The Ramsey method in high-precision mass spectrometry with Penning traps: Experimental results

0.59 MB·English

by  S. George

The Ramsey method in high-precision mass spectrometry with Penning traps:
Experimental results
S. George1,2,∗ K. Blaum1,2, F. Herfurth1, A. Herlert3, M. Kretzschmar2,
S. Nagy2, S. Schwarz4, L. Schweikhard5, and C. Yazidjian1
1GSI, Planckstraße 1, 64291 Darmstadt, Germany
2Johannes Gutenberg-Universit¨at, Institut fu¨r Physik, 55099 Mainz, Germany
3CERN, Division EP, 1211 Geneva 23, Switzerland
7
0 4NSCL, Michigan State University, East Lansing, MI 48824-1321, USA and
0 5Ernst-Moritz-Arndt-Universit¨at, Institut fu¨r Physik, 17487 Greifswald, Germany
2 (Dated: February 2, 2008)

n ThehighestprecisionindirectmassmeasurementsisobtainedwithPenningtrapmassspectrom-
a etry. Most experiments use the interconversion of the magnetron and cyclotron motional modes
J of the stored ion due to excitation by external radiofrequency-quadrupole fields. In this work a

2 new excitation scheme, Ramsey’s method of time-separated oscillatory fields, has been successfully
2 tested. It has been shown to reduce significantly the uncertainty in the determination of the cy-
clotronfrequencyandthusoftheionmassofinterest. Thetheoreticaldescriptionoftheionmotion

] excitedwithRamsey’smethodinaPenningtrapandsubsequentlythecalculation oftheresonance
h
lineshapesfordifferentexcitationtimes,pulsestructures,anddetuningsofthequadrupolefieldhas
p
been carried out in a quantum mechanical framework and is discussed in detail in the preceding
-
m articleinthisjournalbyM.Kretzschmar. Here,thenewexcitationtechniquehasbeenappliedwith
theISOLTRAPmass spectrometer at ISOLDE/CERN for mass measurements on stable as well as
o
short-livednuclides. Theexperimentalresonancesareinagreementwiththetheoreticalpredictions
t
a and a precision gain close to a factor of four was achieved compared to the use of theconventional
. excitation technique.
s
c
i PACSnumbers: 07.75.+h,21.10.Dr,32.10.Bi
s
Keywords:
y
h
p
I. INTRODUCTION motions of confined ions in a Penning trap was first put
[
forward at ISOLTRAP by G. Bollen and coworkers in

1 1992 [9] and later tested at the SMILETRAP experi-
The mass and its inherent connection with the atomic
v
ment[10]. Atthistimethecorrecttheoreticaldescription
6 and nuclear binding energy is an important property of
of the observed line-shapes was not available. Instead,
3 anuclide. Thus,precisemassmeasurementsareeminent
2 for various applications in many fields of physics [1, 2]. the observations were discussed qualitatively in terms
1 The required precision of the atomic mass depends on of the Fourier transform of the applied pulse sequence.
0 the physics being investigated. For radionuclides, which However, a study of the precision gain as well as on-
7 line mass measurements with the new excitation scheme
often have half-lives considerably less than a second, it
0 ranges from δm/m= 10−5 to below 10−8 and for stable were not performed. In the preceding paper [11] a the-
/
s nuclides evendownto δm/m=10−11. Since the interest oretical description of the obtained lineshape has been
c in masses of short-lived and stable nuclides arises from developed in a quantum mechanical framework. In the
i
s a wide range of applications with different requirements following, the corresponding experimental investigations
y on the accuracy, a continuous development of e.g. new and the gain in linewidth and precision of the “Ramsey
h
method” will be presented.
p iondetectionandpreparationtechniques iscarriedonat
: several facilities world-wide [3].
v
In 1949 N.F. Ramsey improved the molecular-beam
i
X magnetic-resonancemethodofI.I.Rabi[6,7]byapplying
r oscillatory fields to the transversing molecular beam in
a spatially separated regions [8]. This replacement of the
uniformly applied field led to a linewidth reduction to
60% and thus to a more precise determination of the II. THEORETICAL OVERVIEW
resonance frequency. For a detailed description see [8]
and references therein.
In high-precision mass spectrometry the idea to use For a quick orientation we review here in a non-
time-separatedoscillatoryfieldstomanipulate the radial technical manner the physical assumptions forming the
basisofthe theoreticalmodel,theapproximationsneces-
sary for analytical calculations, and the results that are
mostimportantformassspectrometry. Detailsarefound
∗Correspondenceto: [email protected] in the preceding article [11].
