Doctoral Thesis

Isotope Shift and Hyperfine Structure in Yb I
by Atomic-Beam JLaser Spectroscopy

Wei-G-uo JIN

Depa.rtment of Physics, Faculty of Science,
Hiroshima UniYcrsity, Higashi-Hiroshilna 724, JAPAN(October, 1991)

Contents

Abstract

1. IntroJuction
2. Experimental
2.1 Experimental setup
2.2 Population of the metastable state of Yb I by an electric discharge
3. Theoretical Description
3.1 Isotope shift
3.2 Derivation of changes in mean square nuclear charge radii 8 <r2> and deformation parameters 8</3 2 >3.3 Hyperfine interact.ion
3.4 Crossed-second-order (CSO) effect
4. Experimental Results
5. Analysis and Discussion
5.1 Nuclear parameter A
5.2 8<r2> and 8</3 2 >5.3 Single-elect.ron IIF paralneters for the 4J 14 6s6d configuration

5.4 J and term dependences of IS and CSO effects for the 3 DJ and 1 D2 states of
4f 14 6s6d6. Conclusion
Acknowledglnent
References

Abstract

Isotope shifts of 9 transitions from the ground and metastable states in Yb I
were measured for 7 stable isot.opes wi th A= 1G8- 176 by means of atomic-beam laser
spectroscopy. The electric discharge was used to obtain the intense population of
the metastable state in Yb I. IIyperfine constants A and B of 12 levels for 171 Yb
and 173Yb were deternuned. Reliable values of nuclear parameters A were obtained
and changes in mean square nuclear charge radii 5<1'2> were deduced using boththe two-parameter model and the numerical analysis. Deformations for Yb stable
isotopes and the odd-even staggering effect for odd-N isotopes were discussed. Hyperfine constants A and B were analyzedl with the effective operator procedure andthe single-electron hyperfine parameters of 6s and 6d electrons were derived for the
Yb I 4f 14 6s6d configuration. For the 3 D J and 1 D2 states of 4f 14 6s6d, J and tenn
dependences of isot.ope shifts were obt.ained and t.he parCU11eters of the crossed-secondorder effect were derived to be Z6d/).. = 18(6) !\IHz/fn12 and g2(6s, 6d)/A = -1120(60)MHz/fm2.

1

1. Introduction
Isotope shift (IS) is the energy shift of an optical transition between different isotopes for one element. \Vhen a neutron is added or taken out from a nucleus, thereduced mass of atomic electron, the correlation between electrons and the charge
distribution of the nucleus are changed. This results in the changes of kinetic energy
and potential energy of electrons which correspond to the isotope shift. Hyperfine
structure (HFS) originates from the magnetic dipole and electric quadrupole interaction between electrons and the nucleus, since the nucleus has a magnetic dipole andelectric quadrupole moments. Because isotope shift has an order of magnitude of 10 3
MHz which is comparable to HF split ting of 10 2 -10 3 MHz, studies of IS and HFS are
always connected.
Measurements of isotope shift and hyperfine structure yield the informations not
only for the nucleus but also for the electronic property of the atom. From the isotope
shift, the information about the nuclear charge distribution and the nuclear shape canbe extracted . The important parameters related to the charge distribution are changes
in mean square nuclear charge radii 5<7'2> and deformation paralueters 5<f3 2 >. Fromthe hyperfine structure, the nuclear spin, the nuclear magnetic dipole moment p, andthe nuclear electric quadrupole mOluent q are derived. The paranleters of the nucleus
5 < 1'2>, P, and Q are the basic ones for the nuclear ground state and comparisonof these parameters with theoretical calculations could give insight and infornlation
about nuclear structure. The IS and HFS also give out the electronic information suchas the radial integral < 1'-3>, the elec tron density at the nucleus, the configuration
mixing and the electron screening effect. These effects related to behavior of electrons
are of great help to atoluic physics. Therefore, study of IS and HFS is a unique way,compared with the nuclear reactions such as electron scattering, inelastic scatteringof proton and a-particle and Coulomb excitation, to deepen the understanding of thenucleus and electrons.
Studies of IS and HFS have a long history. The conventional optical spectroscopy(e.g. Fabry-Perot interferometer) was widely used in the early days. But obtained
values have the uncertainty of several % and less values exist for isotopes with smallabundance. Early works of IS and HFS were reviewed by Heilig and Steudel,1 and2Childs. 2 It was the CW tunable dye laser that revolutionized the optical spectroscopy
with its high resolution and high sensitivity3 from the early 1970. The fluctuation oflaser frequency is as small as several MHz and this leads to an energy resolution of

8v /v :S;10-7. The power of the dye laser beam reaches to several watts, thus atomic
beam can be as small as 10 3 atorns/s for laser induced fluorescence measurements.
The values of IS and HFS are obtained with the uncertainty of an order of 0.1 %by the laser spectroscopy. Measurements are also possible even for isotopes withvery small abundance. Studies across long isotopic chain are possible for stable andunstable isotopes by this technique and this is particularly superior to other methods
for nuclear study, for example, nuclear reaction and Coulomb excitation etc., whereenriched isotope target is always necessary. Studies of IS and HFS for the stableand unstable 4 ,5 nuclei have been extensively done by the laser spectroscopy in recent
years and were recently reviewed by Otten. 6 Particularly, many works 7 - 16 have been
carried out for the stable isotopes of the ra.re-earth elements by the atomic-beam laser
spectroscopy because the beam of the stable isotopes is easily obtained. Very reliable
informations can be extracted for stable isotopes. To get the information about theelectronic property, the atomic-beam laser spectroscopy is usually used.
The nuclear parameters 8<r2> and 8<(32) are extracted from the isotope shift.
The general trends of 8<r2> were already known for most of stable isotopes by theconventional optical spectroscopy,l but the uncertainty is more than 10 %. Since values of 8<r2> can be evaluated by using, for example, the liquid-drop or droplet model,
more precise values of 8<r2> are needed to check these nuclear models. The nuclear
deformation has been found to be mainly quadrupole and higher order deformations
have been understood very little. The defofll1ation parameter 8<(32) deduced fronl IS
is contributed from not only quadrupole but also higher order deformations. Thus, informations about nuclear quadrupole and higher order deformations can be obtained
from IS. Recently, strong octupole defonnation was found for Ra isotopes fronl IS
measurements. 4
From HFS, one can determine HF coupling constants A and B which describe
the magnetic dipole and electric quadrupole interactions, respectively. Hyperfine consta.nts A and B were reported for the ground state and some low-lying levels. 17 Re-

3cently, preCIse values of A and B have been determined for high-energy levels ofsome rare-earth elements by the laser spectroscopy.lO,18 HF constants A and Bare
contributed from each electron of open atomic shells. To study the behavior and contribution of a single electron for the HF illteraction, the single-electron HF parameteris introd uced. The effec ti ve operator procedure 19 is used to extract the single-electron
hyperfine parameters from the measured values of A and B. Such studies were well reported for 4f, 5d, 6s and 6p electrons of the rare-earth elements and recently reviewed
by Pfeufer. 20
Isotope shift mainly depends on the electronic configuration and should be dependent on electronic term and angular mornentum J just like the atomic level energy.
In 1976, Bauche and ChalTIpeau 2l attributed term and J dependences of IS to theso-called crossed-second-order (CSO) effect of IS and refined parametric descriptionof the CSO effect. The CSO effect is the effect of configuration mixing from very-far
atomic levels. The effect of the configuration mixing is very important one for theoretical calculations and is actually related to the behavior of the electron. Particularly, it
is only the CSO effect which can give information about the mixing effect from very far
configurations. Investigations of the CSO effect were reported for some configurationsof low-energy levels for 4f and 5d electrons of the rare-earth elements. 16 ,22-25
Ytterbium with the aton1ic number 2'=70 is the last even-Z element in the rareearth region and has closed 4f and 6s subshells in the ground state. The groundconfiguration 4f146s 2 only yields one state 1So (the ground state). AtOIIDC spectruln
shows an alkaline-earth like character when 4f-shell is not broken and also becomescomplicated when 4f electrons are excited. Because there are no other states « 10 4
cm- l ) near the ground state which are thermally populated, less transitions exist in
the visible wavelength region of the dye laser and reported studies of IS and HFS
are less. Fron1 the standpoint of nuclear side, stable isotopes of Yb have the neutron
number N =98- 106 and lie almost in the middle of two closed shells (N =82 and

126). For Yb stable isotopes, Clark et al. 13 measured isotope shifts of one transitionby the atomic-bealTI laser spectroscopy and derived nuclear parameters A related
to 5<r2> by combining the optical IS with the electronic and muonic X-ray IS's.
However, such analysis seems to have sorne alnbiguities because the definition of the4nuclear parameter of optical IS differs slightly from that of the muonic X-ray IS.
Baumann and Braun 26 measured HF constants A and B for the 4f 14 6s6d 3 D J and
1D2 states of 171 Yb and 173Yb by means of two-step laser excitation, but the results
have large uncertainties. Moreover, single-electron HF parameters for 6d electron are
not reported. Concerning the CSO effect, no report can be found for the 6d electronofYb.
We have measured IS and HFS for the stable isotopes 14 - 16 of Nd, Sm, Gd, Dy
and Er. The nuclear deformation has been examined for the nuclei with the neutron
mumber N =82-102. To study the nuclear deformation of Yb and extend our systematic study up to N =106, we carry out measurements of IS and HFS in Yb I. Anotherpurpose of this study is to measure IS and HFS for excited upper configurations andto investigate the HF interaction and the CSO effect for the 6d electron. For the
study of upper levels, transitions from nletastable states are chosen and an electric
discharge is used to populate metastable states. This thesis will describe details about
experimental setup of laser spectroscopy and method of electric discharge as well as
methods of analysis for IS and HFS. We try to extract reliable values of 8<r2> for the
stable isotopes of Yb by means of atomic-beam laser spectroscopy and to examine thedeformation of these isotopes. We can obtain HF constants A and B, and term andJ dependences of IS. Moreover, single-electron HF parameters and CSO parameters

can be determined for 6s and 6d electrons. Discussion about experimental results willbe also given.

