High resolution spectroscopy of the hydrogen atom - III. Wavelength comparison and Rydberg constant determination

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Submitted on 1 Jan 1990

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Wavelength comparison and Rydberg constant

determination

J.C. Garreau, M. Allegrini, L. Julien, F. Biraben

To cite this version:

High

resolution

_spectroscopy

of the

_hydrogen

atom

III.

_{Wavelength comparison}

and

_Rydberg

constant

determination

J.C. Garreau

_(*),

M.

_{Allegrini(**),}

L. Julien and F. Biraben

Laboratoire de

_{Spectroscopie}

Hertzienne de

_{l’ENS(***),}

Université Pierre et Marie _Curie, 4

_place

Jussieu, Tour _12, E0175252 Paris Cedex _05, France.

(Received

2

_February

1990, revised 15 June _{1990, accepted} 27 June

₁₉₉₀₎

Résumé. - Cet article est le dernier d’une série de trois donnant une

_description

détaillée de notre

mesure récente de la constante de

_Rydberg.

Nous _{y décrivons}

_{soigneusement}

la méthode de mesure

des

_longueurs

d’onde. Les

_longueurs

d’onde des trois transitions A deux

_photons

_2S-8D, 2S-10D et

2S-12D dans

_{l’hydrogène}

et le deutérium sont

_{comparées à}

celle d’un laser He-Ne stabilisé sur l’iode,

laser

_qui

est actuellement la meilleure référence dans le domaine

_optique.

Cette

_comparaison

est

faite à l’aide d’un étalon

_Fabry-Perot

de

_grande

stabilité. Les

_{déphasages à}

la réflexion sont

_pris

en

compte en

_changeant

la

_longueur

de cette cavité

_("méthode

des miroirs

_virtuels").

Pour mesurer le

déphasage

de Fresnel nous étudions la différence de

_fréquence

entre le

_premier

mode transverse et

le mode fondamental de cette cavité. A

_partir

des six

_longueurs

d’onde mesurées, nous déduisons une

nouvelle valeur de la constante de

_Rydberg

R~ = _109737,

315709(18)

cm-1,

la

_plus

précise à

l’heure

actuelle. A

_partir

des

_{déplacements isotopiques}

entre

_{l’hydrogène}

et le deutérium nous déduisons

aussi une valeur _pour le _rapport de masse entre le _proton et l’électron:

_mp/me

= 1836.15259

(24).

Abstract. 2014 This

paper is the last of a series

_giving

a detailed

_description

of our recent determi-nation of the

_Rydberg

constant. Here we

_carefully

describe the

_wavelength

_comparison

_procedure.

The

_wavelengths

of the three

_two-photon

transitions _{2S-8D, 2S-10D,} and 2S-12D in

_hydrogen

and deuterium are

_compared

to that of an

_{I2-stabilized}

He-Ne laser which is the _present best reference

in the

_optical

domain. The

_key

of this

_comparison

is a

_{high stability Fabry-Perot}

_cavity

etalon. The reflective

_phase

shifts are taken into account

_by

_changing

the

_length

of this

_cavity

_("virtual

mirrors

method").

In order to measure the Fresnel

_phase

shift we

_study

the

_frequency

difference between the first transverse mode and the fundamental mode of the

_cavity.

From the six measured

_wavelengths

we

deduce a new value for the

_Rydberg

constant, R~ =

109737.315709(18)

cm-1,

_currently

the _most

precise

one. The H-D

_isotopic

shifts have been also measured and a value for the

_{proton-to-electron}

mass ratio is deduced:

_mp/me

=

1836.15259(24).

Classification

Physics Abstracts

35.10 - 06.20J - 32.80K

1. Introduction.

This _paper is the last of a series of three

_(referred

as _papers

_{I, II}

and

_{III), giving}

a

_complete

description

of our recent determination of the

_Rydberg

constant

_{[1] .}

Our method is to measure

the

_wavelengths

of the 2S-nD

_Doppler-free

two

_photon

transitions in an atomic beam of

_hydrogen

or deuterium n =

8, 10, 12.

