Great feeling

Walking Ifenwithout

machines

Sunday Jan 26, 2007

VUV photon absorption in warm dense matterJ. Meyer-ter-Vehn and A. Tronnier

Max-Planck-Institute for Quantum Optics, Garching, GermanyV. Krainov (MIFT, Moscow)

0.5

1

1.5

2

0.5

1

1.5

20

51015

11016

0.5

1

1.5

20

51015

11016

/F

T TB Fk T

1 s,effv T

Al (solid density)

DESY VUV-FEL (FLASH) parameters 2006

1.5 *1012 photons= 32.3 nm (38 eV)

40 -50 fsfocus: 20 – 35 m

presently

implies

Pulse energy: ≈10 µJPulse power: 250 MWIntensity: ≤ 1014 W/cm2

heats metals up to ≈ few eV

38 eV photons

Al foil

Major problem for DESYpeak intensity experimentsis presently to reduce thefocal spot.

Now campaign for 2 m focus!

VUV-photon interaction model:Collisional absorption and photo-ionisation

Dispersion relation

2T rk k ic

Light intensity

I ~ |E |2 ~ exp(-x)

Dielectric function for quasi-free electrons (Drude model)

2

2

11

1 /p

Tei

collision frequency e <<

2

2 2 2

1

1

p

pc

What we used so far at MPQ

Laser absorption ( = 0.8 m) in Al versus

intensity (temperature)

Expt: Price et al.,PRL 75, 252 (1995)

Eidmann et al., PRE62, 1202 (2000)

Al refl.

Photon absorption by inverse Bremsstrahlung

x

p

p

Z

Start from spontaneous Bremsstrahlung emission rate

( , )spe ip p p

dR n

d

Do not forget about stimulated emission :

1( )tot sh

pe peR R n

stimulatedemission

22 30

2

8 ph

Ecn

Photon number per quantum state

2

3 2

( ) /( ', ) 64 '

3 ( ') (1 exp( 2 '))(exp( 2 ) 1)

d F dd p p p

d c p p p

Use differential cross-section (Sommerfeld, atomic units):

/Z p

24 /( )pp p p

221 /p pp

Absorption by Inverse Bremsstrahlung

Z x

p

p

( , )aR p p

Photon absorption rate

1

( , ) ( )

( , ) ( )

ph

aphsp

e

E E

E E

p p dZ d

p p dZ d n

R pn

R p

Ratio of absorption and emission rate only depends on final state densities:

Total absorption rate:

22 30

2

8 ( , ) ( , )( , )tot

t a i

p p p pp p

Ec d dw R n p p

d d

stim. emission

2 2p p

absorption2 2p p

Parameter plane of radiative Coulomb scattering

(I) Fast electrons (Born approx.)

2 2

30

3

2

3

it

Z n Ew

p

(II) Slow electrons (p3/Z>1) V.Krainov, JETP 92, 960 (2001)

2 2 30

3 3

2

3lni th

tth

Z n E pw

p Z

(III) Slow electrons (p 3/Z<1) V. Krainov, J.Phys.B33, 1585 (2000)

2 202/33

11.65 i

t

Z n E

pw

Z

p

Z

2p

3

1p

Z

3

1p

Z

fastslow

electrons

high low frequency

(I)

(II)(III)

1

2 2 30

3 3

2

3lni th

tth p

Z n E pw

p Z

Spitzer

result

Fermi average <1/p> including Pauli blocking

3

3

2 3 1ln

12( ) 1 ( ) 1 F

e

yF

y zF

zp d p e

en h p

Tf f h e

T

p

p

Integration over Fermi sphere ( , , ):/ By k T / Bz k T 2 / 2F B F Fk T p m

h

F

Collision frequency (slow electrons):

33 2

0 3

2/3

2 2 1.32/ ( , )e th

th

eff t

Zn pE

p Zv w F T h

( , ) 2

th F

F

p pF T h

p p

1/ 23

min , 1 for T 04 F F

T h

T

1 for T

( ) /( ) 1 1Bk Tf e 3

32( )( ) e p

hn f d p

Collision frequency (interpolated model)

3

3

2/31.32

22 2 ( , )ln 1e th

th

eff

Zn pF T h

p Zv

max( , )p

Fermi correction

0.5

1

1.5

2

0.5

1

1.5

20

51015

11016

0.5

1

1.5

20

51015

11016

/ FT T

B Fk T

effv

Al (solid density)

h=1 eV

h=10 eV

h=100 eV

(>p)

log

eff

1 0.1 10 100 1000

Te (eV)

Maximum Temperature (Te) and Transmisson vs Intensity

100 fs pulse of38 eV photons

50 nm Al foil

FLA

SH

20

06

550 eV

FLA

SH

20

06

Heating and expansion of 50 nm Al foil at 1015 W/cm2

38 eV photonsAl foil

Density Electron temperature

Al (T=300 K)100 nm

XUV

VUV transmission data as function of photon energyin cold Al compared with present model

L edge

plasmafrequency

presentmodel for T=300 K

Keenan et al. (2002)

Rus exp. 2006

DESY exp. 2006

DESY exp.

