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Stopping, straggling and inner-shell
ionization within the shellwise local plasma
approximation
Stopping, straggling and inner-shell
ionization within the shellwise local plasma
approximation
C. C. Montanari and J. E. Miraglia
Instituto de Astronomía y Física del Espacio (IAFE)
andDepartamento de Física, Facultad de Ciencias Exactas y
Naturales, Universidad de Buenos Aires (UBA), Buenos Aires, Argentina.
CAARI 2010-Fort Worth
r
Shellwise local plasma approximation (SLPA)
v
r
r
Free electron gas of local density
Shellwise local plasma approximation (SLPA)
v
r
r
Free electron gas of local density
Inputs: densities and binding energies, shell to shell
Shellwise local plasma approximation (SLPA)
v
r
r
Free electron gas of local density
Inputs: densities and binding energies, shell to shell
Dielectric response for each nl-shell, independent shell approx.
Shellwise local plasma approximation (SLPA)
v
r
r
Free electron gas of local density
Inputs: densities and binding energies, shell to shell
Dielectric response for each nl-shell, independent shell approx.
Perturbative limitValidity limits ZP < ZT
intermediate to high impact energies,
Shellwise local plasma approximation (SLPA)
v
r
eion vv
)E,k(q,ωε nlF,
kF nl
(r)Enl
o
oooo
Dielectric response function
)(,, rkk)k(q,ωε FFF k
F nl(r)o
oooo
Shellwise local plasma approximation
Lindhard (1954), e-e correlation to all orders ZP to first order
Levine & Louie (1982), energy gap Enl , shell to shell response, satisfies f-sum rule
BoundFEG SSS
...... 541 dfsBound SSSS
nlnlnl
kvjPnl
j Erkrdd
kdk
vZ
S,,,
1Im
2
002
2
j=0, ionization cross sectionj=1, stopping cross section (SCS); j=2, square straggling (2)
• Bound nl-shells
• total
Calculation
Stopping
Energy loss Straggling
Ionization of inner shells
SLPA Results
10 100 1000 1E40.01
0.1
1
E (keV/u)
2/
2
B
Kawano (1988) Friedland (1981) Nomura (1976) Kido (1991) Kido (1987)
H + Cu
1 10 100 1000 100000
10
20
30
40
50
60He in Al
Sto
ppin
g cr
oss
sect
ion
(10
-15 ·e
V·c
m2 /a
tom
)
E (keV/u)
SLPA SRIM2003 Eppacher (1992) Harrys (1975) Mertens (1979) Santry (1980) Comfort (1966) Feng (1975) Shchuchinsky (1984) Andersen (1977) Martinez-Tamayo (1996) Kreussler (1982) Schulz (1982)
100 101 102 103 104 1050
5
10
15
20
25
Shiomi (1994) Valdes (1994) Bichsel (1992) Sakamoto (1991) Ishiwari (1988) Semrad (1986) Khodyreb (1984) Sirotonin (1984) Bauer (1984) Semrad (1983) Kido (1983) Mertens (1982) Mertens (1980) Bednyakov (1980) Izmailow (1980) Paul (1935-1979)