In

paper 1

[2],

we have described the

_principle

of our

_experiment

and the

_{experimental procedure}

used to observe lines as narrow as 600 kHz. In _paper II

_[3],

we have

carefully analyzed

the

_lineshape

of the

_two-photon

transitions. As a result we know the atomic

frequency positions

with

_respect

to a _{very stable reference}

_{cavity (FPR).}

We descnbe now the measurement of the

_Rydberg

constant. In section 2 we

_present

the

wave-length

measurements, and in section 3 we descnbe the détermination of the

_Rydberg

constant.

2.

_Wavelength

measurement.

2.1 STANDARD LASER - The transition

_wavelength

is measured

_by

interferometric

_techniques

based on the

_comparison

of our

_dye

laser

_wavelength

with

thè

one of an

_{l2 2013}

stabilized He-Ne

laser at 633 nm. This He-Ne laser is the best reference in the

_optical

domain since its

_frequency

has been measured with

_respect

to the cesium clock

_[4].

In fact we have two

_12-

stabilized He-Ne lasers which are

_{continuously compared}

and

_running

with an

₁₂

_cold-finger

_temperature

of 16° C

and a modulation

_peak-to

_{peak amplitude}

of 5 MHz. In order to check these reference

_lasers,

we have made a

_{frequency comparison}

with a standard He-Ne laser of the Institut National de

Métrologie (INM).

This

_comparison

was made _every

_day

at the

_beginning

and at the end of the

experiment

The

_frequency

différence value between the INM laser

_(stabilized

on the d

_hyperfine

component

of the

127 I2

11 -

_{5R(127) transition)}

and ours one

_(stabilized

on the f

_component)

was 26.195

₍₅₎

MHz

_(mean

of the data obtained

_during

about a

_{month). 1àking}

into account the

value for this différence

_(26.224

MHz

_[5]

_),

we deduce that our laser was +29 kHz shifted with

respect

to the INM one. The INM laser has been

_previously

_compared

with the one of the Bureau International des Poids et Mesures

_(laser

BIPM

_2).

With

_respect

to this laser the INM laser was

-23 kHz shifted

_{[6] .}

_Finally

our

_{12-stabilized}

laser is +6 kHz shifted with

_respect

to the BIPM

one. We have assumed that the

_frequency

of the BIPM laser

_(stabilized

on the i

_component)

is as

measured

_by

D.A.

_Jennings

et aL

_[4]

with a 1.6 x

10-10

relative

_uncertainty:

1àking

into account the value for the

_i-f

_{splitting (138.892}

MHz

_[5]

_),

we deduce the

_frequency

value for

_{the f}

_component

of our

_{Iz-stabilized}

He-Ne laser:

We have used this value for the

_wavelength

measurements and for the

_Rydberg

constant determi-nation.

2.2 INTERFERENCE APPARATUS The

_{key element}

for the interferometric

_comparison

is a

non-degenerate

Fabry-Perot

etalon

_(FPE)

_kept

under a vacuum lower than

10 - 6

mbar. In these condi-tions the error due to the index variation between the

_{compared wavelengths}

is reduced to about

10-15.

This

_{optical cavity}

is built with two silver coated

_mirrors,

one fiat and the other

_spherical

(R

= ₆₀

cm),

in

_optical

contact to a zerodur rod. Its finesse is - 60 _at 633 _nm and - 100 _at 778

Fig.

1. -

Experimental

set-up for the

_{wavelength comparison.}

Figure

1 shows the

_{expérimental}

_set-up

for the

_wavelength

_comparison.

As

_explained

in _paper 1

[2] ,

the

_dye

laser

_frequency

is known with

_respect

to a reference

_{Pabry-Perot cavity}

_(FPR)

which is locked to an

_{I2-stabilized}

He-Ne laser. The

_dye

laser

_frequency

is

_precisely

scanned

_by

means of an

acousto-optic

modulator in

_conjunction

with a

_{computer-controlled frequency}

_synthesizer.

With

this

_apparatus

we have

_precisely

measured the

_frequency

_différence

_f2p

between the

_two-photon

transition

_frequency

and the

_Airy

_peaks

of the reference

_cavity

_{[3] .}

_Figure

2 shows the relative

position

of the atomic

_signal

with

_respect

to the

_{Airy peaks}

of the reference

_cavity

FPR and of the

_Fabry-Perot

etalon FPE. For the

_{wavelength comparison}

we have to measure the

_frequency

difference

AR

between the

_{iodine f}

_component

and the closest

_peak

of the FPE. For that _purpose

an

_auxiliary

He-Ne laser is mode-matched and locked to the FPE

_cavity.

