03-04-trans_normal - Αντιγραφή.pdf

dσ(k, ϑ, ϕ)

dΩ=

N

N0

− 2

2µ∇2u(r) + V (r)u(r) = Eu(r)

(∇2 + k2

)u(r) =

2µV (r)

2u(r) , k2 =

2µE

2

u(r) = eikz + f (k, ϑ, ϕ)1

reikr , r → ∞

dσ(k, ϑ, ϕ)

dΩ=

N

N0= |f (k, ϑ, ϕ)|2

V (r) = V (r)uklm(r) = Rkl(r)Y

ml (ϑ, ϕ)

d2Rkl

dr 2+

2

r

dRkl

dr+

[k2 − 2µ

2V (r) − l(l + 1)

r 2

]Rkl = 0

uk(r , ϑ) =∑∞

l=0 ClRkl(r)Pl(cos ϑ)

Rkl(r)

V (r) r0 V (r) 0 r ≥ r0 r ≥ r0

d2Rkl

dr 2+

2

r

dRkl

dr+

[k2 − l(l + 1)

r 2

]Rkl = 0 , r ≥ r0

!"! : Rkl(r) = Al jl(kr) + Blnl(kr) , r ≥ r0

"# r → ∞ $: jl(kr) =sin(kr − 1

2lπ)

kr, nl(kr) = −cos(kr − 1

2lπ)

kr Rkl(r)

Rkl(r) =Al

cos δl︸︷︷︸Cl

1

krsin(kr − 1

2 lπ + δl) = Cl1

krsin(kr − 1

2 lπ + δl) , r → ∞ %

Rkl(r) = & & & V (r) r → ∞!' # &

uk(r , ϑ) =∞∑l=0

Cl1

krsin(kr − 1

2 lπ + δl)Pl(cos ϑ) , r → ∞ (

) # & δl = δl(k) & l* & &!

) # & & Schrodinger # # & & &

uk(r , ϑ) =∞∑l=0

Cl1

krsin(kr − 1

2 lπ + δl)Pl(cos ϑ) , r → ∞ +

uk(r) = eikz + f (k, ϑ)1

reikr , r → ∞ ,

) & # & -& & $ & & r → ∞! "#&

∞∑l=0

Cl1

krsin(kr − 1

2 lπ + δl)Pl(cos ϑ) = eikz + f (k, ϑ)1

reikr .

"# / & ##& & #& Legendre

eikz = eikr cos ϑ =∞∑l=0

i l(2l + 1)1

krsin(kr − 1

2 lπ)Pl(cos ϑ) , r → ∞ 0

sin t = 12i

(eit − e−it) $ . & & #& !

f (k, ϑ) =1

2ik

∞∑l=0

(2l + 1)(ei2δl − 1)Pl(cos ϑ) 1

f (k, ϑ) =1

k

∞∑l=0

(2l + 1) eiδl sin δlPl(cos ϑ)

& & & &

dσ

dΩ= |f (k, ϑ)|2 =

1

k2

∣∣∣∣∣∞∑l=0

(2l + 1) eiδl sin δlPl(cos ϑ)

∣∣∣∣∣2

σ(k) =4π

k2

∞∑l=0

(2l + 1) sin2 δl

) $ & & & &# 2 &# 2 #&&- 2!

• ' 2&& & #& 2&& # &# && !• 3) #& #& #2 4 #&&& &# &# δl !• & # 4 & 2 # & #& 4 &4!

r0 #&-& # & $& &*&

|l| = pb, b < r0

#& p & & & b # * & &!

r0

b>r0

b<r0

"# |l |2 = √

l(l + 1) p = k $&

√l(l + 1) = kb → b =

√l(l + 1)

k

:

• 2 & &# b > r0!

• $ & &# b ≤ r0!• 5 $ & & & l ≤ lmax √

lmax(lmax + 1) kr0 %

• 6) r0 #& k #& & & & & # 4& & & #& !• 5 #& $ & & & l = 0 2 4 & !"# P0(cos ϑ) = 1 & #& & & 2

f (k, ϑ) 1

keiδ0(k) sin δ0(k) ,

dσ

dΩ 1

k2sin2 δ0(k) (

#& $ & & # && &# ϑ!

!