52. Experimental
2.1 Experimental setup
The experimental setup is shown sche:m atically in Fig. 1. An Ar-ion laser (Spectraphysics 2016) with 5-W power was used to pump a ring dye laser (Spectra-physics
380A) which gives out the continuously tunahle laser light. Dyes of Rhodamine 6G
and 110 were used and a wavelength range of about 530- 610 nm was available. The
typical power of the dye laser was about 100 m\V during the experiment and thefluctuation of the frequency was about 5 MHz. The continuous scanning of the laser
frequency over 30 GHz was available.

Dye laser
Spherical
m irro r

Ar-ion laserbeam~-7--~~~~r---~:--~I----~~

Laser beam

Slits

Microcomputer

SA ~' _ _ _ _ _~>IOSCiI10-1. scope.

Fig. 1. Genera.l layout of the experimental setup. Abbreviations are as follows:
PI,,!: phot.omultiplier. Amp.: amplifier. TSCA: timing single channel analyzer. FPI:
Fabry-Perot int.erferometer. SA: spectrum analyzer. PD: photodiode. MCS: multichannel sca.ler.

6

The laser beam was collimated by four slits of 3 mm in diameter in the verticaldirection. The ato111ic beam was ejected from a s111all hole (0.5 mm in diameter) of theoven and collimated wi th a second sli t of 2 mm in diameter at a distance of 20 cm fromthe oven in the horizotal direct.ion. The collimation ratio is 100 and this means thatthe loss of atomic beam is about 10 4 by t. he second slit. The laser beam was crossed
orthogonally with the atomic beam and the Doppler broadening was greatly reduced.
The residual Doppler broadening is about 1011Hz. Increase of the collimation ratio
would reduce the Doppler broadening bu t yield larger loss of the atomic beam. The
resonant fluorescence from the atomic beam, induced by the laser beam, was focussed
by a spherical mirror and was detected by a photomultiplier (PM). The solid angle forthe photon detection was enlarged to 1.41T sr by the spherical mirror with a diameterof 8 cm and a radius of curvature of 6 cm.. The photomultiplier (Hamamatsu 643-02)
was cooled down to -20 °C by the Peltier effect to reduce the dark current and used in
the single photon counting mode. Many barnes were placed between the atomic beamoven and the laser atomic-beam interaction zone to red uce the radiation background.
Then the signal-to-noise ratio was typically 2000 . The vacuum of the chamber for theinteract.ion and measuring zone was kept at about 10- 6 torr by a diffusion pump anda liquid-nitrogen trap and that for the atomic oven zone was kept at about 10- 5 torr
by another diffusion pump and a liquid-nitrogen trap.
For the relative frequency calibration of measured spectra, we measured the transmit ted light of the laser through a 25-cm confocal Fabry-Perot interferOlneter (FPI)
with a free spectral range (FSR) of 297.7:::1:0 .2 MHz. The value of 297.7±0.2 MHz wasdetermined by measuring the HF split.tings of 139La which were precisely determinedby Childs and Goodman 27 wit.h the laser-rf double resonance technique. The signal
was amplified and di crilninated by a tirning single channel analyzer (TSCA). The
fluorescence fro111 the ato111ic beam and the transilnitted light of the FPI were simultaneously recorded with a dual-input 111ultichannel scaler (MCS). The MCS with
32768 x 2 channel melnories was controlled by a lnicrocomputer and worked with 50
channels per second. The scanning speed of the laser was about 100 11Hz per second
and the data taking t.ilne for one spectrurn was about 3 min. The absolute wave number of the transi tion was determined by lneasuring the emission spec tra of an 12712

7

cell since the rotational spectra of 12712 rnolecule were already well measured. 28 The
transmitted light through a spectrum analyzer (SA) wit.h FSR of 2 GIIz was detectedby a photodiocle (PD) and the frequency scanning of t.he dye laser was monitored
wi th the oscilloscope.

2.2 Population of the metastable state in Yb I by means of an electric
discharge
Populations of various states in the atonlic bealTI are given by the Boltzman distribu tion. Only low-lying levels are thennally populated for the temperature below
2000 °c. To study the transitions from upper-lying metastable states, populations ofsuch metastable states are necessary. Electric discharge is a convenient and uniquemethod to populate metastable states and has been used 8 ,9,29 for Ba, Sm and Eu, etc.
However, details of processes to populate metastable states are not known to us andno reports have been found for Yb in the literature. For studies of IS and HFS of themetastable states and higher excited levels of Yb I, we constructed an a.tomic beamoven of a discharge type. Details on the popula t ion of lnetastable st.ates of Yb I byelectric discha rge were ex ami ned. 30
Figure 2 shows the atomic bealn oven Qf a discharge type. A molybdenum crucible(8 mm in diameter and 20 mm in length) loaded with Yb (about 0.3 g) is mounted

•

Mo

~ To
~ ST

Fig. 2. Atomic beam oven of the[2J BN

discharge type. VF : voltage of the cruci ble fila.ment, Vc: cathode heating voltage, VD : discharge voltage. ST and BN
1cm

L-J

are the abbreviations of sta.inless steel
and boron ni tride, respectively.

8

on top of a tantalum rod. An atomic beam of Yb is ejected from a small orifice witha diameter of 0.5 n1m by means of resistance heating of a crucible filament. Boronnitride (BN) is used as insulators. The oven temperature up to 2000 °C is easily
obtained with the help of tantal urn and stainless steel heat shields. The temperature
T of the crucible is measured by a Pt-Rh thermocouple. The discharge cathode is a

Ta wire of 0.5 mm in diameter in a 5-turn helical coil of 3 mm in diameter, 4 mm
in length and is placed at about 7 mm from the crucible in the beam direction. The
crucible itself serves as a discharge anode, and electric discharge is produced aroundthe orifice of the crucible.
The dependence of the discharge current ID on the crucible temperature T, on
the cathode current Ie and on the discharge voltage VD is shown in Fig. 3. Figure
3 (a) shows that the discharge current is very small at the crucible temperatureof 600°C and drastically increases when the crucible temperature reaches 700 °C(corresponding to the vapor pressure of about 2.5 torr). Temperature above 700°C is
not necessary because the discharge becolnes unstable above 700°C. Stable discharge
is easily attained at the cathode heating current of 7 A, as shown in Fig. 3 (b) . Higher\1

500 -400

(a)

~

<l:

5 300 -•
Ie = 7 A•
Yo =20 V•••a

200 100 -

o

•

•
•
•
•
•

I
I
..... 1,··1"
550 600 650 700 750
T (OC)

I

(c)

( b)
~

~

T =690°C
Vo= 20 V

fo-

fo-

•

•

•

•

r-

•

l-

•

~

•
•

fo-

I

4

...

•I

I ...

I

I

...

7

8 9

, 6

C:'

Ie (A)

T= 690°C
le=7 A

I

I

\

l-

•

-

•
•
I

•• 1

••

I

••
• ••

I

1

•

I

o 10 20 30 40 50
Vo (V)

Fig. 3. The dependence oftlle discharge current JD on the vRrious paran1eters: (a)the crucible t.en1pera t.ure T, (b) the cathode cu rrent Jc, (c) t.he discharge voltage VD .

9cathode-heating current caused the discharge at lower temperature but produced
strong background for the laser-induced fluorescence measurenlents. The cathode withthe heating current of up to 7 A yielded almost no such background. Figure 3 (c) showsthat the discharge current is not very sensitive to the discharge voltage in the range of15- 50 V. A discharge voltage over 50 V is too high and causes the discharge to become
unstable. Thus, it was found that the conditions to maintain the stable discharge are
650- 700 °C for the crucible tenlperature, about 7 A for the cathode current and 20- 50
V for the discharge voltage.
The relative population of the metastable state of Yb I was studied by measuring laser-induced fluorescence of two optical transitions. The simplified energy level
scheme 31 and the studied transitions are shown in Fig. 4. The 555.65 nm transitionis from the 1 So ground state of the 4f 14 6s 2 configuration to the 17992 cm- 1 3 PI stateof the 4f 14 6s6p configuration. Since the ground state of Yb I is 1 So, the 25860 cm- 1(7/2,3/2)s (expression of jj coupling scheme: (jl, j2)J) state of the 4f 13 5d6s 2 configuration is a metastable state. The 585.45 nm transition is from this metastable state
to the 42936 cnl- 1 (7/2,3/2)s state of the 4f 13 5d6s6p configuration. For the measurement of the 555.65 nm transition from the ground state, no discharge was needed andthe oven tenlperature was about 570 °C. This tenlperature corresponds to the vapor
pressure of about 0.2 torr or 10 14 atoms/s from the oven.

5 ~L.113Sd6s6p

T(~'!)5

4 r-

Lf)

~
Lf)

'I

E

u
~

0

>-

O'l
L-

~

~(B)5

3-

L.113Sd6s 2

L.IIL. 6s6p

2-

T

1 e-

I.D

(lJ

c

w

Jp
,

Lf)

..n

Lf)
Lf)

Or-

Fig. 4. The energy level schemeI

and the two transitions (wavelength

discharge

in nm) in Yb I to measure the pop-~'50

ulation of the lnetastable state by

L.IIL. 6s 2

the electric discharge.

