Interaction of Charged Particles with Matter · dr r Z dr r Z r d rr e r Vr edr r r e eZ ... eigen...

W. Udo Schröder, 2009

Nuc

lear

Def

orm

atio

ns2

Coulomb Fields of Finite Charge Distributions

|e|Z

e

θ

z

( )rρ ′

r ′

r r r′ −

Coulomb interactionnucleus - e

( )3( )Nucleus

eV

Z rr r

re

rd

ρ ′⋅′∫ ′

=−

Expansion of 1r r for r r−′ ′−

( ) r rr r r r r r rrr r r

θ θ−− −− ′ ′ ′ ′ ′ ′− = − = + − = + −

1 21/2 1/21 2 2 2 12 cos 1 ( 2cos )

«1

Certain regions of Segré chart: Non-spherical nuclei in ground states (different types of deformation). Excitation allows for many types of deformation.

Detailed shape “invisible” at r (everything looks like a point charge)

e

θ

z

( )rρ ′

r ′

r r′ −

( ) ( )3; : 1Charge distribution r Norm d r rρ ρ′ ′ ′ =∫


Nuc

lear

Def

orm

atio

ns3

Coulomb Fields of Finite Charge Distributions

( )rCoulomb Potential V r Ze d r

r rρ ′

′= ∫ ′ −

2 3( )

1 21 1

1 ( 2cos )For r r

r rr

r rr rexpand

θ−

−′′ +

′− = −

′

x x x x− ⋅ ⋅ ⋅+ = − + − + ⋅ ⋅ ⋅

1 32 21 1 3 1 3 51 1 .....2 2 4 2 4 6

1 2

0

2 1c1 1 ...os 3cos 1 (cos )12

r Pr rrr r

r rr r

θ θ θ∞−

= −

′ ′ ′ = + + ′− ∑

+ =

( ) ( ) ( )20 1 2

11, cos cos , cos 3cos 12

Legendre Polynomia

P P

ls

P θ θ θ θ≡ = = −

Recovered expansion of symmetric angular shape in terms of

|e|Z

e

θ

z

( )rρ ′

r ′

r r r′ −

( )

( )

( )

( ) ( )

er r

e

Z

Z d r r

Z d r r

Z

r dr r

e rV r e d r

r r

e e Z

e e Z

e e Zr ed r Qr

rr

ρ

ρ θ

θ

ρ

ρ

⋅

′

′ ′

′⋅′= =∫ ′ −

+ ←

+ ←

∫

⋅

′ ⋅ − +

′ ′∫

←∫

+

′ ′

3

2

2

2

2 23

3 2

3

23 3

1

cos

1

(

3cos 1

)

2......


Nuc

lear

Def

orm

atio

ns4

Multipole Expansion of Coulomb Interaction

Monopoleℓ = 0

Dipole ℓ = 1Quadrupole ℓ =2

Point Charges

Nuclear distribution

|e|Z

e

θ

z

( )rρ ′

r ′

r r r′ −

Different multipole shapes/ distributions have different spatial symmetries


Nuc

lear

Def

orm

atio

ns5

Parity: A Quantal Symmetry

Symmetric nuclear shape symmetric Symmetry of H( )rρ

ˆ ˆ ˆˆ ( ) ( )ˆ ˆ, 0 E

Parity Op r r H r H r

H simultaneous eigenfunctions

eratorπψ

Π→ − Π = −

→ Π = →

3(cos ) (( ) cos ) ( )n nnl

n neM dr P rr r rP

ρθ θψ ψ∗ ∝ ∝ ∫

both even or odd

n even π = +1n odd π = -10el

nM for n odd= =

If strong nuclear forces conserveparity

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )2 2

ˆ

1ˆ1

ˆ ˆ

E E E E

E E E

E E E EH r E r and

r r r r

evenr r r

od

r r

d

π π π π

π π

π π

π

π π

ψ ψ ψ π ψ

ψ π ψ

ψ ψ π

π

ψ ψ

ψ

Π = − → − = ⋅

+Π = = → =

= ⋅ Π ⋅

−

=

All odd electrostatic moments = 0

Simultaneous eigen

functions


Nuc

lear

Def

orm

atio

ns6

Restrictions on Nuclear Field

Expt: There are no nuclei with non-zero electrostatic dipole moment

Consequences for nuclear Hamiltonian (assume some average mean field U for each nucleon i):