V (r) < 0 & &# 2! # V (r) > 0 & &# ! 7 # & & &# 2!

0

(a)

Λυση για V=0

r

δ/k

0

(b)

r

δ/k Λυση για V=0

φ(r) = rR(r) a V = 0 b V = 0

"

&& & &

Rkl(r) =

R

(i)kl (r) r < r0

R(o)kl (r) r > r0

+

R(i)kl =& & & &!

R(o)kl =& # &$ #& & &!

' R(o)kl &

R(o)kl (r) = Al jl(kr) + Blnl(kr) = Al

[jl(kr) +

Bl

Alnl(kr)

]= Cl [cos δl jl(kr) − sin δl nl(kr)] r > r0 ,

#& BlAl

= − tan δl Al

cos δl= Cl ! ' # & R

(o)kl (r)

dR(o)kl (r)

dr= Cl

[cos δl

djl(kr)

dr− sin δl

dnl(kr)

dr

]r > r0 .

5 $ & R(i)kl (r) # Schrodinger 0 ≤ r < r0

#&&-& # !) &# & # 2 $ & # 4& & & r = r0

R(i)kl (r0) = R

(o)kl (r0),

dR(i)kl (r)

dr

∣∣∣∣∣r=r0

=dR

(o)kl (r)

dr

∣∣∣∣∣r=r0

' & 4 $ & 2# 4& &

1

R(i)kl (r)

dR(i)kl (r)

dr

∣∣∣∣∣r=r0

=1

R(o)kl (r)

dR(o)kl (r)

dr

∣∣∣∣∣r=r0

' 4 , . r = r0 # # $

1

R(i)kl (r)

dR(i)kl (r)

dr

∣∣∣∣∣r=r0

=cos δl

djl (kr)dr

− sin δldnl (kr)

dr

cos δl jl(kr) − sin δl nl(kr)

∣∣∣∣∣r=r0

0

/ & & & & 2&

βl = βl(k) =r0

R(i)kl (r)

dR(i)kl (r)

dr

∣∣∣∣∣r=r0

1

& & & & & & $ 0 cos δl & # & tan δl # # &

tan δl =r0

djl (kr)dr

− βl jl(kr)

r0dnl (kr)

dr− βl nl(kr)

∣∣∣∣∣r=r0

=kr0

djl (ρ)dρ

− βl jl(ρ)

kr0dnl (ρ)

dρ− βl nl(ρ)

∣∣∣∣∣ρ=kr0

#

tan δl =r0

djl (kr)dr

− βl jl(kr)

r0dnl (kr)

dr− βl nl(kr)

∣∣∣∣∣r=r0

=kr0

djl (ρ)dρ

− βl jl(ρ)

kr0dnl (ρ)

dρ− βl nl(ρ)

∣∣∣∣∣ρ=kr0

) δl(k) 4 # k r0 βl # & & & &! / #&2& ρ = kr0 1 Bessel Neumann $

jl(kr0) (kr0)l

(2l + 1)!!, nl(kr0) −(2l − 1)!!(kr0)

−l−1

djl(ρ)

dρ

∣∣∣∣ρ=kr0

l(kr0)l−1

(2l + 1)!!,

dnl(ρ)

dρ

∣∣∣∣ρ=kr0

(2l − 1)!!(l + 1)(kr0)−l−2

#& (2l + 1)!! = 1 · 3 · 5 · · · (2l + 1)! 8 $ &

tan δl (kr0)2l+1

(2l + 1)!!(2l − 1)!!

l − βl

l + 1 + βl, kr0 1

6) & kr0 #& kr0 → 0 2 $&βl = βl(k) βl(0), tan δl sin δl δl

&# $ δl cl(kr0)2l+1, cl = 1

(2l+1)!!(2l−1)!!l−βl (0)

l+1+βl (0)

δl cl(kr0)2l+1, cl =

1

(2l + 1)!!(2l − 1)!!

l − βl(0)

l + 1 + βl(0)

5& & k → 0 $& δl → 0 l = 0, 1, 2, . . .!

7& & k → 0 & # &##&& & 4 #& & 2 && !

$ 2& - # & δl # &$ & & #& & & & l(l + 1)/r 2 & Schrodinger#& #&- #& & & & “22&” # & & &!