H + Cu
Energy (keV)
Sto
ppin
g c
ross
sect
ion
(1
0-1
5 eV
cm
2 /ato
m)
FEGBound
10 100 1000 1E40.01
0.1
1
E (keV/u)
2/
2
B
Kawano (1988) Friedland (1981) Nomura (1976) Kido (1991) Kido (1987)
H + Cu
1 10 100 1000 100000
10
20
30
40
50
60He in Al
Sto
ppin
g cr
oss
sect
ion
(10
-15 ·e
V·c
m2 /a
tom
)
E (keV/u)
SLPA SRIM2003 Eppacher (1992) Harrys (1975) Mertens (1979) Santry (1980) Comfort (1966) Feng (1975) Shchuchinsky (1984) Andersen (1977) Martinez-Tamayo (1996) Kreussler (1982) Schulz (1982)
100 101 102 103 104 1050
5
10
15
20
25
Shiomi (1994) Valdes (1994) Bichsel (1992) Sakamoto (1991) Ishiwari (1988) Semrad (1986) Khodyreb (1984) Sirotonin (1984) Bauer (1984) Semrad (1983) Kido (1983) Mertens (1982) Mertens (1980) Bednyakov (1980) Izmailow (1980) Paul (1935-1979)
H + Cu
Energy (keV)
Sto
ppin
g c
ross
sect
ion
(1
0-1
5 eV
cm
2 /ato
m)
FEGBound
101
102
2 3 4 5 6 7
data: Singh et al, Phys ReV A 74, 052714 (2006)
data: Singh et al, Phys ReV A 74, 052714 (2006)
SLPA
E( MeV/amu)
C+4 + Sb
101
102SLPA
O+q + Sb
q = 4 - 8
2 3 4 5 6 7101
102
data Tribedi, Phys ReV A 64, 012718 (2001)
KI (
ba
rn)
F+q + SbSLPA
Relativistic atoms
Wave functions and binding energies Dirac equation
GRASP, HULLAC
1s 2s 2p- 2p+ 3s 3p- 3p+ 3d- 3d+ 4s 4p- 4p+ 4d- 4d+ 5s 4f- 4f+ 5p- 5p+ 5d- 5d+ 6s+ 6p- 6p+101
102
103
104
105
106
107
1s 2s 2p- 2p+ 3s 3p- 3p+ 3d- 3d+ 4s 4p- 4p+ 4d- 4d+ 5s 4f- 4f+ 5p- 5p+ 5d- 5d+ 6s+ 6p- 6p+
101
102
103
104
105
106
107
Au x 102
Pb x 104
Bi x 106
Bin
din
g E
ne
rgie
s (a
.u.)
Exp. Williams (solids) STO (no relat) GRASP (relat) HULLAC (relat)
Au
Pb x 102
Bi x 104
Fig. 1 Montanari et al
100 101 102 103 104 1050
10
20
30
40 Paul, compil. (1935-1959) [5] Paul, compil. (1960-1969) Paul, compil. (1970-1986) Semrad (1987) Semrad (1989) Ogino (1988) Eppacher (1992) Sakamoto (1991) Shiomi-Tsuda (1994) Martinez-Tamayo (1996) Valdes (1993) Valdes (2000) Möller (2002)
SC
S
(10-1
5 eV
cm
2 /ato
m)
Energy (keV)
H+ + Au
Bound
FEG
total
SRIM
Stopping Power of protons in very heavy atoms ( 73< Z <84 )
102 103 1040
5
10
15
20
25
30
35
40
Bethe limit (I=837 eV)
Sirotinin (1972) Luomajنrvi (1979) Chumanov (1979) Sirotinin (1984)
H+ + W
Sto
ppin
g c
ross
sect
ion (
10-1
5 eV
cm
2/a
tom
)
Energy (keV)
FEG
SLPA
Bound
Bethe limit (I=727 eV)
Best
SRIM08
Au
-3.25
-4.13
-12.5
0
4f 5/2
5s
4d 3/2
EF4f 7/2
4d 5/2-11.8
-3.11
Au
-3.25
-4.13
-12.5
0
4f 5/2
5s
4d 3/2
EF4f 7/2
4d 5/2-11.8
-3.11
Independent shell approximation
Screening among electrons-correlation
Same shell? Binding energy?