2

1

2 22 2 2 2

2 2

1

1 1 1

ˆˆ ( )2

ˆˆ ˆ , 0( )

ˆ( ), 0

( ) ( ) (| |

ˆ ˆ,

)

: ( )

0A

ii

i i

i ii i

Ai

i

A A Ai i i

i i i

i i

pH U r parity conserving

m

Since p px x

and therefore U r

U r U r U r

Arbitrary r r

H

U

=

=

= = =

= + →∑

∂ ∂ = − = − → Π = ∂ ∂ −

Π =∑

→→

= − =∑ ∑ ∑

=

Π =

( ) (|(

|))

i i

iU r CentralU

Potentir r

aU

l−

=

=

Average mean field for nucleons conserves π U= inversion invariant, e.g., central potential

n nInteraction B d Eε µ= −


Nuc

lear

Def

orm

atio

ns7

Neutron Electric Dipole Moment

- + B n n

nn

B E

EB d

dε µ ω

ε µ ω

+ +

− −

= + =

= − =

2

2

2

2

Overall charge qn = 0, but possibly small dn ≠ 0.CP and P violation could explain matter/antimatter asymmetry.

Measure NMR HF splitting for E B↑↓

Transition energies∆ω=4dnE

s

nd

B

E

B = 0.1T, tune with Bosc B. E = 1MV/m ω = 30Hz spin-flip

ultra-cold (kT~mK) neutrons

Ekin=10-7eV, λ =670Åneutrons in magnetic bottleguided in reflecting Ni tubes

PNPI (1996): dn < (2.6 ± 4.0 ±1.6)·10-26 e·cm

ILL-Sussex-RAL (1999): dn < (-1.0 ± 3.6)·10-26 e·cm

dn experimental sensitivity/upper lim.From size of neutron (r0≈ 1.2fm): expect dn 10-15 e·m.So far, only upper limits for dn

Experimental Results for n Electric Dipole Moment


8N

ucle

ar D

efor

mat

ions Atomic beam experiments:

199 28

205 27

d( Hg) 2.1 10 ecmd( Tl ) 1.6 10 ecm

−

−

< ×

< ×

I.B. Khriplovich et al., CP Violation without Strangeness: Electric Dipole Moments of Particles, Springer-Verl., New York (1997)


Nuc

lear

Def

orm

atio

ns9

Intrinsic Electric Quadrupole Moment

Consider axially symmetric nuclei (for simplicity), body-fixed system (’), z =z’ symmetry axis

θ’

z

r ′

( )( )( )

3 2

2 2

20

3

( ) 3cos 1

3) co( s

Q eZ d r r r

rr reZ d r θ

ρ θ

ρ

′ ′ ′ ′= ⋅ −∫

′ ′ ′′ ′∫ −= ⋅

Sphere:

sphr x y z Qz= + + = → =2 2 2 20

2 3 0

Q0 measures average deviation from spherical shape.

2 20cos 3 z rz r Qθ′ ′= ⋅ → = −


Nuc

lear

Def

orm

atio

ns10

Collective and s.p. Deformations

Q0>0 “prolate” Q0<0 “oblate”

collectivedeformation

cigarsingle holearound core

zz z z

collectivedeformation

discsingle particlearound core

z

b a

Planar single-particle orbit: 20 3spQ eZ z= ⋅ { }2 2r eZ r− = − ⋅

Ellipsoidprincipal axes a, b

( )

( )2 2 20

; ; :22 45 5

coll

a b RR R b aR

Q eZ b a eZ R

δ

δ

+ ∆= ∆ = − =

= ⋅ − = ⋅ ⋅

Deformation parameter δ


Nuc

lear

Def

orm

atio

ns11

Spectroscopic Quadrupole Moment

Body-fixed {x’, y’, z’}, Lab {x, y, z}Symmetry axis z defined by the experiment

2 20 3Q eZ z r′= − Intrinsic Q moment

What Q is measured in the Lab system?(Transformation of variables, c.f. Segré)

( ) ( )( )

2

0 2

2 20

13 ....... 3co

c s

s

o )

1

(2

z

z

Q Q P

Q eZ z r Q eZ θ

θ

= − = = −

= ⋅ finite rotation through θ

Measured Q depends on orientation of deformed nucleus w/r to Lab symmetry axis. define Qz as the largest Q measurable.

