THE UNIVERSITY OF MICHIGAN The Foundations of Neutron Transport Theory R. K. OSBORN S. YIP October 1963 ANN ARBOR THE FOUNDATIONS OF NEUTRON TRANSPORT THEORY R. K. Osborn S. Yip Department of Nuclear Engineering University of Michigan ACKNOWLEDGEMENTS We thank Professor G. C. Sommerfield for his perusal of the manuscript and his helpful criticisms thereof. One of us (s. Y.) gratefully acknowledges the University of Michigan Institute of Science and Technology for a post-doctoral fellow- ship and the Michigan Memorial-Phoenix Laboratory for hospitality during the course of this work. iii Page Missing in Original Volume TABLE OF CONTENTS Page ACKNOWLEDGEMENTS. ........................................... , iii ACKNOLEDGMENS .... ..... ..... ..... ..... ..... ..... ..... .0 . ..... iii L ISTO SYMO LS. O . ..... ...... ..... ...... ...... ...... .. .... vi I INTRODUCTION...................... .................. 1. II A TRANSPORT EQUATION IN COARSE-GRAINED PHASE SPACE......... 9 A. Some Basic Formalism... ............................. 10 B. A Kinetic Equation for F(X,K,t). ..... ..... ............. 15 C. The Streaming Termm.................................. 26 D. The Collision Terms............... ... .......... ...... 29 E. Effect of an External Field............................ 36 III NEUTRON-NUCLEAR INTERACTIONS: MAINLY NUCLEAR CONSIDERATIONS.......... ..... ... ..... .. 41 A. Formal Development of the Transition Probability........43 B. Radiative Capture............ ..... ......... ....... 58 C. Elastic Scattering..................................... 73 D. Fission and Inelastic Scattering.......................80 E. The Neutron Balance Equation in Continuous Momentum Space ......................................... 86 IV NEUTRON-NUCLEAR INTERACTIONS: MEDIUM EFFECTS........... 93 A. Transport in an Ideal Gas.............................. 96 Radiative Capture.................................... 97 Elastic Scattering................................... 103 B. Transport in Crystals................................. 111 Radiative Capture.................................... 112 Elastic Scattering ................................... 118 The Thermal Average.................................. 134 A. Neutron Thermodynamics............. ......... 143 B. Higher Order Neutron Densities - Particularly the Doublet Density.......... ...................... 156 V Page Missing in Original Volume LIST OF SYMBOLS In the following some of the more frequently used mathematical symbols are defined. Whenever possible the equation in which the symbol is first introduced is given and the reader is referred to the text for definition. Some symbols are used for more than one quantity, but their meanings should be clear from the context. A - Atomic mass A(K'k';Kk) - Scattering "frequency", Equation (5.2) a - "Free-atom" scattering length a+(X,K, s), a(X,K, s ) - Creation, destruction operators for neutrons at (X,K,s) - Scattering lengths, see Equation (4.71), a 2 _ coh inc (4.72), and (4.76) 2 BA+1 - (m+MA-MA+l) c B ( ',) B (P' ,P),B (P" P P') - Fission frequencies, Equations (2.80), o-- - a a (5.43b), and (5,43c) D - Density operator, Equation (2.29); also see Equation (4,45) D -(t)- Density matrix, Equation (2.28) 2 E, EK - Neutron energies, my2/2 or -h K2/2m E(X,x) - Step function, Equation (2.2) ER - Nuclear recoil energy, Equation (4..53) E - Photon energy, -hcK EKk, Er - Relative energies, Equation (4.4) EA - External energy of mass A nucleus in k external state k A E - Internal energy of mass A nucleus in internal state a vii A+1 E - Excitation energy of a-th level in mass A+l compound nucleus EA+1 A+1 A+1 - E -B - EA+1 + s + BA+1 a a (n) - Coarse-grained neutron singlet density, F(X,K, t) or F11 (_K- t) Equation (2.27) Fs (XK,t) - Spin-dependent coarse-grained neutron density, Equation (3.65) F (X, _,t) - Coarse-grained photon density, Equation (3.67) (a) F (X,K,t) - Coarse-grained singlet density for a 1 - particles, Equation (5.44a) F2 (X,K,X',K',t) or - Coarse-grained neutron doublet density, Fn)(X,K,X',K_',t)2 Equation (2.70) (a) - Coarse-grained doublet density for a F2 (x,K,X',K',t) particles, Equation (5.44b) - Coarse-grained cross density for neutrons and a particles, Equation (5.44c) f(X,K,t) - Neutron density, Equation (2.30) Density of fA(X,kt) mass A nucleus, Equation (4.27) S(P ' - P) - Scattering frequency, Equation (2.80) D (P-4P' - Detection frequency, Equation (5.l45c) G(Z) - Resolvent operator, Equation (3.2) G(r,t) - See Equation (4.78) H - Hamiltonian of system which interacts with the neutrons H' - See Equat ion (2. 