10--- 15 ~-----------------,Mo

176Yb 174Yb 172Yb*~ 10r-

OJ
C

OJ

u

~ 5u
(j)

~ 00 11!1 ~ I ,I ~L, ,-"--,,I..----J
I~.,.---JL'---.,---J,'""l~i
I

LL

23456

7

Laser frequency (GHz)

Fig. 5. The laser-induced flllore~cence spectrum of the 585.45 nln t.ransition. The
peaks of even-mass isotopes are labelled wi t h t he isotope symbol. Peaks lnarked with(*) are the hyperfine peaks of 1i1 Yb and other peaks are t.he hyperfine peaks of 173Yb.
The nleasured fluorescence spectrum of the 585.45 nln transition is shown in Fig. 5.
Not only the peaks of even-ma~s isotopes of 168- 176 Yb bu t the IIF peaks of 171 Yb (nuclear spin 1= 1/2) and 173Yb (1=5/2) are well resolved, except the accidental overlapof 168Yb and one HF peak of 171 Yb. The discharge current was controlled by changing
the cathode heating current. The cathode current had no influence on the populationof the metastable state. The crucible temperature and the discharge voltage were
fixed at 690 °C and 23 V, respectively. The fluorescence spectra of the 585.45 nm
and 555.65 nm transitions were lneasurcd at different discharge currents . The output
power of the dye laser was kept below 50 /-L \V by nleans of a neutral density filter.
As a result., the power broadening and the saturation effects were negligible. The
counts at all of the peaks in each spectrun1 were integrated and the background wassu btracted. Thus, the dependence of the relative population of the metastable state
and the ground state on the discharge cnrrent was obtained from the 585.45 nm and
555.G5 nnl transitions, respectively. Results are shown in Fig. 6 (a). In the sanle way,
the dependence of the relative population of the metastable state on the discharge

11

voltage was also obtained and is shown in Fig. 6 (b), where the crucible temperatureand the discharge current were fixed at G90 °C and 200 rnA, respectively.
Figure 6 (a) shows that the populat.ion of the Inetast.able stat.e increases gradually
until 300 rnA and reaches sat.uration around GOO rnA. The population of the groundstate decreases with the discharge current and reaches about 30% above 500 rnA.
This means that atoms pOPlilating in tlle ground st.ate in the absence of discharge are
excited to metastable states by the discharge. Results for the metastable state andthe ground state are consistent. Because the lowest excited levels 31 lie above 17000
cm- 1 , almost all atoms populate in the ground state in the absence of discharge forthe temperature below 700°C. Thus, it could be concluded that at least about 30%
of atoms remain in the ground state and up to 70% of atOlns can be populated in allmetastable states by the discharge. It is also shown in Fig. 6 (a) that the discharge
current of 200-400 rnA is best and can ]produce a relative population of GO-90% of
the metastable state. Figure 6 (b) shows t.hat the population of the n1etastable stateis less sensitive to the discharge voltage in the range of 20-50 V.
~----------------------~~-------------------------.

1.0

(a)

0

c

o

o

o

0·8

o

•

:J
Q

•

• •

- (b )•-•

T= 690 O(• o

OJ> 0 ·4
ro

+-'

•

<l.J

ex 0·2•••••••••••••

Vo= 23 V

•

•

•

o

oQ 0·6

o

•

•

• •

T =690to

10= 200 rnA

o

o

o

o

•
~~

o 100 200 300 400 500 600 0
10 (mA)__~t____~I___~I___~I___~1

10

20

30
Vo (V)

40

50

Fig. 6. (a) The dependence of the relative populat.ion of the (7/2,3/2)s state of13

2

4f 5d6s (closed circles) and t he ground st.a t.e (open ci rcles) on the discharge current I D . Normalizations were carried out at a discharge current of GOO rnA for themetastable state and at t.he absence of the discharge for the grouno state. (b) The
dependence of the relative population of t.he met.astable stat.e on the discharge voltage

VD . Normalization was carrieJ out at a dilschcuge voltage of 50 V.

12In SUlnma.ry, the best conditions to obtain intense population in the metastablestates a.re as follows: 6.50- 700

0

C for the crucible temperature, above 7 A for thecathode current, 20- 50 V for the discharge voltage and 200- 400 rnA for the discharge
current . The population of the metast.able st.a t.e of Yb I st.rongly depends on the
discharge current and is less sen sitive to t he discharge voltage above 20 V. The population of all nletastab1e states up to about 70% can he obtained by the discharge.
This method can also be applied for the laser spectroscopy of metastable states ofother elements .

13

3. Theoretical Description
3.1 Isotope shift
The observed isotope shift bVi, between isotopes with nl(lSS nlllnber A and A' for atransition i, is the sum of the normallnass shift (N11S), t.he specific 111aSS shift (SMS)
and the field shift (FS). Then 5Vi can be written as 1 ,(1)where the normal mass shift (NMS) bViNMS is caused by the change in the reduced
mass of electron and nucleus and is calculated easily. The specific mass shift (SMS)bViSMS

is caused by the correlation between electrons, and its calculation is very

difficult partic ulady for rare-earth elements and yields only qu ali tati ve agreement
wi th experimental resul ts.

A'-A
(2)
,
AA'
A' - A(3)
bViSf\,IS = A1iS r-.ls,
AA'
where A1iNMS and Af iSMS are the factors of N1IS and SMS, respectively. The factor
bViNMS =A{NMSlvfim1s

has the siIl1ple expression,

AliN r-. rs ==Vi1836 .105,(4)

where Vi IS the transition frequency. For a pure ns 2 - nsnp transit.ion, the SMS is
negligibly small and the senliell1pirical evaluation is usually used,bVSMS =

(0 ± O.,s)bVNMS.

(5)

The field hift (FS) bViFS is caused by the change in t.he nuclear charge distribution
and is given in fir t-order perturbation t.heory by(6)
where the nuclear parameter A is related to the change in mean square nuclear charge
radii b<1'2> and higher order contributions.

(7)14

where the expansion coefficients C l , C 2 and C 3 are calculated by Seltzer. 32 The relativistic correct.ion fact.or f(Z) of t.he FS depends on t.he at.Olnic nun1ber Z and isgiven by Ahmad et a/. 4 as follows,f( Z) -

Cunif'i;""'

Lm

Cn~_ T2nA-23n-l(A'_A)'(8)

c. 2n+.\ 0

where A = ~(A + A') and ro=1.2 fIll. The isotope shift constant Cunif is recalculated
recently by Blundell et al. 33 for the nucleus of a unifornlly charged sphere. Equation(8) differs from the conventional definition of f(Z) in ref. 1 because the higher ordercontributions of 8<r4> and 8<r 6 > were omitted in ref. 1
The electronic factor Ei is related to the change of the nonrclativistic electron
charge density 8 I 'ljJ( 0) I; at the nucleus,(9)
where ao is the Bohr radius. According to t.he senlienlpirical evaluation,1 8 I 'ljJ(0) I:
can be deternlined from the charge densi ty I 'ljJ(0) I~& of the s elect.ron,

8 I 'ljJ(0) I~&-np(31 'ljJ(0) I~&,(10)

8 I V(0) 1~,2-n3np,I 'ljJ(0) l;t3'(11)where {3 and, are the screening factors and values of (3=1.12 and ,=0.73 were
estimated for the rare-eart.h elelnent.s. l Thus, E n &2- nmp is writt.en as(12)
wi th the defini tion of E( ns),

(13)
The elect.ron density I 'ljJ(0) 1~3 of the s electron can be determined empirically for
an alkalilike atoln. This can be done from t.he fine structure level scheme by theGoudsmi t- Fernli-Segre fornllIl a,6

(14)

15

where na and dn a/ dn are the effective quantum nunlber of the ns electron and its
derivative, respectively. Za is the out.er charge and is taken to be 1 for neutral atonls,
2 for singly ionized atOlns, etc. I 'IjJ(O) I~., is also detennined [rOIn the Fernli-contact
parameter an., of the hyperfine structure,6
')

an.,

I 'IjJ(O) I~$= ~f.LBI'N(/,. /I)Fr(Z)(I- 5)(1- c)'(IS)

where I'.n and IlN arc the Bohr and nuclear Inagnctons, respectively. I is the nuclear
spin and ~l is the nuclear magnetic dipole moment. Fr (Z) is the relativistic correction
factor. (1 - 8) and (1 - c) arc the Breit-Crawford-Schawlow and the Bohr-Weisskopf
correction factors, respectively.

3.2 Changes in mean sqnare nuclear charge radii b<r2> and nuclear
deformation parameters b <(32 >A. Two-parameter model

The nuclear parameter). can be derived from the field shift. The calculations of
the higher order tenns 8 <1'4> and f, < 1'6 > arc necessary to obtain changes in meansquare nuclear charge ra.dii 8<1'2> from 'x. Such calculation can be done by using thetwo-parameter Inodel which was proposed by Ahnlad el al. 4 In this model, the termof 6< 1'2n > is di vided into spherical and defoflnation prlrts,('

u < l'

2n

('

>= u

< l' 2n >sph +U(' < l' 2n >def,(16)

with the assunlption of a unifornl charge distribution with a sharp cutoff a.t the nuclear
surface . The following relation was deduced,') ')8 < l' 2 n >= ') Sn+ 3 R_71--8< 1'-') > sph + -43n R-'1) n 5 < (3 2 > ,
1
_n

(17)

~

,,,,here R~ == ~ <1' 2 >sph. Values of <r 2 >sph and 6<r 2 >sph for a spherical nucleus canbe calculated by using a droplet mocIe1. 3 4! For 71= 1, eqnation (17) reads('

U< l' 2 >== U(' < l' 2 >sph +-S < r 2 >sph U(' < (32 > .
47r

16

(18)

Equation (18) has been used for many years in the nuclear spectroscopy. By using eq.

(17), values of 0<1'2> and 0<f3 2> are related to the value of A as follows:
A- (x - y)o < 1'2 >spho < 1'2 >

(19)

l+y

A - (1 + :r)o < 7,2 >spho< f32 >7r(1 + y) < 1'2 >sph

5
4

(20)
'

x

10(2 R2
5C3 R4
7C l 1 + 3Cl 1)

(21)

y

2~-R~

C" ') + 3-R4.
C3

(22)

C1

1

C1

1

B. Numerical analysis

Recently, \Vakasugi et al. 14 proposed a. direct nUl11erical analysis of 0<1'2> insteadof t.he two-parameter Inodel. In such numerical anrtlysis, the Fernli charge distribution
with parameters of the nuclear deformation f3, the diffuseness s and the hollow depth
w is assumed,p( 1', B) = flo 1

1 + 7l'( 17(8))2C- 17 (8)) ,+erp - t -(23)

for the well defonned nucleus. The constant Po is the charge density at the nuclear
center, t is related to the diffuseness paranleter s as t=s/4ln3. R(B) is written as

R(B) = R 2 (1 + f3 Y20) ,

(24)

f32
R2 = Ro/(l + 47T

(25)

J,

where Ro=1'oAt and Y20 is the spherical hC1f1110nics . Thus, the term 0<1'2n> is calculated by using the following relation,o<Jop(T,B)r 2n dV

2n
l'

>=

J p (1', B) dV

.