! " Breit-Wigner

9 & $ ! ) &# & # $

tan δl (kr0)2l+1

(2l + 1)!!(2l − 1)!!

l − βl

l + 1 + βl%

: & #& E = Er $

l + 1 + βl(Er ) = 0 (

tan δl(Er ) = ∞ δl(Er ) =

π

2

5 # # & l− # &- && E = Er & & & & !9 & $ % & #& & && Er ! 7& # l + 1 + βl(E) Taylor # &$ & & E = Er

l + 1 + βl(E) = l + 1 + βl(Er ) + (E − Er )∂βl(E)

∂E

∣∣∣∣E=Er

+ · · ·

& && $ (

l + 1 + βl(E) (E − Er )∂βl(E)

∂E

∣∣∣∣E=Er

8 & βl(E) # $ $ % & & # &

l − βl

l + 1 + βl=

2l + 1

l + 1 + βl− 1 2l + 1

[βl(E)/∂E ]E=Er

1

E − Er− 1

2l + 1

[βl(E)/∂E ]E=Er

1

E − Er

' # & # E & && & #& & & &! /24 & & # && $ % & & 2&

Γ = − 2(kr0)2l+1

[(2l − 1)!!]2l

[∂βl/∂E ]E=Er

+

$ # &

tan δl(E) − Γ/2

E − Er,

"# & & l− & & & &

σl(E) =4π

k2(2l + 1) sin2 δl =

4π

k2(2l + 1)

tan2 δl

1 + tan2 δl

&2 , &

σl(E) =4π

k2(2l + 1)

Γ2

4(E − Er )2 + Γ2.

' σl(E) Breit-Wigner! l− &* & # &$ * && E = Er !' σl(E) $ & & & E = Er

σl,max =4π

k2(2l + 1)

σl,max/2

σl,max

σ

EEr -Γ/2 Er Er +Γ/2

E = Er ± Γ2#

σl(Er ± Γ

2) =

1

2σl,max

7& 2& Γ #& & $ + & & width σl(E) & & & # &$ & &&!8#& $ 2 l & & & #& βl(E) && 2& ! ∂βl/∂E < 0 #& & & Γ $ Γ ≥ 0!

" $ # $ ,

tan δl(E) − Γ/2

E − Er

2 & # E < Er E = Er $ E > Er ! "# Γ ≥ 0 # E < Er $ E = Er tan δl(E) δl(E) & 4 # #& 4 && $

tan δl(Er ) = ∞ , δl(E) =π

2,

∂δl(E)

∂E

∣∣∣∣E=Er

> 0

σl(Er ) =4π

k2(2l + 1)

& &*& E > Er tan δl(E) δl(E) &

−∞ < tan δl(E) < 0

π

2< δl(E) < π

l=1

EE0E

r

l=2

l=0

π/2

π

συντονισµός όχι συντονισµός

δl

l = 0, 1 2

δl(E) # & (π, 0)! #& E0 > Er δl(E) # π

2

&$ && # # & δl(E)$ ∂δl(E)/∂E |E=Er

< 0! 7 # # & & 5$ % #& $ # # 4 l = 0, 1 2!5& # $ & & l = 1 E = Er ! ' E = E0 &$ &!

!

r0

περιοχή

ενεργειών

συντονισµού

Veff

(r)

-V0

κεντρόφυγο φράγµα

E

r

Veff (r) = V (r) + l(l+1)r2

V (r) ! "# V0 " r0$ % " & !

Veff (r) = V (r) + l(l + 1)/r 2

& l & Veff (r) #& # “ ” #& & # & & &! 5 && & $& !

"

' # && &&2 2 2& $! ) 2 N(t) && #& #& & $2& $ & t &&

N(t) = N(0) e−t/τ

#& τ & & $ & - &&! ) & $ & - & #& Γ && $# &&

∆E · ∆t

' # && && ∆E Γ4 & $ & - ∆t τ !

d2φkl

dr 2+

[k2 − 2µ

2V (r) − l(l + 1)

r 2

]φkl = 0 , k2 =

2µE

2

r = 0 V (r) r → 0 1/r 2

φkl(r) = r−l φkl(r) = r l+1

H − Kr Lennard-Jones

V (r) = ε

[(α

r

)12

− 2(α

r

)6]

, ε = 5.9 meV, α = 3.57

! "

r = 0 φ(r) = e−w/r5

w !

