Proc Natl Conf Theor Phys 36 (2011), pp 274-277 POLARIZATION VECTOR OF SCATTERING NEUTRONS IN CRYSTAL WITH MAGNETIC HELICOIDAL STRUCTURE PHAM THI THU HA, NGUYEN DINH DUNG Department of Physics , College of Science, VNU Abstract In this note, the differential magnetic scattering cross-section of polarized neutrons in crystal with magnetic Helicoidal structure is studied In order to study this problem the method of nuclear optics of polarized matter have been used We obtained the differential magnetic scattering cross-section of polarized neutrons and the change of neutron polarization I INTRODUCTION In order to study the crystal structure the method of nuclear optics of polarized matter have been used This method has been used to study polarization of atomic nucleus and correlation function of nuclear spins in works [1,2,3,4,5] In this note, we study the differential magnetic scattering cross-section of polarized neutrons and polarization vector of scattering neutrons in crystal with magnetic Helicoidal structure II THE DIFFERENTIAL CROSS-SECTION OF MANEGTIC SCATTERING OF POLARIZED NEUTRONS IN CRYSTAL WITH MEGNETIC HELICOIDAL STRUCTURE Suppose there is a stream of polarized neutron falling on the magnetic Helicoidal structure crystal that has polarized electrons The differential magnetic scattering crosssection per unit solid angle, per unit energy, is given by [1]: d2 σ m2 = dΩdEp′ (2π)3 p′ p +∞ ∫ { ⟨ ⟩} i dte (Ep′ −Ep )t Sp ρσ ρe Vp+′ p Vp′ p (t) (1) −∞ 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 = 12 (I⃗ + p⃗0⃗σ ): polarization vector of neutron We only consider potential of magnetic interaction.The matrix element of transition is given by: 1∑ 4π ⃗ ⃗j , ⃗σ − (⃗e⃗σ )⃗e) r0 γ Fj (⃗q)ei⃗qRj × (S (2) Vp′ p = − m j ⃗ j - location vector of nucleus j Where: R ⃗q = p⃗ − p⃗’- scattering vector ⃗e = qq⃗ : unit scattering vector − → Sj : spin of lattice point j POLARIZATION VECTORS OF SCATTERING NEUTRONS IN CRYSTAL 275 ⃗s: spin of coming neutron Zj i⃗q⃗r (⃗s S⃗ ) ∫ ∑ e υ υ j Fj (⃗q) = ψj∗ Sj (Sj +1) ψj dτj υ ψj : wave function of electron in atom j ⃗sν : spin of electron in atom j In addition, use the following notations: ⃗j , ⃗σ − (⃗e⃗σ )⃗e) Lj = (S ⃗ j = (S ⃗j − (⃗eS ⃗j )⃗e) M We can calculate expression (1) for magnetic Helicoidal crystal structure: d2 σ m2 = dΩdEp′ (2π)3 p′ p +∞ ∫ i dte (Ep′ −Ep )t Sp {ρσ ρe ⟨A⟩}Xjj ′ (⃗q, t) (3) (4) (5) −∞ where: ( A= 2π r0 γ m )2 ∑ Fj (⃗q)Fj ′ (⃗q)Lj (0)Lj ′ (t) jj ′ ⃗ Xjj ′ (⃗q, t) = ⟨e−i⃗qRj (0) ei⃗qRj ′ (t) ⟩ ⃗ where we have used: [ ] ⃗1 ×M ⃗ p⃗ sp {(⃗ p⃗σ ) L1 L2 } = i M ( ) ⃗ 1M ⃗2 Sp { L1 L2 } = M We can calculate trace in (5) and obtain: ( )2 ∑ 2π sp {ρσ ρe ⟨A⟩} = r0 γ Fj (⃗q) Fj ′ (⃗q) T1 m ′ (6) (7) (8) jj d2 σ 2 p′ = r γ dΩdEp′ 2π p where: T1 = +∞ ∫ ∑ i dte (Ep′ −Ep )t Fj (⃗q) Fj ′ (⃗q) T1 Xj ′ j (⃗q, t) −∞ (9) jj ′ {⟨−−−→−−→ ⟩ ⟨[−→ −−→ ]⟩ →} Mj (0)Mj ′ (t) + i Mj (0) × Mj ′ (t) − p0 In the magnetic Helicoidal crystal structure, vector of spin of atom can be described by formula: − → ⃗ ⃗ − ⃗ ⃗ ⃗ + + Seik0 Rj m ⃗ (10) S j = Se−ik0 Rj m 2 Where: m ⃗+=m ⃗ x + im ⃗ y; m ⃗−=m ⃗ x − im ⃗ y; m ⃗ x and m ⃗ y are unit vectors along axis x and y µ ⃗ =m ⃗ − (⃗em) ⃗ ⃗e µ ⃗± = m ⃗ ± − (⃗em ⃗ ± ) ⃗e 276 PHAM THI THU HA, NGUYEN DINH DUNG And m ⃗ is a unit vector along the symmetric axis of the magnetic Helicoidal crystal structure − → Corresponding to (10), M j can be expanded by the following formula: ] −→ [ −i⃗k0 R⃗ j + ⃗ ⃗ − Mj = S e µ ⃗ + eik0 Rj µ ⃗ (11) Using (8) and (11), we obtain: Sp {ρσ ρe ⟨A⟩} = π2 2 ∑ r0 γ S Fj (⃗q) Fj ′ (⃗q) T2 m2 ′ (12) jj T2 = [( ) [ + ]] −i⃗k µ ⃗ +µ ⃗ − + i⃗ p0 µ ⃗ ×µ ⃗− e ( ⃗ j −R ⃗ R ) j ′ + [( ) [ + − ]] i⃗k0 µ ⃗ +µ ⃗ − − i⃗ p0 µ ⃗ ×⃗ µ e ( ⃗ j −R ⃗ R ) j ′ After lengthy calculation we have got the differential scattering cross-section of polarized neutrons in crystal with magnetic Helicoidal structure as follows d2 σ 2 p′ = r γ dΩdEp′ 8π p +∞ ∫ ∑ i Fj (⃗q) Fj ′ (⃗q) Xjj ′ (⃗q, t) T2 dte (Ep′ −Ep )t S (13) jj ′ −∞ III POLARIZATION VECTOR OF SCATTERING NEUTRONS IN CRYSTAL WITH MAGNETIC HELICOIDAL STRUCTURE → Polarization vector of scattering neutrons in crystal − p can be calculated by the following formula [1,2]: { ⟨ ⟩} i +∞ ∫ → dtSp ρσ ρe Vp+′ p − σ Vp′ p (t) e (Ep′ −Ep )t −∞ p⃗ = +∞ (14) { ⟨ ⟩} i ∫ (Ep′ −Ep )t + dtSp ρσ ρe Vp′ p Vp′ p (t) e −∞ where we used: (−→−→) ) −→ −→ (−→→) (−→− → → − → p + M1 → p M2 − − p M1 M2 Sp {(− p→ σ ) L1 − σ L2 } = M1 M2 − [ ] ⃗1 ×M ⃗2 Sp {L1⃗σ L2 } = −i M We can calculate the numerator in the formula (14) { ⟨ ⟩} 4π ∑ Sp ρσ ρe Vp+′ p⃗σ Vp′ p = r02 γ Fj (⃗q) Fj ′ (⃗q)Xj ′ j (⃗q, t) T3 m ′ jj where: ⟨[−→ −−→ ]⟩ ⟨−→⟩ (⟨−−→ ⟩ ) → T3 = −i Mj × Mj ′ (t) + Mj Mj ′ (t) − p0 + (⟨−→⟩ ) ⟨−−→ ⟩ ⟨−→−−→ ⟩ − → + Mj → p0 Mj ′ (t) − − p0 Mj Mj ′ (t) (15) (16) (17) POLARIZATION VECTORS OF SCATTERING NEUTRONS IN CRYSTAL 277 Then the polarization vector of scattering neutrons for magnetic Helicoidal crystal structure is given by: ⟩} π { ⟨ ∑ 2 = Sp ρσ ρe Vp+′ p⃗σ Vp′ p r γ S Fj (⃗q) Fj ′ (⃗q)T4 (18) m2 ′ jj where: { ( − ) [ − ] − ( ) ( + − )} ) ( ⃗ j −R ⃗ ′ −i⃗k0 R j + T4 = µ ⃗ µ ⃗ p⃗0 − i µ ⃗ ×µ ⃗ +µ ⃗ µ ⃗ p⃗0 − p⃗0 µ ⃗ µ ⃗ e ) ( { +( − ) [ − ] ( + ) ( + − )} i⃗k0 R⃗ j −R⃗ ′ j + µ ⃗ µ ⃗ p⃗0 − i µ ⃗ ×µ ⃗+ + µ ⃗− µ ⃗ p⃗0 − p⃗0 µ ⃗ µ ⃗ e Using (11),(13) and (14), we can calculate the polarization vector of neutron in magnetic Helicoidal crystal structure, which is given by: +∞ ∫ ∑ i dte (Ep′ −Ep )t S Fj (⃗q) Fj ′ (⃗q)Xjj ′ (⃗q, t) T4 + p⃗ = −∞ +∞ ∫ −∞ + + jj ′ ∑ dte (Ep′ −Ep )t S Fj (⃗q) Fj ′ (⃗q) Xjj ′ (⃗q, t) T2 (19) i jj ′ IV CONCLUSION We obtain the analytical expression for: i) The differential magnetic scattering cross-section of polarized neutrons in crystal with magnetic Helicoidal structure ii) The polarization vector of magnetic scattering in crystal with magnetic Helicoidal structure For these expression, one can get useful information of lattice spin correlation function In the limit of unpolarized neutrons, we rederived the Idumov-Oderop result[2] REFERENCES [1] [2] [3] [4] [5] V G Baryshevsky, Nuclear Optics of Polarized Matter (1976) Minsk I A Idumov, R.Oderop, Magnetic neutron optics (1966) Science, Moskow P Mazur, D L Mills, Phys Rev 26 (1981) Nguyen Dinh Dung, VNU J Sci (1993) 56 Ly Cong Thanh, Nguyen Thi Khuyen, Nguyen Dinh Dung, VNU J Sci Mathematics Physics T.XXII, N.0 AP (2006) 154156 Received 30-09-2011 ... −∞ III POLARIZATION VECTOR OF SCATTERING NEUTRONS IN CRYSTAL WITH MAGNETIC HELICOIDAL STRUCTURE → Polarization vector of scattering neutrons in crystal − p can be calculated by the following formula... Mj ′ (t) (15) (16) (17) POLARIZATION VECTORS OF SCATTERING NEUTRONS IN CRYSTAL 277 Then the polarization vector of scattering neutrons for magnetic Helicoidal crystal structure is given by: ⟩}... obtain the analytical expression for: i) The differential magnetic scattering cross-section of polarized neutrons in crystal with magnetic Helicoidal structure ii) The polarization vector of magnetic