The true mystery of the world is the visible,

The true mystery of the world is the visible, not the invisible Oscar Wilde

K

Cu

Na

Outline of lectures (I)

• L1: Probing the nuclear fingerprint on the atom

1.1 Hyperfine structure

1.2 Isotope shifts

• L2: Doppler-free laser spectroscopy

2.1 Introduction to lasers and laser spectroscopy

2.2 The crossed beams methods and fast beams collinear laser spectroscopy

2.3 The structure of K isotopes & the shell model

2.4 The N=60 region and nuclear deformation

Outline of lectures (II)

• L3: Efficiency, sensitivity and precision

3.1 Selective Radioactive Ion Beam production 3.2 In-source resonance ionization spectroscopy - ISOL (hot cavity approach) - IGISOL (gas cell approach) 3.3 In-gas jet spectroscopy & CRIS 3.4 Towards the superheavy elements 3.5 229Th (a nuclear clock) Outlook

THEORY • H. Kopfermann, Nuclear Moments, Academic Press, New York (1958) • W.H. King, Isotope Shifts in Atomic Spectra, Plenum Press, New York

(1984) • G.K. Woodgate, Elementary Atomic Structure, McGraw-Hill (1970) • D.M. Brink and G.R. Satchler, Angular momentum, Oxford Science

Publications, third edition (1994)

LASERS • J. Wilson and J.F.B. Hawkes, Lasers Principles and Applications,

Prentice Hall Europe (1987)

LASER SPECTROSCOPY • W. Demtroder, Laser Spectroscopy Basic Concepts and Instrumentation, 3rd

Edition Springer-Verlag (2003)

RESONANCE IONIZATION • V.S. Letokhov, Laser Photoionization Spectroscopy, Ac. Press, (1987)

Between 1995 and 2010 (B. Cheal and K.T. Flanagan,

J. Phys. G 37 (2010) 113101)

Before 1995 (J. Billowes and P. Campbell,

J. Phys. G 21 (1995) 707) Unpublished

Submitted to PPNP (Campbell, Moore, and Pearson 2015)

Previous laser spectroscopic reviews

What are limits of nuclear existence?

Do new forms of nuclear matter exist?

Are there new forms of collective motion?

Does the ordering of quantum states change?

Key questions in nuclear science

Between 1995 and 2010 (B. Cheal and K.T. Flanagan,

J. Phys. G 37 (2010) 113101)

Before 1995 (J. Billowes and P. Campbell,

J. Phys. G 21 (1995) 707) Unpublished

Submitted to PPNP (Campbell, Moore, and Pearson 2015)

Previous laser spectroscopic reviews

Recent work mostly by

collinear laser spectroscopy

Recent work mostly by resonance

ionization spectroscopy (RIS)

N=4

N=2

N=3

N=1

Balmer Series

Lyman Series

N=∞

A historical note on atomic spectroscopy

• 1704: release of Newton’s ”Opticks”. Sun’s light can be dispersed into a ”spectrum”

L

S

J=L+S

• Increasing the resolution by a factor of ~5000 reveals

a fine structure splitting of hydrogen

Increasing in resolution

λ=656.279 nm (N=3→N=2)

F=J+I

2P3/2

2P1/2

2S1/2

Fj

Fi

• A further factor of 1000 in resolution reveals a finer

splitting due to the coupling of the nucleus with the

electronic orbital

→ Hyperfine structure

Hyperfine interaction = the interaction of nuclear magnetic and electric moments with electromagnetic fields (produced at the nucleus

by the orbiting electrons)

Lets consider the effect on an atomic orbit of spin J

J

Nuclear spin

I

Electron spin

Hyperfine interactions in free atoms

e

I

J

F

The atomic and nuclear spins

couple to form the total

angular momentum

F = I + J

Each state J has

several F-states:

States of the same I and J but coupled to different angular momenta F will have slightly different energies

Nuclear magnetic moment

Contributions from orbiting charge and intrinsic spin

Magnetic (dipole) moment

The magnetic dipole moment of a state of spin I = expectation value of the z-component of the dipole operator μ :

The magnetic moment (or g factor) therefore tells us about the valence nucleon orbits and couplings. What about filled shells?