# !;

#

#

/ & # & & “&&”

d2φkl

dr 2+

[k2 − 2µ

2V (r) − l(l + 1)

r 2

]φkl = 0 , k2 =

2µE

20

#& # &# # & $& Rkl(r) =φkl(r)

r"# 0 - # 4 # & & & 2& #& $ &#&& & #&&& & &4 $& #& & #&& $ &!) 2 & φkl(r)

φkl(r)|r=0 = 0 1

φkl(r) = kr(Al jl(kr) + Blnl(kr)) , r ≥ r0

' &# & &!/ 2& Bl/Al = − tan δl

φkl(r) = Clkr(cos δl jl(kr) − sin δl nl(kr)) , r ≥ r0

#& δl # & l− & &!

2 # δl & !"! 0 $ &#&4 2&& 2 & NDSolve Mathematica! / #& & # & 4 24 1 # $4 4!

' $ 2 φkl(r = 0) = 0!' $ 2 #& 2 2 φkl(r = h) # # “” !7& & r = h & & & r = 0!

; !"! 0 $ & & r = r0 $& & 2 # & φkl(r) ## & 2 # & & 2 && r = r0 #& & 2 !

' “” φkl(r)|r=h #& &#&#& # 2 Cl #& # &#&#& & !"! 0 &&!: # # && & # # & &l(l+1)

r2 #& & & & &#&$& && 2 !/ & & & r = 0 & # & r → 0 #& # 1/r 2 #& 2 # & # !"! # &$ & &r = 0 #& φkl(r) = r l+1! #& φkl(r = h) = hl+1 #& h & “&& ”!

52 $ &#&& $ & 2 # 4& #& 2 !"! 0 #& &# & 2 & r1 ≥ r0 r2 > r1 ≥ r0! & !"! 0 $ & & r = r2 $-& &

φkl(r1) = Clkr1 (cos δl jl(kr1) − sin δl nl(kr1) )

φkl(r2) = Clkr2 (cos δl jl(kr2) − sin δl nl(kr2) )

4& Cl δl ! &

φ(1)kl = Cl r1

(j(1)l − tan δl n

(1)l

), Cl = Clk cos δl

φ(2)kl = Cl r2

(j(2)l − tan δl n

(2)l

), φ

(1)kl = φkl(r1) , φ

(2)kl = φkl(r2)

' & & # & Cl tan δl # 03-phase-system.nb

tan δl =G j

(1)l − j

(2)l

G n(1)l − n

(2)l

, #& G =r1 φ

(2)kl

r2 φ(1)kl

Cl =n

(1)l φ

(2)kl r1 − n

(2)l φ

(1)kl r2

(j(2)l n

(1)l − j

(1)l n

(2)l ) r1r2

%

' # && - δl & [−π2, π

2]!

' & δl #&#& & π & &#&& # - # 2 & & & &!

7& #&#& & π & #& # && & & 2 4 & φkl(r)

&$ & 2 & & φ(0)kl = krjl(kr) #& -& r < r0!

< - # &&$ & r1 r2! / & r2 #& & &r1 $ & 2 # 2 # & & 2 & & # && $ & & ! / # & #& # # & $ &#&&&!

! 5 2 # && & $ &#&& & “"” & & 4 & 2 # & & #& & & #& & 2&! & 2 $& & 2& #&&&!: # =& && !"! &!

/ & #& & & 2 & L0 E0&$ r = rL0 E = EE0 0

d2φkl

dr 2+

[C(E − V (r)) − l(l + 1)

r 2

]φkl = 0

2& r → r E → E

d2φkl

dr 2+

[C(E − V (r)) − l(l + 1)

r 2

]φkl = 0 (

#& 4 r E V (r) 2 C

C =2µE0L

20

2=

2(µc2)E0L20

(c)2

C =2µE0L

20

2=

2(µc2)E0L20

(c)2

5 /& > $ &#&& & & nm A & & eV ! ) &$ & : > fm MeV

1 nm = 10 / = 10−9 m = 10−7 cm, 1 fm = 10−15 m = 10−13 cm

1 eV = 1.602177 · 10−19 J, 1 MeV = 106 eV

"#

mec2 0.511 MeV = 5.11 · 105 eV, c = 197.33 MeV fm = 1973.3 eV /

2 C & & #

µ me , [L0] = /, [E ] = eV C = 0.26246 µ me , [L0] = nm, [E ] = eV C = 26.246 µ mp

2 mn

2, [L0] = fm, [E ] = MeV C = 0.0241

$ Mathematica %

' & # & & #& & !

