TRANSITION RADIATION AND IONIZATION
ENERGY LOSSES OF HIGH-ENERGY
‘HALF-BARE’ ELECTRON
S.V. Trofymenko
Akhiezer Institute for Theoretical Physics of NSC
‘Kharkov Institute of Physics and Technology’
Kharkov, Ukraine
™N F Shul’ga,
™N.F.
Sh l’ S.V.
S V Trofymenko,
T f
k V.V.
V V Syshchenko
S h h k // JETP Lett.
L (2011)
™N.F. Shul’ga, S.V. Trofymenko // Chapter in the Book “Solutions and Applications
off Scattering,
S tt i
Propagation,
P
ti Radiation
R di ti andd Emission
E i i off Electromagnetic
El t
ti Waves”,
W
”
InTech (2012)
™N F Shul
™N.F.
Shul’ga
ga, SS.V.
V Trofymenko // Phys.
Phys Lett.
Lett A (2012)
1
COHERENCE LENGTH. ‘HALF-BARE’ PARTICLE
Ultra relativistic case ( γ >> 1 ):
incident particle
‘half-bare’
interaction
bremsstrahlung
atoms
lC ≈ 2γ 2 / ω
the frequency
q
y ω appears
pp
on
distance
lC ≈ 2γ 2 / ω from
the interaction area
ε
For electron energy ≈ 100 GeV
boundary between
two substances
γ – Lorentz-factor
q
y
ω – radiated frequency
p of Fourier-components
part
p
is absent in
the field around the particle –
the particle is ‘half-bare’
EL F
E.L.
Feinberg//
i b // Sov.
S Phys.
Ph JETP,
JETP 1966
lC ≈ 2γ 2 / ω
lC
≈ 10
meters
transition
radiation
MANIFESTATION OF ‘HALF-BARE’ STATE OF
ELECTRON IN ITS BREMSSTRAHLUNG
1) Coherent bremsstrahlung
d / dω
dI
BH
2) Landau-PomeranchukMigdal effect (suppression of
Bethe-Heitler spectrum al low
f
frequencies)
i )
LPM
ω
3) Ternovsky-Shul’ga-Fomin
Ternovsky Shul’ga Fomin
effect (modification of LPMeffect in thin layers
y of substance))
Observed in CERN NA63 experiment:
H.D. Thomsen, K.K. Andersen, J. Esberg, H. Knudsen, M. Lund, K.R. Hansen,
U I Uggerhøj
U.I.
Uggerhøj, et.
et al.,
al (2009).
(2009)
U. Uggerhøj : ‘… we have seen the ‘half -bare’ electron !’
3
AIM OF THE PRESENT PAPER
Search for manifestation of ‘half-bare’
state of electron in its:
1) Transition radiation
2) Ionization energy losses
ELECTRON ‘UNDRESSING’
BY INSTANT SCATTERING
z'
‘half-bare’
electron
‘t
‘torn-away’
’ field
fi ld
0
z
R = ct
The total field for t > 0 :
r
r
r
ϕ ( r , t ) = θ ( r − t )ϕ vr ( r , t ) + θ (t − r )ϕ vr ' ( r , t )
A.I. Akhiezer, N.F. Shul’ga // High Energy Electrodynamics in Matter, 1996
N.F. Shul’ga, V.V. Syshchenko, S.N. Shul’ga // Phys. Lett. A, 2009
TRANSITION RADIATION BY
‘HALF-BARE’
HALF BARE ELECTRON
z'
metallic plate
ƒ suppression of
radiation for
z '0
z '0 << 2γ 2 / ω
ƒ period of oscillations Λ =
4π
ω (ϑ 2 + γ −2 )
z
R = ct
ƒ the oscillations can
be observed for
ω / Δω
z '0 <
2π
Δω (ϑ 2 + γ −2 )
– detector resolution
Transition radiation by ‘half-bare’ electron:
dε
e2
ϑ2
= 2
dω do π (ϑ 2 + γ −2 )2
