MAGNETIC SCATTERING OF POLARIZED NEUTRONS AND POLARIZATION VECTOR OF SCATTERING NEUTRONS IN FERROMAGNETIC CRYSTALS

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 Sc

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:

d2σ dΩdE p ′ = m

2

(2π)3~5

p ′

p

∫

−∞

dte i~(E p′ −E p )r Sp

{

ρ σ ρ e

⟨

V p+′ p V p ′ p (t)

⟩}

(1)

where: ρ σ - density matrix of spin of neutrons; ρ e - density matrix of spin of electrons

E p ′, E p - energy of coming neutrons and scattering neutrons

ρ σ = 1

2(⃗ I + ⃗ p0⃗ σ)

⃗0: polarization vector of neutron

We consider to magnetic scattering of neutron, therefore we only consider potential of magnetic interaction:

V p ′ p=− 4π~2

m r0γ

1 2

∑

j

F j (⃗ q)e i⃗ q ⃗ R j × (⃗S j , ⃗ σ − (⃗e⃗σ)⃗e) (2)

where: ⃗ R j - location vector of nucleus j

⃗

q = ⃗ p − ⃗p’- scattering vector

⃗ e = ⃗ q/q - unit scattering vector

−

→

S j - spin of lattice point j

F j (⃗ q) =∫

ψ ∗

j

Z j

∑

υ

e i⃗ q⃗ rµ(⃗ sµ ⃗ Sj )

S j (S j+1) ψ j dτ j

ψ j - wave function of electron in atom j

⃗ µ - spin of electron µ in atom j

We denote:

L j = ( ⃗ S j , ⃗ σ − (⃗e⃗σ)⃗e) (3)

⃗

Then we have

d2σ dΩdE p ′ = m

2

(2π)3~5

p ′

p

∫

−∞

dte~i (E p′ −E p )t Sp {ρ σ ρ e ⟨A⟩}X jj ′ (⃗ q, t) (5)

where:

A =

(

4π~2

m r0γ

1 2

)2∑

jj ′

F j (⃗ q)F j ′ (⃗ q)L j (0)L j ′ (t)

X jj ′ (⃗ q, t) = ⟨e −i⃗q⃗ R j(0)e i⃗ q ⃗ R j′ (t) ⟩

In addition, we have:

1

2Sp {L1L2} =(M ⃗1M ⃗2)

(6) 1

2sp {(⃗p⃗σ) L1L2} = i[M ⃗1× ⃗ M2

]

Using (3),(4),(6) and (7), we can calculate trace in (5) and obtain:

Sp {ρ σ ρ e ⟨A⟩} = 4π2~4

m2 r0 γ2∑

jj ′

F j (⃗ q) F j ′ (⃗ {⟨−−−→

M j(0)−−→

M j ′ (t)

⟩

+ i ⟨[−→

M j(0)× −−→ M j ′ (t)

]⟩− →

p0

}

(8)

We now calculate expression (8) for ferromagnetic crystal:

In this case, one has:

−

→

S j = S j z m + ⃗ 1

2S

+

j m ⃗ −+1

2S

−

where: ⃗ m ± = ⃗ m

x ± i⃗m y ; ⃗ m x ; ⃗ m y are unit vector along axis x and axis y

⃗

m = [ ⃗ m x × ⃗m y]

Corresponding to (9), ⃗ M j can be written in the form:

−→

M j = S j z ⃗ µ + 1

2S

+

j ⃗ µ −+1

2S

−

j ⃗ µ+ (10)

⃗

µ = ⃗ m − (⃗e⃗m)⃗e; ⃗µ ± = ⃗ m ± −(⃗ e ⃗ m ±)

For the Heisenberg model, we have:

⟨−→

M j(0)

⟩

=

⟨

⃗

M j (t)

⟩

For ferromagnetic crystals, cross correlation functions are equal to 0:

⟨

S j z (0)S ±

j ′ (t)

⟩

=

⟨

S ±

j (0)S j z ′ (t)

⟩

=

⟨

S ±

j (0) S ±

j ′ (t)

⟩

and:

−

→ µ2

= 1− (− → e − → m)2

;

[− →

µ+× − → µ −]

=−2i (− → e − → m) − → e ;(− → µ+− → µ −)

= 1 + (− → e − → m)2

(14)

Using (10),(13),(14), we obtain:

⟨ ⃗ M j (0) ⃗ M j ′ (t) ⟩ =⟨S j z (0)S j z ′ (t)

⟩ [

1− (⃗e⃗m)2]

+ 14

⟨

S+j (0)S −

j ′ (t)

⟩ [

1 + (⃗ e ⃗ m)2

] +

+1 4

⟨

S −

j (0)S j+′ (t)

⟩ [

1 + (⃗ e ⃗ m)2

]

(15)

and:

i ⟨[M ⃗ j(0)× ⃗ M j ′ (t)

]

⟩⃗p0 = 1 2

{⟨

S −

j (0)S j+′ (t)

⟩

(⃗ e ⃗ m) (⃗ e− → p

0)−⟨S j+(0)S −

j ′ (t)

⟩

(⃗ e ⃗ m) (⃗ e− → p

0) } (16) Inserting (15) and (16) into (8), we obtain:

Sp {ρ σ ρ e ⟨A⟩} = 4π2 ~ 4

m2 r0 γ2∑

jj ′ F j (⃗ q) F j ′ (⃗ q)

{⟨

S z

j (0)S z

j ′ (t)

⟩ [

1− (⃗e⃗m)2]

+

1

4

⟨

S j+(0)S −

j ′ (t)

