Sizes. W. Udo Schröder, 2007 Nuclear Sizes 2 Absorption Probability and Cross Section Absorption upon intersection of nuclear cross section area j beam

Sizes

Nuclear SizesNuclear Sizes

W. Udo Schröder, 2007

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Absorption Probability and Cross Section

Absorption upon intersection of nuclear cross section area j beam current areal densityA area illuminated by beamL = 6.022 1023/mol Loschmidt# NT # target nuclei in beamMT target molar weightT target densitydx target thickness[]=1barn = 10-24cm2

Targetdx

Incoming

0N j A

Transmitted

0xN N e

# absorpti

T

on

T

P

per

nuclei exposed

to beam

LA

nucleusd

dxM A

x

Mass absorption coefficient dN = -Ndx

0 0 1 xabsN N N N e

abs

abs

T

T

T

L AxM

NN N x

A

N j current densN ity j

00

Thin target, thickness x

abs

nucl

NN j

elementary absorption cross section area per nucleus

Illuminated area A

Nucleus cross section area


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Target n Detector Electronics DAQ

(Pu-Be) n Source

Size Information from Nuclear Scattering

Basic exptl. setup with n source: Count

Target in/target out

d from small accelerator (Ed100 keV): T(d,n)3He En 14 MeV

J.B.A. England, Techn.Nucl. Str. Meas., Halsted, New York,1974

22

4.5

14-MeV neutr n 1.2

5 /

o

dBroglie wave length

c mc E

AE MeV fm

fm

AR

1 3 3

30.17

3

.

4A A

A

A

AA nuc

R A V

lconst

V

R A

fm

Amp/Disc

Cntr

Experiment (approx. analysis)

Equilibrium matter density 0


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Interaction Radii

scattering

16O scattering

12C scattering

P.R. Christensen et al., NPA207, 33 (1973)

D.D. Kerlee et al., PR 107, 1343 (1957)

el Ruthd d

d d

dDistance of closest approach scatter angle


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Electron Scattering

a

b

Detector

r

ik

fk

2 ( )

:

i f

i f

a ba b k

a k r k b k r k

k k r k q r k

Momentum transfer q

phase difference of elementary waves

3 2

( , ) ( , , ) ( )

2 exp ( )

2

( , ) ( , , ) ( )

el pi i n n

n

pi n n

n

i f

el pf n f n n

n

r t k r t r

ik r i t r

p p p k

r t k r t r

( , ) ( , , ) ( )

exp ( )

( , , ) ( ) exp

el pf n f n n

n

pf n n n

n

el pf n n n

nik

r t k r t r

ik r i t ik r

k r t r

Impulse Approximation: Whole is sum of parts, no interactions among parts.

Incoming plane wave= approximation to particle wave packet

Center of nucleus r=0

probability amplitude for proton n


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Momentum Transfer and Scatter Angle in (e,e)

ik

fk

q

Scattering angle determines momentum transfer

2 sin( 2) ( ) !

e A fi Lab com

q k q q

m m k k k

2

2

2 2

2

2

2

1

( , ) ( , ) exp ( , )

exp ex ( ) exp

( ) e

p

exp xp ( ) exp

i fi

n nn

n

fi n nnf

i

n n n

f

n n

dr t r t ik r i t r t

d

ik r i t ik r i t f r ik

f ri ik f r ikq r

q


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Separation of Variables

Point nucleus: a=b, n=0

2 22

0 0( )i fi f

nff PN Rutherfon

PNrd nf r

d df Zf

d d

iq rn n

n Nucleus

iq rn

Nucleus

i f

f

r nuclear

charge de

df r ik d r f Z r

d

d

Zn

ri

rs

fty

2

223

0

30

2 ( )

( ) exp ( ) e

( ) e

2( )i fi f

ff PN

d dF q

d d

Scatter cross section for finite nucleus = cross section for point-nucleus x form factor F of charge distribution


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Mott Cross Section for Electron Scattering

222 2E pc mc

2

2

1.952 102 ( / ) 1 1.0222

e

e

c fm

K MeV K MeVK K m c

In typical nuclear applications, electron kinetic energies K » mec2 (extreme) relativistic domain

(100 ) 2e MeV fm e- = good probe for objects on fm scale

Ruth

Mott Ruth

Ruth

ddd d

d d dd

21

2 2

0

cos ( )2( )

1 sin ( )2

Obtained in 1. order qu. m. perturbation theory, neglects nuclear recoil momentum.