Protons: gl = +1 gs = +5.586

Neutrons: gl = 0 gs = -3.826

E = - m . Be = - m Be cos q

The interaction energy depends on angle θ

J

Nuclear spin

I

Electron spin

Magnetic dipole interaction

Since and

then the interaction Hamiltonian can be expressed as

m

The different energy shifts of the different F-states are then

where

Be can be calibrated by measuring the energy shifts for an isotope of a known magnetic moment.

Spectroscopic quadrupole moment

The electric quadrupole moment provides a measure of the deviation of charge distribution from sphericity:

I

q

m

Z

this assumes a well-defined deformation axis

Using angular momentum algebra, we get

Experiments measure the maximum ”projection” of the intrinsic quadrupole moment along the quantization axis

Note for nuclear spin I=0 and I=1/2 the spectroscopic quadrupole moment vanishes even if the intrinsic shape is deformed. The intrinsic moment can in turn be related to the quadrupole deformation parameter β2

Oblate Spherical

Prolate

In a uniform electric field the energy of an electric quadrupole moment is independent of angle and there is no quadrupole interaction. However, there is an angle-dependence in an electric field gradient.

Quadrupole deformation

β2 -ve +ve 0

Electric quadrupole interaction

E = ¼ e Q0 VJJ P2(cos q) J

Nuclear spin

I

Electron spin

Electric field gradient along J-direction due to atomic electrons.

Energy shifts of the F-states are then

where

The hyperfine factor “B” is measured by experiment

The electric field gradient VJJ may be calibrated with an isotope with known Qs

The original fine structure level E(J) is perturbed so that the final energy due to hyperfine effects:

Summarizing the hyperfine splitting

E(J)

5/2

3/2

1/2

5/2 A

3/2 A

-B

+5/4 B

1/4 B

5/2 A + 5/4 B

3/2 A - 9/4 B

)1()1()1( where

)12)(12(2

)1()1()1(

2)( 4

3

JJIIFFC

JIJI

JJIICCBC

AFE

201Hg

Nuclear spin I=3/2

Atomic spin J=1 nucleusat field magnetic)0(

,)0(

e

eI

B

JI

BA

m

Access to nuclear spin I (number of HF components) and µI

The original fine structure level E(J) is perturbed so that the final energy due to hyperfine effects:

Summarizing the hyperfine splitting

E(J)

5/2

3/2

1/2

5/2 A

3/2 A

-B

+5/4 B

1/4 B

5/2 A + 5/4 B

3/2 A - 9/4 B

)1()1()1( where

)12)(12(2

)1()1()1(

2)( 4

3

JJIIFFC

JIJI

JJIICCBC

AFE

201Hg

Nuclear spin I=3/2

Atomic spin J=1

nucleusat field magnetic)0(

,)0(

e

eI

B

JI

BA

m

Access to nuclear spin I (number of HF components) and µI

nucleusat

gradient field electric)0(

),0(

JJ

JJs

V

VeQB

Access to Qs

But...

All you actually need is,

82

But...

All you actually need is,

82 Page

JU FU I FL JL 1

(2FL+1) (2JL+1) (2Fu+1) { } Intensities:

Splittings:

Only 3 transitions allowed (no F=0 → F=0)

In practise: example of K

• The nuclear spin can be unambiguously determined from the number of HF components (relative spacings and intensities)

• Selection rules obeyed, ΔF=0, ±1, F=0 → F=0 forbidden

49K I=1/2

51K I=3/2

J. Papuga et al., PRL 110 (2013) 172503

Frequency (MHz)

102

101

100

99

98

96

94

93

92

95

97

(Pauli 1924!!!)

B Cheal et al, PLB 645 (2007) 133

= Frequency difference in an electronic transition between two isotopes

Isotope 1

Isotope 2

Isotope shifts of electronic transitions

The Nobel Prize in Chemistry 1934 was awarded to Harold C. Urey "for his discovery of heavy hydrogen".

H. Urey (1932): Splitting in the Balmer series in natural hydrogen

→ ”heavy” hydrogen (deuterium!)

