3] and Rothenstein and Dagan [4].A general quantum formalism due primarily to Van Hove [5] expresses the DDCS ofpotential scattering by a bound nucleus as a Fourier transform of a 2-point correlationfunction. Kazarnovski et al. [8] and Word and Trammell [7] extended the Van Hovetheory to study resonant processes. The resonant DDCS becomes a Fourier double-Laplace transform of a 4-point correlation function.In the case of a harmonic crystal, the numerical computation of the multifoldintegral is difficult because of the highly oscillating behavior of the 4-point correlationfunction. A simplified multiphonon model, known as the ”uncoupled phononapproximation” (UPA) was proposed by Naberejnev [9]. However, the validity ofthis approximation is questioned especially within the short time range (case of hightemperature for instance). The present work suggests a new approximation that givesa correct limit at short time and provides a simple formula to compute the scatteringkernel.
2. General formalism
When the scattering amplitude is independent of incident neutron energy, it is knownsince the work of Van Hove [5] that the scattering DDCS is related to the Fouriertransform of a two point correlation function χ ( τ ) : d σd Ω dE f = σ p π k f k i F { χ ( τ ) } (∆ E ) . (1)∆ E = E f − E i is the energy transfer with E i and E f the energy of the neutronbefore and after the scattering event. The Fourier transform is defined as F { χ ( τ ) } ( x ) = 12 π Z ∞−∞ e − ixτ χ ( τ ) dτ. (2) F { χ ( τ ) } (∆ E ) is the usual scattering function S ( α, β ) = S ( ~ ∆ k, ∆ E ) whichdepends on the momentum transfer ~ ∆ k = ~k f − ~k i where ~k i and ~k f are the initial andfinal neutron wave vector respectively . The theory was later generalized to treat thecase of resonant scattering [7]. In the incoherent approximation, the scattering crosssection includes the resonant, the potential-resonant interference and potential terms. pproximate Model of Neutron Resonant Scattering in a Crystal L F of a 4-point correlation function χ ( τ, t, t ′ ): d σd Ω dE f = σ m π Γ n Γ k f k i Γ L F { χ ( τ, t, t ′ ) } (∆ E, − i ( E i − E ) + Γ / , + i ( E i − E ) + Γ / . (3) E , Γ n and Γ are the energy, the neutron and total width of the resonance. σ m = 4 πg Γ n /k i Γ and g is the usual statistical spin factor. The Fourier-double-Laplacetransform is defined as L F { f ( τ, t, t ′ ) } ( x, z , z
2) = 12 π Z ∞−∞ Z ∞ Z ∞ e − ixτ e − z t e − z t ′ f ( τ, t, t ′ ) dτ dtdt ′ . (4)Similarly, the resonant-potential interference term is expressed as a Fourier-single-Laplace transform of a 3 point correlation function χ ( τ, t ).A quantum calculation shows that χ ( τ, t, t ′ ) is a function of time-dependantdisplacement operators: χ ( τ, t, t ′ ) = < exp (cid:16) − i~k i .~r ( τ + t − t ′ ) (cid:17) exp (cid:16) i~k f .~r ( τ + t ) (cid:17) × exp (cid:16) − i~k f .~r ( t ) (cid:17) exp (cid:16) i~k i .~r (0) (cid:17) > T . (5)Note that χ ( τ ) = χ ( τ, ,
0) and χ ( τ, t ) = χ ( τ, t, < X > T denotes the thermalaverage at temperature T . In this paper, time τ , t and t ′ are expressed in unit of energyand ¯ h = 1. The right hand side of Equation 1 and 3 should be multiplied by (cid:16) M + mm (cid:17) which is implicitly assumed in the paper. M and m are the mass of the target nucleusand incident neutron and A = M/m respectively . The scattering is assumed isotropicin the center-of-mass frame and multiple scattering effects are neglected (single collisionapproximation). Equation 3 assumes that the Hamiltonian of the target and compoundnucleus is the same. This condition neglects the mass change of the target nucleus whenthe neutron is absorbed and re-emitted:
