            ω≡E
e
−E
′
e
 Obzornik mat. ﬁz. 55 (2008) 2
Sodobne meritve elektromagnetnih lastnosti protonov
 ~ q≡~ p
e
−~ p
′
e
 (E
e
,~ p
e
) (E
′
e
,~ p
′
e
)  E
e
 E
′
e
 θ
e
 E
e
≈|~ p
e
| E
′
e
≈|~ p
′
e
| |~ q|
2
=E
2
e
+E
′2
e
−2E
e
E
′
e
cosθ
e
 q
μ
≡ (ω,~ q) Q
2
=−q
μ
q
μ
=~ q
2
−ω
2
> 0 ω ~ q E
e
 E
′
e
 θ
e
 Q
2
≈ 4E
e
E
′
e
sin
2
(θ
e
/2)   α
2
  α =e
2
0
/4πε
0
~c≈ 1/137 b) a)
e p e p
g
g
g
e p
g
D
c)
e p
d)
g
g
*
*
*
*
*
  Δ γ
∗
 ω ~ q  GeV fm fm Q
2
→ 0 54–63
Simon Širca
 ℄ dσ
dΩ
e
=
dσ
Mott
dΩ
e
1
1+τ
h
 
G
p
E

2
+
τ
ε
 
G
p
M

2
i
, ε τ = Q
2
/4M
2
 M ≈ 938MeV/c
2
 Q
2
 ε τ  G
p
E
(Q
2
) G
p
M
(Q
2
) θ
e
 sin
−4
(θ
e
/2) 1/2 cos
2
(θ
e
/2) G
p
E
(Q
2
) G
p
M
(Q
2
)  +1μ
B
= e
0
~/2M μ
p
≈ +2,79μ
B
  ℄ Q
2
= 0  +1e
0
 μ
p
 G
p
E
(0) = 1 G
p
M
(0) =μ
p
 Q
2
> 0 E
e
 θ
e
 Q
2
 τ/ε  
G
p
M

2
  
G
p
E

2
  Obzornik mat. ﬁz. 55 (2008) 2
Sodobne meritve elektromagnetnih lastnosti protonov
 Q
2
 Q
2
 Q
2
 Q
2
6= 0 ρ(~ r)  V(~ r)∝ρ(~ r) M∝
Z
ψ
∗
e
′(~ r)V(~ r)ψ
e
(~ r)d
3
~ r≈
Z
e
−i~ p
′
e
~ r
V(~ r)e
i~ pe~ r
d
3
~ r =
Z
e
i~ q~ r
V(~ r)d
3
~ r.
 F(~ q
2
)  F(~ q
2
) =
Z
ρ(~ r)e
i~ q~ r
d
3
~ r,
 |M|
2
 

F(~ q
2
)


2
 54–63
Simon Širca
  G
p
E
 G
p
M
 GeV   ρ
ch
(r) =
2
π
∞
Z
0
dkk
2
j
0
(kr)G
p
E
(Q
2
)(1+τ)
λ
E
,
μ
p
ρ
m
(r) =
2
π
∞
Z
0
dkk
2
j
0
(kr)G
p
M
(Q
2
)(1+τ)
λ
M
,
 k
2
=Q
2
/(1+τ) (1+τ)
λ
E
 (1+τ)
λ
M
    λ
E
 λ
M
 λ
E
=λ
M
= 2 ℄ Q
2
  Q
2
 G
p
E
(Q
2
) = 1−
1
6
Q
2


r
2
E

p
+··· 

r
2
E

p
=−6
dG
p
E
dQ
2




Q
2
→0
≈ 0,69 fm
2
. q


r
2
E

p
≈ 0,83 ℄ Obzornik mat. ﬁz. 55 (2008) 2
Sodobne meritve elektromagnetnih lastnosti protonov
  P
l
 P
t
  μ
p
G
p
E
(Q
2
)
G
p
M
(Q
2
)
=−
P
t
P
l
E
e
+E
′
e
2M
tan
θ
e
2
. G
p
E
(Q
2
)/G
p
M
(Q
2
) P
t
/P
l
  Q
2
 ρ
ch
(r) ρ
m
(r)  ℄  ~
F =mω
2
~ x =
e
0
~
E mω
2
  ~ p
e
= e
0
~ x = (e
2
0
/mω
2
)
~
E ≡ α
~
E   α ≈ 10
−28
m
3
 54–63
Simon Širca
 E = 10kV/cm x≈ 10
−6
 ℄  ~
E ~
H
 ~ p
e
=α
p
~
E ~ p
m
= β
p
~
H fm α
p
≈ 10
−18
α≈ 10
−3
fm
3
  ~ p
e
 z ΔW
E
=−
X
X6=p
|hp|−b p
ez
E|Xi|
2
M
X
−M
p
=−
E
Z
0
~ p
e
d
~
E =−
1
2
α
p
E
2
.
 P
X
|XihX| = 1 |pi |Xi ℄  ΔM
Xp
≡M
X
−M
p
≈ 700MeV α
p
≈
2
P
X
hp|b p
ez
|XihX|b p
ez
|pi
ΔM
Xp
=
2
ΔM
XN
hp|e
2
0
1
3
r
2
|pi =
2e
2
0
3ΔM
Xp


r
2
E

p
.
 

r
2
E

p
≈ 0,69 fm
2
  α
p
≈ 10· 10
−4
fm
3
  β
p
≈ 2·10
−4
fm
3
    Obzornik mat. ﬁz. 55 (2008) 2
Sodobne meritve elektromagnetnih lastnosti protonov
  ~
E ~
H   ~ p
e
 ~ p
m
 ~ω ~ω
′
  θ  ℄ dσ
dΩ
=
dσ
Powell
dΩ
−ωω
′
e
2
0
M

ω
′
ω

2

α
p
+β
p
2
(1+cosθ)
2
+
α
p
−β
p
2
(1−cosθ)
2

, ω ω
′
  ω
′
=ω

1+
ω
M
(1−cosθ)

−1
 ω ω
′
 θ α
p
 β
p
  (μ
p
−1) ℄ M e
0
 ℄ ω→ 0 lim
ω→0
dσ
Powell
dΩ
=
r
2
0
2
 
1+cos
2
θ

,
 r
0
 M  α
p
 β
p
 θ = 0
◦
 α
p
+β
p
 θ = 180
◦
 α
p
−β
p
  α
p
 β
p
 θ = 90
◦
 α
p
 β
p
 ℄ α
p
= (12,0±
0,6)· 10
−4
fm
3
 β
p
= (1,9∓ 0,6)· 10
−4
fm
3
 54–63
Simon Širca
 α
p
+β
p
=
1
2π
2
∞
Z
prag
σ
tot
(ω)
ω
2
dω = (13,9±0,3)·10
−4
fm
3
.
  1/ω
2
   hr
2
α
i
p
∝−6
 
dα
p
/dQ
2

 Q
2
→ 0  Q
2
> 0,5 (GeV/c)
2
 ℄ αp βp Q
2
 Obzornik mat. ﬁz. 55 (2008) 2
Sodobne meritve elektromagnetnih lastnosti protonov
   ℄  hr
2
α
i
p
 α
p
(Q
2
) β
p
(Q
2
) ℄   ℄   ℄   ℄ GEp
/GMp
 ~ ep → e~ p Q
2
= 5.6 GeV
2
 ℄    ℄    ℄    ℄   ℄     Q
2
  Q
2
= 0.92 2
      54–63