/ & #2 4 2 & && # !

7& # & & # & & & Ze # &= screening & # Z & ! # " $% & Coulomb '1/r(!

8 # & & & & Lenz-Jensen #& 2$ &#&4 & # #& Thomas-Fremi &! 5 # & & & & & &

V (r) =Ze2

re−x(1 + x + b2x

2 + b3x3 + b4x

4) +

#&

x = 4.5397Z 1/6r 1/2, e2 = 14.409, b2 = 0.3344, b3 = 0.0485, b4 = 0.002647

7& eV /! 7& #& 2 2## r0 = 2 /!

!

0.5 1 1.5 2r @fiD

−300

−200

−100

100

200VeffHrL @eVD

l=0

l=1

l=2

l=3

Veff (r) = CV (r) + l(l+1)r2 l '

Z = 20

7& # #& 2& & && 0.5 eV ≤ E ≤ 1 KeV # & & & # & +!) &# # #& && Z E #&&- &# & 2 & & & & ϑ# 0o 180o 10o !' #& #& &&2& & #&& # # Mathematica ? "

6) # # &#2& & =& #& & 4 #& $ &#&2 & # #4 & !• 5 $ & # & & 04-electron-scat.nb & -& & 2 c = 0.26247 r0 Ze Bessel Neumann l &2 4 Bessel Neumann #& & & BesselJ[n, z] BesselY[n, z]!7& # 2 #& & $! /& & & & Ze & - 2 4 iEn &#& 2 #&&-& & 2! & & & # &$ 4 & 2& k 4 & & 2& l lmax kr0! #& #&#& #&&& #& Legendre & & # lmax × 19 #& 2 # $ #& Legendre ! lmax # & l 19 & 2 4 θ &#& 2 #&& & &!5 $ & -& & $ + $# ( &2 & NDSolve! '& & - & l & $& &rsmall #& # 2 & & rsmall l+1

(l + 1)rsmall l &$ # rmax! ' ## 2 & # # & 2& k!

• 7& & & # & & & & $& & Do For! ) $& 2 iEn #= 4 & $& 2 lmax(E) #= 2 #&&- & 2 l = 0, 1, 2, . . . , lmax(E)!: # & $& & -& & 2 &$ # sigmaTotal sigmalE! 7 &$ & # 4& & & & & ! 7 &$ & & # iEn× 3 & 4 &4 &4 l = 0, 1, 2 # &# !/ & -& & 2 rstart[l] & & &#&& 22& & rsmall !"! & l! / & 2 rsmall $ $& &#&& & #& # &- $ #& 4 r !5 # 4 & & & $& & -& & k24 r1 r2 &#& 2 2 !"! & l 2#&&& 2 #& &! 7& & r1 && 4 & & r2 2 & # #& & & & # & r1! 7& & & # $ λ = 2π/k = 2π

√CE !

7 r1 r2 & -& # $ r1 = r0, r2 = r1 + 0.9πk!

5& & r1 r2 # # # & #= # jl nl #& & kr 4& 2 #& & & & 2π/k!

#

8 & $& & -& & & && # &#& NDSolve #& !"! 6" -& & 0 ≤ r ≤ r2 E l #& #& & #&&& #& G = G(r1, r2) $ !

5 # & #&&-& & jl(kr) nl(kr) r1 r2! 8 &2 4 4 & G tan δl

$ &#&4 $ # # & l− &&!

8 &2 & δl #&&- & & l− & & &#& l < 3 #&2 & # sigmalE! 8 # 2 4 &4 4 &4 & l & & !