The beat

_frequency

be-tween the

_auxiliary

and the standard He-Ne laser is measured

_by

a

_frequency

counter. We have

also to measure the

_frequency

difference

_AIR

between the

_{two-photon signal}

and the closest

_peak

NIR

of the FPE

_cavity.

We have

_AIR

=

f2p

+

fFP, where

ffp

is the

_frequency

difference between

a

_{given Airy peak}

of the reference

_cavity

FPR and one of the etalon

_cavity

FPE

_(Fig.

_2).

In order

to measure the

_{frequency ,}

_fFp,

the

_dye

laser is scanned to the résonance with the etalon

_cavity

FPE,

and mode-matched to this

_cavity.

Its

_frequency

is locked to the etalon

_{cavity through}

the

computer.

The

_dye

laser

_frequency

is modulated

_(about

1 MHz

_peak-to-peak

at 1.2

_kHz)

and the modulation of the transmitted beam is detected

_by

a lock-in

_amplifier.

After conversion

_by

an

_{analog-digital}

_converter, the

_signal

is read

_by

the

_computer,

which drives the

_{acousto-optic}

modulator in order to cancel the

_output

of the lock-in

_amplifier.

_During

the measurement

_(about

2

_minutes),

the

_computer

records

_{simultaneously}

_{f FP}

_(twice

the

_frequency

of the

_{acousto-optic}

fluctuation of the

_length

of this

_cavity

and

_fFp

and

AR

fluctuate. These fluctuations are correlated

and,

in order to eliminate

_them,

the

_computer

calculâtes the

_{quantity :}

Fig.

Z - Relative

_position

of the

_{two-photon signal}

with _respect to the

_{Airy peaks}

of the reference

_cavity

and of the etalon

_cavity.

This

_quantity

is

_independent

of the

_length

fluctuations of the etalon

_cavity

FPE and

_corresponds

to the

_ffp

value obtained

_{when AR -}

_{OR (Aor is}

a fixed

_value,

close to the

_experimental

one and

arbitrarily fixed).

1b obtain a

_precise

_wavelength

_measurement, it is _necessary that the

_frequencies

of the

_dye

laser

and of the

_auxiliary

He-Ne laser coincide with the resonance

_frequencies

of the etalon

_cavity

FPE. lb take into account the

_intensity

variation due to the

_gain

curve of the

_auxiliary

He-Ne

laser,

we

monitor the difference between the transmitted beam

_through

the etalon

_cavity

and a reference beam

_having

the same

_intensity.

We have tested the

_efficiency

of this

_{procedure by locking}

the

auxiliary

He-Ne laser to two

_adjacent

_peaks

NR

and

NR - 1

of the etalon FPE.

_By

this means we

obtain a measurement of the free

_spectral

_range of this

_cavity

_(frequency

_measurement)

which can

be

_compared

with the most

_precise

value obtained

_by

_{interferometry (wavelength}

_{measuremènt).}

As

_explained

in the

_following,

we have used two etalon

_{cavity lengths,}

10 cm and 50 cm. The

comparison

between the two free

_spectral

_range measurements is

_given

in table I. A

_systematic

discrepancy

appears in the short etalon measurements. In this case we reach the limit of the

differential method used to eliminate the

_gain

curve effect because the two

_{Airy peaks}

of the

etalon

_cavity

are in the

_wings

of this curve. In our

_wavelength

measurements we have used the

mean of the two results obtained when the

_auxiliary

He-Ne laser is locked to the

NR

and

NR -

1

peaks.