Berkeleytables

2B

DC 3 1/31.44

i

k Tme

n

13 1DC 10 s

Temperature (K)

Au

Exp

13 1DC 10 s

Temperature (K)

Ag

Exp

DC limit -> 0 of compares surprisingly well with data

obtained from measured electric and thermal conductivities

eff

Conclusions

• For VUV interaction with warm dense matter, the the Drude-Fermi description appears to be adequate, when combined with radiative Coulomb scattering appropriate for`slow electrons´, as derived by Krainov.

• A simple analytical expression is derived for the collision frequency, covering the full temperature range from solids to plasma and photons ranging from optical to X-rays.

• It is in reasonable agreement with existing data.

First Transmission ExperimentFirst Transmission Experiment

Measured by Measured by Sokolowski-Sokolowski-Tinten, Tinten, Tschentscher Tschentscher Krzywinski, Juha, Krzywinski, Juha, Sobierajski et alSobierajski et al

Deduced absorption length in cold AL: L=130 nm

Light propagation in plasma

p

k

p2 = 4e2n/m

plasma frequency

2 = p2 /(1+ie/) + k2 c2

dispersion relation

EL=E0 exp{ikr-it}

plane wave

(k2 - 2/c2) EL = (4i/c2) j

Maxwell equation

j = -enu

-iu = -(e/m) EL - eu

electron current

du/dt collisions

lightpropagates

reflection

0.5 1 1.5 2

11015

21015

31015

41015

51015

61015

71015

0.5

1

1.5

2

0.5

1

1.5

20

51015

11016

0.5

1

1.5

20

51015

11016

h=1 eV

h=10 eV

h=100 eV

log

eff

1 0.1 10 100 1000

Te (eV)

0.5 1 1.5 2 2.5 3

14.5

15

15.5

16

16.5

10 10001001

heV)

log

eff

300

K Te = 10 eV

Te = 1 keV

12

32 2

02/33

1 / 3

2 / 3

2 4

315 3

1 i

t

Z n Ew

pZ

Inverse bremsstrahlung rate for slow electrons (Krainov,J.Phys.B33,1585(2000))

12

32 2

02/33

1 / 3

2 / 3

2 4 2

3 215 3

1 i th

tth

Z n E pw

p pZ

1.32 ( , )F T h

( , ) 2

th F

F

p pF T h

p p

1/ 23

min , 1 for T 04 F F

T h

T

1 for T

Absorption by Inverse Bremsstrahlung

Z x

1p

2p

1 2( , )aR p p

x

1p

2p

Z

Photon emission rate (spontaneous)

2 1 2( , )spe ip p

dR p m n

d

Photon absorption rate

Total photon emission rate

1( )tot sh

pe peR R n

stimulatedemission

22 30

2

8 ph

Ecn

Photon number per quantum state

Rates are related by detailed balance

2 2

1 1

1 2

1

2 1

2

1

( , ) ( )

( , ) ( )

ph

aphsp

e

E E

E E

p p dZ d

p p dZ d n

R pn

R p

1 2 2 1 2 1 1 2( , ) ( , ) ( , )tot sp spa ph e ph ep p p p p p p pR n R n R

absorption stim. emission

total absorption

Total photon emission rate

1( )tot sh

pe peR R n

stimulatedemission

22 30

2

8 ph

Ecn

Photon number per quantum state

Photon absorption by inverse Bremsstrahlung

Photon absorption rate:

'eppB uStimulated emission rate:

'appB u

20/ 2 / 8du E

(radiation energy density)

Detailed balance(Einstein relations)

3 2' 0

3' 8

epp

pp

B u c E

A d

' 'e app p pB B

2

3 2

( ) /64 '( ', )

3 ( ') (1 exp( 2 '))(exp( 2 ) 1)

d F dp dd p p

c p p p

Spontaneous photon emission rate:

' ( ', )pp iA pn d p p

Bremsstrahlung cross-section:

XFEL Heating with 3 keV photons

Specific energy deposited:

e(J/g) = (cm2/g) I(W/cm2) (s)

100fsI = 1014 -1018 W/cm2

3.1 keV photons

e = 104 -108 J/gcm2/ggold opacity

1 keV 5 Gbar

Berkeley X-ray data : Transmission vs. photon energy

20 nm Cu 20 nm Ag 20 nm Au

Collosional absorption wave vector (from dispersion relation)

light intensity

absorption coefficient (e)

2

2 2 2

1

1 /

pe

pc

2

2

11

1 / 2p

re

k k ic i

I ~ |E |2 ~ exp(-x)

Unique feature: Explore absolute peak of photo-absorption in solid-density matter

ph

oto

-ab

sorp

tion

photo-ionisationLaser

Electrons

nc

10 100 10001h (eV)

as a function ofphoton energy

ph

oto

-ab

sorp

tion

10 100 10001T (eV)

plasmasolid

TTFermi

as a function oftemperature

Foils as Ultra-Fast Switches

1810 W/cm2h = 30 eV

50 nm Al foil

1.0

0.0

0.5

transmission

absorptionreflection

6 fs

0 10 20time (fs)