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Elastic (e,e) Scattering Data

R. Hofstadter, Electron Scatt. and Nucl. Struct., Benjamin, 1963 J.B. Bellicard et al., PRL 19,527 (1967)

X 10

X 0.1

3-arm electron spectrometer (Univ. Mainz)

dd/d/d diffraction patterns diffraction patterns1st. minimum q()4.5/R


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0

Fourier Transform of Charge Distribution

rR

Homogeneous

sharp sphere

r Rr

r R

0

0

q r qr q z

cos ||2 cos

2

0

0

( ) sin ( )

2 ( )

4( ) sin( )] )[ (

iqr

iqr iqr

F q r dr d d r e

e edr r r

i

r

qr

F q dr qrq

r

Generic Fourier transform of f:

f r dq f q qr

0

2( ) ( ) sin( )

r r dq q F q qr

2

0

1( ) ( ) sin( )

2

Form factor F contains entire information about charge distribution

0( )

1 r C ar

e

Fermi distribution , half-density radius C diffuseness a

R

C

4.4a

C is different from the radius of equivalent sharp sphere


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1

Nuclear Charge Form Factor

3( ) ( ) eiq rn

NucleusF q d r r

q momentum transfer

Form factor for Coulomb scattering = Fourier transform of charge distribution.

r-Distributionr-Distribution Function Function (r)(r) Form FactorForm Factor -Distribution-Distribution

Point 1 constant

Homogeneous sharp sphere

0 for r R=0 for r >R

oscillatory

Exponential exponential

Gaussian Gaussian

1( )

4r

22 2

21

a

q

3

8a ra

e

3

3 sin cos( )

( )

qR qR qR

qr

2 23 222

2

a ra

e

2

2exp

2

q

a


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2

2 2 4 4

22

0

1 1( ) 1

6 120

6q

F q q r q r

dFr

dq

Model-Independent Analysis of Scattering

3 51 1sin( ) ( ) ( ) ....

3! 5!qr qr qr qr

2 2 4 4

0 0 0

2 2 24 4( ) 4 [ [ [

6 1( )] ( )] )

0(

2]r r r rF q dr q drr q dr rr r

r 2 mean-square radius of charge distribution

rR

2 235

r R Equivalent sharp radius of any (r): eqR r

25

:3

Interpretation in terms of radial moments of charge distributionExpansion:

=1


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Nuclear Charge Distributions (e,e)

R. Hofstadter, Ann. Rev. Nucl. Sci. 7, 231 (1957)

t=4.4a

C: Half-density radiusa: Surface diffusenesst: Surface thickness

Leptodermous: t « C

Holodermous : t ~ C

0( )1 exp

Fermi Distribution

rr C

a


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Charge Radius Systematics

1 3( ) ( ) 1.23

0.54 . (

2.4 . )

equR A R A A fm

a fm const small isotopic

t fm const effect

1 3

2 1 30 0

2

:

( ) 1.07 0.54 .

0.94

53equ

Charge distributions

C A A fm a fm const

r r A r fm

Homogeneously charged sphere

R r

0(charge) decreases for heavy nuclei like Z/A for all nuclei:

0(mass) = 0.17 fm-3 = const. 1014 g/cm3


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Muonic X-Rays

Negative muon:- e- m = 207me

Replace electron by muon “muonic atom”

Bohr orbits, a = ae/207

107 times stronger fields

(r)

r

2e

2 2710 e

(r)

1) X-ray energies 100keV–6 MeV

2) Isomeric/isotopic shifts Eis

3d2p

1sEis(1s)

Eis(2p)

r VCoul(r)

En

22 2

0

22 2

0

4 ( ) ( )

