Stopping, straggling and inner-shell ionization within the shellwise local plasma approximation

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