How to control or determine orientation of nuclear Q?

Nuclear spin to symmetry axis, no quantal rotation about z’

No spin, no alignment control

z

θ


Nuc

lear

Def

orm

atio

ns12

Angular Momentum and QQz =maximum measurable maximum spin (I) alignment

( ) ( )2 2 2 2 ˆˆ ˆ ˆ ˆ3 3z zm II I I

I IQ eZ z r z r Q

m I m I== − = − ∝

= =

{Legendre polynomials}= complete basis set

2

0( , ) ( , ) (cos )

II Id Pφ ψ θ φ ψ θ φ α θ∗

=⋅ = ∑∫

232 2

0( ) (cos ) ( ) sin (cos ) (cos )

Iz I IQ d r r P r d P Pψ θ ψ α θ θ θ θ∗

=′ ′ ′∝ = ⋅ ⋅∑∫ ∫

2δ=

0

22 ,2 0 1z

for any

Q I

Q

0 for I → == <

Spins too small to effect alignment of Q in the lab.

z ISpinI couples with I to L =2

L

32( ) (cos ) ( ) . .z I I IQ eZ d r r P r nucl wave functψ θ ψ ψ∗′ ′ ′∝ =∫

Clebsch-Gordan Coefficient (spin coupling)


Nuc

lear

Def

orm

atio

ns13

Vector Coupling of Spins

20

1ˆ 3cos 12z m III I

I IQ Q Q

m I m Iθ

== = −

= =

I≠0:

( )( )

1 cos cos1

IIm I I I

I Iθ θ= = + → =

+

( ) 0 (2 12

0 ,1 2)1

0z for IIQ QI

−= ⋅

+= =

( ) ( )23 ( 1)

( )2 1

Iz I z I

m I IQ m Q m I

I I− +

= ⋅ =−

mI

z

θ ( )1I I +

I

Any orientation

quadratic dependence of Qz on mI

“The” quadrupole moment


Nuc

lear

Def

orm

atio

ns14

Electric Multipole Interactions

Inhomogeneous external field torque on deformed nucleus. orientation-dependent int. energy WQExamples: crystal lattice, heavy ion fly-by

E U= ∇

z

E+∆E

E

F+∆F

F

( )2 2 2

2 2 20 2 2 20

0 0 0

1( ) ...2

U U UU r U r U x y zx y z

∂ ∂ ∂ = + ∇ + + + + ∂ ∂ ∂

0Taylor expansion r center of nucleus=

2 2

2 , ,ij ii j i

U U x x y zx x x

δ ∂ ∂

= = ∂ ∂ ∂

Axial symm. field (z)

( )

3

2 2 22 2 2

0 2 2 200 0 0

( ) ( )

...2

W eZ d r r U r

eZ U U UeZU eZ U r x y zx y z

ρ= ⋅ =∫ ∂ ∂ ∂ = + ∇ + + + +

∂ ∂ ∂

Monopole WIS

Dipole 0Quadrupole WQ

no mixed derivatives

WIS: isomer shift, WQ: quadrupole hyper-fine splitting


Nuc

lear

Def

orm

atio

ns15

Electric Quadrupole Interactions

2 22 2 2

2 20 0

2QeZ U UW x y z

x z

∂ ∂ = + + ∂ ∂

Maxwell Equs. 4 (0) 0U πρ∆ = =

2 2

2 20

2

200

12

U U Uzx y

∂ ∂ ∂ ∂

∂− ∂

=

=2 2 2

2 2 20 U U U

x y z

∂ ∂ ∂= + + ∂ ∂ ∂

No charge at center

{ }

{ }

22 2 2

20

22 2 2

20

24

24

QeZ UW x y z

z

eZ U r z zz

∂ = − + + ∂

∂ = − − + ∂

2

20

(4

) )( z IIQeZ U

zQ mmW

∂⋅ ⋅

∂ =

Field gradient · spectroscopic quadrupole moment mI

2

axial symm=Uzz

2 23 z r= −


Nuc

lear

Def

orm

atio

ns16

Quadrupole Hyper-Fine Splitting of γ Lines

Inhomogeneous external electrostatic field aligns Q via nuclear spin I,Measure interaction energies WQ (mI) (possible for I >1/2). Quadrupole hyper-fine splitting of nuclear or atomic energy levels