⎧
⎫
⎡ ω z ' 0 −2
2 ⎤
(
γ + ϑ )⎥ ⎬
2 ⎨1 − cos ⎢
⎣ 2
⎦⎭
⎩
FERMI AND BETHE-BLOCH FORMULAE
I fi it medium
Infinite
di
Bethe-Bloch formula ( γ ≤ I / ω p ):
dε
γ
= 2 ln
dz
v
bI
ω p2 e 2
r
v
Fermi formula ( γ > I / ω p):
dE / dz
v
dε
= 2 ln
dz
v
bω p
ω p2 e2
Fermi
(density effect)
γ > I / ωp
ln I / ω p
ln γ
γ
– electron Lorentz-factor
I –
ωp–
mean ionization potential
plasma frequency
7
THIN LAYER OF SUBSTANCE
Bethe-Bloch and Fermi formulae are valid in boundless
homogeneous substance
Garibian G.M.// JETP, 1959
Sørensen A. // Phys.Rev.A, 1987
Total absence of the density effect in
thin plates:
a ≤ I /ω
2
p
Particle energy loss:
Δ
E=
ω p2 e 2
v
2
a ln
γ
bI
for
1≤γ < ∞
a
8
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM
ρ
substance
vacuum
z
9
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM
ρ
substance
vacuum
z
Fourier component of the total field (γ >> 1 ,ω >> ω p ):
⎧ ⎡
⎤
ω
i z
2
∞
q
z
⎪
e
1
1 ⎥ iω z − 2ω
ev
⎪ ⎢
ρ r
2
⎥e
Eω ( r ) = 2 ∫ dqq J1 ( qρ ) ⎨ ⎢
−
+
2
2
2
ω
ω
ω
v0
2
2
2
2
⎥
⎢
⎪ q + ωp +
q + 2
q + 2
2
⎥
⎢
γ
γ ⎦
γ
⎪⎩ ⎣
J1 ( x) – Bessel function
transition radiation
Coulomb
field
⎫
⎪
⎪
⎬
⎪
⎪⎭
10
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM
ρ
substance
vacuum
z
For
z → 0:
ω
2
2
⎛
⎞
i z
2
ω
ω
e
2 ⎟ v
2
⎜ρ
Eωρ ( ρ ) =
+
ω
K
+
ω
e
1
p
p
2
2
⎜
⎟
v γ
γ
⎝
⎠
suppressed frequencies
ω ≤ γω p
electron is ‘half-bare’
K1 ( x ) – modified Bessel function
11
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM
ρ
substance
vacuum
z
For z > 2γ / ω :
2
2e ⎧⎪ ω ⎛ ω ⎞ i v z
e
E (ρ , z) =
K
ρ
e
+
F (ρ / z)
⎜
⎟
⎨
1⎜
⎟
v ⎪⎩ γ ⎝ γ ⎠
r
ρ
ω
ω
iω r
⎫⎪
⎬
⎪⎭
F – definite function of ρ / z
r = ρ 2 + z2
12
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM
ρ
substance
vacuum
z
z1
For z > 2γ / ω :
2
2e ⎧⎪ ω ⎛ ω ⎞ i v z
e
E (ρ , z) =
K
ρ
e
+
F (ρ / z)
⎜
⎟
⎨
1⎜
⎟
v ⎪⎩ γ ⎝ γ ⎠
r
ρ
ω
ω
iω r
⎫⎪
⎬
⎪⎭
F – definite function of ρ / z
r = ρ 2 + z2
13
IONIZATION ENERGY LOSSES OF
‘HALF-BARE’
HALF-BARE ELECTRON
Total ionization per unit path ( q0 >> ω p >> I / γ ):
dε
=
dz
η p2 e 2 ⎧⎪ q0 vγ
v
2
⎨ln
I
⎪⎩
+ ln
ω p vγ
I
+
+ Ci (λγ ) − cos λ pCi (λ p + λγ ) − sin λ p Si (λ p + λγ ) +
⎫⎪
+ λγ Si (λγ ) + cos λγ ⎬
⎪⎭
where:
λ p = z1ω p2 / 2 I
λγ = Iz1 / 2v 2γ 2
η p – plasma frequency of the plate
I – mean ionization potential
v 2 dε
η p2 e 2 dz
IONIZATION ENERGY LOSSES
OF ‘HALF
HALF-BARE
BARE’ ELECTRON
(from Fermi to Bethe-Bloch formula)
Bethe-Bloch + radiation
Bethe-Bloch
Fermi
ω vγ
q0vγ
+ ln p
I
I
q vγ
ln 0
I
ln
ln
q0
ωp
γ2/I
0
z1
For ε = 100 GeV
z
Possibility
P
ibili off ‘half-bare’
‘h lf b ’ electron
l
experimental observation
γ2
I
≈ 10 meters !