Incertainty in energy
SLPA
iii r
v
tE
1
Au
-4.13
-12.1
0
5s
EF
4f
4d
-3.17
101 102 103 104 1050
10
20
30
40
Eppacher (1992) Bichsel (1992) Sakamoto (1991) Ogino88 Sakamoto (1986) Sirotonin (1984) Ishiwari (1984) Sörensen (1973) Sirotonin (1972) Bader (1956) Green (1955)
SRIM 2006
H + Pb
SC
S (
10-1
5 eV c
m2 /a
tom
)
Energy (keV)
FEG
Bound
100 101 102 103 1040
10
20
30
40
50
Eppacher (1995) Valdes (1993) Ogino (1988) Krist and Mertens (1983a) Krist and Mertens (1983b) Eckardt (1978) Arkhipov (1969) Green (1955)
SRIM 2006 DFT, Q=0.25, rs=2.17
H + Bi
SC
S (
10-1
5 eV
cm
2 /ato
m)
Energy (keV)
FEG
Bound
10-2 10-1 100 101
0
1
2
3
4
5S
topp
ing
num
ber
v2/(v2
0Z
T)
W with SLPA Au with SLPA Bi with SLPA Pb with SLPA SRIM08 for H+W Bethe, I(Au)=790 eV Bethe, I(W)=727 eV Bethe,I(Pb)=823 eV
Energy loss straggling of protons in very heavy atoms ( 73< Z <84 )
101 102 103 104100
101
102
100
101
102( 2 /
B
2 ) Z
T
Energy (keV)
FEG
Bohr limit
H+ + W
Bound
total
101 102 103 104
10-1
100
1
10
100
N
O M
feg
H + Bi Num
ber of active electrons
2 / B
2
Energy (keV)
Lantchner (2001) Eckardt (1978) Chu (1976) [7]
total
L
101 102 103 104
0.1
1 Num
ber of active electrons
feg
L
O
M
N
2 / B
2
Energy (keV)
Malherbe (1982)
H+ + Pb
10
100
100 101 102 103 104 105
10-2
10-1
100
2 /
B2
Energy (keV)
Besenbacher (1980) Alberts (1983) Eckardt (2001) Andersen (2002) Möller (2008) Chu (1976) [7]
Ntotal
M
L
Nu
mb
er o
f active
ele
ctron
s
feg
5s
H + Au
1
10
100
Inner-shell ionization of in Relativistic atoms
GRASP, HULLAC
20 40 60 80 1000
2
4
6
8
10
L-shell ionization of Au by He2+
Cro
ss s
ect
ion
(kb
arn
)
E (MeV)
Exp: Hardt et al Phys Rev A 14 137 (1976)
He2+ + Au
SLPA
10 20 30 400
1
2
3
4
5
M-s
hell
ioni
zatio
n X
Sec
tion
(Mba
rn)
E (MeV)
Czarnota et al (2009) Phys. Rev. A 79, 32710
O8+ + Au
Oq+ with q=3-6
SLPA
SCA
ECPSSR
20 40 60 80 1000
2
4
6
8
L-shell ionization of Pb by He2+
Cro
ss s
ectio
n (k
barn
)
E (MeV)
Exp: Hardt et al Phys Rev A 14 137 (1976)
He2+ + Pb
SLPA
10 20 30 400
1
2
3
Oq+ with q=3-6
SCA
ECPSSR
SLPA
Cro
ss s
ect
ion
(M
ba
rn)
E (MeV)
Czarnota et al (2009) Phys. Rev. A 79, 32710
O8+ + Bi
M-shell ionization
Concluding remarks Concluding remarksSLPA:
Ab-initio calculation (bound electrons)
Independent shell approximation
includes electronic correlation
Input just densities n(r) and binding energies
good for DFT and QCh
Fast calculation (PC), the same for 4f, 3d o 2p
Limits
Perturbative first order in ZP
Independent shells vs screening among shells
Locality
Future
Complex elements, molecules, clusters
Non perturbative calculation
Semilocal approximation
Screening among different FEG
Acknowledgements Acknowledgements
Darío Mitnik
Claudio Archubi
Nestor Arista
Juan Eckardt
Moni Behar
Lokesh Tribed
Helmut Paul
Instituto de Astronomía y Física del Espacio, Buenos Aires, Argentina
Insttuto Balseiro and Centro Atómico Bariloche, Argentina
Universidad Federal de Rio Grande do Sul, Porto Alegre, Brazil
Tata Institute of Fundamental Research, Mumbai, India
Thank you!
Buenos Aires, Argentina
Stopping, straggling and inner-shell
ionization within the shellwise local plasma
approximation
Stopping, straggling and inner-shell
ionization within the shellwise local plasma
approximation
C. C. Montanari and J. E. Miraglia
Instituto de Astronomía y Física del Espacio (IAFE)
andDepartamento de Física, Facultad de Ciencias Exactas y
Naturales, Universidad de Buenos Aires (UBA), Buenos Aires, Argentina.