2
A. Penning traps with azimuthal quadrupole n +1
+
excitation
n=3
z
1. The ideal Penning trap
hω
n=2 z
TheelectromagneticfieldconfigurationofanidealPen- hω z
+
ningtrapconsistsofastronghomogeneousmagneticfield
B0 = B0ez in the axial direction and an electrostatic nz=1
quadrupole field E = Φ derived from the potential
0
Φ0(x,y,z) = 2z02U+r02 ·(−2z∇2−x2 −y2). The motion of a n+ nz=0 n-=0
single ion of mass m and electric charge q in this elec- n-=1 hω-
tromagnetic field is determined by the field parameters n=2
-
B0 and U, or equivalently, by the cyclotron frequency n-=3
ν = ω /2π = qB /(2πm) and by the axial frequency
c c 0
ν = ω /2π with ω = 4qU/(2z2+r2), where r and
z z z 0 0 0
z0 denote the inner radipus of the ring electrode and the FIG. 1: Energy level scheme of the harmonic oscillators for
distance of the endcap electrodes from the trap center, a spin-less charged particle in an ideal Penning trap. ω+ is
respectively. The dynamics of a single ion in an ideal the modified cyclotron angular frequency, ωz is the angular
Penning trap is described by three uncoupled harmonic frequency, and ω− is the magnetron angular frequency. n+,
oscillators: the oscillator of the cyclotron motion with nz,andn− denotethecorrespondingquantumnumbers. The
the frequency ω = 1(ω + ω2 2ω2), the inverted total energy is given by the sum of the energies of the three
+ 2 c c − z independent harmonic oscillators. The contribution of the
oscillator of the magnetron mpotion with the frequency
invertedmagnetronoscillatorisnegative. Zeropointenergies
ω = 1(ω ω2 2ω2), and the axial oscillator with
− 2 c− c − z of the oscillators havebeen subtracted.
the frequencypωz. It is useful to introduce the abbre-
viation ω = ω2 2ω2, so that ω = 1(ω ω ).
1 c − z ± 2 c ± 1
With appropriaptely chosen canonical coordinates qk, pk
(k =+, ,3) the Hamiltonian for the motion of a single
−
spinless ion can be written [11] as -U
q
H = ω 1(q2 +p2) ω 1(q2 +p2)
0 +· 2 + + − −· 2 − −
+ω 1(q2+p2). (1)
z· 2 3 3
Canonical coordinates offer an easy access to the quan-
tized version of the theory. Annihilation and creation r
+U 0 +U
operators for the oscillator quanta are defined by q q
1
a = (q +ip ),
k √2~ k k
1
a† = (q ip ), (k =+, ,3) (2)
k √2~ k− k −
-U
q
with commutation relations [a ,a† ] = 1, [a ,a† ] = 0,
± ± ± ∓
and [a ,a ]=0. The Hamiltonian then takes the form
± ∓
FIG. 2: Radial segmentation of thering electrode (top view)
H = ~ω (a† a + 1) ~ω (a† a + 1) fortheapplication of radiofrequencyfields. A predominantly
0 + + + 2 − − − − 2 quadrupolar field can be generated by applying a radiofre-
+~ω (a†a + 1). (3)
z 3 3 2 quency between pairs of opposing electrodes of the four-fold
segmented ring electrode.
The quantizedversionof the theory provides a clear pic-
ture of the energy level scheme associated with the ion
motion in an ideal Penning trap (see Fig. 1). Therefore,
itisvaluablefortheidentificationoftheinteractionwith
the external rf-quadrupole field that is used to convert
the magnetron motion into the cyclotron motion. by avoiding large distances of the ion from the trap cen-
Real Penning traps generally possess some anhar- ter. The theoretical discussion assumes also that all an-
monic perturbations that introduce nonlinear terms into harmonic perturbations and couplings to the axial oscil-
the equations of motion and couple the three motional lator mode can be neglected. Therefore the description
modes, thus preventing us from finding analytical solu- concentratesexclusivelyonthe cyclotronandmagnetron
tionsfortheionmotion. Thisproblemcanbeminimized motional modes.
3
2. The ideal Penning trap with azimuthal quadrupole familiar from the study of magnetic-resonance or the
excitation quantum theory of two-level systems. An important re-
sult is that the interconversion of the magnetron and
The application of a radiofrequency potential to the cyclotron motional modes by a quadrupole field of fre-
four-fold segmented ring electrode (see Fig. 2) adds to quency νd is periodic with the ‘Rabi frequency’ νR =
theHamiltonian(3)newtermsthatareperiodicwiththe ωR/(2π)= (g/π)2+(νd νc)2.