M >> m . In the case where the target nucleus behaves like a free gas, the displacement operatoris proportional to the momentum operator r ( t ) = r (0) + p (0) t/M . After averaging overa Maxwellian distribution of momentum at temperature T , χ within the free gas model(FGM) becomes χ ( τ, t, t ′ ) = exp { − i M [ τ ( ~k f − ~k i ) + ( t − t ′ ) k i ] − kT M [ − ~k i ( t − t ′ ) + τ ( ~k f − ~k i )] } . (6)When the target nucleus is bound to a crystal lattice, the 4-point correlationfunction is more complex. Under the harmonic approximation, the Bloch theorem [9]transforms χ into pproximate Model of Neutron Resonant Scattering in a Crystal χ ( τ, t, t ′ ) = exp {− ρ ( ~k i , ~k i , − ρ ( ~k f , ~k f , }× exp { ρ ( ~k i , ~k f , − t ′ ) + ρ ( ~k f , ~k i , + t ) + ρ ( ~k f , ~k f , τ ) }× exp { ρ ( ~k i , ~k i , τ − t ′ + t ) − ρ ( ~k i , ~k f , τ − t ′ ) − ρ ( ~k f , ~k i , τ + t ) } . (7)For a cubic lattice with a phonon density of states ρ ( ω ), ρ ( ~k, ~k ′ , + t ) = ~k.~k ′ M Z ∞ dω ρ ( ω ) ω [ coth ( ω/ kT ) cos ( ωt ) − isin ( ωt )] . (8)In the short time approximation for τ , t , and t ′ the DDCS can be simplified using ρ ( k, k ′ , + t ) ≈ ρ ( k, k ′ ,
0) + ~k.~k ′ ( − kT eff M t − it M ) , (9)with kT eff = 12 Z ∞ dωρ ( ω ) ωcoth ( ω/ kT ) . (10)Plugging 9 into 7, we get the free gas formula 6 with T eff instead of T .Consequently, the crystal-lattice model lead to the free gas model, with an effectivetemperature, when τ , t , and t ′ are small (short time approximation). This is the sameeffective temperature as Lamb [12] derived for the Doppler broadening of a captureresonance in a solid in the weak binding limit. The UPA approximation proposed in [9] was an attempt to compute the resonantscattering kernel for a harmonic crystal. It neglected t and t ′ in the coupling terms ρ ( ~k i , ~k i , τ − t ′ + t ), ρ ( ~k i , ~k f , τ − t ′ ) and ρ ( ~k f , ~k i , τ + t ) in Equation 7 and applied ashort time approximation to ρ ( ~k i , ~k f , − t ′ ) and ρ ( ~k f , ~k i , + t ) ‡ . The correlation functionbecomes χ UP A ( τ, t, t ′ ) = exp (cid:16) ρ ( ~ ∆ k, τ ) − ρ ( ~ ∆ k, (cid:17) × exp ~k i ~k f [ − kT eff M ( t + t ′ ) − i ( t − t ′ )2 M ] ! . (11)The integral over τ and t, t ′ can be performed separately. Since the first exponentialis the Van-Hove function χ ( τ ), the differential cross section can be factored out as d σd Ω dE = 14 π k f k i × S ( ~ ∆ k, ∆ E ) × ˜ σ ( ~k i , ~k f ) . (12)The detailed mathematical expression for the so-called UPA cross section ˜ σ ( ~k i , ~k f )can be found in [9]. The UPA model separates the phonon dynamic and the nuclear ‡ A refinement of the model called MUPA (modified uncoupled phonon approximation) computed ρ ( ~k i , ~k f , − t ′ ) and ρ ( ~k f , ~k i , + t ) with a discrete phonon spectrum pproximate Model of Neutron Resonant Scattering in a Crystal τ , t , and t ′ becomes χ UP A ( τ, t, t ′ ) = exp { − i M [ τ ( ~k f − ~k i ) +( t − t ′ ) ~k i ~k f ] − kT eff M [ ~k i ~k f ( t + t ′ )+ τ ( ~k f − ~k i ) ] } , (13)which differs markedly from the required equation 6 discussed in the previoussection. Consequently, the DDCS within the UPA approximation does not give thecorrect limit for small t and t ′ .