We estimate the error in the control

_loop

of the He-Ne laser

_frequency

on the short etalon

cavity

to be ± 9 kHz

_(half

the difference between the two free

_spectral

_range

_{measurements).}

1àking

into account the virtual mirrors

_procedure,

which will be

_explained

in the

_following,

this

error introduces an

_{uncertainty of 4.7 x}

10-12

in the final result. Measurement with the

_{long cavity}

etalon

_always

involves an

_{Airy peak}

at the

_top

of the He-Ne

_gain

curve and our differential method

corresponding

to an

_uncertainty

of 1.3 x

10-12

in the final result. The same kind of error can

exist for the

_dye

laser

_{frequency servo-loop.}

Nevertheless these errors can be

_neglected

because

i)

the finesse of the etalon

_cavity

is

_larger

in the infrared domain than in the red one;

_ü)

there is

no

_dependence

of the

_dye

laser

_intensity

versus the

_dye

laser

_frequency

_(we

have verified that the

signal

at the modulation

_frequency

of the

_dye

laser

_frequency

is

_negligible).

Table I. Free

_spectral

_range measurment

_of

the etalon

_cavity

FPE.

1

2.3 VIRTLJAL MIRRORS METHOD. - The

_frequencies

of the red and infrared radiations inside the etalon

_cavity

FPE are determined

_by

the resonant condition:

where L is the

_{cavity length,}

N is an

_integer

_numbers

is the reflective

_phase

shift for the

_light

of

_{frequency v}

and 03A6 the Fresnel

_phase

shift. The Fresnel

_phase

shift

_{depends only}

on the

_cavity

geometry.

Its measurement will be described in the

_following.

The reflective

_phase

_{shift 1/;}

due

to the mirror

_coating

is

_{wavelength dependent}

and is eliminated

_by

the method of virtual mirrors

by using alternately

two rods of différent

_{lengths (10}

cm and 50

_cm)

for the etalon

_cavity.

This

method is

_explained

in detail in reference

_{[7] .}

The basic idea is that the reflective

_phase

_shift

is

_independent

of the

_{cavity length.}

We can write the resonant condition

₍₁₎

for the

_long

and the short

_{cavity (labelled 1}

and s

_{respectively)}

and for the red and infrared radiations

_(R

and IR

_labels)

which are

_compared.

For

_example,

we have for the

_long

etalon and for the infrared

_frequency:

where

_bIR

is the fractional order relative to the

_{higher frequency Airy peak.}

It is the ratio between

the

_{frequency splitting}

_AI

_IR

_{(see Fig. 2)}

and the Cree

_spectral

range c .

From the

_{equations (2)}

2L

relative to the short and

_long

etalon and to the two

_radiations,

we can eliminate the two

_cavity

lengths

and the reflective

_phase

shifts

_03C8IR

and

03C8R.

We obtain the ratio between the two

frequen-cies :

conséquent

changes

of

_1PR

and

_1PIR.

The difference between the two reflective

_phase

shifts

_OR

and

_ÇIR

can be considered as a variation 6L of the

_cavity

_optical

_length

for the red and infrared radiations. For the infrared

_radiation,

the mirrors are more reflective than for the red one and the

cavity

optical length

is shorter. From our

_wavelength

measurements with the 10 cm étalon

_cavity

FPE,

we can deduce 6L at the

_beginning

and at the end of the

_experiment.

Ibe results are

_given

in table II. As the

_hydrogen

and deuterium

_wavelengths

are _very close we have taken the mean of the

_hydrogen

and deuterium results relative to the same 2S-nD

_two-photon

transition. We have observed a

_slight

variation of 6L

_(about

_0.03.nm)

_during

the

_experiment

and we take this effect into account in the

_uncertainty

of our final result.

Table II. - Variation

_{of the}

_cavity

_{optical length}

between the red and

_infrared

radiation at the

begin-ning

and at the end

_{of the}

_experiment.

2.4 FRESNEL PHASE SHIFT MEASUREMENT. - For a

_{plané-spherical}

_Fabry-Perot,

the Fresnel

phase

shift

_depends

on the

_length

of the

_cavity

and on the curvature R of the

_spherical

mirror

_{[8] :}

We have determined the Fresnel

_phase

shift

_{by measuring}

the

_frequency

interval between the

fundamental

_TEMoo

mode and the first transverse mode

TEMo1

_(or

TEM,O)

of the

_cavity.