−
driving frequency ν =ω /2π and have a phase χ . The p
d d d
leadingterminamultipolarexpansionisthequadrupole
contribution (x2 y2)cos(ω t+χ ), highermultipoles 3. Ion trajectories
d d
∝ −
mustbe minimized asthey introduce non-linearitiesinto
the equations of motion. In the quantized version of the While the viewpoint of quantum mechanics was very
theory x and y are components of the position operator helpful for the identification of the relevant interaction,
of the ion and can be expressed in terms of the creation itisingeneralsufficientfortheapplicationtomassspec-
andannihilationoperatorsofthe modifiedcyclotronand trometrytoconsidertheionmotionasaclassicalmotion,
magnetron motional modes. It then becomes apparent i.e. following trajectories of macroscopic scale. To work
thatthequadrupolecontributionactuallydescribesthree out this aspect expectation values of the quantum me-
physical processes: Most importantly the absorption of chanical operators are taken with respect to quasiclas-
an electromagnetic field quantum ~ω with simultane- sical coherent oscillator states. Thus, the annihilation
d
ous conversion of a magnetron oscillator quantum ~ω− operators a+(t) and a−(t), obtained as solutions of the
into a quantum of the cyclotron oscillator ~ω , together Heisenberg equations of motion, are translated into two
+
with the reverse transition. This process is dominant if complex functions α+(t) and α−(t) that are denoted as
thedrivingfrequencyν approximatelysatisfiesthereso- the ‘complex oscillator amplitudes’ of the cyclotron and
d
nancecondition~ω ~ω +~ω =~ω . Itis described magnetron oscillators at time t. The complex conjugate
d + − c
by an additional term≈in the Hamiltonian functions α∗(t) and α∗(t) correspond to the respective
+ −
creation operators. The explicite solution of the initial
H (t) = ~g e−i(ωdt+χd)a†(t)a (t) value problem is [11]:
1 + −
(cid:16)
+e+i(ωdt+χd)a† (t)a (t) . (4)
− +
(cid:17)
The real coupling constant g has the physical dimension
ofa frequency. It is proportionalto the amplitude ofthe
rf-quadrupolefield,butitsvalueisalsoinfluencedbyde-
tails of the trap geometry. As shown later, it determines
the conversion time τ = π/2g, which is defined as the
c
time required for the full conversion of a state of pure
magnetron motion into a state of pure cyclotron motion
by a quadrupole field at the resonance frequency ω (see
c
final results in Eq. (9)). For each magnetron quantum
~ω that is annihilated (created) a cyclotron quantum
−
~ω is simultaneously created (annihilated). The inter-
+
action obviously conserves the total number of quanta
N = N +N present in the system and has no sim-
tot + −
ple description in a purely classical picture. The other
two physical processes described by the quadrupole con-
tribution are the absorption (emission) of a field quan-
tum~ω withsimultaneouscreation(annihilation)oftwo
d
quanta of the modified cyclotron oscillator (driving fre-
quency ω 2ω ), and the absorption (emission) of a
d +
field quantu≈m ~ω with simultaneous annihilation (cre-
d
ation)oftwoquantaofthe magnetronoscillator(driving
frequency ω 2ω ) [12]. These two processes are neg-
d −
≈
ligible if ω ω .
d c
≈
The ideal Penning trap with quadrupole excitation is
now described by the total Hamiltonian H(t) = H +
0
H (t). The resulting Heisenberg equations of motion
1
for the operators a (t) and a (t) are linear and time-
+ −
dependent, they permit a general, exact solution. The
equationsand their solutions correspondclosely to those
4
(a) (b)
ω t δ ω t
α (t) = e−i(ω++δ/2)t cos R +i sin R α (0)
+ +
(cid:20)(cid:18) 2 ω 2 (cid:19)
R
2g ω t
i sin R e−iχdα (0) , (5)
−
− ω 2 (cid:21)
R
2g ω t
α (t) = e+i(ω−+δ/2)t i sin R e+iχdα (0)
− +
(cid:20)− ω 2
R
ω t δ ω t
R R
+ cos i sin α (0) . (6)
−
(cid:18) 2 − ω 2 (cid:19) (cid:21)
R
In these equations δ = ω ω is the detuning of the
d c
−
driving quadrupole field, ω = 4g2+δ2 the Rabi fre- FIG. 3: Conversion of pure magnetron motion to pure cy-
R
quency, χd the phase of the qpuadrupole field at time clotron motion by the action of an azimuthal quadrupolar
t = 0, and α (0) = α (0) exp[ iχ ] the initial values field with the cyclotron frequency νc = ν+ +ν−. Part (a)
± ± ±
| | ∓ and (b) show the first and second half of the conversion, re-
ofthecomplexoscillatoramplitudes. Forexample,foran
spectively. The solid circle in (a) indicates the initially pure
initial state of pure magnetronmotion or pure cyclotron
magnetron motion.
motion they are α (0)=0 or α (0)=0, respectively.
+ −
Thepracticalimportanceofthecomplexoscillatoram-
plitudes for our applications lies in their relation to the
instantaneous radii for the cyclotron and the magnetron τ (a)
motion, 1
s
t
2~ 2~ ni τ τ τ (b)
R (t) = α (t) , R (t) = α (t) . u 1 0 1
+ rmω1| + | − rmω1| − | b.
(7) r
a
Relatingthecomplexoscillatoramplitudestotheoriginal / τ τ τ τ τ (c)
e 1 0 1 0 1
Cartesian coordinates and velocities, an explicite para- d
metric representation of the ion trajectories in the xy- u
t
plane can be derived pli τ τ τ τ τ τ τ (d)
m 1 0 1 0 1 0 1
x(t)+iy(t) = e−2iδt 2~ cosωRt +i δ sinωRt A t / ms
rmω ·(cid:20)(cid:18) 2 ω 2 (cid:19) 0 100 200 300
1 R
e−iω+tα (0)+e−iω−tα∗(0)
· + −
(cid:0)ie−iχd 2g sinωRt (cid:1) FIG. 4: Excitation schemes: (a) standard excitation, (b) ex-
− · ω 2 citation with two 100 ms Ramsey pulses, (c) excitation with
R
three 60 ms Ramsey pulses, and (d) excitation with four 40
e−iω+tα (0) e−iω−tα∗(0) . (8)
− − + ms Ramsey pulses. The excitation amplitude is chosen in a
(cid:0) (cid:1)(cid:3) way that at νc one full conversion from pure magnetron to
A graphical representation of an ion trajectory calcu-
purecyclotron motion is obtained. Thusthesum of thegrey
lated by this approach is shown in Fig. 3. For a three-
colored areas is identical in all four schemes.
dimensionalrepresentationonehastoadd,ofcourse,the
oscillatory axial motion.