CAARI 2010-Fort Worth
n2fn
1d
gap
kF nl
(r)
nl
o
oo
oogap
kF nl
(r)
nl
o
oo
oo
Independent Shell approximation - SLPA
o
o
o
o
oo
o
o
o
o
o
o
o
oo
oo
o
oo+O
1 2 3 4 5 60
10
20
30
40
0
10
20
30
0
10
20
LPA
F + Cu
+6 +7
Tribedi (2003)
E (MeV/u)
LPA
O + Cu
+4 +5 +6 +7
Kadhane (2003)
Sec
ción
efic
az d
e io
niza
ción
de
la c
apa
K (
10-2
1 cm
2 )
ECPSSR
LPA
C + Cu
+4 Kadhane (2003)
102 103 1040
5
10
15
20
25
30
35
40
Bethe limit (I=837 eV)
Sirotinin (1972) Luomajärvi (1979) Chumanov (1979) Sirotinin (1984)
H+ + W S
topp
ing
cros
s se
ctio
n (1
0-1
5eV
cm
2 /ato
m)
Energy (keV)
FEG
SLPA
Bound
Bethe limit (I=727 eV)
Best
SRIM08
100 101 102 103 104 1050
10
20
30
40 Paul, compil. (1980-1986) Semrad (1989) Ogino (1988) Eppacher (1992) Sakamoto (1991) Shiomi-Tsuda (1994) Martinez-Tamayo (1996) Valdes (1993) Valdes (2000) Möller (2002)
SC
S
(10-1
5 eV
cm
2 /ato
m)
Energy (keV)
H+ + Au
Bound
FEG
screening
100 101 102 103 1040
10
20
30
40
50
screening
G H Og88 Kt83a Kt83b Eckardt Ar69 Gr55
separate +- shells
H + BiS
top
ping
Cro
ss S
ect
ion
(10-1
5 eVcm
2 /ato
m)
Energy (keV)
FEG
Bound
W
-2.77
-1.35
0
5s
5p 3/2
5p 1/2-1.66
4f 7/2-1.15
-1.23 4f 5/2
Straggling
10 100 10000.0
0.5
1.0
B
H + Si
Arbo (2002) Kido (2001) Ikeda (1996) Kido (1987)
E(keV/amu)
LPA
10 100 1000 100000,0
0,5
1,0
B
E(keV/amu)
H + Cu
Figure 5: Straggling in Copper
Kido (1991) Kawano (1988) Kido (1987) Friedland (1981) Nomura (1976)
LPA
10 100 1000
0.0
0.5
1.0
1.5
H + C
Tosaki (2005) Konac (1998) Shchuchinsky (1984) in Yang (1991)
E (keV/amu)
B
LPA
10 100 1000 10000
0.0
0.5
1.0
B
H + Al
Eckardt (2001) Kido (1987) Kido(1986) Kido (1983) in Yang (1991)
E(keV/amu)
LPA
e2B
2
0
2 N4,)W(d
Stopping W(ω(ωωdωdE/dlS 1
εnlg
10 100 10000
50
100
150
H
He
Li
E (keV/u)
Sto
pp
ing
cro
ss s
ecti
on
(eV
cm
2 /ato
m)
this work Ogino 1988 Vakevainen 1997 Martinez-T 1996 Lantschner 2004 Lantschner 2004 Martinez-T 1995Khodyrev 1984Luomajarvi 1979
ZP on Zn
X-section atom/,)(Wd nl0
nl
nlg
50000 100000 150000103
104
105
106
K
-sh
ell i
on
izat
ion
(m
b)
E( keV)
F+q + Sb
Exp. Phys. Rev A 64, 012718 (2001)
Levine
Lindhard
Resumé
Advantages of the SLPA:
1- e-e correlation to all order
2- Just the electron densities & binding energies.
Do not need the continuum. Good for DFT used in QCh.
3- Cartessian coordinates. Not needed central potential
4- Projectile classical trajectory selfconsistent (e impact)
Disadvantages
1- First order in the projectile charge
2- It is local
3- It is a model. No perturbative series to follow
Future Developmens
1- Heavy atoms f-shell , molecules & clusters
2- Atom-atom antiscreening (= collision of two FEG)
3- Improve the Local hypothesis by extending to
momentum space. Intense activity in QCh