(26)

The reduced gamma transition probability B(E2) is relat.ed to the charge distribution,(27)
The general trends of the parameters sand w were eXrtmined in ref. 14 and the values of ow(A, A - 2)=0.02 bet ween two isotopes with lllass nUlll1 ers of A and A-2

17were found to be a good approxilnation within one elelnent for the rare-earth region.
Therefore, the parameters f3 and s of the charge distribution can be detennined byusing the nuclear parameter ,\ frOIn IS and the B(E2) value from Coulon1b excitationlTIeaSUrelnents. Values of 5<r2n> are derived by using eq. (2G) and not only 5<1'2>
values but also 5<1'4> and 8<7,6> values can be deduced. This numerical analysis
SeelTIS to be more realistic than t.he two-paralneter lTIodcl because the assumption ofa uniform charge distribution is not used in this ntllnerical analysis.

3.3 Hyperfine interaction
The hyperfine int.eract.ion is usually treated by the effective operat.or procedure.
The Hamiltonian for the interaction is expressed b y19.35.36
00

lhIFs =L TK (e) . TK (n),

(28)

K=1

where TK (e) and TK (n) are t.he tensor operators for the electronic and nuclear parts,
respectively. The HP interaction energy E IIFS is written as

E IlFS= < I J FlllIIFsli J F >f. (_I)I+J+F {IJ IJ ~r.\.r} < JIITK(e)IIJ >< 1IITK(n)II1 >, (29)

K=1

.1

where J is the electronic angular mOInenhlln and F = I + J is the total angularmomentum of the atOIn. It is enough only to consider the magnetic dipole and electric
quadrupole interactions for the hyperfine interaction. That makes the sUlnmation
lilnit to 1(=2. By using the \Vigner-Eckart theoren1,37 the reduced matrix element< 1IITJ«(n)III > is related t.o the matrix clement < 11ITK(n)I11 >. Since we have
< 11IT1(n)I11 >= IL and < 11IT2(n)II1 >= ~eQ, E ITFS is expressed as
E- ~CA
3C(C + 1) - 41(1 + J)J(J + 1)
2+
41(21 _ 1)2J(2J _ 1)
B,

IIFS -

where

C = F(F + 1) - I(I + 1) - J(J + 1).

18

(30)

The constants A and B are the nuclear n1agnetic dipole and the electric quadrupole· 1y, 38
coupling constants, respec t lve

It
I

1

A

- [J(J + 1)(2J + 1)J-2 < JIIT1(e)IIJ >,

(31 )

B

2J(2J -- 1)12eQ[(2J + 3)(2J + 2)(2J + 1)P < JIIT2(e)IIJ > .

(32)

The relativistic description is necessary to describe exactly t.he hyperfine interaction
which is treated by using the effective operators T1(e) and T2(e).19
N

T1 (e)>01 - V'1O(s· X C~)l< r~3>12 +s t- < r~3>10] ,
2IIDIIN "[1·~ 1 < r~3
1
lU
1
t
t
1

(33)

i=l

N

< r t~3 >02 +[s t· X (C~X 1~)J2< r:-t 3 >13 +(s-1 X 1.)2
< r:-t 3 >11} ,e "{-C~
~
t
..11
t
t
i=l

(34)
where 1 and s are the electron orbit and spin operators, respect.ively. C 2 is the operatorof second vec tor spherical harmonics . < r- 3> ~ikl is the effective (relati vis tic) radialintegrals, where k, and kl are the ranks of the tensor operat.or TJ\-(e) in the spin
space and in the orbit.al space, respectively, and 111 means the elect.ron orbit. Since
calculations of then1 are difficult, <r- 3>~:ikl are usually t.reated as free paralneters,

(35)
(36)
a~ik l and b~ikl are the single-electron hyper fine parameters to describe the HF const.ants A and B, respectively. Three paramet.ers of a~L a~7, a~S for t.he const.ant A
and three parameters of b~T, b;S, b~} for the constant Bare int.roduced for ea.ch open
electron shell 71/ . Thus, hyperfine constants A and Bare writt.en as 20

A = L{Q~i(J)a.~i + 0'~7(J)a.;3 + a;I~(J)a.;S},

(37)

nl

(38)
nl

where summat.ion ranges over all open electron shells in the relevant. configuration.

a~ikl (J) and j3~ikl (J) are the angular coefficients and depend on J. Calculations

19

of a~ik' and f3~ik, need the state wavefunction (intennediate-coupling wavefunction)
which is a linear cOlllbination of the pure LS coupling state,

IJ >= LO'd LSJ >LS,
LS

(39)

where ai is the coefficient of the intermedia.te-coupling wavefunction. Equations to
calculate a~ik, and f3~ik, for the pure LS coupling state 1 LS J > LS have been given
by Childs for t Wo- 39 and three-electron 10 shells.

3.4 Crosscd-second-order (CSO) effect
In the first-order perturbation theory, the isotope shift depends only on the electronic confignration. J and t.erm dependences of IS result from the crossed-secondorder effect which is also called the far-configuration-mixing effect. The mixings fromthe close-lying levels are treated in the intermediate-coupling wavefunction and llUXings frOlll much higher states are taken into considerat.ion in the crossed-second-ordereffect.
The crossed-second-order con tri bll t.ion to the isotope shift of the pure configuration sta.te with the zeroth-order wavefunct ion V'o and the energy Eo is given by Bauche
and Champeau,21 and is writ.ten as follows:E- ') ~ < 'l/-;oIQI'I/-;x >< 'l/-;xIOI'l/-;o >cso - -'D
E - E
'
x

0

(40)

x

where 'l/-;x and Ex a.re the zeroth-order wave[unctions and the energies of other configurations, respectively. The sum rangps over all states x of all other electronic configurations. Q denotes the opera.tor of the electrostatic interaction or the magnetic
interaction of all electrons.

Qis expressed by
2

Q=L~'

(41 )

~ = l"""' ( t·I··s·
Q
t
t,

( 42)

i>j Tij

for the electrostat.ic interaction',

oJ

,

20

for the spin-orbit interaction which is the nlain contribution in the magnetic interaction. (nl is the spin-or bi t radi ali n tegral. 10 denotes t.he IS operator,~
1 ~ 2
0= 2.M ~Pi'

(43)

l

for the normal mass shift, where Pi is the lTIOnlentmn of the electron and ],,1 is thenuclear mass.
1o=~ p. . p .
A! ~
J'
A

(44)

l

I>]

for the specific nlass shift.
( 45)

for the field shift, where ClI ! is a nuclear-dependent part. For the field shift, theconfiguration which contributes to t.he fielJ shift in eq. (40) is only that of the n's
shell which is exci ted from the core 11 s shell.
The csa contribution between the eletrostatic operator and the isotope shift operator leads to the term dependence of IS in a pure configuration. The term dependenceof IS is described by the parameter gk wit.h the angular coefficient Ck which is thesa me as that of the elect rost at ic exchange integral (S] at er integral )40,41 Gk . VVhereas
the CSO contribution between the nlagnetic operator and the isotope shift operator
leads to the J dependence of IS in a term of the pure conflgurat.ion. The J dependenceof IS is described by the paralTIeter Znl with the angular coefficient Cnl which is thesame as that of the spin-orbit radial integral 4o ,41 (nl. For a pure configllration state,
the isotope shift including the CSO contributions is thus given by

EIS = d + L Ckgk + L Cn lZ 1,
71

k

(46)

nl

where d denotes the first-order isotope shift and only depends on the configuration.
The internlediate-coupling wavefunction of the state is necessary to calculate thecoefficients Ck and Cnl. For aconfigurat.ion Of71/ N (LS')n's(LS), equation (46) is writtenas(47)
where the coefficients C for the pure LS coupling states have the following simple
expressions,42, 1-- )
cLS=S+
( ,
2
21

- -

S'
-

2/ + l'( 48)

c(L 5 = 5' _ ~~),2=

5' + 1

2/ + 1 'where 5' is the tot.al spin coupled by all electrons of the n/ N shell.

22

(49)

4. Experimental Results
The energy level scheme 31 and the H transitions studied are shown in Fig. 7.
The ground st.at.e of Yb I is 1 So and has the closed 4f shell. Thus, the (7/2, 3/2)J(J=2,5) and (7/2,5/2h st.at.es of the 1J135dGs 2 conflgnration are lllctastable states.
The 555.G5 nm transition is from the ground st.at.e to the 4J 14 GsGp 3 P1 state and isa very pure S2 - sp transition. Another 8 transitions are all from metastable states,
where 4 transitions are also S2 - sp transitions and the other 4 transitions relate to the
3 DJ

triplet and 1 D2 singlet states of t.he 4f 14 GsGd confignration. The details about

the relevant level energy and the transition energy are giycn in Table 1.

5

4

-E

'I

u

3

~ 2
'-

OJ
C

W

I

dischargeoFig. 7. A level scheme and st.udied transitions (wavelengt.h in nn1) in Yb I. Configuration, term and elect.ronic angular m,Olnenttlln J of levels are given.

23

tv

~

Transitionenergy (cm- 1 )17992.016620.2
16649.5
16777.6
16873.0
16691.7
17076.1
17111.4
17386.8

Wavelength(nn1)555.65601.51600.45595.87592.50598.93585.45584.24574.994f135d6s 2

4f 13 5d6s 2
4f 13 5d6s 2
4f 13 5d6s 2
4f 13 5d6s 2
4f 13 5d6s 2(7 / 2,3/ 2h(7 / 2,3/ 2h(7 / 2,3 / 2h
(7 / 2,3 / 2)s(7 / 2,5/ 2)s(7 / 2,5/ 2)s\' / ""'V/""J'2(7 / 2,3/ 2h(7 I ') 1/,) 'L

3 r:.,r/gc 2
Ltfl
J.. j
v ..... V v

15023188.5
23188.5
23188.5
23188.5
23188.525859.730524.730524.7oLower level
State
Energy (cm- 1 )4f 14 6s 2
4f 13 5d6s 2

Configuration

14

V0VU.

4f 6s6d
4f146s6d
4f 13 5d6s6p
4f135d6s6p
4f135d6s6p
4f135d6s6p

J..j

Lt f14 g cg,r/4f 14 6s6p
4f 14 6s6d

Configuration

D2

J=5
J=5

(7/2,3/2h(7/2,3/2)s

1

3 D3

.L./~

3 n~

PI
3 Dl
3

17992.039808.7
39838.0
39966.1
40061.539880.3
42935.847636.1
47911.5

Upper level
State
Energy (cm- 1 )Table 1. The wavelength and energy for the transitions studied. The configuration, state and energy of thelower and upper levels of the transitions are included.