3. Proposed Model
Our model seeks to get a factorization of the DDCS similar to the UPA model: d σd Ω dE = 14 π k f k i × S ( ~ ∆ k, ∆ E ) × ˜ σ ( ~k i , ~k f ) . (14)However, in order to get the correct limit at short time, we computed the ˜ σ termusing the short time approximation of the correlation function (Equation 6), withoutthe previous UPA approximation.For values of ~ ∆ k = ~
0, the following notation is used,∆ = vuut kTM k i k f − ( ~k i ~k f ) ~ ∆ k x = 2 [ E i − E − ( E f − E i ) ~k i ~ ∆ k~ ∆ k − ~k i ~k f M ] / Γ . (15)It is found that the differential cross section, including resonant, potential andresonant-potential interference can be calculated almost analytically and can be splitinto nuclear and transfer terms. The details of the demonstration are presented in theAppendix. The final result is: d σd Ω dE = 14 π k f k i × S ( ~ ∆ k, ∆ E ) × [ σ p + σ m Γ n Γ ψ + s σ p σ m Γ n Γ χ ] . (16)For ∆ = 0 and ~ ∆ k = ~ ψ = √ π × Γ∆ K ( x Γ2∆ , Γ2∆ ) χ = √ π × Γ∆ L ( x Γ2∆ , Γ2∆ ) . (17)For ∆ = 0 and ~ ∆ k = ~ ψ = 11 + x χ = 2 x x . (18)where K ( x, y ) and L ( x, y ), are the real and imaginary part of complexcomplementary error function w ( z ) = exp[ − ( x + iy ) ] erf c [ − i ( x + iy )], related tothe classic Voigt functions. K ( x, y ) = ℜ [ w ( z )] = yπ Z ∞−∞ exp( − t )( x − t ) + y dt (19) L ( x, y ) = ℑ [ w ( z )] = xπ Z ∞−∞ exp( − t )( x − t ) + y dt. (20) pproximate Model of Neutron Resonant Scattering in a Crystal ~ ∆ k = ~
0, the same calculations lead to Equation 16 with :∆ = s kT k i M x = 2 [ E i − E − k i M ] / Γ . (21)We recognize the usual Doppler-broadened cross section σ T , so that the differentialcross section becomes d σd Ω dE = 14 π k f k i × S ( ~ ∆ k, ∆ E ) × σ T ∗ ( E ∗ ) , (22)where T ∗ and E ∗ depend on the cosine of the scattering angle µ = cos θ : T ∗ ( µ ) = T k f (1 − µ ) ~ ∆ k E ∗ = E i − ( E f − E i ) ~k i ~ ∆ k~ ∆ k − ~k i ~k f M , (23)and for ~ ∆ k = ~ T ∗ = T E ∗ = E i − ~k i M . (24)Equation 23 is valid only when E i − ( E f − E i ) ~k i ~ ∆ k~ ∆ k − ~k i ~k f M is positive. The previousequation can be further simplified noting that T ∗ ≈ T [(1 + µ ) / E/ (2 E i )] ≈ T (1 + µ ) / − µ >> | ~ ∆ k | / | ~k i | we have ~k i ~ ∆ k~ ∆ k ≈ − / µ . Makingthese approximations, we get a simple formula d σd Ω dE = 14 π k f k i × S ( ~ ∆ k, ∆ E ) × σ T (1+ µ ) / ([ E i + E f ] / − E i µ/A ) . (25)Equation 25 provides a simple way to compute the differential cross section withinthe free gas model. It also gives an approximate way to account for solid state effectsby using the known scattering function S ( ~ ∆ k, ∆ E ) for the harmonic crystal. Note thatEquation 25 was demonstrated for a single resonance and it is not known if this formulacan be generalized to any form of free scattering amplitude.
4. Application to neutron resonant scattering in UO Low-enriched uranium oxide UO is widely used as nuclear-reactor fuel. With thepresent model, the scattering kernel of U in UO has been calculated near the firstresonance at 6.67 eV by numerical integration of the differential cross section overscattering angles. The weighted phonon spectrum for U in UO has been takenfrom the measurement of Dolling et al. [10]. The spectrum features two acoustic modesaround 14 and 21 meV.The S ( ~ ∆ k, ∆ E ) Van-Hove scattering function has been computed using the usualphonon expansion methods. The U resonance parameters evaluated by Moxon andSowerby [11] were used.The present model has been compared with the classic free gas kernel published byBlackshaw and Murray [2] and studied by Ouisloumen and Sanchez [3]. Fig. 1 and 2 pproximate Model of Neutron Resonant Scattering in a Crystal S ( ~ ∆ k, ∆ E ) of the FGM is used in Equation 16, the numerical computationsgive the same results as Ouisloumen and Sanchez. S c a tt e r i ng k e r ne l ( ba r n s . e v − ) Figure 1.