narrowercentralpeakscomparedtothe conventionalex-
B. Excitation schemes citationscheme. Therefore,avarietyofdifferentRamsey
excitationschemeswereinvestigatedwiththeaimoffind-
The precision in the determination of the cyclotron ing the one best suited for precision mass spectrometry.
frequency strongly depends on the width of the cen- In [11] the line shapes are discussed in the quantum me-
tral peak of the resonance. The narrower the reso- chanical formalism as probability distributions for the
nance, i.e. the smaller the full-width-at-half-maximum partialconversionofagiveninitialstateintoastateofcy-
(FWHM), the moreprecisely the center ofthe resonance clotron motion. This general framework permits to take
canbe determined. The pronouncedcentralpeak,which into account different assumptions on the initial state
is clearly distinguishable from the usually smaller side as well as statistical hypotheses, for example on phases.
bands, marks the point of the maximal conversion and A more elementary approach consists in calculating the
thus of the maximal radial energy and the shortest time time development of the radius of the cyclotron motion
offlightfromthetraptothedetector. Ramsey’smethod R (t)fromthegiveninitialconditions,usingEq.(5),(6),
+
of time-separated oscillatory fields promises to lead to and (7).
5
1. The one-pulse excitation scheme It is therefore reasonableto expect that the use of Ram-
sey’s method of separated oscillatory fields will lead to
increased precision in mass spectrometry too.
Intheconventionalone-pulseexcitationschemetheex-
ternal driving field is applied for a certain time interval A symmetric n-pulse Ramsey cycle of a total duration
τ = τtot with constant amplitude, as shown in Fig. 4. τtot consists of n excitation intervals of duration τ1 with
The lineshape represents the probability for the conver- (n 1)waitingintervalsofdurationτ0inbetween,sothat
−
sion of an initially pure state of magnetron motion into the total cycle time is τtot =nτ1+(n 1)τ0. Note that
−
cyclotron motion as a function of the detuning δ of the thenexcitationpulsesoftherf-quadrupolefieldmustbe
quadrupole field. For given values of τ and g it is ob- coherent in phase.
tained in the general formalism of [11] as Assuming that the ion is initially in a state of pure
magnetron motion, the probability for the conversion of
4g2 ω τ magnetron quanta into quanta of the cyclotron motion
F1(δ;τ,g) = ω2 sin2 R2 . (9) bya2-pulseRamseycyclewithdetuningδ =ωd ωc has
R (cid:16) (cid:17) been calculated to be −
Note that the maximum value 1 is reached when the
4g2 δτ
driving field is at resonance, νd = νc, and the dura- F2(δ;τ0,τ1,g) = ω2 (cid:26)cos(cid:18) 20(cid:19)sin(ωRτ1)
tion of the excitation is chosen to satisfy the condition R
gτ = (2n+1)π/2 with n = 0,1,2,.... In different words, δ δτ
0
+ sin
at resonance (δ =0) complete conversionof a pure mag- ω (cid:18) 2 (cid:19)
R
netron state into a state of pure cyclotron motion is
[cos(ω τ ) 1] 2. (11)
achievedforexcitationpulsesofthedurationofthe‘con- R 1 − }
version time’ τ =π/(2g) or odd multiples thereof.
c This result and analogous ones for Ramsey excitation
This result can be verified by application of Eq. (5)
cycles with 3, 4, and 5 pulses can be found in [11].
and(7)withtheinitialconditionα (0)=0andarbitrary
+ If the frequency of the quadrupole field equals the cy-
α (0),andcomputingthetimedevelopmentoftheradius
− clotron frequency ν and the amplitude of the field is
of modified cyclotron motion: c
chosen such that the n excitation intervals exactly add
up to the conversion time τ , i.e. if the coupling con-
4g2 ω τ c
R2(τ) = sin2 R R2(0) = F (δ;τ,g) R2(0). stant g satisfies the relation nτ = τ = π/2g, then the
+ ωR2 (cid:16) 2 (cid:17)· − 1 · − profile function (11) reaches th1e valcue 1 at resonance,
(10)
F (δ =0;τ ,τ /2,g)=1.
2 0 c
With respect to time the shape of the excitationpulse is
Anelementaryderivationofthe2-pulseprofilefunction
rectangular. Thus,infrequencyspaceitisexpectedthat
F inEq.(11)ispossiblebyapplyingEq.(5),(6),and(7)
2
the excitation resembles the intensity (i.e. the modulus
three times successively to the time intervals 0 t τ ,
1
squared)oftheFouriertransformofarectangularprofile, ≤ ≤
τ t τ +τ , and τ +τ t 2τ +τ =τ , with
namely (4g2/δ2)sin2(δτ/2). The actual lineshape, how- 1 ≤ ≤ 1 0 1 0 ≤ ≤ 1 0 tot
the initial condition α (0) = 0 and arbitrary α (0), in
+ −
ever, differs in two important respects from this Fourier
order to compute the endpoint of the time development
transform: (a)Atresonance(δ =0)itisa periodicfunc-
ofthe radiusofthecyclotronmotion,R (τ ). Thefirst
+ tot
tion of τ, describing the periodic conversion and recon-
calculation yields the initial values α (τ ) for the wait-
± 1
versionof the magnetronand modified cyclotron modes,
ing period, the second calculation with g = 0 yields the
F (δ = 0;τ,g) = sin2(πτ/τ ), whereas the intensity of
1 c phasechangeduringthe waitingperiodandthusthe ini-
the Fouriertransformincreasesproportionallyto τ2. (b)
tial values α (τ +τ ) for the second excitation period,
± 1 0
ThecentralpeakisactuallynarrowerthanfortheFourier
the third step finally results in α (2τ +τ ) = α (τ )
± 1 0 ± tot
transform of a rectangle. This can be deduced from the
and thus in a result for the final radius of the cyclotron
position δ of the zero that separates the central peak
0 motion R (τ ). Note that χ in Eq. (5) and (6) has
from the first satellite peak, (δ τ )2 = 3π2 as compared + tot d
0 c to be replacedin the secondand third step by the corre-
to (δ τ)2 = 4π2 for the Fourier transform of the rectan-
0 sponding phases of the quadrupole field, ω τ +χ and
d 1 d
gle, assuming τ =τ .
