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

Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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

9


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

“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


Nu

clear

Siz

es

“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


Nu

clear

Siz

es

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


Nu

clear

Siz

es

2

3

Documents

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