A measured fluorescence spectruln of the 555.65 nm transition from the groundstate 150 of 41 14 6s 2 to the 3 P1 state of 4f 14 6s6p is shown in Fig. 8. The middle partof Fig. 8 is the fluorescence spectrum and lower part is the simultaneously lneasured
transmitted spectrum of FPI. The spectrUI11 of FPI is very simple and the frequency
interval between any two peaks is just eC}llai t.o the FSR of 2a7.7 ± 0.2 11Hz. Little
background is observed for the fluorescence spectrum because t.he teIllperature of the
Yb oven is very low and the residual radiation background from the oven is effectively
F

3/2(~~_______________________~______~ 5/2

J=l

7/2

I

3

4

I

I

J=O----~-------------------------------~----------~ 5/2
J=1---<~__~_________________________________________

~

J=O

1/2

I

I

171Yb

3/2

2

4

I

I 1/2

5000

176Yb

>-

-rl
(f)

17t.Yb

172Yb

4

_ 1

+J

-

4000

C
(l)

+J

C
-rl

3000 f--

-

3

2

(l)

u

c

(l)

u

170

2000 f--

-

Yb

(f)
(l)

L
0

:J
r-1

-

1000 -

U.

168Yb

°

FrHt' ! ! l l I I ! 1l I ! I 1 ! I I ! II !I

I

Laser' frequency

Fig. 8. A fluorescence spectruIl1 (nuddle petrt) for the 4f 14 Gs 2 150 - 4f 14 6s6p 3 P 1transi tion of 555.65 nm. Pea ks of even- A isotopes are la belled wi th the atomic symboland HF peaks are numbered. Assignments of IIF peaks for 171 Yb and 173Yb are shownin the upper part of the figure, where nurnbers of HF transitions correspond to thoseon HF peaks. The lower part of the figure is the siIllultaneously measured transI11itted
spectrum of FPI.

25

Table 2. The natural isot.opic abundance and the nuclear spin of Yb
stable isotopes.

Abundance (%)
Nuclear spin

0.135

3.1

14.4

21.9

1G.2

31.G

12.6

0

0

1/2

0

5/2

0

0

cutted by many radiation shields. The isot.ope abundance 43 of the natural Yb sample
is presented in Table 2 where the nuclear spin is also given. All observed peaks of the555.65 nIll fluorescence spectnllTI were assigned to the peaks of even-A isotopes orthe HF pea.ks of odd-A isotopes. Peaks of all even-A isot.opes were clearly observed
even for the sIllallest abundance isotope of 168Yb. IIF splittings and assignments oflIF peaks are shown in the upper part. of Fig. 8 [or 171 Yb and 173Yb. Since the groundstate is 1 So, no HF splitting exists for the ground state. Thus, the HF transitions arevery simple and HF peaks are less. All IIF peaks were observed except an accidental
overlap of a 171 Yb HF peak wi th a 173Yb IIF peak. The full width at half-maxilTIum(F\VHl\1) oft11e peak is about 17l\IIIz which contains the residual Doppler broadening(a.bout 10 ~vIHz), the frequency fluctllat.ion of the laser beam (about 511Hz) and thenatural width of the upper level. Thus, the natura.l widt.h of the upper 3PI level wasestinlated to be about 13 l\IIIz.
A measured spectrUI11 of the 5G2.50 nI11 transition frOlTI t.he 111etastable state(7/2,3/2h of 4f 13 5dGs 2 to the 1 D2 st.ate of 4f 14 G,56d is shown in Fig. 9 . A constantbackground was produced by the electric discharge. Peaks of all even-A isotopes wereclearly observed . The odd-A isotope 171Yb with t.he nuclear spin 1=1/2 has a simpler
2G

HF splittings than those of 173Yb with 1=5/2. The HF splittings and assignments of
the IIF peaks for 171 Yb and 173Yb are shown in the upper part of Fig. 9. The FWHM
of the peak is about 1611Hz and this corresponds to the natura.l width of the upper
level of about 1111Hz.
Peak centers for the fluorescence and FPI spectra were determined by a leastsquares fitting using a Lorentzian,F (x) =___71_11_ __(50)

rn2 + (x - 7113)2'

where m-1, m2 and m3 are the parameters to be determined. Fittings by eq. (50) wereconcentrated on obtaining enough accuracy of the paralneter 7113 of the peak center.

F~~9/27/2

17:Y=b2~11---rr-11II-r--t-I1--+-1-1II~;~3/2
1 23 4 5 6 7

J

8

11 12

9

15

=2~<1-t-'--t--111-,+-111---.1..-1-1-_I_,_I ~;~~;~
-

F

1/2

J =2--c:r=---+1-r-----.,--rI
I 5/2
3/2
171Yb

10

J=2 ~,~

5000 - 176Yb

174Yb

10

13

I

11.

16

I

I

15

16

_ _ _....J.'___ _ _ _ _ _ _

L_.

5/2
3/2

172Yb

If)

c

(lJ

.......

c(lJ~ 2500 -

170Yb

(lJ

u
If)

<lJ
L-

3

a

:J

LL

1

9

11 12

I

14

I~ I sJ

oo
I"

7 8

6

2

I I 1134

6

Laser frequency

14

.. I
8
10
(G Hz)

168Yb

12

Fig. 9. A ft uorescence spectrun1 for the 4f 13 5dGs 2 (7/2,3 /2h - 4f 14 6s6d 1D2transition of 592 ..50 nIno The upper part of the figure is IIF split.t.ings and assignlnents of IIF peaks for 171 Yb and 1nYb.

27

Since the frequency interval between any two peaks of FPI spect1'uIll is exactly known,
the relation between frequency and channel was obt.ained f1'OIll the FPI spectrum byusing a least squares fitting with a polynon1ial. Thus relative frequencies between
different peaks were obtained for the fluorescence spectrum. From the IIF peaks, theIIF splitting energies were obtained, thus the IIF coupling constants A and B of the
odd-A isotopes were determined bot h for lower and upper levels of t.he transition.
The center of gravity for odJ-A isotopes was al.~ o derived from the IIF peaks and thedetennined IIF constants A and B. The isotope shifts for the even-A isotopes wereobtained from the relative frequencies of the peaks and those of the odd-A isotopes
were deduced from the center of gra\ ity.
The detern1ined II F con~t ants A and B of 12 levels for 171 Yb and 173Yb arepresented in Table 3. Because the nuclea.r spin of 171Yb is 1/2, only the constantA exists for 171 Vb. Previous values 13,26 of 5 levels are also i nel uded in Ta.ble 3 forcompanson. The present values are in agreement wit h the previous values within
uncertainties . It should be stat.ed that uncertainties of the present values for the 3 DJ
and 1D2 states are an order of magnitude smaller than those of the previous ones.
The HF constants A and B of the ot her 7 levels are newly determined.
The measured isotope shifts of the stable isotopes for 9 transit.ions in Yb I are
shown in Table 4. Values of the 8 transitions except the 555 .G5 nnl transition are newly
determined from the present work . The nncert,Rint.y of IS con~ist.s of t h ree parts: (1)the uncertaint.y of the peak center (less than 111Hz), (2) the uncertRinty of the nonlinearity of the laser scanning or the polynOlnial fitting (about 111Hz), and (3) the
uncertainty of the free spectral range of the FPI (about 0.07%) . The uncertainty of
IS for the odd-A i~otopes al 0 includes the uncert.ainty of the IIF constants A and B.
Since measurements were repeated Illore than 10 t.imes, the uncertainty of IS for theeven-A isot.opes is 0.1-0.2% and that for the odd-A isot.opes is 0.3-0.4%.

28

N
CD

3P

17992.023188.525859.730524.739808.7
39838.039880.3
39966.1
40061.542935 .847636.1
47911.5(7/2,3/2h
5
5

1 D2

3 D3

(7/2,3/2h

3 D2

3 Dl

(7/2,3/2h(7/2,3/2h(7/2 ,5 /2hlSlaleorJ

Level(cm- 1 )3957.4(29)1124.64( 40)732.91(59)648 .35(39)-2988.5(25)2628.4( 15)3114.6(16)2026.1(16)-1587.05(92)48.4 7(14)245.46(27)1316.47(99)Presen tPrevious2030( 50)
-1587(20)-2988(25)2635(15)3957.97( 47)

A(171)- 1093.97(71)- 309.64(11)- 201.83( 10)-178.54(7)818.65( 40)-732.49(38)- 860.23(31)-559.91( 47)438.45(37)-13.32(6)-66.92(7)-363 .36(23)Present-826.5(13)-1702 .50(64)-3036.7(16)-2948.6(13)59.3(20)52.5(23)- 786.7(21)139.6(25)142.2(23)-1944.6( 12)-3813.0(18)-3049.2(36)- 109 /1.20(60)- 559(30)440(10)816(20)-725(15)

Present

Previous

110(50)95( 40)80( 40)-827.15(47)B(173)
PreviousA( 173)

Table 3. Determined hyperfine constants A and B (MHz) [rOln the present work. Previous values arealso included, where values for 3 P l state are [rOln ref. 13 and those for the other states are froln ref. 26.

Table 4. Measured isotope shifts of the stable isotopes for 9 transitions in Yb 1.
Transition(nm)

176- 174

174- 172555.65585.45598.93584.24574.99592.50595.87600.45601.51- 953.0( 13)-1057.6( 12)- 1041.2( 13)-1228.1( 13)- 1232.7(14)-1865.4( 17)-1835.7(21)-1826.3(21)- 1839.9(19)- 1001.0(13)-1112.6(12)- 1090.0(13)- 1285. 7( 12)- 1296.5(15)- 1969.9(18)- 1938.5(23)-1928.7(22)- 1941. 9 ( 19)

Isotope shifts (MHz)172- 170
170- 168173- 172171-170-1285.6(14)-1425.1(14)-1396.3(16)-1647.2(16)-1662.4(18)- 2653.6(23)-2606.8(28)- 2595.0(26)-2612.2(24)-444.9( 14)-493.7( 9)- 485.2(11)- 574.2( 8)-577.7(13)-848.2( 16)- 837.9(30)- 812. 7( 18)- 858.2( 15)-458.8(14)- 508.7(16) .