c ω (τ +τ )+χ , respectively. The final result is
d 0 1 d
R2(τ ) = F (δ;τ ,τ ,g) R2(0). (12)
2. The two-pulse excitation scheme and more general + tot 2 0 1 · −
schemes The time development of the ion orbit during the two
excitationperiodsisshowninFig.3aandFig.3b,during
Althoughitisnotobviousonfirstsight,acloseformal the waiting time the ion follows a rosette shaped orbit.
analogy exists between nuclear magnetic resonance on Plotting the profile function (11) with a fixed value of
the one hand and the interconversion of the magnetron thewaitingtimeτ asafunctionofthedetuningδoneob-
0
andcyclotronmotionalmodesofanioninaPenningtrap tains the spectral lineshape of the 2-pulse Ramsey cycle.
due to quadrupole excitation on the other hand. This ItbearsaresemblancetotheFouriertransformofasignal
was shown in [11] using the concept of a Bloch vector. consisting of two rectangular pulses, but as for a single
6
MCP3 or
Channeltron
1.0 (a) 1.0 (b) s320
Percentage of converted quanta0000....2468 Percentage of converted quanta0000....2468 n time of flight / 238000 pPreetnrcanispiinogn
0.0 0.0 a
-16 -12 -8 -4 0 4 8 12 16 -16 -12 -8 -4 0 4 8 12 16 Me260 MCP2
’ / Hz ’ / Hz 39Ca19F+
240
-3 -2 -1 0 1 2 3
1.0 (c) 1.0 (d) (C-1567015.81 Hz) / Hz preparation
Percentage of converted quanta0000....2468 Percentage of converted quanta0000....2468 Petnranping MCP1
0.0 0.0
-16 -12 -8 -4 0 4 8 12 16 -16 -12 -8 -4 0 4 8 12 16 stable alkali
’ / Hz ’ / Hz reference ion
source 2nd pulsed
drift tube
rf trap 1st pulsed
ISOLDE beam
drift tube
(continuous)
FIG. 5: Percentage of converted quanta in the case of a 60 keV 2.8 keV ion
quadrupolarexcitationnearωc asafunctionofthefrequency bunches
detuning δ′ =δ/2π for different excitation schemes shown in HV platform
Fig.4. Thetotalcycletimeofallschemesis300ms. Theline
profile of a one-pulse excitation with 300 ms excitation time
is shown in (a). (b) represents the line profile of a two-pulse
FIG. 6: Schematic drawing of the mass spectrometer
excitation, each of 100 ms duration. The excitation time as
ISOLTRAPincludingtheRFQtrap,thepreparationandpre-
wellasthewaitingtimeofthethree-pulsesexcitationscheme
cision Penning traps, as well as the reference ion source. In
in (c) is 60 ms. In (d) the four 45 ms pulses are interrupted the inset a resonance for 39Ca19F+ ions with an excitation
by40mswaitingperiods. Thecenterfrequency(δ=0)isthe
period of 900 ms is shown. To monitor the ion transfer and
cyclotron frequency ωc for a given ion mass m. to record the time-of-flight resonance for the determination
ofthecyclotronfrequency,micro-channel-plate(MCP)detec-
tors or a channeltron are used.
pulse there are important and characteristic differences.
LineshapesforhigherorderRamseycyclesarecalculated
in an analogous fashion using the results of [11]. In Fig.
rods [18]. Here, the ions are accumulated, cooled, and
5genericresultsaredisplayedforn=1,2,3,4,assuming
bunched. From the Paul trap the ions are guided, after
a total cycle time τ = 300 ms and τ = τ /n for all
tot 1 c
passing two pulsed drift tubes in orderto reduce the po-
excitation schemes. Note that with the n-pulse excita-
tential energy from 60 keV to about 100 eV, to a buffer-
tion scheme a spectral distribution is obtained in which
gas filled cylindrical Penning trap [19]. This preparation
the valley between the first major sideband and the cen-
trapislocatedinasuperconductingmagnetof4.7-Tfield
tralpeakcontains(n 2)smallpeaks,whilethedistance
− strength. Here, the ions are mass-selectively cooled [20]
between the first major sideband and the central peak
with a resolving power of up to 105. Thus, an isobar-
increases with n.
