generaon of plasmons by quantum charged fast oriented …...planar channeling planar channeling -...
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Genera&onofPlasmonsbyquantumchargedfastoriented
par&cle
N.A.KUDRYASHOV,E.A.MAZURNATIONALRESEARCHNUCLEARUNIVERSITY,MOSCOW,RUSSIA
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Specifics of channeling potential in covalence crystals
Standardapproach:U(x)=∫dydz(Ze/r)exp(-r/Ratom) U(x)
x
U
Incovalencecrystals(C,Si)4electronsarefarawayfromnucleusandevenlydistributedU*(x)=∫ dydz(Z-4)e/r)exp(-r/Rion)<U(x)
U(x)
x U*
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Planar channeling
Planar channeling - thick crystal
θ<θL=(2U/E)1/2Lindhard angle E>>mc2;U~20-40eV
d–interplane distance;l–period of trajectory
θ
z
x
d
l ~ d/θL
U(x)
x
U
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Z
Y
X
𝑘 ϕ
𝑃↓𝑖
pPPLAKinematics 𝑃↓𝑖 = 𝑃↓𝑖 (𝑥)+ 𝑃 ↓𝑖⟘ (𝜃)= 𝑃↓0 (1− 𝜃↓0↑2 /2 )𝑖 + 𝑃↓0 𝜃↓0 𝑗
𝑃↓𝑖⟘↑2 /2𝑚 ⪅<𝑈>, 𝑃↓0↑2 𝜃↓0↑2 /2𝑚 ⪅<𝑈>, 𝜃↓0 ≤ 2𝑚<𝑈>/𝑃↓𝑜↑2 = 𝜃↓𝐿↑2
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“ Accompanying” reference (coordinate) system ”
𝑃↓𝑦↑′ = 𝑃↓𝑦 ⇒ 𝑃↓⟘↑′ = 𝑃↓⟘ 𝑃↓𝑧↑′ = 𝑃↓𝑧
𝐸↑′ = 𝐸−𝑉𝑃↓𝑥 /√1− 𝑉↑2 /𝐶↑2 ⇒𝑚↓0 𝑐↑2 + 𝑃↓⟘↑2 /2𝑚↓0
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In “accompanying” system
𝜔= 𝜔↓0 √1− 𝑉↑2 /𝐶↑2 /1+ 𝑉 𝑘 /𝜔↓0 = 𝜔↓0 √1− 𝑉↑2 /𝐶↑2 /1+ 𝑉/𝐶 cosψ ψ=π-ε𝜔= 𝜔↓0 √1− 𝑉↑2 /𝐶↑2 /1− 𝑉/𝐶 + 𝑉/𝐶 𝜀↑2 /2 ⇒𝜔= 𝜔↓0 1/𝛾↑−1 + 𝜀↑2 /2 𝛾
In “accompanying” system
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• Structure of energy bands and radiative transitions of 56-MeV electrons channeled along the (110) plane in Si
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PLASMONDISPERSION
( ) ( ) ( )( )
( )
2
2
222 2
1Im ;, 2
.2
p
p
qq q
qqm
πωδ ω ω
ε ω ω
ω ω
⎡ ⎤− = −⎢ ⎥⎣ ⎦
⎛ ⎞= + ⎜ ⎟
⎝ ⎠
rr r
rr
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Channeling plasmon-photon radiation conditions
Usual spontaneous radiation Plasmon “wings” in radiation
1. Potential in lab. system: U0~20eV;d~0,2-0,3A(Si);in accomponying system (vx = 0): U=U0(E/mc2)
U0d
2. Number of levels: N~Pzmaxd/h~(EU0)1/2d/hc3. Distance between levels in acc. system: ΔE~U/N~(EU0)1/2(h/mcd)
4. Plasmon energy in acc. system: hxomega= 2hxomega0/(mc2/E+Eψ2/mc2)<2hv0E/mc2
5. In resonance conditions radiation “wings” can be very effective:
ΔE~(EU0)1/2(h/mcd)=hv<2hv0E/mc2=2hE/mcλ0
6. Resonance can be reached by correctly orienting laser beam, if: E/U0>(λ0/2d)2~107-8
Ei Ei-ΔE
hv=ΔE hv=ΔE
hv0-hxomega
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Plasmon and photon excitationsby “bound” electrons
𝑃↓𝑖
𝑘↓𝑜
𝑃 P ↓𝑓 𝑘↓
𝑃↓𝑖
𝑘↓𝑜 𝑃
P ↓𝑓 𝑘↓
2
2qm
r
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Generation of longitudinal excitation (plasmon) and a photon by a “bound”
electron 𝐴↓𝑖 (𝑥↓1 )
𝑥↓1