:30- 1369.9(14)-1523.6(16)- 1489.6(17)-1756.8(21)-1769.2(18)- 2843.3(25)- 2802.5(27)-2784.6(27)-2806.6(27)-499.4(19)- 591.5( 17)-596.4(27)-876.5(17)-864.9(30)-847.1(22)-888.4(21)5. Analysis and Discussion
5.1 Nuclear paralTIctcr A
1

To analyse the isotope shift, t.he modified isotope shift and I\ing plot are usuallyused. The 11loclified IS bviod is defined asAA'- A' _ A

8v~od = (8v · - 8v ''' ' ~ ''IC:;) l

t

mn

~

2
176. 174'(51 )where the N1'lS is subtracted in the equation. The relation between 8vimod and 8vjlOd
is expressed as

E·
mod
8 Vi
= Et. 8 vjlOd + (..i\l iS r--fS J

E·

2

Et. Aljs t\IS) 176 . 174

(52)

J

The plot of bviod versus 8vrod is called a King plot which shows a linear relation inabsence of the second-order HF perturba tion.
The 555. G.5 n 111 t ra nsi t.ion was used as a reference for the l\ing plot, since it is
known to be a pure 52 - 5p transition., King plots of the 585.4.5 nm, 584.24 nm
and 592.50 nm transitions are shown in Fig. 10. The 598.93 nm, 574.90 nm and595.87 nIn transitions were not included in Fig.I0 because t.he King plots of these
3 transitions have the same tendencies as the 3 transi tions shown in Fig. 10. Very
good linear relations were found for these 6 transitions. However, for the 600.45 nm
and 601..51 nnl transit.ions, the isot.ope shift

for the odd-A isotopes were found todeviate la.rgely [roD1 t.he linear relation. Deviations of bv imod fron1 t.he linear relation
for these 2 transi tions are shown in Fig.l1. Figure 11 shows that deviations for theeven-A isotopes are zero within experimental uncertainties and deviations for the oddA isotopes are about 4011Hz for the GOl. :SI nn1 t.ransition and about - 30 MHz for the600.45 nlll t.ransition, respectively. Such deviations lllay be attributed to the secondorder HF pertllrbation. 44 \Ve tried to explain the deviat ions considering second-order
HF perturbations between t.he 3 Dl and ~I D2 states. As a. result, the deviat.ions were
found to derive 11lainly frOlTI the two states but could not be explained cOlTIpletely.
This llleans that. t.he possibilit.y of the perturbations from ot.her stat.es cannot be
excluded. Furt.her ITIeaSllrernents are needed to make det ailed analysis of second-order
HF perturbat.ions.
31

o~--------------------------~--~

II--;:; -10
:r:
:L
N

o

'0

o

581..21. nm

E.-

~ -20

-30L-------~~-------~~------~

o

-10
-5
I:V
mod
(2
)
l)
555.65 nm 10 MHz-15

Fig. 10. A King plot of the isotope shifts for the 3 transitions in Yb 1. The 555.65
nm transition is taken as a reference. Lines arc results by the least squares fitting andexperimental uncertainties are within closed circles.

--..
N

I

2:

50~

'-'"

c
0

•

600.1.5 nm

A

601.51 nm

25~

ro
(lJ
LL-

ro
(lJ
c

0

....

••

.....
.....

..........

:= -25 0L--c -50 0

~

I

I

I

I

I

~

N

0

co

N

~

r-..

'I

I'

ro

r-..

(lJ

~

'>
0

+

•

E

I"

~
I

~

~

~

r-..

I"

N

~

0

~

~

I
0

r-..

I'
.

r::
~

Isotope pa ir

Fig. 11. Deviations of the experiment.al modified IS from the linear relation in
the King plot for the 600.45 nm and GOl.51 nm transitions.

32

To deduce FS and nuclear parameter A [ro m the observed IS, the semiempiricalevaluation 1 of bvs~rs = (0 ± 0 .5 )bvNl\fS is often used [or a pure s2 - sp transit ion.
The 555.G5 nm tran sition [rolll t he grou n d sta t.e can be considered as a pure S2 - sp
transition and the senuempirical evaluation of S11S was adopt.eel. The electronic factor
Ei = E( ns), was obtained by using the Gs electron dcnsi ty E( Gs) = 0.457 and the2screening ratio, = 0.73. 1 The value of f(Z) = 36.7 GITz/[n1 was ext.racted by using
2
the IS constant recalculat.ed by Blundell e l al.33 Thus, the value of Fi = 12.2 GHz/fn1
was obtained with 5% uncertainty for the 555.65 nm transition. The values of M iSMS
and Fi [or the other 8 transitions were deternlined [rom t.he intercepts and slopes of theKing plots . Results o[ lvfiSMS and Fi for the 9 t.ransitions are given in Table 5 in which
the S11S bViSt\IS and the FS bViFS are also included for the isotope pair of 176- 174. For
the 4 transitions o[ 4f13 5d6s 2 - 4f13 5d6s op, S11S's are nearly zero wit hin uncertainty
and this shows that these tran sitions are ahnost pure s2 - sp transitions. Values ofFi

are slightly larger than that of t.he 5S5.G5 nn1 t.ransit.ion and this is because theTable 5. The SMS factor M iSMS and the electronic factor Fi for the
9 transitions studied . The NMS bViNMS, the SMS bViSMS and the FS
are also given for the isotope pair of 176-174. The systeluatic
uncertainty of 5% from the semiempirical estimation is not included inbViFS

the uncertainty of Fi .
Transition

M iSMSFi(nm)(10 GHz)(GHz/fln2)

555.65bViNMS(MHz)bViSMS
(MHz)

bViFS
(MHz)

- 12.25190(10)- 972(10)598 .93- 17(10)-13.17( 7)18-11(12)- 1048(12)585.457(10)- 13.62( 7)

18

4(12)

-1080(12)584 .24- 23(10)- 15.49( 8)18- 15(14)-1231(14)574.99- 9(11)- 15.71( 8)18- 6( 14)-1245(14)592 .50736(12)- 29.78( 9)18482(25)- 2365(25)595 .87
600.45720(14)- 29.27(10)18470(25)- 2324(25)714(16)- 29 .10(11)18466(25)- 2311(25)601.51722(25)- 29.33(11)18471(25)- 2329(25)33screening effect by electrons of 4f 13 5d shells is weaker for the 6s electron than that14by the electrons of 4f14 shell. For the 4 t.ransi tions of <1f 13 5dGs 2 - 4f 6s6d, SMS's
2

are about 470 MHz for a pair of 176- 174 and values of Fi are ahout 30 GHz/fm • The
G00.45 nm and GOl.S1 nIn transitions were not. llseJ for t.he derivation of A because theisotope shifts of odd-A isotopes for these 2 transit.ions deviate from linear relations.
Averaged values of A were obtained from the other 7 t.ransitions.
Derived relative and absolute values of A are shown in Table 6. The relative value
is independent of the electronic factor E;; f(Z) and the uncertainty is less than 0.6%
which is due to the uncertainty of IS. The uncert.aint.y of the absolute value is about 5%
which is lnainly due to the uncertainty of Fi . Previous values from optical/ electronic
X- ray 45 and muonic X-ra.y46 isot.ope shifts are also included. The A values frOlTI opticala.nd Inuonic X-ray data in Table G were converted froln 8<r2> values by multiplying
0.95 which is the correction factor for the higher order contributions. The present values agree wit.h the optical values and are systema.tically smaller than the electronic

Table 6. Relative and absolute values of A froln the present work.
Previous values are included for comparison.

Present

Absolute A (fm2 )Optical a Electronic
X-raybMuonicX-rayc1.0

0.0794( 40)

0.083( 12)0.103(12)0.102( 4)174-172

1.0492(59)0.0833(42)0.087( 14)0.141(13)0.114(3)172- 170
170-168
173- 172

1.3423(69)1.4305(73)0.4678(28)0.1066(5~1)

0.110(15)0.163(19)0.138(5)0.1136( 57)
0.0371(19)0.122(18)0.039( 10)0.050(27)

171-170

0.4821(29)0.0383( 19)0.039( 10)0.077(32)

174-173

0.5815(32)0.0462(2~1)

0.091(33)0.066(5)172-1710.8601(41)0.0683(34)0.086(14)0.075(5)Isotope
palI

Relative A

176-174

aref. 1.
bref. 45 .
cref. 46.

34

and muonic values. In ref. 13, Clark et al. c0l11bined the optical IS of the 555.65 nm
transition with the electronic and muonic: X-ray IS's and derived A using S:rvfS valuesof 200- 300 11Hz for the 555.G5 nnl transition. This seClns unreasonable for a quite
pure s2 - sp transition. 31 It should be noted that the values from the electronic X-ray
IS have large uncertainties and "alues frorn the 111110nic X-ray IS are model-dependent.

Changes in 111ean square nuclcar charge radii 8<r2> and nuclcar defonnation
parameters 8<(32) have been derivcd for lG8- 176 Yb from values of A by using thetwo parameter model and are given in Table 7. Using the opt.ical values of A and theB(E2) values from Coulol11b excitat.ion experinlents, we deduccd values of 8<r2>,
8<r4> and 8<r6>. Results of 8<r2>, b<r 4> and 8<r6> are also given in Table 7

Table 7. Changes in mean square nuclear charge radii 8<r2>, nuclear
deformat.ion paralneters 8<(32), and higher order tenns 8<1,4> and 8<r 6 >for the staLle isotopes of Yb.
Two-parameternl0delIsotope

Numerical analysisb<r2>

8<(32)

8<r2>

8<r4>

b<r 6>(f1n2)(10- 3 )(f1n2)(10 fn14)(10 3 fn1 6 )176-174

0.0833(43)-2.20(3G)

0.08<18(46)0.734(40)0.554(30)

174-172
172-170
170-168

0.0875( 45)0.1124(57)0.1200(61 )-1.80(41)0.52(53)0.0888(46)0.1135(66)0.1210(72)0.731(38)0.915(54)0.514(27)0.617(36)0.984(59)0.687(41)173-172
171- 170

0.0389(20)o.0,101 ( 21 )-1.35(18)-1.23(lG)pau

1.24(57)

35

In which only values for even-A isotopes are presented because the reliable B(E2)values 47 are available only for the even-·A isotopes. From Table 7, it can be seenthat the 8<r2> "alucs from the two-paranlct.er nl0dcl are in agreenlent with thosefrom the nunlcrical analysis . This means that. the two-paramcter I1lodel is a good
approxinlation to extract 8<r2>.
C0l11bining the prcsent results for Yb st.able isotopes wit.h Ollr previous results

14

,15

for Nd, Sm, Gd, Dy and Er stable isotopes, we obtained the systelnatic trends of

8<r2> and 8<(32) for the neutron number of N=82- 10G. Figure 12 shows the sysYb
1.0

Er

~ 0.5
~
v

to

~Yb

00

0.0

Oy

~ 0.5
/0,

N

L..

