Sizes. W. Udo Schröder, 2008 Nuclear Sizes Absorption Probability and Cross Section Absorption upon intersection of nuclear cross section area j beam.

Sizes

W. Udo Schröder, 2008

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Absorption Probability and Cross Section

Absorption upon intersection of nuclear cross section area sj beam current areal densityA area illuminated by beamL = 6.022 1023/mol Loschmidt# NT # target nuclei in beamMT target molar weightrT target densitydx target thickness[s]=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 : m dN = -N mdx

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 s

2


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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 r0

3


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Interaction Radii

a 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

4


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Elastic Electron Scattering

a

b

Detector

l

nr

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 rel ph

Momentum transfer q

ase

r

phase difference of elementary waves relative to center of nucleus

3 2

1,..,

( , ) ( , , ) ( )

2 exp ( )

2

( , ) ( , , ) ( ) ( )

el pi i n n

n

pi n n

n

i f

Zel p pf n f n n nn

n n n

r t k r t r

ik r i t r

p p p k elastic scattering

r t k r t r r

1,..,( , ) ( , , ) ( ) ( )

exp ( ) ( )

( , exp, ) ( ) ( )

Zel p pf n f n n nn

n n n

p pi n n n nn

n n n

el p pi n n nn

nn

n n

r

ik

t k r t r r

ik r i t ik r r

k r t r r

Incoming plane wave= approximation to

particle wave packet

Center of nucleus: r=0

probability amplitude for proton n

l

Impulse Approximation for interaction:

0ˆ ( )eN n

nH f r r

5

: nindependent r r

r


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Momentum Transfer and Scatter Angle in (e,e)

ik

fk

q

/2q/2q

Scattering angle q determines momentum transfer

2 sin( 2) ( ) !

e A fi Lab com

q k q q

m m k k k

i fi eN fi n f

nf

p el p pi n n n f n n nn n

nn n n

pi n i n n n

n

pn n

f

dr t H r t r t f r r r t

d

ik r i t r f r r k r t r r

ik r i t ik r i t r f r

22

0

2

0

*

0

(

ˆ( , ) ( , ) ( , ) ( ) ( , )

exp ( ) ( ) ( , , ) ( ) ( )

exp (e ) )xp (

r f rn n rn

r r

nn

n n n n n nn nn n

n

n n

rik

f r ik f r ir rk

2

2 2

) ( )0

2

1

ex

exp

( ) exp ( ep 0 ) xp

q

<bra* | ket>

f0 x density of proton n at rn

6

n

integration overr and all r r


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Separation of VariablesPoint nucleus (PN): a=b, jn=0 determine scaling factor Z protons

2

0

2

20( )

i fi f

nff PN Mottn n

nsame

PNrn

d dfr Zf

df

d

22

30

232

0

( ) exp ( ) ( ) e

( ) e

iq rn n n n

n Nucleus

i

i

q rn

f

Nucle

rnf

us

d

d

Z

f r ik r d r f Z r

d r rf

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

( )r normalized

nuclear charge

density distribution

Finite nucleus: integrate over space where proton wave function are non-zero

Strength of Coulomb interaction same for each proton

7


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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 (b =v/c)

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

Ruth

Ruth

Mott Ruth d

dd

d

d

d

dd

2 2

0

21

cos (( )

12

)

)

sin (2

Obtained in 1. order quantum mechanical perturbation theory, neglects nuclear recoil momentum.

check non-relativistic limit

8


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Elastic (e,e) Scattering Data

R. Hofstadter, Electron Scattering and Nuclear Structure, Benjamin, 1963 J.B. Bellicard et al., PRL 19,527 (1967)

X 10

X 0.1

3-arm electron spectrometer (Univ. Mainz)

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

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Fourier Transform of Charge Distribution

r

rR

Homogeneous

sharp sphere

r Rr

r R

0

0

q r qr q z

cos | |iqr

d

iqr iqr

F q r dr d d e

e edr r

i r

r

rq

2 cos

cos

2

0

( )

)

( )

2 (

sin

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 r, half-density radius C diffuseness a

R

C

4.4a

C is different from the radius of equivalent sharp sphere Req

rq r qrrF dq

0( )]

