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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
W. Udo Schröder, 2008
Nu
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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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
Nu
clear
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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
W. Udo Schröder, 2008
Nu
clear
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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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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
W. Udo Schröder, 2008
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