ψ↓𝑖 (𝑥↓1 )
𝑆↑(𝑒) ( 𝑥↓1 , 𝑥↓2 )𝐴↓𝑓 (𝑥↓2 )
𝑥↓2
ψ↓𝑓 (𝑥↓2 )
𝐴↓𝑓 (𝑥↓1 )
𝑥↓1
ψ↓𝑖 (𝑥↓1 )
𝑆↑(𝑒) ( 𝑥↓1 , 𝑥↓2 )
𝐴↓𝑖 (𝑥↓2 )
𝑥↓2
ψ↓𝑓 (𝑥↓2 )
𝐴↓𝑖 (𝑥)= 𝐴↓𝑖 (𝑟)𝑒↑−𝑖𝜔↓𝑖 𝑡 = 𝑒↓𝑖 /√2𝜔↓𝑖 𝑒↑𝑖( 𝑘↓𝑖 𝑟 − 𝜔↓𝑖 𝑡) 𝐴↓𝑓 (𝑥)= 𝐴↓𝑓 (𝑟)𝑒↑𝑖𝜔↓𝑓 𝑡 = 𝑒↓𝑓 /√2𝜔↓𝑓 𝑒↑−𝑖( 𝑘↓𝑓 𝑟 − 𝜔↓𝑓 𝑡) ψ↓𝑖 (𝑥)= ψ↓𝑖 (𝑟 )𝑒↑−𝑖𝐸↓𝑖 𝑡 ψ↓𝑓 (𝑥)= ψ↓𝑓 ( 𝑟 ) 𝑒↑−𝑖𝐸↓𝑓 𝑡
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GeneraXonofscalarexcitaXonsinthecrystalbyfastchargedparXcles
( ) ( ) ( ) ( ) ( ) ( )0 0 4 4if if ifdw j x j x x x d xd x dν∗ ′ ′ ′= <Φ Φ∫∫
( ) ( ) ( ) ( ) ( )( ) ( )( )( )( )
( ) ( ) ( )
0 0
24 4
2 2
exp exp
4 4exp , , .
if if ifq BZ q BZ GG
ee
w j x j x i q G r i q G r
ei i t t d xd x S q G q Gq G q G
ω
π πω δ ω
∗
′ ′∈ ∈
′ ′ ′ ′= + − ×
′ ′ ′ ′× + − + −⎡ ⎤⎣ ⎦′ ′+ −
∑ ∑ ∑ ∑∫∫rr
r rr r r r
r rr rr rr r
( )( )
( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )
3 22
30
12
2 2 2 2
11 3 222 2 2 2 2 221 2
2 2
Im , , , ,4 4 exp1 exp
exp ( ).
x xz z
x x y zx
x y z
x z y z z y y
dE d d q mdq C Pdx EP
q q q qe dx x iq x xq q q q
dx x iq x x m P P m P q P q
κκκ
κ κ
κ κ
ωωπ υπ
ε ωπ π ψ ψωβ
ψ ψ δ κ κ ω
∞
′
− +∞
′−∞
+∞
′−∞
′− = ×
′× ⋅ ×
′ + + − −
′ ′ ′ ′ ′× − + + − − + − + − − −
∑∫ ∫ ∫
∫
∫
r
LossofenergychanneledparXclesuponexcitaXonofthecrystal
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OrientaXondependenceofthebandstructureofrelaXvisXcpositronsinsinglecrystal
( ) ( )4 ,0,0
1,22 cos 4 .nn
V x V V Gx=
= + ∑
( ) ( )2
2 2 cos2 0,U a q S U SS
∂ + − =∂
%
2 2 2 2 02 2 2
2 2 2
8 / 4 ;2
; 2 ; 2 / ;2
Gмин
EVa E G cc G
EVq S Gx G G dc G
π
⊥= −
= = ≡ =
% hh
hSquaresmoduleofeven(a)andodd(b)wavefuncXonsofpositronswithanenergyof28MeVintheplanarchannel(110)inasinglecrystalSi.(BD1)-firstdeepsub-barrierlevel;(Bd2)-thesecondlevelinthemiddleofthechannel;(BD3)-thefirstabove-barrierzone(level)(zone);(Bd4)-secondabove-barrierzone.
OrientaXondependenceoftheprobabilityofthepopulaXonofthelevelsofthetransversemoXonofthepositronlforevenandoddlevelsdependingontheangleofincidenceofthepositroninacrystalrelaXvetotheplane(110)intheSi.Angleθismeasuredinreciprocallafcevectors.LevelsoforientedparXclesarenumberedwithindexbd.Onthex-axisthewavevectorisspecifiedasafracXonofthereciprocallafcevector,withfiveofthereciprocallafcevectorsatenergy28MeVcorrespondapproximatelytoanLindhardtangleofincidenceofthepositron.
ThebandspectrumtransversemoXonofafastorientedparXcleintheapproximaXonofasinusoidalcrystalpotenXal(bandspectrumallocatedto25MeVenergyparXcles).Lowerzoneborderandtheupperborderofzonesvspulse(indimensionlessunits)forthecrystallographicplane(110)insiliconSi.
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Orientation dependence Squared modulus of the matrix element of the transition
positron to the even and odd levels depending on the angle of incidence of the positron in a crystal.