Comparison of scattering kernels calculated with the present formalism(black line) and the free gas model of Ouisloumen and Sanchez [3] (grey line) at incidentneutron energy of 6.674 eV. The dotted line is the kernel with a static model. pproximate Model of Neutron Resonant Scattering in a Crystal S c a tt e r i ng k e r ne l ( ba r n s . e v − ) Figure 2.
Comparison of scattering kernels calculated with the present formalism(black line) and the free gas model of Ouisloumen and Sanchez [3] (grey line) at incidentneutron energy of 6.520 eV. The dotted line is the kernel with a static model and thedashed line is the scattering cross section ( σ s /
100 in barns). pproximate Model of Neutron Resonant Scattering in a Crystal
5. Conclusion
An approximate formula is proposed to treat solid state effects in neutron-crystalinteractions. The DDCS is the product of a Doppler-broadened scattering cross sectionand the usual Van-Hove scattering function. d σd Ω dE = 14 π k f k i × S ( ~ ∆ k, ∆ E ) × σ T (1+ µ ) / ([ E i + E f ] / − E i µ/A ) . (26)This formula gives the correct free-gas limit for short-time range (high-temperaturecases). A rigorous calculation of the DDCS within the crystal lattice model is still anopen issue. The present work develops the case of a single isolated resonance but furthergeneralizations to multilevel forms of collision matrix may be possible.In reactor applications, the model predicts small solid-state effects in the scatteringof neutrons in UO at 300K, contrary to previous studies using the UPA approximation.Measurements of the secondary spectrum of neutrons scattered elastically in resonanceswould be valuable to check existing scattering models. Acknowledgments
It is a pleasure to acknowledge fruitful discussions with R. Dagan, C. R. Lubitz, and A.Santamarina.
Appendix A. Derivation of Equation 16
For ~ ∆ k = ~ χ in Equation 6 can be transformed into χ = exp { − i M [ τ − ~k i ~ ∆ k ( t − t ′ ) ~ ∆ k ~ ∆ k + ( t − t ′ ) ~k i ~k f ] − kT M [ τ − ~k i ~ ∆ k ( t − t ′ ) ~ ∆ k ~ ∆ k − k i k f − ( ~k i ~k f ) ~ ∆ k ( t − t ′ ) ] } . (A.1)Using the variable u = τ − ~k i ~ ∆ k ( t − t ′ ) / ~ ∆ k , we recognize χ the pair correlationfunction for a free gas: χ ( u ) = exp { − i M u ~ ∆ k − kT M u ~ ∆ k } . (A.2)Therefore, the Fourier transform of χ (integration over τ ) will lead to the S ( ~ ∆ k, ∆ E ) terms with a phase factor to account for the change of variable τ → u .The DDCS takes the form of Equation 22 with˜ σ = σ m Γ n Γ Γ Z ∞ Z ∞ e − zt e − z ∗ t ′ φ ( t − t ′ ) dtdt ′ , (A.3) pproximate Model of Neutron Resonant Scattering in a Crystal z = − i (cid:18) E i − E − ( E f − E i ) ~k i ~ ∆ k~ ∆ k − ~k i ~k f M (cid:19) + Γ2 and φ ( t ) = exp − kT M k i k f − ( ~k i ~k f ) ~ ∆ k t . (A.4)To compute the double Laplace transform, the following identity is used: Z ∞ Z ∞ e − zt − z ∗ t ′ φ ( t − t ′ ) dtdt ′ = 1 z + z ∗ Z ∞ e − zt φ ( t ) dt + Z ∞ e − z ∗ t φ ∗ ( t ) dt. (A.5)Equation A.3 becomes˜ σ = σ m Γ n Z ∞ e − i (cid:16) E i − E − ( E f − E i ) ~ki ~ ∆ k~ ∆ k − ~ki~kf M (cid:17) t + Γ t e (cid:18) − kT M k i k f − ( ~ki~kf )2 ~ ∆ k t (cid:19) dt + c.c. (A.6)c.c denotes the conjugate complex. This equation is then reduced into a singleintegral which can be further simplified using Z ∞ e − iat − bt dt = r πb exp − a b erf c [ ia √ b ] (A.7)We recognize in A.6 and A.7 the real of the complex complementary error function.Equations 16, 17 and 20 are obtained in a similar way when the potential and potential-resonant interference terms are accounted for. References [1] Wigner E P and Wilkins E J 1944 AECD-2275, Clinton Laboratory[2] Blackshaw G L and Murray R L 1967
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