ically clean ion cloud is obtained, which is transferred
to the precision Penning trap. This hyperbolic Penning
trap [14], which is placed in a second superconducting
III. IMPLEMENTATION AT THE MASS
magnetof5.9-Tfieldstrengthwithafieldhomogeneityof
SPECTROMETER ISOLTRAP
10−7to10−8withinonecubiccentimeterandatemporal
stability of δB 1 2 10−9/h,providesthe possibility to
A. Experimental setup B δt ≈ ·
resolve even low-lying excited nuclear states [21, 22] and
serves for the actual mass measurement of nuclides with
Thetriple-trapmassspectrometerISOLTRAP[13,14] half-lives even below 100 ms [23, 24].
installed at the on-line facility ISOLDE/CERN [15, 16] In order to manipulate the ion motion in the precision
is dedicated to high-precision mass measurements of ra- trapbytheapplicationofexternalrf-fields,theringelec-
dioactivenuclides. Itreachesarelativemassuncertainty trode of the trap is four-fold segmented, as illustrated in
of below 10−8 [17]. A Paul trap and two Penning traps Fig.2. First,themagnetronradiusofallionsisincreased
are the main parts of the apparatus as shown in Fig. 6. viaa dipolarexcitationatthe magnetronfrequency [25].
Theionsfromthecontinuous60-keVradioactivebeamof If required, there is the possibility to remove unwanted
ISOLDEorfromthestablealkaliionsourcearecaptured contaminationsvia amass-selectivedipolarexcitationat
in-flight by a gas-filled linear Paul trap with segmented thecorrespondingmodifiedcyclotronfrequency. Thereby
7
the ions’ cyclotron radius is increased until the ions hit
the electrode and are lost. A quadrupolar excitation at

νc convertsthemagnetronmotionintocyclotronmotion. 1.05
Finally, the ions are ejected out of the trap and pass the
1.00
gradient of the magnetic field, which interacts with the
0.95
orbitalmagneticmomentoftheionsandacceleratesthem
0.90
in the axial direction to the detector. This acceleration
is proportionalto the strength ofthe ions’ magnetic mo- 0.85 n=4
ment, i.e. to the radial energy obtained by the excita- y 0.80
tion. The time of flight after ejection from the trap to 0.75
the detector is measured. This procedure is repeated for 0.70 n=3
differentfrequencies ofthe quadrupolarexcitationin the
0.65
precisiontraparoundtheexpectedvalueofthecyclotron
0.60 n=2
frequency. By the determination of the mean time of
0.55
flight for the different excitation frequencies a time-of-
0.0 0.2 0.4 0.6 0.8 1.0
flight resonance curve is recorded (see inset of Fig. 6). x
Forappropriateexcitationparameterstheminimumtime
of flight is measuredat the cyclotronfrequency [26]. For
reference measurements, i.e. to calibrate the magnetic
field, ions with well-known mass from a stable alkali ion FIG. 7: Reduction of the width of the central peak due to
source are used. the Ramsey method for constant cycle time τtot. For sym-
The experimental standard deviation σ(ν ) of the cy- metrical n-pulse excitation a Ramsey cycle is composed of n
c
clotron frequency ν is a function of the resolving power excitation pulses of duration τ1 = τc/n, where τc is the con-
c
versiontime,separatedby(n 1)waitingperiodsofduration
of the precision trap, i.e. the quadrupolar excitation −
time Tq, and the total number Ntot of recorded ions. τv0a.riaTbhleusyt=heδt(ont)aτl cy/c(πle√t3im)eisisaτmtoeta=surτecf+or(nth−e F1)Wτ0H.MThoef
The resolving power is Fourier limited by the duration | 0 tot |
the central peak for n-pulse excitation relative to the corre-
of the quadrupolar excitation, which itself is limited by
sponding quantity for 1-pulse excitation, x = (n 1)τ /τ
the half-life of the ion of interest in case of short-lived − 0 tot
is the fraction of the total cycle time spent during waiting
radionuclides. An empirical formula [27] describes this
periods.
relation:
σ(ν ) 1 c
c
= , (13) apuremagnetronstateintoapurestateofcyclotronmo-
ν ν √N T
c c tot· q tionoccurs,i.e. thehighestpossibledegreeofconversion.
Asuitableparametertodescribethewidthofthecentral
where c is a dimensionless constant. In a large number
peakisthepositionδ(n)ofthezeroseparatingthecentral
ofmeasurementswithcarbonclusterstheconstantcwas 0
peak fromthe firstside band. For a one-pulse excitation
determined for the ISOLTRAP mass spectrometer to be
c=0.898(8)[17]. (δ0(1)τtot)2 = 3π2. Therefore, for an n-pulse excitation
the variable y = δ(n)τ /(π√3) = δ(n)/δ(1) represents
| 0 tot | | 0 0 |
the ratio of the zeros for n-pulse and 1-pulse excitation.
To a very good approximation this variable equals the
B. Reduction of the line-width
ratio of the full-width-half-maximum of the two excita-
tion schemes. In Fig. 7 these ratios have been plotted
Havingdiscussedthestandardquadrupolarexcitation,
for n = 2,3,4 as a function of x = (n 1)τ /τ , i.e.