V
,\::,

Gd

Sm

z

~ 0.5
v

00

2.0

z

~ 0.5
Nd

N

'-

v

;:; 1.5

00

E

1.5

1

z

~ 1.0

Nd

v

00

z

~
N~

1.0
0.5

v
00

0.5
0.0
0.0

82

86

94

90

98

102

106

Neutron number N

Fig. 12. SystelTIatics of 8<r2> for the stable isotopes of Nd, Sm, Gd, Dy, Er
and Vb. Normalizations were carrried out at the isotope with the smallest neutron
nUlnber N for each element. Da hed line is the change of 8<r2> for a spherical nucleus
calculated by the droplet lTIodel.

36tematic change of 8<1'2> for the neutron number of 82- 106, where values of 8<1'2>
are nonnalized to 0 at the isotope with the smallest neut.ron number for each element. The dashed line is the value of 8<1'2> for a spherical nucleus calculated byusing the droplet mode1. 34 Fron1 Fig. 12, it can be seen that the values of 8<1'2>change rapidly at N =88- GO and the change of 8<1'2> becon1es aln10st parallel to thedashed line above .N=9'1. This drastical change of 8<1'2> at N=88-GO is clearly seenin Fig. 13 which shows the systematic change of 8<1' 2>N-2,N. It is also seen from Fig.

13 that the values of 8<r 2>N-2,N reduce very slnoothly above N=94. The systematic

0.5

N

E

z.

0

Nd

I:::.
0

Sm
Gd

¢

Oy

0

Er

•

Yb

2x6<r 2 >N-l,NN

I

for odd-N

Z
/\
N

L...

v

0.2

c.()

0.1

°B~2--~--~--~--~--~--~94--~--~98--~---1~0-2--~--1~0-6~
Neutron number

N

Fig. 13. Dependence of b<r 2>N-2,N on the neutron nnmber J.l for the stable
isotopes of Nd, Sm, Gd, Dy, Er and Yb . Values of 8<1'2> are connected with lines
only to guide the eye .

37change of 8<{32>N-2 ,N is shown in Fig. 14 for N=82- 10G. Figure 14 shows that theval ues of 8 <(32 > are very large around N =88- GO and becOlne small above N =94. The
va.lues of 8<{32> are nearly zero around N=100 - 102 and are negative at N=104 and106. That is, the values of 5<{32> reduce slowly from positive to negative. This means
tha.t the nucl e ar shape changes frOIn spherical (vibrat.ional) to defonned one aroundN =88- GO and t he nuclei wi t.h }\T= 100- 102 have very sta ble CIl1 adrupole deformation.
It is also shown that deformation reduces a little at N=10 ,1 and 106.

4
0

Nd

A

Sm

,......,.

0

~
0

0

Gd
Oy

0

Er

3

~

"""""

2

z.

• Vb

z

2 X 0([32 >N-1.N

C'j'J

A

N

~

v

for 0 dd-N

1

toO~---------------+·~-r~~-T~~----~-1~-----~~-----~~-----~-~~~---~~---~~---~

82

86

94

90

98
Neutron number

102

106

Fig. 14. Systenlatic trends of 8<{32>N-2,N for Nd, SIn, Gd, D y, Er and Yb stable
isotopes. Valu es of 8<{32> are connected wit. h li nes on ly to guide the eye.

38

It has been kno\\'n from the early days of IS llleasurements that the value of<1'2> for an odd-N nucleus is slllaller than the average value of <1'2> for its even-N
neighbors for the general case. This effect is called the odd-even staggering effect6

and is described by the staggering paramleter I as follows:
28 < 1'2 >N-1,N(53)

1= 8 < 1'2 >N-1,N+1'

where N is odd. The regular odd-even staggering effect is expressed by I < 1 (recently
the irregular staggering effect of II

> 1 \vas also observed 4). The staggering effect of< 1 is qualitatively explained by the pailring effect but no quantitative theory exists.
FrOlTI the present and previous values of 8<1'2>, the odd-even staggering pa.ram-

eters I were derived for the st.able odd-lv isotopes of Nd, SIll, Gd, Dy, Er and Yb.
Results are given in Table 8. All values of I are SlTIaller t.han one and this shows
the regular staggering effect. Because t.he paralneter I is derived from the ratio of

8<1'2>, the uncert.ainty of I is independent of the electronic fa.ctor Fi and derives
from the uncert.ainty of t.he relat.ive A. The dependence of I on the neutron nUlllber

Table 8. Odd-even staggering parameters I for the stable odd-N isotopesof N d, Sm, Gd, Dy, Er and Yb.
NeutronIsotopenumber83

14,3Nd

0.847(31)

85

145Nd

0.763(32)

87

149Sm

0.6053( 43)

91

155Gd

0.9780( 48)

93

157Gd

95

161 Dy0.3824(33)0.5333(77)

97

163Dy

0.6400(85)

99

167Er

0.6895(70)

101

171Yb

O. 7135( 58)

103

173Yb

0.8891(76)3DN is shown in Fig. 1.5. It is seen frOln Fig. 1.5 that the value of , reduces with theneutron number N for N =83- 87 and increases for N =03- 103. A large value of,I"V 1

is found at 1\T=91. This lneans that t.he staggering effect becOlnes st.ronger, with N,near the spherical region (N =82) and becomes weaker for the well Jeformed nucleus(above N=9 /1). The large value of I at N=01 shows t h at the nuclear shape change
at about N =00 is elonunant and the staggering effect is small.

1.0
(1-

<lJ
-+oJ

Q;

E
rU
rU

0

Nd

n. 0.5

~

Sm

CJ)

0

¢

Gd
Oy

0

Er

1-

c

1-

<lJ
CJ)
CJ)

• Yb

rU

~

if)

0

80

90

100
Neutron number N

110

Fig. 15. Dependence of the oelJ-even st.aggering parameter, on the neutron
number N for the stable odJ-N isotopes of Nel, SIn, Gd, Dy, Er and Yb.

40

5.3 Single-electron HF parameters for the 4f 14 6s6d configuration
To obtain the single-electron HF paran1eters of 6s and 6d electrons , we carriedout the paralnetric analysis of t he effective operator procedure for t.he HF constants

A and B of the 3 D J and 1 D2 st.ates of the 4f 14 Gs6d configuration . Paralneters of the
4f electron need not to be considered for the Yb I 4f116sGd con figuration, because
electrons of t.he closed 4f shell couple to t.he 150 tern1.:n The parameter a~~ for the

HF constant A was omit ted because ai~ is known to be very small (rvag~/50) .16 Thus,
both three single-electron UF parameters of ag~, a~~ and a~~ for the HF constant

A and bg~, b~~ and b~~ for the UF const.ant. B were taken into considera.tion for the
4f14 Gs 6d configu ration.
(54)
(55)
where the coefficients a~ikl and (3~ikl were calculat.ed by using the simple intermediatecoupling wavefunctions 26 described below. States of 3 D1 and 3 D3 are considered as
pure L5 coupling st.at.es but l11ixing exists between 3 D2 and 1 D 2 •
(56)
(57)
where 0'1 = 0.926 26 and 0'2 = (1 -ai)1/2.

The calculated coefficients a~~, a~:/I and(3~:/1 are given in Table 9 for the states ofJDJ and ID 2 ·k ,kl· t s a 10.s' a k,kl an d f3 6d
Ta ble 9. Cal cu1ate d angu 1ar coe ffi Clen
6d
6for the 3 D J and 1 D2 states of the 4f 14 6s6d configuration.

State

aID

OI
a 6d

l2
a 6d

(302

(313

63

(311

3 D1

- 0.2500

1.50000.50000.20000.0490- 0.0200

3 D2

0.21430.85710.28570.32560.1000- 0.0000

3D3I D2

0.1667- 0.1310

0.6667- 0.0952

0.5714- 0.0350

0.9762- 0.1191

0.5306- 0.0300

0.13330.0667

6d

41

6d

6d

The HF constants A of 171 Yb and constants A and B of 1 73 Yb for the 3D J and
1D2 states of 4f 14 6s6d were fit.ted by eqs. (5 /1) and (55). The obtained single-electron
HF parameters are given in Table 10. The value of a.~~ for the 173Yb I 4f 14 6s6p
configuration obtained by Clark ct al. 13 is also included in Table 10, and the absolutevalue is somewhat. slnaller than the present result for the 4f 14 Gs6d configuration. The
reason is that the screeni ng effect by the IGp ekc t.ron in the 1f14 GsGp configuration isstronger for the 6s electron t.han that by the 6d electron in t.he 4f 14 Gs6d configuration.

It can also be seen from Table 10 that the contribution of the 6s electron is dominant
for the HF constant A.
Deviations of fit.t.ed values by eqs. (5<1) and (55) from experimental values of A
and B are shown in Fig. 16. Only deviat.ions for 173 Yb are shown in the figure sincedeviations for 171 Yb have the same t.endency as those for 173Yb. Figure 16 showsthat deviations of A and B are nearly wit.hin uncert.ainties. This lneans the fit isreasonable. Although the second-order lIF pert.urbat.ion was found for t.he 3D1 and3

D2 states of 4f 14 6s6d from the deviation of the isotope shifts from the King plot,

influence of such perturbat.ion on IIF constants A and B could not be observed at
the present uncertainty.

Table 10. Single-electron hyperfine paralneters (MHz) of the 4f 14 6s6d
configuration in Yb 1. Reference value of a~~ is also given for the 173Yb I
4fl46s6p configuration.
Isotope

aOl

6d

12
a 6d

lO
a'6.

l71Yb
l73Yb

10.88(76)-4.81(26)60.6(34)-28.49(88)12143(5)-33130(1)

-2815

a

aref. 13.