42) In(x) - Modified Bessel function of order n and argument x viii K - Neutron wave vector, //; also a particle wave vectors - A set of J neutron wave vectors {KJ {K } ({K} ' ) - A set of J wave vectors which contains JK (does not contain) the wave vector K {Ks} - Wave vector and spin labels of the J neutrons produced by fission k - Quantum label of nuclear external state, Equation (3.38) k - Wave vector of a nucleus, p/t ktB - Boltzmann's constant L Cell length in coarse-grained configuration space - Superscript or subscript denotes a particular nucleus t(v'-> v) - Fission frequency, Equation (3.99) M - Mass of the nucleus m - Neutron mass mL - Mass of L-th nucleon N(X,K,s), N'(X,K,s) - Neutron occupation number at (X,K,s) in the state In>, In'> - Number of mass A nuclei in the spatial cell NA(X) centered at X n - Nuclear density n > - Neutron state, Equation (2.18); system state, Equation (3.35) - Total number operator P - See Equation (4.45) P - Neutron momentum; momentum of a~ part icle ix Pk,P(k),D (t) - Distribution of external nuclear states ~-kk L L - Nucleonic momentum conjugate to Q - Neutron momentum transfer divided by T1 RT(X,K) - Total reaction rate Rs(X,K), Rs(X,K'---K) - Scattering reaction rate, see Equations (3.95) and (3,97) - Position vector of center-of-mass of 2-th nucleus R - Reduced transition matrix, Equation (3.33) Rnn' r - Reduced reaction matrix with neutron and nn'l photon number dependence extracted, Equation (3.51) L r - Position of L-th nucleon in £-th nucleus c c n'n kf' X,kK~s r , ,kKs'Rc(X,K) - See Equations (3.42), (3.51), (3.52), and (3.94) Rk a~k~({Ksjj), riik({Ks - See Equations (3.81), (3.82), and RF(X,K), RF(X,K'-_) (3.96), (3.98) Es E s - See Equations (3.68) and (3.69) Rk'K's',kKs' k 'K's',kKs I.s Is r - See Equations (3.87) and (3.88) k'K's'c'X',kKs C_ 5 - Neutron spin orientations label sa - Shift function, Equation (3.49) U' - Reduced potential for elastic potential scattering by £-th nucleus, Equation (3.73) U1 - See Equation (4.19) UN2 (uN) - Nuclear matrix element for emission (absorption) of a neutron by 2-th nucleus - Nuclear matrix Oa~"(a"0 ) element for emission (absorp- tion) of a photon by 2-th nucleus, Equation (5.55) x V - VN+V7 V - Nuclear velocity vN - Neutron-nuclear interaction 7 V - Photon-nuclear interaction - Neutron velocity v0 (r), v1 (r) - Potential functions for elastic potential scattering, Equation (3.71) - Transition probability per unit time, nn (T) Equation (2;.56) WC wsGwsL ,WF - See Equation (3.39) f'n' f' n'n n'n K' K-+K'' s-- - - - _F - See Equations (2.65), (2.67), (2.73), and WK-4{2KJ (2.71.) - Position vector locating the center of a spatial cell x - Equilibrium position vector of the 2-th nucleus in a crystal. z - See Equation (4..5); see also Equation (5,23); also as complex variable Greek Letters a - Quantum label of nuclear internal state 1 - (kBT) ra - Width function, Equation (3.50) r(R) (p(N)) - Partial radiation (neutron) width for a-th level (Z) - Width and shift function 7n - EK - EK' xi 8(x - x') - Dirac delta - Kronecker delta. 8x ,K(X - x') - See Equation. (4.45) 9 - Scattering angle (laboratory) - Photon wave vector x - Photon polarization; also as effective range in neutron-nuclear interaction - See Equation (3.72) x0' l'1 - Reciprocal neutron wave vector, K~ - Reduced mass; also as chemical potential, Equation (5.6) p (X, K) - Neutron number operator, Equation (2.16) Ec - J-9F'EsEt - Macroscopic cross section for capture, detection, fission, scattering and total reaction - Microscopic cross section a(E -4E',©) - Energy and angle differential scattering cross section, Equation (4.36) oP ar l - Microscopic cross section for potential, s' s' s resonant, and interference scatterings (p(X,K,x) - Cell function, Equation (2.1) x - See Equation (4.25) - Oscillator frequency n nn - See Equation (3,17) xii I. INTRODUCTION Theoretical studies of neutron distributions(l) are usually based - more or less directly - upon the transport equation, ( + v V + v ) f (x, v, t) = fdv'vt E' f(x, v', t))'(v' -v) + S(x, v, t). (l.) Here E is a macroscopic cross-section (or probability per unit path for small paths for a binary interaction), )' is a scattering frequency, and S is a neutron source which may or may not depend upon f. The distribu- tion function, f(x, v, t), is the neutron singlet density in phase space or the expected number of neutrons per unit phase volume to be found at the phase point (x, v) at time t. The theoretical defense of this equation is generally couched in phenomenological, though plausible, terms.(1) However, its usefulness as a description of neutron distributions has received so much experi- mental verification that little interest has been generated for the exploration of the logical basis for its phenomenological derivation.* Nevertheless, such an exploration seems desirable - if only on esthetic grounds, or "for the sake of completeness." Thus in this work such a study is initiated. The axiomatic basis for the theoretical examination of macro- scopic systems is usually taken to be the Liouville equation. Although *See, for example, the determination of neutron age in water. (2) -l- -2- there is a deceptively classical appearance to Equation (1.1), it turns out to be practically necessary to start from the quantum Liouville equation.