Weak “hf” splitting mI2

of nuclear and atomic levels in Uzz≠0

splitting of γ emission/absorption lines

dN/dEγ

Eγ

Uzz=0

Estimates atomic energies ~ eVatomic size ~ 10-8cmpotential gradient Uz ~ 108V/cmfield gradient Uzz ~ 1016V/cm2

With Q0 ~ 10-24 e cm2

WQ ~ 10-8 eV small !

mI=±2

mI=±1

I=0

I=2

mI=0

mI=0E2

ground stateUzz=0 Uzz≠0

Excited state

isomer shift

centroid


Nuc

lear

Def

orm

atio

ns17

Experimental Methods for Quadrupole Moments

Weak “hf” splitting WQ of nuclear and atomic levels in Uzz≠0

splitting of X-ray/ γ emission/absorption lines

Measurable for atomic transitions with laser excitations

nuclear transitions with Mössbauer spectroscopy

Muonic atoms:

107 times larger hf splittings WQ with X-ray and γ spectroscopy

scattering experiments Uzz(t)

Nuclear spectroscopy of collective rotations model for moment of inertia

( )2

0

2

12

: ( )2

II I

Q E

Lanthanides10 - 20 keV

Actinides

+→ ℑ → =

ℑ ℑ

g.s. I=0I=2I=4

I=6

I=8

. .

. .

0Qℑ →


Nuc

lear

Def

orm

atio

ns18

Collective Rotations

43 5

RR

πβ ∆=

z

b a β : deformation parameter

Nuclei with large Q0 (lanthanides, actinides) collective rotations

Wood et al.,Heyde

;2

a bR R a b+= ∆ = −

20

20

2. . .: 1 0.3159:8

rig

irr

rigid body m o i MR

hydro dynamical MR

β

πβ

− ℑ = +

− ℑ =

( )2

12IRotational and inversion symmetry ev II Ien E→ = +

ℑ

0023 1 0.16

5Intrinsic quadrupole moment Q eZ R β

πβ= +

20 0( , ) 1 ( , )R R Yθ φ β θ φ = +

Rotational Spectra of Actinides


Nuc

lear

Def

orm

atio

ns19

215 18

2keV≈ −

ℑ


Nuc

lear

Def

orm

atio

ns20

Systematics of Electric Quadrupole Moments

2Q R

RZRδ ∆

∝ =

odd-Nodd-Z

Q(167Er)=30R2

Prolate

Oblate

8 20 28 50 82 126

Q<0 : e.g., extra particle around spherical core. pattern recognizable17

29 5163 123

8209

8 3, , ,O Cu Sb Bi

Q>0 : e.g., hole in spherical core pattern not obvious. If such nuclei exist, weak effect of hole for Q

27 55 115 176 16713 25 49 71 68, , , ,Al Mn In Lu Er

Somewhat tightly bound nuclei are also spherical (Q = 0):“Magic” numbers N or Z = 8, 20, 28, 50, 82, 126, …

Mostly prolate (Q>0) heavy nuclei

Not a necessary condition for Q = 0


Nuc

lear

Def

orm

atio

ns21

Q0 SystematicsMøller, Nix, Myers, Swiatecki, Report LBL 1993

Q0 large between magic N, Z numbersQ0≈0 close to magic numbers

Interaction of Charged Particles with Matter · dr r Z dr r Z r d rr e r Vr edr r r e eZ ... eigen...

Documents

Transcript of Interaction of Charged Particles with Matter · dr r Z dr r Z r d rr e r Vr edr r r e eZ ... eigen...