In-deed,

the resonance condition for the

_TEMn,m

mode is

_{[8] :}

We can deduce

In

_principle

_TEMO,

and

TEMIO

are

_{degenerate by}

_{symmetry. However,}

due to

_{imperfections}

in the _{mirror curvature,}

_they

are not

_completely

_degenerate.

_Depending

on the

_alignment

of the

modes which

_give

different results for the relative

_frequency

of the first transverse mode and the

TEMoo

mode. We have made measurements for various

_alignments

and the result is shown in

figure

3,

which shows the

_dependence

of this

_frequency

différence _upon the orientation of the two transmitted

_light

_spots

of the

TEMol

mode. The Fresnel

_phase-shift

that we need is then

_{given by}

the mean value of the curve of

_figure

3

_[9].

We have made this measurement for each

_step

of the

experiment (10

cm, 50 cm and 10 cm

_etalon)

with the He-Ne laser. For the 50 cm

_etalon,

we have also made the measurement with the

_dye-laser

at 780 nm. The results are

_given

in table III. We have observed a

_significant

variation between the two 10 cm etalon measurements. This is due to

the fact that we do not use

_exactly

the same

_points

of the mirrors in the two cases. On the other

hand,

the two 50 cm etalon measurements are _{in very}

_good

_{agreement :}

the variation with the

wavelength

of the size of the laser beam inside the

_cavity

has a

_negligible

effect.

Thble III. -

Frequency difference

(MHz)

between the

_fundamental

mode

_TEMoo

and the

_first

trans-verse mode

_TEMm.

10 cm etalon

_(first

_measurment, 633

nm)

: 200.776

(67)

50 cm etalon

_{(633 nm)}

: 109.851

(13)

50 cm etalon

_{(780 nm)}

: 109.853

(13)

10 cm etalon

_(last

measurement, 633

_{nm) :}

201.024

₍₄₉₎

Fig.

3. -

Frequency

difference between the first transverse modes TEMO, and TEM1 o and the fondamental mode _TEMoo vs. the orientation of the two _spots of the transverse modes.

2.5 RESULTS. 2013 We used

_{équation (3)}

to détermine the infrared radiation

_frequency.

In the

two

_{preceding paragraphs}

we have described the measurements of the Fresnel

_phase

shift 03A6 and

-stabilized He-Ne laser and the six

_two-photon

transition

_wavelengths

which are measured.

Indeed,

taking

into account our earlier determination of the

_Rydberg

constant

_[11],

these

_wavelengths

are

known with a relative

_uncertainty

better than

10-9.

Since the

_uncertainty

in a fractional order

measurement is about

10- 4

in the case of the 50 cm etalon and is smaller for the 10 cm

_etalon,

and

since the etalon

_length

is

_initially

known with a relative

_uncertainty

of about

_10-3,

the fractional

order data for these seven

_wavelengths

are sufficient to determine

_{unambiguously}

the

_integer

number N. The values of N are

_given

in table IV

Table IV Values

_of

the

_integer

numbers N.

As we have two sets of data for the 10 cm

_etalon,

we can

_{apply equation (3)}

twice

_{(firstly :}

first 10 cm data and 50 cm

_data;

_{secondly :}

50 cm data and last 10 cm

_data).

The results for the six studied transitions are

_given

in table V The second column

_gives

the difference between the second and the first determinations. This difference is due to the

_ageing

of the mirrors. The third column

_gives

the mean of the two results. This is the value that we have

_adopted.

The

quoted uncertainty

does not include the reference laser

_uncertainty.

It is the

_quadratic

sum of :

i)

the

_uncertainty

of the

_comparison

between the He-Ne lasers

_(about

_10-11),

_ii)

the

_uncertainty

of the atomic

_{frequencies (statistical}

_uncertainty

and

_uncertainty

of the theoretical

_lineshape

as

explained

in

_{[3] ),}

_iii)

the

_uncertainty

of the Fresnel

_phase

shift measurements

_(about

1.2 x 10-11 ),

iv)

the statistical

_uncertainty

of the fractional order measurements and

_v)

the

_uncertainty

due to

the

_ageing

of the mirrors

_(half

the value of the first column of Tab.

_V).

A more detailed

_analysis

of the uncertainties will be

_given

in section 3.

3.

_Rydberg

constant detennination.