0 tot
what is the advantage of Ramsey’s method of time- −
the percentage of the total cycle time τ spent during
tot
separated oscillatory fields? In many experimental sit-
waiting periods. From the graph it is obvious that with
uations, the total time available for a complete mea-
60% the two-pulse excitation scheme offers the largest
surement cycle has an upper limit, for example due to
width reduction relative to a one-pulse excitation. This
the lifetime of the radioactive species under investiga-
excitationschemeisthereforefavoredforthe application
tion. Thus, a precision gain simply by increasing the
in high-precision mass spectrometry.
waiting time τ may not be feasible due to experimen-
0
tal limitations. Therefore, different excitation schemes
are compared with respect to the predicted width of the
central peak, assuming that a total time τ is available C. Time-of-flight detection technique at
tot
ISOLTRAP
to performone complete Ramsey measuringcycle. For a
symmetric n-pulse excitationit is τ =nτ +(n 1)τ ,
tot 1 0
−
where τ denotes the duration of an excitation interval As described above, the frequency determination via
1
and τ the duration of a waiting interval. For τ = τ /n a time-of-flight detection technique [5] is based on the
0 1 c
and an excitation at resonance a complete conversion of interaction of the magnetic moment of the orbiting ion
8

6 B1
T
d /
el 4
c fi
eti
n
g
a 2
M
B2
0
10 20 30 40 50
Axial position / cm
FIG.8: Themagneticfieldamplitudefromtheprecisiontrap
totheendoftheconversionsection. Betweenthetwomarked
pointstheamplitudedecreases about 90% of itsorigin value.
FIG.9: Calculatedtime-of-flightcyclotronresonancesfordif-
ferent excitation schemes. The total excitation and waiting
with the magnetic field gradient (Fig. 8). Thus, for a time in the precision trap is 300 ms. The time of flight of
detailed comparison with experimental data it is neces- a one-pulse excitation with 300 ms duration is shown in (a).
sary to convertthe theoreticalline shapes for the energy (b) shows the time of flight of a two-pulse excitation, each
conversion into time-of-flight spectra, which will be dis- of the pulses being 100 ms long. The three pulse excitation
(c) is done by three 60-ms excitation periods and two 60-ms
cussed in the following. The kinetic energy in the radial
waiting periods. In (d) thefour 45-ms pulses are interrupted
motions is predominantly due to the cyclotron motion;
by40-ms waiting periods.
the contributionof the magnetronmode is negligible be-
cause of ω ω :
− +
≪
the equations of conversion (see Eq. (9) and (11) and
E (ω ) = Ekin(ω )+Epot(ω )
r d r d r d Ref. [11]) are shown in Fig. 9. In each graph the time of
1
= m R2(τ ,ω )ω2 +R2(τ ,ω )ω2 flight of the ions from the trap to the detector is plotted
2 · + tot d + − tot d − as a function of the frequency detuning δ′ = δ/2π with
(cid:0) (cid:1)
1
−2m·ω+ω− R+2(τtot,ωd)+R−2(τtot,ωd) respect to the cyclotron frequency νc.
(cid:0) (cid:1)
1
m R2(τ ,ω )ω2. (14)
≈ 2 · + tot d + IV. RESULTS
Here, R (τ ,ω ) are the radii of the two radial modes
± tot d
Toconfirmthe calculations,i.e. to determinethe line-
after the quadrupolar excitation has been applied with
width reduction and, most importantly, to specify the
the frequency ω . The magnetic moment of an ion with
d
precision gain due to the Ramsey excitation method,
kinetic energy E (ω ) in a magnetic field B~ =B ~e can
r d · z morethan300time-of-flightresonancecurveswithdiffer-
be written as ~µ(ω ) = [E (ω )/B]~e . The interaction
d r d z ent excitation schemes were recorded with the Penning
with the gradient of the magnetic field causes an axial
trap mass spectrometer ISOLTRAP. The ion species for
force F~z(ωd) = −~µ(ωd) ·∇B~z on the ion, which leads all measurements was 39K+ provided by the stable al-
to a reduction of the time of flight from the trap to the
kali off-line ion source. Eachresonance consists of about
detector. This time of flight can be calculated with [26]
2500 ions in order to have identical statistics. To min-
imize ion-ion interactions only time of flight measure-
z1 m 12 ments with at most five ions in the trap were taken into
T(ω )= dz ,
d
Zz0 (cid:26)2[E0−q·V(z)−µ(ωd)·B(z)](cid:27) account. The efforts were concentrated on the specifi-
(15) cation of the uncertainty in the frequency determination
for different excitation schemes. Figure 10 shows time-
where E is the total initial energy of the ion, V(z) of-flight cyclotron resonances for the one-, two-, three-,
0
and B(z) are the electric and magnetic fields, respec- and four-pulse excitation scheme. A fit of the theoreti-
tively, along the way from the trap to the detector. At cally expected line shape (solid line) to the data points
ω = ω the magnetic moment is maximal and thus allowsthedeterminationoftheFWHMandthecyclotron
d c
the time of flight minimal. Typical theoretical time- frequency ν along with its uncertainty δν . To perform
c c
of-flight cyclotron-resonance curves for different excita- thesefitsthestandardevaluationprogramofISOLTRAP
tion schemes using the radial energies calculated from [17] was extended in order to analyze the measured cy-
9

300
s s 300
me of flight / 226800 me of flight / 222468000 4 ttwwoo--ppuullssee eexxcciittaattiioonn,, ttoott==360000 mmss
Mean ti 222400 -(1a2) -8 -4 0 4 8 12 Mean ti 220200 -1(2b) -8 -4 0 4 8 12 Hz 3 two-pulse excitation, tot=900 ms
300 ( 233141 6 Hz) Hz 300 ( 233141 6 Hz) Hz M /
s s H
Mean time of flight / 222224680000 (c) Mean time of flight / 222220246800000 (d) FW 12
-15 -10 -5 0 5 10 15 -12 -8 -4 0 4 8 12
( 2331416 Hz) Hz 2331416 Hz) Hz
0 100 200 300 400 500 600 700 800
Waiting time 0 / ms
FIG. 10: The measured mean time of flight as a function of
the quadrupolar excitation frequency with predicted curves
fitted to the data. Here 39K+ ions from the stable alkali ion FIG. 11: The full-width-at-half-maximum (FWHM) values
sourcewereused. (a)Resonanceoftheconventionalone-pulse ofthetime-of-flight cyclotron resonances as afunction ofthe
excitationschemewith300msexcitationtime,(b)atwo-pulse waitingtimeforthetwo-pulseexcitationschemeisgiven. The
excitation scheme with two times 100 ms excitation and 100 total cycle times are τtot = 300 ms (squares), 600 ms (cir-
mswaitingtime. (c)threepulses,each60ms,interruptedby cles), and 900 ms (triangles). The bold solid lines are the
twowaitingperiodsof60ms,(d)four45msexcitationpulses theoretically calculated values. Since the determination of
and three waiting periods of 40 ms. the FWHM was performed manually from the fit curves, the
error bars are conservatively estimated to be 0.05 Hz.