/# jl(kr) nl(kr) & & & & #&&-& 2 Cl $ &#&4 $%! 6" #& 2 & r < r0 r > r0

0 45 90 135 180θ @degreesD

11.011.021.031.041.05

σH

θL

ê

σH

0

L

0 45 90 135 180E=0.1 eV, Z=50

0 45 90 135 180θ @degreesD

0.80.9

1

1.1

1.2

σH

θL

ê

σH

0

L

0 45 90 135 180E=1. eV, Z=50

0 45 90 135 180θ @degreesD

0

0.2

0.40.6

0.8

1

σH

θL

ê

σH

0

L

0 45 90 135 180E=43. eV, Z=50

σ(θ)/σ(0) =dσ(θ)dΩ

/dσ(0)dΩ

! Z = 50

) E = 0.1 eV, E = 1 eV E = 43 eV & && #& & 2 2& $ = &$!

: & $ & & $ & &# θ! 5 & 2 & & 4

dσ(0)

dΩ≤ dσ(θ)

dΩ≤ 1.05

dσ(0)

dΩ, θ ∈ [0o , 180o ]

5 $ & l = 0 & &

dσ(θ)

dΩ 4π

k2sin2 δ0, k =

√CE

6)& $-& & l > 0 & & $- # θ # & # & 5$&!

E = 1 eV 2k 1/r0 # $ #& # θ!

5 E = 43 eV #& #& 2 2 = 2 θ = 0o # Pl(cos 0)@ & 2 #& # $ & 2 ! 5 # $ & #& & & ! / # & $ # = diffraction form!

0 2 4 6 8 10Energy @eVD

20

40

60

80

100

120

σ

la

to

t

@

fi

2

D

Z=20

0 10 20 30 40 50Energy @eVD

0

2

4

6

8

10

12

σ

la

to

t

@

fi

2

D

Z=50

0 20 40 60 80 100Energy @eVD

10

20

30

40

σ

la

to

t

@

fi

2

D

Z=70

! Z = 20, 50, 70

7& $ 4 # & $& & # &-& & & & Z !/ & $ σtotal(E) # &-& && # # & & #&& & &! #& E = Er #&& 2 # &- &!

Z = 20 # &- & & E 1.5 eV Z = 50# &- & E 20 eV Z = 70 # &- && E 10 eV & E 70 eV!

#4& #& - & # # &# & & # ! ) # 4& # & & $& & #& 5$4 & & !

0 2 4 6 8 10Energy @eVD

6

8

10

12

σ

l

@

fi

2

D

l=0 Z=20

0 2 4 6 8 10Energy @eVD

020406080

100120

σ

l

@

fi

2

D

l=1 Z=20

0 2 4 6 8 10Energy @eVD

00.20.40.60.8

11.2

σ

l

@

fi

2

D

l=2 Z=20

0 10 20 30 40Energy @eVD

0

0.5

1

1.5

σ

l

@

fi

2

D

l=0 Z=50

0 10 20 30 40Energy @eVD

00.250.5

0.751

1.251.5

1.75

σ

l

@

fi

2

D

l=1 Z=50

0 10 20 30 40Energy @eVD

0

2

4

6

8

10

σ

l

@

fi

2

D

l=2 Z=50

0 20 40 60 80 100Energy @eVD

0

5

10

15

20

25

σ

l

@

fi

2

D

l=0 Z=70

0 20 40 60 80 100Energy @eVD

05

1015202530

σ

l

@

fi

2

D

l=1 Z=70

0 20 40 60 80 100Energy @eVD

0

0.5

1

1.5

2

σ

l

@

fi

2

D

l=2 Z=70

! l = 0 1 2 Z = 20 50 70

/# & 5$ 0 # & σ1(E) Z = 20 Z = 70 # &- && # #& #& # &- & & # & & & & Z !5 # # & Z = 50 & & σ2(E) #& &# &- # #& #& # &- & &!6" & $ 4 # & 5$& 0 σ0(E) Z = 20 Z = 70 # E 0!/ $& & Z *&&* & # $ #& !

! 04-electron-scat.nb δl(E) l = 0, 1 2 Z = 20, 50 70 ! $ % &!

σl(E) %

! H − Kr Lennard-Jones

V (r) = ε

[(α

r

)12

− 2(α

r

)6]

, ε = 5.9 meV, α = 3.57 ,

! H Kr !$ 04-electron-scat.nb

03-04-trans_normal - Αντιγραφή.pdf

Documents

Transcript of 03-04-trans_normal - Αντιγραφή.pdf