4( ) sin([ )

1

0


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Nuclear Charge Form Factor

iq rn

NucleusF q d r r

q momentum transfer

3( ) ( ) e Form factor for Coulomb scattering = Fourier transform of charge distribution.

r-Distribution Function r(r) Form Factor q-Distribution

Point 1 constant

Homogeneous sharp sphere

r0 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

1

1


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q

dFr

dq

22

0

6

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

r

rR

2 235

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

Interpretation in terms of radial moments of charge distributionExpansion:

=1

F q q r q r 2 2 4 41 1( ) 1

6 120

eqR r

25:

3

1

2


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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

Rz(H) = (0.85-0.87) fmRz(He)= 1.67 fm

Density of 4He is 2 x r0 !

1

3


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Charge Radius Systematics

equequR A R A r A A fm

a fm const small isotopic

t fm const effect

1 3 1 30( ) ( ) 1.23

0.54 . (

2.4 . )

r

rms rmso

equ

Charge distributions heavy solid

C A A fm a fm const

r r A r fm

Homogeneously charged sphere

R r

1 3

0

2 1 30

2

( ) :

( ) 1.07 0.54 .

0.94

:

53

r0(charge) decreases for heavy nuclei like Z/A for all nuclei:

r0(mass) = 0.17 fm-3 = const. 1014 g/cm3 (r0=1.07 fm)

Note: Slightly different fit line, if not forced through zero.

1

4


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Muonic X-RaysEffect exists for also for e-atoms but is weaker than for muonsNegative muon:m- e- mm = 207me

Replace electron by muon “muonic atom”

Bohr orbits, am = ae/207

107 times stronger fields

r(r)

r

2e

2 2710 e

r(r)

1) X-ray energies 100keV–6 MeV

2) Isomeric/isotopic shifts DEis

DEis(2p)

22 2

0

22 2

0

4 ( ) ( )

4 ( ) ( )

is

is ex gs

E Ze dr r r r r

E Ze dr r r r r

D

D

point nucleus

Excited ground nuclear state

Finite size

3d2p

1sDEis(1s)

r VCoul(r)

En

Point Nucleus

1

5


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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

1

6


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Mass density distribution:

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

Mass and Charge Distributions

Charge density:

C A A fm

a fm

R A A fm

1 3

1 3

( ) 1.07

0.54

( ) 1.21

N fm nucleons 30 0.17

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

A ZA

r rZ

1 3( ) 1.23

0.55Z

Z

R A A fm

a fm

1 3( ) 1.1

0.55Z

Z

C A A fm

a fm

Parameters of Fermi Distribution

r r A r fm

t a fm

2 1 30 0; 0.94

2 ln9 2.40

1

7

Coul

Coul Coul

eZE sharp sp

Co

hereR

bE E

R

ulomb self energy

20

0 2

35

51 ( ) ..

2


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Leptodermous Distributions

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

ff r

r Ca

0( )1 exp

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

Profile 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

Leptodermous Expansion in (b/R)n

Fermi Distribution (a C)

1

8


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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)

1

9


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“Interaction Radii for Exotic Nuclei

Derive sR =sTotal - selastic

sR =:p[RI(P)+RI(T)]2

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

=(N-Z)/2Kox Parameterization: Interaction Radius

vol p T surf p T cm P TR R A A R A A E r f A A 1 3 1 3int 0( , ) ( , , ) ( , )

2

0


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“Halo” Nuclei

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

6He - 8He mass density distributions

Experiment: dashed, Theory (fit):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

Parameterization:

2

1


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Neutron Skin of Exotic (n-Rich) Nuclei

8Henn

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) !

rms 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 eF

Plausible because of weaker nuclear force133Cs78 stable, normal n-skin thickness, tn ~0.1fm181Cs126 unstable, significant n-skin, tn ~ 2 fm

Can one actually make 181Cs, role of outer n ??

Are there p-halos ? Not yet known.

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

Which n Orbits?

DRrms =Rnrms - Rp

rms

2

2


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Sizes. W. Udo Schröder, 2008 Nuclear Sizes Absorption Probability and Cross Section Absorption upon intersection of nuclear cross section area j beam.

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