4 ( ) ( )

is

is gs ex

E e dr r r r r

E e dr r r r r

point nucleus

ground excited nuclear state


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Charge Radii from Muonic Atoms

Engfer et al., Atomic Nucl. Data Tables 14, 509 (1974)

1 3( ) 1.25R A A fm

Energy/keV

E.B. Shera et al., PRC14, 731 (1976)

2p3/2 1s1/2

2p1/2 1s1/2

Sensitive to isotopic, isomeric, chemical effects


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7 Mass density distribution:

except for small surface increase in n density (“neutron skin”)

Mass and Charge Distributions

1 3( ) 1.1

0.55Z

Z

C A A fm

a fm

1 3( ) 1.23

0.55Z

Z

R A A fm

a fm

Charge density:

1 3

2 1 30 0

1 3

( ) 1.07

0.54

; 0.94

( ) 1.21

2 ln9 2.40

C A A fm

a fm

r r A r fm

R A A fm

t a fm

30.17N fm nucleons

Constant central density for all nuclides, except very light (Li, Be, C,..)

A ZA

r rZ


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Leptodermous Distributions

C = Central radiusR = Equivalent sharp radiusQ = Equivalent rms radiusb = Surface width

0

(

( )1 ex

)

p

Fermi Distribution

ff r

r Ca

a C

R.W. Hasse & W.D. Myers, Geometrical relationships of macroscopic nuclear physics, Springer V., New York, 1988

( ) ( )g r df r dr

2

0

22 2

0

( ) 1 ( ) ...

5( ) 1 ( ) ...

2

( )3

bC dr g r r R

R

bb dr g r r C R

R

b a a C

20

0 2

35

51 ( ) ..

2

Coul

Coul Coul

Coulomb self energy

eZE sharp sphere

Rb

E ER

leptodermous


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Studies with Secondary Beams

Produce a secondary beam of projectiles from interactions of intense primary beam with “production” target projectiles rare/unstable isotopes, induce scattering and reactions in “p” target

Tanihata et al., RIKEN-AF-NP-233 (1996)


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“Interaction Radii for Exotic Nuclei

Derive R =T - el,

R =:[RI(p)+RI(T)]2

Tanihata et al., RIKEN-AF-NP-168 (1995)

=(N-Z)/2int

1 3 1 30

( , ) ( , , )

( , )

vol p T surf p T cm

P T

R R A A R A A E

r f A A

Kox Parameterization:


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“Halo” Nuclei

From p scattering on 11Li extended mass distribution (“halo”). Valence-neutron correlations in 11Li: r1 = r2 = 5fm, r12 = 7 fm

6He - 8He mass density distributions

Experiment: dashed, Theory:solid

9Li n

n11Li2 2

2 2 2 23 2 3 2 3 2 5 2

2 3( ) exp( ) exp( ) ( )

23ci ni

iN Nr r

r Ar B r ba a b b

11 : 3, 6,

0, 2,

1.89 , 3.68

0.81, 0.19

cp cn

np nn

Li N N

N N

a fm b fm

A B

, .i n p

Korshenninikov et al., RIKEN-AF-NP-233, 1996

tn


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Neutron Skin of Exotic (n-Rich) Nuclei

8He n

n

Qrms (4He) = (1.57±0.05)fm

Qrms (6He) = (2.48±0.03)fm

Qrms (8He) = (2.52±0.03)fmV(8He) = 4.1 x V(4He) !

matter radii

D.H. Hirata et al., PRC 44, 1467(1991)

Thick n-skin for light n-rich nuclei: tn ≈ 0.9 fm (6He, 8He)

Relativistic mean field calculations: tn F

133Cs78 stable, insignificant n-skin, tn ~0.1fm181Cs126 unstable, significant n-skin, tn ~ 2fmCan one make 181Cs ??p-halos ? Coulomb barrier keeps p together, expansion could reduce it

Tanihata et al., PLB 289,261 (1992)


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End of End of Nuclear SizesNuclear Sizes

Documents

Sizes. W. Udo Schröder, 2007 Nuclear Sizes 2 Absorption Probability and Cross Section Absorption upon intersection of nuclear cross section area j beam