⟩ [

1 + (⃗ e ⃗ m)2

] +14

⟨

S −

j (0)S j+′ (t)

⟩ [

1 + (⃗ e ⃗ m)2

] +

1

2

⟨

S −

j (0)S j+′ (t)

⟩

(⃗ e ⃗ m) (⃗ e− → p0)−1

2

⟨

S j+(0)S −

j ′ (t)

⟩

(⃗ e ⃗ m) (⃗ e− → p0)}

Finally,we obtain the differential magnetic scattering cross-section of polarized neutron in ferromagnetic crystal

d2σ

dΩdE p′ = 2π1~r02γ 2 p p ′

+∫∞

−∞ dte

i

~(E p′ −E p )t ∑

jj ′ F j (⃗ q) F j ′ (⃗

{⟨

S j z (0)S j z ′ (t)

⟩ (

1− (⃗e⃗m)2) +14

⟨

S j+(0)S −

j ′ (t)

⟩ [(

1 + (⃗ e ⃗ m)2

)

− 2 (⃗e⃗m) (⃗e− → p0)

] +

1

4

⟨

S −

j (0)S j+′ (t)

⟩ (

1 + (⃗ e ⃗ m)2

)

+ 2 (⃗ e ⃗ m) (⃗ e− → p

0) }

X j ′ j (q, t)

III POLARIZATION VECTOR OF MAGNETIC SCATTERING

NEUTRON IN FERROMAGNETIC CRYSTAL

Polarization vector of magnetic scattering neutron in crystal is described by formula:

⃗

P =

+∫∞

−∞ dtSp

{

ρ σ ρ e

⟨

V p+′ p − → σ V

p ′ p (t)

⟩}

e~i (E p′ −E p )t

+∫∞

−∞ dtSp

{

ρ σ ρ e

⟨

V p+′ p V p ′ p (t)

⟩}

e~i (E p′ −E p )t

(17)

Denominator is calculated in section I We now need to find the numerator of (17) We can show:

1

2Sp { L1⃗ σL2 } = −i [ ⃗ M1× ⃗ M2 ] (18) 1

2Sp {(− → p − → σ ) L

1− → σ L2} = −→ M1(−→

M2− → p)+(−→

M1− → p ) −→

M2− − → p (−→

M1−→

M2

)

(19) Using (18) and (19), we obtain:

Sp

{

ρ σ ρ e V p+′ p ⃗ σV p ′ p (t)

}

= 4π2~ 4

m2 r0 γ2∑

jj ′ F j (⃗ q) F j ′ (⃗

[

−i ⟨[−→

M j(0)−−−→ ×M

j ′ (t)

]⟩

+⟨−−−→

M j(0)⟩ (⟨−−→

M j ′ (t)

⟩− →

p0

) +(⟨−−−→

M j(0)

⟩− →

p0) ⟨−−→

M j ′ (t)

⟩

− − → p0⟨−→

M j(0)−−→

M j ′ (t)

⟩]

X j ′ j (q, t)

Using (12),(15) and (16) for ferromagnetic crystal, we obtain:

Sp

{

ρ σ ρ e V p+′ p V p ′ p (t)

}

=

4π2 ~ 4

m2 r0 γ2∑

jj ′ F j (⃗ q) F j ′ (⃗

{

2S2(T )− → µ (− → µ − → p

0) −⟨S j z (0)S j z ′ (t)

⟩ (

1− (− → e − → m)2)− →

p0

]

−

−1

4

⟨

S j+(0)S −

j ′ (t)

⟩ [(

1 + (− → e − → m)2)

−

→ p0− 2 (− → e − → m) − → e]−

−1

4

⟨

S −

j (0)S j+′ (t)

⟩ [(

1 + (− → e − → m)2)

−

→ p0+ 2 (− → e − → m) − → e]}X

j ′ j (q, t)

So, polarization vector of magnetic scattering neutron in ferromagnetic crystals can be described by the following formula:

−

→ p = − → p1 + − → p2

+∫∞

−∞ dtSp

{

ρ σ

⟨

V p+′ p V p ′ p (t)

⟩}

e~i (E p′ −E p )t

(20)

Where:

⃗1 = 12

∞

∫

−∞ dt.e

i

~(E p′ −E p)∑

jj ′ F j (⃗ q) F j ′ (⃗ q) ×

×[⟨S j+(0)S −

j ′ (t)

⟩

(− → e − → m) − → e −⟨S −

j (0)S j+′ (t)

⟩

(− → e − → m) − → e]X

j ′ j (q, t)

⃗2 = ∫∞

−∞ dt.e

i

~(E p′ −E p )t∑

jj ′ F j (⃗ q) F j ′ (⃗

{

2S2(T )− → µ (− → µ − → p

0)−⟨S j z (0)S j z ′ (t)

⟩ (

1− (− → e − → m)2)− →

p0

−1

4

⟨

S j+(0)S −

j ′ (t)

⟩ (

1 + (− → e − → m)2)− →

p0 −1 4

⟨

S −

j (0)S j+′ (t)

⟩ (

1 + (− → e − → m)2)− →

p0

}

X j ′ j (q, t)

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] V G Baryshevsky, Nuclear Optics of Polarized Matter (1976) Minsk.

[2] P Mazur, D L Mills, Phys Rev B 26 (1981) 51-57.

[3] I A Idumov, R Oderop, Magnetic neutrons optics, Moskow, Science, (1996)

[4] Nguyen Dinh Dung, Vestnic BGU 1 (1987) 61-62.

[5] 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.