3.1 DETERMINATION OF THE

_{2S1/2 - nD5/2}

ATOMIC SPLITTING. - Th détermine the

2S1/2

-nD5/2

atomic

_splitting

from the

_frequency

measurements

_reported

in table

_V,

we have to make several corrections. These corrections are

_given

in table VI.

i)

Second order

_Doppler

_effect.

This correction has been discussed in _paper II

_{[3] .}

As we have

measured the

_velocity

distribution in the metastable atomic

beam,

we can calculate

_precisely

the

second-order

_Doppler

effect for

_hydrogen

and deuterium.

ii)

_2Si/2

hyperfine

splitting.

As we have measured

_only

the most intense

_hyperfine

_component,

’Iàble V -Resonance frequencies of

the

_two-photon

transitions

_(column

₃₎

and

_{difference (D)}

be-tween the last and the

_first

measurements

_(column

_2).

1àble VI. -

_{Corrections for}

the second-order

_{Doppler effect}

and the

_{hyperfine splittings}

_(MHz).

is _very well known for

_hydrogen

_[12]

and deuterium

_[13]

and the

_{corresponding}

corrections are

+ 44.389 MHz and + 13.641 MHz

_{respectively.}

üi)

nD5s/2

hyperfine splitting.

We do not resolve the

_hyperfine

structure of the

_nD5/2

state. This

_splitting

can be calculated with the Fermi formula

_{[14] .}

For

_example

we obtain 142.6 kHz for the

_8Ds/2

level in

_{hydrogen .}

As the initial level of the

_two-photon

transition is a

_particular

hyperfine

level

_(for

_{instance F;}

= 1 for

hydrogen),

the intensities of the two

_hyperfine

transitions

2S1/2(Fi

=

1) -

nDs/2(FJ)

(Ff

= 2 and

Ff

= 3 for

hydrogen)

are not

_proportional

to the

degeneracy

2Ff

+ 1 of the excited

_hyperfine

level but to :

3.2 CALCULATION OF THE RYDBERG CONSTANT. - The corrections

_given

in table VI lead to

six values of the

_{2Si/2 - nD5/2}

_{splittings corresponding}

to the six transitions

_{investigated.}

We

have used the

_experimental

value of the

_hydrogen

2S

_Lamb

shift

_(1057.851(2)

MHz

_{[15] )}

and the theoretical value

_(1057.229

_MHz)

of the deuterium one, calculated

_using

reference

_[16],

to deduce

the

_{2Pi/2 - nDS/2}

_frequencies.

We have then

_compared

these

_frequencies

to those

_reported

_by

Erickson

_[17]

after correction for the 1986 recommended values of the fine structure _constant,

the

_{proton/electron,}

and the

_{deuteron/proton}

mass ratios

_{[18] :}

The measured

_frequencies

lead to six

_independent

determinations of the

_Rydberg

constant

given

in table VII. The

_quoted

uncertainties do not include the reference laser

_uncertainty.

As

can be seen from this

table,

our six

_independent

measurements of

Roc

are

_quite

self-consistent.

1àking

into account the reference laser

_uncertainty,

our final result is:

Uble VII. - Determination

_{of the Rydberg}

constant.

The

_precision

is 1.7 x

10-1°

and 4.3 x

10-11

with

_respect

to our reference laser. This result

is

_slightly

smaller

_(about

4.4 x

10-11)than

our

_preliminary

result

_published

in 1988

_{[19] .}

This is

due to three reasons:

i)

The second-order

_Doppler

effect correction has been calculated with the measured

_velocity

distribution and not with

_{an a priori}

one

_(correction

of about -0.6 x

10-11)

ii)

The

_Rydberg

constant calculation was

made

with an old value of the fine structure constant

a-1 =

137.03604

_(correction

of about -1.3 x

10-11)

Table VIII.

_Comparison

with other recent measurements.

Thèse values are all corrected

_using

the new definition of c and the

_experimental

ratio

_{me /mp.}

In table VIII our result is

_compared

to our

_previous

one

_[11]

and to those obtained

_by

different

groups since 1978 from excitation either of the Balmer-a and

-,Q

transitions

[20-24]

or of the 1S-2S transition

_[25-28].