42

02
b6d

l3
b6d

ll
b6d

311.6(76)-432(35)-446( 49)

2 l-

• A

N

:r:

A

L

-

I-

0

0

ro -1
>
(];

0

- 10

+'~

U

0
+oJ

-

N

:r:

B

,f

<J:

c

- 20

I

0

m

0

c
0

4

t

L

+oJ

- -10 .~
>

<lJ

-2 -

- -20
I

I

3

I

3

02

0

I

03

State

Fig. 16. Deviations of fitted values IOf HF constants A and B [or 173Yb fromexperinlental values for the 3 DJ anJ 1D2 states of the ·jf 14 6s Gd configuration .

5.4 J and term dependences of IS and CSO effects for the 3D J and1 D2 states of 4f116s6d

For the 3 DJ and 1D2 states of 4f14GsGd, we introduce the residual isotope shift
T res of a state which is a difference in the level IS relative t.o that of the 3 Dl state .
The N1tIS is subt.ract.ed in T res . Values of T res were obt ained for the 3 D J and 1D2
states from the 4 transitions of <lf135dGs 2 -4f 14 GsGd which share the saIne lower state(7/2,3/2h . Results are shown in Table 11 where only values for even-A isotopes are
given, because t.he isotope shifts for odd-A. isotopes deviate from linear relations asshown in §5.l. J and term dependences can be found [or the 3 DJ and 1D2 states fromvalues of T res in Table 1l.
As discussed in §5.3, there is a mixing between 3 D2 and 1 D2 states . Thus, themeasured values of T res are written as

(58)
(59)
where the coefficients c and CGd were calculated by using t.he intermediate-coupling
wavefunction 26 and are presented in Table 12 for the 3 D J and 1 D2 sta tes of 4f 14 6s6d.

43

~

..;:.

Average

176- 168176- 174
176- 172176- 170

Isotope
pau

0
0
0
0
65.9(98)

26.8( 46)43.9(72)

1"r-{29\l') . u~)16.4(99)

12.5(72)

4.0(28)7.3(48)-132.8(98)- 54.0( 44)- 95.8(70)- L.

()5 .O\L.
o/()6\)

Residual isotope shift (MHz)
3D 2
3Dl
1 D2
3D3

- 431(39)- 313(28)- 86( 11)
- 177(18)g2(Gs,6d)(MHz)

6.6( 40)2.9(19)5.0(29)16(11\.1. \.1.1)

(MHz)

Z6d

- 112( 6)18( 6)

17(10)- 113(12)- 109(12)- 116(12)20(1 4)18(12)
19(11)(MHz/fm 2 )Z6d / )..

1f18(1t:\
\.lU}
- .1V

g2(6s,6d)/)"(10 MHz/fm 2 )Table 11. Residual isotope shifts and CSO paranleters for the :I DJ and lD2 states of the
4f146s6d configuration.

Table 12. Calculated angular coefficients c and C6d for the 3D J
and 1 D2 states of the 4f146s6d configuration.
State

c

- 0.1000- 1.5000- 0.0428- 1.2858- 0.1000
1.00000.24280.7858The CSO parameters g2(GS, Gd) and Z6d were obtained from T res of 3D3 and 1 D2
and results are also shown in Table 11. T res of t.he 3D2 stat.e could not be reproduced byeq. (59). This suggests that the configuration mixing exi ts for this level. "N1ixing withthe (7/2,3/2h st.at.e of the 4f135d6sGp configurat.ion is possible because the energy ofthe state is 30880 CIll- 1 which is close to t he energy of 30838 cm- 1 of t.he 4f146s6d
3

D2 state. To COlnpare the parameters g2(6s, Gd) and ::6d for different isotope pairs,

these values are di vicIed by nuclear para lnet.ers A and results are gi ven in Table 11.
Values of g2(6s, 6d)/ A and Z6d/ A are constant within uncertainties as seen in Table 11.
This means t.hat the FS has dOlninant cont.ribution in t.he CSO effects both for theelectrostatic and Illagnetic i nteract.ions. The averaged values of - 1120( 60) 11Hz/ fm 2for g2(6s, 6d)/A and 18(6) ~ITIz/fn12 for Z6d/A were ohtained.
For cOlTlparing the present value of Z6d wi th the Z5d values of other elelnents, weused the normalizeJ paranleter =711,norm = Znl/A(nl. The present value of Z6d,norm for the
Yb I 4f 14 6s6d configuration was obtained and is given in Table 13 which also includes
7

val ues of ZSd,norm for the Gel I 4f 5dGs 2 con figu ration (our previous val ue 16 ), ZSd,norm
for the Gd II 4f 7 5d6s configuration and ZSd,norm for t.he Eu I 4f 7 5d6s configuration.
From Table 13, it is seen t.hat. the paralneters of Z6d,norm and ZSd,norm are constantwithin experinlent.al uncert.aint ies. It seeIllS that values of Zn l ,norm are constant not
only for the 5c1 electrons 16 but also for the Gd electrons for t.he rare-eart.h elenlents.
For the paranlet.er ;:4j of some rare-earth eleillent.s, values of Z4j,norm were shown toincrease wit.h t.he atOlnic numher as the spin-orhit. integraJ25 (41' However, values of theparanlet.ers Z5d,norm for different. elements seem to scatt.er for the ground configuration(5d + 6s)N in the series of 5ci clelnent s. 53

4.5

Table 13. Comparison of the pr esent value of the CSO parameter Z6d,norm with theprevious va.lues of Z5d,norm'
Configurati on

Electron shellnl
Yb I 4f 14 6sGd
Gd I 4f 7 5d6s 2
Gd II 4f5d6s
Eu I 4f 7 5d6s

6d
5d
5d
5d

A
(fI112)

Znl(11Hz)

42(11)a
43 .6(59)b

O.265(10)CO.552(25)daref. 48, bref. 22, cref. 14,
dref. 49, eref. IG, !ref. 50,
9ref. 51, href. 52.

4G

Znl/ A(1{II z/ fl112 )(nl
1(C111- )Znl ,norm(10- 6 fm- 2 )18( 6)
134( 4)e63!

9 .5(32)710(64)96.29(60)

158(42)79( 11)832(28)h389 a

6.3( 17)6.77(97)

6. Conclusion
To get informat.ion about nuclear and elect.ronic properties, we measured isotope
shifts and hyperfine structures in Yb I by means of atonuc-beanl laser spectroscopy.
The experilnental setup of atonlic-beam laser spectroscopy was shown to have enoughenergy resolution and counting efficiency to record IS and IIFS spectra. Isotope shifts
and hyperfine strnctures of a transitions were 1l1easured for the stable isotopes with
A=168- 176. One transition is directly from t.he ground state and the other 8 transitions are from metastable states which were measured for the first time. The electric
discharge was developed to popula te the rnetast.a ble sta tes in Yb I. Det.ailed conditions
to maintain the stable electric discharge were exalnined. Dependences of populations
for the ground and 1l1etastable states on the discharge current and voltage were investigated . Intense population of the nletast.able stat.es were achieved up to 70% by
the electric discharge.
From HFS, HF constants A and B of 12 levels were detennined for 171 Yb and 173Yb
of which 7 levels were newly deternlineJ. For the 3 D J and 1 D2 states of the 4f 14 6s6d
configuration, IIF const ants A for 171 Yb and A and B for 173Yb were analyzed withthe effective operat.or procedure and single-electron HF paraDleters a~~, a~~, a~~ andb~~, b~~, b~~ of 6s and 6d electrons were derived . The 6s electron was found to havedominant contribution to the constant A. The parameters of Gd electron were deduced
for the first t.ilne .
From IS, reliable nuclear parameters )\ were obtained and found to be in agreementwith the values fronl the previous optical IS and systell1at.ically slnaller than the values
from electronic and nluonic X-ray IS's . Changes in mean square nuclear charge radii

8<r2> were deduced using both the two-paralneter model and the nllJnerical analysis.
The values obtained with these two methods agreed quite well. Higher order terms

8<r4> and 8<r 6 > were also determined with the nUDlerical analysis for the evenA isotopes. It was found that Yb jsotopes wit.h A=170 and 172 have very stable
quadrupole deformation and deformation reduces a little for the isotopes with A=174
and 176. The odJ-even staggering effect was discussed for the odd-N isotopes.
For t.he 3 DJ and 1 D2 st.ates of the 4f l4l 6s6d conflgurat.ion, J and tenn dependences47

of IS were exalnined and they were attributed to the csa effect. csa paranleters ofg2(6s,6d)/,X= - 1120(GO) ~vlIIz/fn12 and Z6d/'x = lS(G) 11TIz/fn12 were derived for the
first time. The value of Z6d/'x was compared wi th the values of 5d electrons of otherelelnents a.nd was found to be proportional to the spin-orbit int.egral (nl just as thatof 5d electrons. The field shift was found to be dominant in the csa effect both forthe electrostatic and magnetic interactions .
For the 3 D1 and 3 D2 states of the 4f14GsGd configurat.ion, isotope shifts related tothese two states were found to largely deviate from the King plot. 'vVe concl uded that
this deviation is caused by the second-order IIF perturbat.ion . However, influence of
this perturbation on IIF constants A and B was not ohservcd within present uncertaint.y. Since othcr levels should be involved into the second-order HF perturbation,
detailed analysis of this deviation for the 3 Dl and 3 D2 states of 4fl46s6d was notdone at the present work.

48

Acknowrledgment

I would like to express my sincere appreciations to Professor Yasukazu Yoshizawa
for his valuable advice and strong encouragenlent, and to Professor Ichta Endo for
his useful guidance and critical reading of this thesis. I a.m particularly grateful to
Dr. Takayoshi Horiguchi for his cont.inuous support and detailed suggestions. I am
also indebted to many students of our group, Dr. 11asanori \Vakasugi, 11r. Takashi
Hasegawa and ~ls . \Vei Yang. I would like to t hank all lTIenlbers of DepartInent of
Physics, Faculty of Science for their hospit.alit.ies during my study a.nd life in Hiroshima. I greatly appreciate the fellowship frOln the ~Ifinistry of Education, Science
and Culture of Japan during nly stay at Hiroshinla U niversit.y.

4G

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