Proc Natl Conf Theor Phys 36 (2011), pp 278-282 MAGNETIC SCATTERING OF POLARIZED NEUTRONS AND POLARIZATION VECTOR OF SCATTERING NEUTRONS IN FERROMAGNETIC CRYSTALS NGUYEN THANH NGA, NGUYEN DINH DUNG Department of physics, college of Science, VNU Abstract In this note, the magnetic scattering of polarized neutrons on ferromagnetic crystals is studied In order to study this problem the method of nuclear optics of polarized matter has been used We obtained the analytical expression for the differential magnetic scattering cross-section of polarized neutrons and polarization vector of magnetic scattering neutrons in ferromagnetic crystals I INTRODUCTION In order to study of crystal structure, the method of nuclear optics of polarized matter have been used This method have been used in works [1,2,3,4,5] In this note, we study the differential cross-section of magnetic scattering of polarized neutrons and polarization vector of magnetic scattering neutrons in ferromagnetic crystal We showed that, they have important information about correlative function of electron lattice nodes spins II THE DIFFERENTIAL MAGNETIC SCATTERING CROSS-SECTION OF POLARIZED NEUTRONS IN FERROMAGNETIC CRYSTALS Suppose there is a stream of polarized neutrons falling on the ferromagnetic crystals that have polarized electrons The differential cross-section of magnetic scattering per unit solid angle, per unit energy, is given by: +∞ ∫ { ⟨ ⟩} i d2 σ m2 p′ (Ep′ −Ep )r + ′ p (t) = dte Sp ρ ρ V V (1) ′ σ e p pp dΩdEp′ (2π)3 p −∞ where: ρσ - density matrix of spin of neutrons; ρe - density matrix of spin of electrons Ep′ , Ep - energy of coming neutrons and scattering neutrons ρσ = (I⃗ + p⃗0⃗σ ) p⃗0 : polarization vector of neutron We consider to magnetic scattering of neutron, therefore we only consider potential of magnetic interaction: 4π 1∑ ⃗ ⃗j , ⃗σ − (⃗e⃗σ )⃗e) Vp′ p = − r0 γ Fj (⃗q)ei⃗qRj × (S (2) m j MAGNETIC SCATTERING OF POLARIZED NEUTRONS 279 ⃗ j - location vector of nucleus j where: R ⃗q = p⃗ − p⃗’- scattering vector ⃗e = ⃗q/q - unit scattering vector − → Sj - spin of lattice point j Zj i⃗q⃗r (⃗s S⃗ ) ∫ ∑ e µ µ j Fj (⃗q) = ψj∗ Sj (Sj +1) ψj dτj υ ψj - wave function of electron in atom j ⃗sµ - spin of electron µ in atom j We denote: ⃗j , ⃗σ − (⃗e⃗σ )⃗e) Lj = (S ⃗ j = (S ⃗j − (⃗eS ⃗j )⃗e) M (3) (4) Then we have d2 σ m2 = dΩdEp′ (2π)3 where: ( A= p′ p +∞ ∫ i dte (Ep′ −Ep )t Sp {ρσ ρe ⟨A⟩}Xjj ′ (⃗q, t) (5) −∞ 4π r0 γ m )2 ∑ Fj (⃗q)Fj ′ (⃗q)Lj (0)Lj ′ (t) jj ′ ⃗ Xjj ′ (⃗q, t) = ⟨e−i⃗qRj (0) ei⃗qRj ′ (t) ⟩ ⃗ In addition, we have: ( ) ⃗ 1M ⃗2 Sp { L1 L2 } = M (6) [ ] ⃗1 ×M ⃗ p⃗ sp {(⃗ p⃗σ ) L1 L2 } = i M (7) Using (3),(4),(6) and (7), we can calculate trace in (5) and obtain: {⟨−−−→−−→ ⟩ ⟨[−→ } −−→ ]⟩ − 4π 2 ∑ → ′ ′ ′ Sp {ρσ ρe ⟨A⟩} = M (0) × M (t) p r γ F (⃗ q ) F (⃗ q ) M (0) M (t) + i j j j j j j m2 ′ jj (8) We now calculate expression (8) for ferromagnetic crystal: In this case, one has: − → 1 ⃗ − + Sj− m ⃗+ Sj = Sjz m ⃗ + Sj+ m 2 where: m ⃗±=m ⃗ x ± im ⃗ y; m ⃗ x; m ⃗ y are unit vector along axis x and axis y (9) m ⃗ = [m ⃗x×m ⃗ y] ⃗ j can be written in the form: Corresponding to (9), M −→ 1 Mj = Sjz µ ⃗ + Sj+ µ ⃗ − + Sj− µ ⃗+ 2 (10) 280 NGUYEN THANH NGA, NGUYEN DINH DUNG where: ( ±) µ ⃗± = m ⃗ ± − ⃗em ⃗ ⃗e µ ⃗ =m ⃗ − (⃗em) ⃗ ⃗e; (11) For the Heisenberg model, we have: ⟩ ⟨−→ ⟩ ⟨ ⃗ j (t) = Sjz µ Mj (0) = M ⃗ (12) For ferromagnetic crystals, cross correlation functions are equal to 0: ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ Sjz (0)Sj±′ (t) = Sj± (0)Sjz′ (t) = Sj± (0) Sj±′ (t) = (13) and: ) ( [ ] − + →− 2 →+ − → − → → → − − → → → µ − µ = + (→ e− m) = −2i (− e− m) − e; → µ = − (− e− m) ; − µ ×→ µ (14) Using (10),(13),(14), ⟨ we obtain:⟩ [ ] ⟨ ⟩[ ] ⃗ ⃗ ⟨Mj (0)Mj ′ (t)⟩ = Sjz (0)Sjz′ (t) − (⃗em) ⃗ + 14 Sj+ (0)Sj−′ (t) + (⃗em) ⃗ + + ⟩[ ] 1⟨ − Sj (0)Sj+′ (t) + (⃗em) ⃗ (15) and: ⟩ ⟨ ⟩ } [ ] {⟨ − − → ⃗ j (0) × M ⃗ j ′ (t) ⟩⃗ Sj (0)Sj+′ (t) (⃗em) ⃗ (⃗e→ p0 ) − Sj+ (0)Sj−′ (t) (⃗em) ⃗ (⃗e− p0 ) i⟨ M p0 = (16) Inserting (15) and (16) into (8), we obtain:{⟨ ⟩[ ] ∑ ⃗ + Sp {ρσ ρe ⟨A⟩} = 4πm2 r0 γ Fj (⃗q) Fj ′ (⃗q) Sjz (0)Sjz′ (t) − (⃗em) jj ′ ⟨ ⟨ ⟩[ ] ⟩[ ] 2 + − − + 1 S (0)S S (0)S (t) + (⃗ e m) ⃗ (t) + (⃗ e m) ⃗ + + j j′ j′ ⟨ j ⟨4 ⟩ ⟩ } − → S − (0)S + (t) (⃗em) ⃗ (⃗e→ p ) − S + (0)S − (t) (⃗em) ⃗ (⃗e− p ) j j′ j j′ Finally,we obtain the differential magnetic scattering cross-section of polarized neutron in ferromagnetic crystal {⟨ ⟩( ) +∞ ∫ i (Ep′ −Ep )t ∑ 2 p′ d2 σ z (0)S z (t) ′ (⃗ = dte F (⃗ q ) F r γ q ) S − (⃗ e m) ⃗ ′ j j j j dΩdEp′ 2π p −∞ jj ′ ⟨ ⟩ [( ) ] − + 41 Sj+ (0)Sj−′ (t) + (⃗em) ⃗ − (⃗em) ⃗ (⃗e→ p0 ) + ⟨ ⟩( ) } − → S − (0)S + (t) + (⃗em) ⃗ + (⃗em) ⃗ (⃗e p ) X ′ (q, t) j j′ jj MAGNETIC SCATTERING OF POLARIZED NEUTRONS 281 III POLARIZATION VECTOR OF MAGNETIC SCATTERING NEUTRON IN FERROMAGNETIC CRYSTAL Polarization vector of magnetic scattering neutron in crystal is described by formula: { ⟨ ⟩} i +∞ ∫ → dtSp ρσ ρe Vp+′ p − σ Vp′ p (t) e (Ep′ −Ep )t −∞ P⃗ = +∞ (17) { ⟨ ⟩} i ∫ (Ep′ −Ep )t + dtSp ρσ ρe Vp′ p Vp′ p (t) e −∞ Denominator is calculated in section I We now need to find the numerator of (17) We can show: ⃗1 ×M ⃗2 ] Sp { L1⃗σ L2 } = −i [ M (18) ) −→ (−→−→) −→ (−→→) (−→− → → − → (19) p + M1 → p M2 − − p M1 M2 Sp {(− p→ σ ) L1 − σ L2 } = M1 M2 − Using { (18) and (19), }we obtain: [ ⟨[−→ −−−→ ]⟩ ∑ Sp ρσ ρe Vp+′ p⃗σ Vp′ p (t) = 4πm2 r0 γ Fj (⃗q) Fj ′ (⃗q) −i Mj (0)×Mj ′ (t) jj ′ ⟨−−−→⟩ (⟨−−→ ⟩ ) (⟨−−−→ ⟩ ) ⟨−−→ ⟩ ⟨−→ −−→ ⟩] − → → − + Mj (0) Mj ′ (t) p0 + Mj (0) − p0 Mj ′ (t) − → p0 Mj (0)Mj ′ (t) Xj ′ j (q, t) Using { (12),(15) and}(16) for ferromagnetic crystal, we obtain: Sp ρσ ρe Vp+′ p Vp′ p (t) = { ⟨ ⟩( ) ] → → → → − − 4π 2 ∑ F (⃗q) F ′ (⃗q) 2S (T )− µ (− µ− p ) − S z (0)S z (t) − (→ e→ m) − p − r γ j j jj ′ ⟨ ⟩ [( + − 1 Sj (0)Sj ′ (t) m2 0 j j′ ) ] → → → − − − + (− e− m) − p0 − ( → e→ m) → e − ⟨ ⟩ [( ) ]} → − − → → → − 14 Sj− (0)Sj+′ (t) + (→ e→ m) − p0 + ( − e− m) − e Xj ′ j (q, t) So, polarization vector of magnetic scattering neutron in ferromagnetic crystals can be described by the following formula: − → → p1 + − p2 − → (20) p = +∞ { ⟨ ⟩} i ∫ (Ep′ −Ep )t + dtSp ρσ Vp′ p Vp′ p (t) e − −∞ Where: ∫∞ ∑ i dt.e (Ep′ −Ep ) Fj (⃗q) Fj ′ (⃗q) × p⃗1 = 12 jj ′ [⟨ −∞ ⟩ ⟨ ⟩ ] − → → → → → × Sj+ (0)Sj−′ (t) (→ e− m) − e − Sj− (0)Sj+′ (t) (− e− m) − e Xj ′ j (q, t) { ⟨ ⟩( ) ∫∞ ∑ i → → → → − − dt.e (Ep′ −Ep )t Fj (⃗q) Fj ′ (⃗q) 2S (T )− p⃗2 = µ (− µ− p0 ) − Sjz (0)Sjz′ (t) − (→ e→ m) − p0 −∞ jj ′ ⟨ ⟩( ⟨ ⟩( ) ) } − → → → → → e− m) → e− m) − p0 − 14 Sj− (0)Sj+′ (t) + (− p0 Xj ′ j (q, t) − 41 Sj+ (0)Sj−′ (t) + (− 282 NGUYEN THANH NGA, NGUYEN DINH DUNG IV CONCLUSION In this note, we obtain the analytical expressions for: i) The differential magnetic scattering cross-section of polarized neutron in ferromagnetic crystals ii) For the above formulas one can set the information about the lattice spin correlation functions In the limit of unpolarized neutron we recover the result of Idumop-Oderop[3] REFERENCES [1] [2] [3] [4] [5] V G Baryshevsky, Nuclear Optics of Polarized Matter (1976) Minsk P Mazur, D L Mills, Phys Rev B 26 (1981) 51-57 I A Idumov, R Oderop, Magnetic neutrons optics, Moskow, Science, (1996) Nguyen Dinh Dung, Vestnic BGU (1987) 61-62 Luong Minh Tuan, Nguyen Thi Thu Trang, Nguyen Dinh Dung, VNU JOURNAL OF SCIENCE: Mathematics-Physics T.XXII, N0 2AP (2006) 178-181 Received 30-09-2011 ... j′ jj MAGNETIC SCATTERING OF POLARIZED NEUTRONS 281 III POLARIZATION VECTOR OF MAGNETIC SCATTERING NEUTRON IN FERROMAGNETIC CRYSTAL Polarization vector of magnetic scattering neutron in crystal.. .MAGNETIC SCATTERING OF POLARIZED NEUTRONS 279 ⃗ j - location vector of nucleus j where: R ⃗q = p⃗ − p⃗’- scattering vector ⃗e = ⃗q/q - unit scattering vector − → Sj - spin of lattice point... NGA, NGUYEN DINH DUNG IV CONCLUSION In this note, we obtain the analytical expressions for: i) The differential magnetic scattering cross-section of polarized neutron in ferromagnetic crystals ii)