Since our 1986 measurement there is a _very

_good

_agreement

which is illustrated

in

_figure

4.

3.3 UNCERTAINTIES ANALYSIS. - The various contributions to the total

_uncertainty

are listed

in table IX. In this table the first two items do not come from our

_experiment

and

_give

the

_present

limitation to the method

_(about

1.9 x

10-11 ) .

For

_hydrogen

the 2S Lamb shift

_experimental

un-certainty

is 2 kHz

_[15].

_{Nevertheless,}

_taking

into account another less

_precise

measurement

_[29]

(1057.845 (9)

MHz),

we have

_adopted

a 10 kHz

_uncertainty

for the

_hydrogen

2S Lamb shift

mea-surement. As the Q.E.D.

uncertainty

of the

_2Pi/2

level _energy calculation is about 3 kHz

_[17],

the total

_uncertainty

due to the Lamb shift and the _energy level calculations is about 1.5 x

10-11,

which is the value indicated in table IX. For deuterium the Lamb shift calculation

_uncertainty

is = 14 kHz

_(quadratic

sum of the 9 kHz _Q.E.D.

_uncertainty

and of the 11 kHz nuclear size

_uncertainty)

and the total Lamb shift and _energy calculations

_uncertainty

is = 1.8 x

10-11.

In table IX the other items

_depend

_upon our

_experiment.

The statistical

_uncertainty

is the standard deviation of our six

_independent

results. The Stark

effect,

second-order

_Doppler

effect and theoretical

_lineshape

uncertainties have been discussed in _paper II

_[3].

In

_comparison

with

our

_previous

measurement

_[11]

the

_improvement

in the Stark effect

_uncertainty

is checked

_by

the

_study

of the 2S-20D

_two-photon

transition which

_gives

an _upper limit

₍₂

_mV/cm)

to the

_stray

electric fields. The uncertainties due to the

_ageing

of the

_mirrors,

to the Fresnel

_phase

shift

mea-surement and to the

_comparison

between the He-Ne lasers have been discussed in this _paper.

Finally

the more

_{important uncertainty}

is due to the standard laser

_(1.6

x

10-1°) . With

_respect

Fig.

4. -

Comparison

of our result with recent

high

precision

measurements of the

Rydberg

constant.

TABLE IX. Uncertainties

_(part

in

_{1011 ).}

3.4 ISOTOPIC SHIFTS. - Our

_experiment

also

_provides

three

_isotopic

shifts which are

_given

in

table X. The

_quoted

uncertainties are much smaller than those on each

_{261/2 - nD5/2}

measure-ment, since various

_systematic

errors cancel. From these three

isotopic

shifts,

we can deduce three

_independent

measurements of the

_{proton/électron}

mass ratio. Our mean value :

_mP/me

=

1836.152

₅₉₍₂₄₎

is in excellent

_agreement

with the more

_precise

one

_mp/me

=

1836.152701(37)

of Van

_Dyck

et aL

_[30].

The

_uncertainty

of our measurement is the rms sum of the

_following

(6.3

x

10-8)

and second-order

_Doppler

effect

₍₂

x

10-8).

_Figure

5 shows the

_comparison

of our

result with other recent measurements

_[30-33].

’Tàble X. -

Isotopic shifts.

Fig.

5. -

Comparison

of our result with recent determinations of the

_{proton/electron}

mass ratio.

4. Conclusion.

We have made a new measurement of the

_Rydberg

constant, the most

_precise

one at the

_present

time. It is limited

_by

the

_precision

of the

_wavelength

and

_frequency

standard in the _{visible range.}

This shows that the

_present

realization of the meter is no

_longer

_satisfactory

in the

_optical

domain

and that more

_{precise frequency}

standards are

_required.

Acknowledgements.

The authors are indebted to Prof. B.

_Cagnac

for much fruitful advice

_during

this

_experiment.

We

thank Drs R Juncar and Y. Millerioux from the Institut National de

_Métrologie

for the loan of a

etalon. This work is

_{partially supported by}

the Bureau National de

_Métrologie.

One of us

_(JCG)

is

_grateful

to the Brazilian Financial

_Agency

CAPES for a

_scholarship.

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