15
CERN NA63 EXPERIMENT (2010)
a >> I / ω p2
K.K. Andersen, J. Esberg, K.R. Hansen, H. Knudsen, M. Lund, H.D. Thomsen, U.I. Uggerhøj,
et. al., NIM B (2010).
CONCLUSIONS
‘Half-bare’ state of electron should be manifested in its transition
radiation and ionization losses as well as in bremsstrahlung:
™Dependence of electron transition radiation characteristics
on distance between the plate and the scattering point
™Existence of transition pprocess in which ionization losses of
the particle are defined by the mechanism of restoration of the
field around it (not by absorption)
™ Gradual increase of electron ionization losses in thin plate
from the value with density effect (Fermi formula) to the value
without it (Bethe-Bloch formula) supplemented by
contribution to ionization from transition radiation
17
IONIZATION OF SUBSTANCE BY EXTERNAL
FIELD
ω0 = I
&x& + βx& + ω02 x =
(
e C
Eρ + EρF
m
Δ
Energy transfer to a harmonic
oscillator by external field:
2
2
ε
r
e
=
Eω0 ( r )
2m
)
x (t )
ρ
z
JJ.D.
D JJackson
k
// Classical
Cl i l
electrodynamics, 1999
ω0 – oscillator’s own frequency
Total ionization per unit path:
dε
e2
ρ r 2
=n
dρ 2πρ Eω0 ( r )
∫
dz
2m 0
∞
18
REBUILDING OF THE FIELD AND
IONIZATION LOSSES
rebuilding of the field L ≈ γ / I
2
z
change of ionization
losses
L ≈ I / ω p2 ≈ absorption length
I – mean ionization potential
19
IONIZATION ENERGY LOSSES OF
‘HALF-BARE’
HALF-BARE ELECTRON
Assumption: q0 >> ω p >> I / γ
Ω
For
dε
q0
pe
= 2 ln
dz
v
ωp
2
z→0
2
Ω
For I / ω << z << 2γ / I
2 2
q0 zω p
dε
pe
= 2 lln
dz
v
2I
For z ≥ 2γ / I
2 2
ω p vγ ⎫
dε
q0vγ
pe ⎧
= 2 ⎨ln
+ ln
⎬
I
I ⎭
dz
v ⎩
2
p
2
Ω
2
F (ϑ ) =
ω 2p
ϑ
ω 2 (ϑ 2 + ω 2p / ω 2 + 1 / v 2γ 2 )(ϑ 2 + 1 / v 2γ 2 )
20
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM AT ULTRA HIGH
ENERGIES
Total field:
r rC r F
E=E +E
Boundary conditions:
(E
(E
ρ
+ EρF )
C
z
+ EzF )
C
diel
= (EρC + EρF )
vac
z =0
diel
z =0
z =0
vac
)
= ε (E zC + EzF )
rF
+ divE = 0
z =0
Fourier expansion of the free field:
r
e
Eρ (r , t ) =
∞
2
dqq
∫ J 1 (qρ ) ×
π v0
⎡
v
v
2
2
2
2
q
q
1
+
ω
ε
−
ε
+
ω
ε
−
⎢
ω −q
ω
ω
× ∫ dω
−
⎢
2
2
2
2
2
2
ω
ω
2
2
2
2
ω
ε
ε
ω
−
q
+
−
q
⎢ q +
−∞
−
q
εω
+
−
ω
⎢⎣
v2
v2
r
r
q – component of vector k orthogonal to z axis
+∞
2
2
⎤
⎥ i
⎥e
⎥
⎥⎦
ω 2 − q 2 z −iω t
21
EVOLUTION OF THE FIELD AROUND THE
ELECTRON IN VACUUM AT ULTRA HIGH
ENERGIES
Own Coulomb field of the particle:
r
∂
EρC ( r , t ) = −
∂ρ
e
ργ
2
−2
+ ( z − vt )
2
=
e
π
∞
+∞
dω ∫ dq
∫
v
−∞
0
q 2 J 1 ( qρ )
q2 +
ω
v 2γ 2
2
ω
e
i z
v
Total field:
Eρ = EρC + EρF
NIM B, Volume 268, Issue 9, 1 May 2010, Pages 1412
1412–1415
1415
22