AAAA ''

Nuclear radius: few × 10-15 m Atomic radius: few × 10-10 m

Point nuclear charge: Coulomb potential (-1/r)

Potential is slightly deeper for the smaller isotope: s-electrons more

tightly bound

The nuclear volume effect (field shift)

The finite spatial extent (volume) of the nucleus gives an electrostatic potential difference to that of the Coulomb potential

– this perturbs the electron wavefunction Ψe(r)

V(r

)

r

V(r)

r

Isotope shift of atomic level

The nuclear volume effect II

rdrrVreE eefs

3

0)()()(

The difference in this energy shift between two isotopes can be expressed as

where δV(r) is the change in Coulomb potential between two isotopes. We assume constant electron density in the region of the

nucleus and arrive at the change in frequency of a transition:

',22

0

2', )0(

6

AA

e

AA

fs rh

Ze

Change in electron density at the nucleus between the two atomic levels comprising the

transition (electronic part)

Difference in the mean

square nuclear charge radius

(nuclear part)

The Finite Nuclear Mass Effect

i e

inkinetic

m

p

M

PE

22

22Kinetic energy

(nucleus + electrons)

In the Centre of Mass frame, the nucleus has a momentum equal and opposite to that of the

sum of the electrons

i

in pP

ji

ji

i

inuclear ppM

pM

E ˆˆ1

ˆ2

1 2This leads to a nuclear kinetic energy:

recoil correlations

”Unfortunately” this is an effect which we must take account of, even though it has no ”nuclear physics” interest…

E

One can finally solve the Schrodinger equation to calculate the energy change between two isotopes A and A’ (and the transition

frequency)

Normal and specific mass shifts

ji

ji

i

i

u

ms pppAA

AA

mE ))ˆˆ(2ˆ(

'

'

2

1 2

Normal mass shift • due to the finite

nuclear mass • easy to calculate

Specific mass shift • due to electron correlations • no analytical expression

• calcuable only for 2- and 3- electron systems

'

')(

AA

AAKK SMSNMSSMSNMSms

1/A2 A>>1

Isotope shifts in practise

IS = FS

2δ r

MS +

|e(0)|2 6hε0

THEORY EXPERIMENT

Ze2

To evaluate IS data:

- mass data from Atomic Mass Evaluation (2012) - SMS either calculated (ab-initio, MBPT, coupled cluster) or evaluated

via non-optical data (elastic e scattering, muonic atom X-rays) - Field shift factor from non-optical, semi-empirical, atomic theory

(accurate to 15%)

Cheal, Cocolios and Fritzsche, PRA 86 (2012) 042501

To summarize the effects so far…

Isotope (A-1)

Isotope A

Isotope (A-1)

Isotope A

F=5/2

F=3/2

F=1/2

Point nucleus + Finite size + Magnetic + Electric dipole quadrupole

Example: J=1, I=3/2

+ higher order multipoles

(too small to consider in laser measurements)

These energy shifts of may be only a few parts per million of the energy of an optical atomic transition. Optical techniques provide the sensitivity and precision required to measure these effects.

qm coseB )(cos4

120 qPVeQ JJMass shift + Field shift

The important message(s) from Lecture 1

Sizes

Spins

Qs

μ

Model Independent

(measured)

Ground-state spin assignments underpin all excited levels. We measure them directly!

F. Charlwood et al, Hyp. Int. 196 (2010) 143

1974

Mn58 3+,4+

65 s

Mn58 2+,3+

65 s

Mn58 3+ (0+)

3.0 s 65 s

b – 3.8, .. g 810.8, 1323.1, 459.2,…

E 6.25

Mn58 3+ (0)+

3.0 s 65 s

b – 6.1, .. g 1447-, 2227

b – 3.8, .. g 810.8, 1323.1, 459.2,…

E 6.25

Mn58 1+ (4)+

3.0 s 65 s

b – 6.1, ..

g 1446.5,

2433.1,

…

b – 3.8, ..

g 810.8,

1323.1,

459.2,

…

IT 71.78

E 6.25

Mn58 (1)+ (4)+

3.0 s 65 s

b – 6.1, ..

g 1446.5,

2433.1,

…

b – 3.8, ..

g 810.8,

1323.1,

459.2,

…

IT 71.78

E 6.25

1980 1988

1996 2002 2010

End of Lecture 1