Angle θ is measured in reciprocal lattice vectors. Levels of oriented particles are numbered with index bd. On the x-axis of the wave vector is specified as a fraction of the
reciprocal lattice vector, with five of the reciprocal lattice vectors at energy 28 MeV correspond approximately to an
Lindhard angle of incidence of the positron.
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( ) ( ) ( )0
32 2 2 2
0 0 00
2 4 exp 1 /2
i p ipLf L pLθ θ θ θ θ θ θπθ
⎧ ⎫⎡ ⎤≈ − − −⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
( ) 02 / / 1pa R Rπ >⎡ ⎤⎣ ⎦ %
( ) ( ) ( ) ( ) ( )2 3
12
2 Im , , expG
dE Ze d qr d qv q q G qv iGrdl q ω
π ε ω δ ωυ
−⊥− = − −∑∫
0 0
0 0
22 2 2 2
0 /2 0
2/ sin / / sin / 0.046
pL
f dx x dx dx x x
θ θ
θ θ
θ πη σ σ π π
σ
− ∞ ∞⋅= = ≈ ≈ ≈∫ ∫ ∫
V
V
V
VolumecaptureoffastchargedparXclesinabentcrystal
GraphoftheeffecXvepotenXalprofileinacurvedcrystalnearthepointoftrajectoriescirculaXonoffastchargedparXcles
SchemaXcoftheexperimenttocaptureintochannelingstateinavolumeofacurvedcrystal:1protonbeam;2-inputendofthecrystal;3outputendofthecrystal;4trappingregionintochannelinginthecrystal;L-lengthalongthebeam
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ENERGYLOSSESOFTHEFASTCHARGEDPARTICLEWITHTHEACCOUNTOFTHE
POTENTIALCONFIGURATION
( )0
2 2 1 , ,3
dE Ry ug udl a
απ α
=
( ) ( )2 3, ln 3 / / ;g u n uα π α= 2/ ; / .F Fu V V l Vα= = h
23 32 10n см−≈ ⋅62 / 1,6 10F F элV mε≈ ≈ ⋅ 2
0 / 10Fu V V≈ ≈
( )( ) ( )2 2/ / 1/137 10 1.Fl c c V= ≈ ≈h2 / Fl Vα = h
( ) 3 0/ 10 / .dE dl эВ A−:
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MoXonofthefastparXcleinthebentcrystal
( ) ( ),= −∇ +rrr r r&&
r cr Kmr U r F r
( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )( )
2 30
2
0 01
2
Im , , exp ,
K
G
dE Ze d qF r r d q v tv qdl
q q G qv t iGr t
π ω
ε ω δ ω
⊥⊥ ⊥ ⊥
−
= − = ×
× + −
∫ ∫∑r
rr r r r rr h
r rr r rr r
( ) ( ) 2 3/ / .cr êèëümr U r r F r M mr⊥= −∂ ∂ + +r r&&
,rmr Kr F F= − − +&&
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MoXonofthefastparXcleinthebentcrystal
( ) ( )..
2 2 30 / sin / / ,rFr U ma r a M m r
m= − +% % % r R r′= +%
/r r a′=
( ) ( ) ( )32 202 / sin 2 2 / / ,rFra U ma r R a M m ra R
mπ π π= + − + +&&
20/ 2T ma Uπ=
( ) ( ) ( )2
32 2002
0
/ 2 2 / sin 2 2 / / ,/ 2
rma U Fra U ma r R a M m ra Rma U m
π π π ππ
= + − + +&&
( ) ( )3sin 2 2 / / / 1r r R a b H ra Rπ π= + − + +&&
02rab FUπ
=
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PARAMETERS2
01 /2aH mv URπ
=
20/ 1mv U ?
2 2 8 70/ / 9 10 / 25 4 10 1mv U mc U eV eV≈ = ⋅ = ⋅ ?
7 19 10 7 103 5
190
10 1.6 10 3 10 10 3 10 1 10 2 102 6.28 24 1.6 10 6.28 24 6.28 8rab FUπ
− − −− −
−
⋅ ⋅ ⋅ ⋅ ⋅ ⋅= = = = ⋅ = ⋅⋅ ⋅ ⋅ ⋅ ⋅
/ 1a R =
10 7 23 10 4 10 1.2 10H − −= ⋅ ⋅ ⋅ = ⋅
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Volumecapture
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VolumereflecXon
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RESUME1.The theory of the generation of the single-particle and collective excitations in a crystal by a fast charged quantum oriented relative
to the crystallographic axes particle is constructed. 2.The dependence of the intensity of the generation of excitations in
the crystal depending on the level of the transversal both sub barrier and over barrier movement is obtained.
3. It is shown that the loss of transverse energy by the fast quantum charged particle moving in the potential of the curved crystal is
leading to the effect of the volume capture of such particles in the crystal.
4. Mathematical modeling of the various modes both of the volume capture and volume reflection of fast charged particles in a bent
crystal is carried out.