±
clotron resonances using the Ramsey method.
TABLEI:ThemaximumandminimumexperimentalFWHM
The fit results concerning the FWHM are presented in
of the different excitation schemes for different cycle times
Fig. 11 and Fig. 12. Figure 11 shows results obtained
are given. In addition the reduction gain is calculated. For
withatwo-pulseexcitationschemeofdifferentoverallcy- furtherexplanation see text.
cle times (τ = 300 ms, 600 ms, 900 ms). The FWHM
tot
is given as a function of the waiting period. Due to field number cycle time max. FWHM min. FWHM reduction
inhomogeneities and ion-ion interactions the data points of pulses τtot Hz Hz gain %
areshifted slightly to higher FWHM values comparedto 2 300 4.1 (0.1) 2.6 (0.1) 37.3 (2.0)
theory. Similar results are shown in Fig. 12 for differ- 2 600 2.1 (0.1) 1.3 (0.1) 38.1 (4.0)
ent numbers of excitation pulses, where the total cycle 2 900 1.4 (0.1) 0.9 (0.1) 35.7 (5.9)
time inthe precisiontrapisconstantτ =300ms. The 3 300 4.1 (0.1) 3.0 (0.1) 26.5 (2.2)
tot
experimental values are on average 0.1 Hz higher than 4 300 4.1 (0.1) 3.3 (0.1) 20.5 (2.7)
the theoretical ones. This line-broadening effect is due
to the electric and magnetic field imperfections and ion-
ion interactions, which were not taken into account in
the calculationsdescribedabove. A significantreduction reduction of close to 40% of the normal line-width is
of the FWHM for shorter excitation pulses with longer observed, similar to the results in the original work of
waiting periods in between can be observed. Ramsey[8]. Thereductioninline-widthisespeciallyim-
In Tab. I the experimental results are summarized. “cy- portant in context with the achievable resolving power
cle time” is the duration of the total cycle. The maxi- R = m/∆m = ν/∆ν. As theoretically predicted, the
mum FWHM is the one of the standard one-pulse reso- largest possible reduction is obtained by a two-pulse ex-
nance curve. The minimum FWHM is measured using citation scheme. However, the relative gain in reduction
the longest possible waiting time, i.e. shortest possi- depends only weakly on the total cycle time τ (see
tot
ble excitation time, which is determined by the maximal Tab. I and Fig. 11).
possibleamplitudeofthequadrupolarexcitationfieldre- In Fig. 13 the experimental uncertainty δν of the
c
quiredtoobtainonefullconversionfrompuremagnetron measured cyclotron frequencies for different numbers of
topurecyclotronmotion. ThelastcolumnofTab.Igives pulsesanddifferentlengthofthe totalcycletime isplot-
the maximum line-width reduction (reduction gain= ted versus the waiting time τ . Each data point repre-
0
(max. FWHM - min. FWHM)/max. FWHM) using sents the mean value of three to ten individual measure-
the different Ramsey excitation schemes. A remarkable ments. The uncertainty δν decreases with increasing
c
10

4.4
two-pulse excitation two-pulse excitation
three-pulse excitation (a) three-pulse excitation
4.0 four-pulse excitation 25 four-pulse excitation
z
M / H 3.6 mHz 20
FWH 3.2 / C15
2.8
10
2.4
0 50 100 150 200 250 300
0 50 100 150 200 250
Waiting time 0 / ms
Waiting time / ms
0

(b)
two-pulse excitation, tot=300 ms
FIG. 12: The full-width-half-maximum values of the time- 25 two-pulse excitation, tot=600 ms
two-pulse excitation, tot=900 ms
of-flight cyclotron resonances as a function of the sum of all
waiting periods τ =P τi for two, threeand four excitation
0 i 0 z 20
pulses. The total cycle time is τ =300 ms. The solid lines H
tot
m
are thetheoretically calculated FWHM values.
/ C15
10
waiting time. This is expected due to the decreasing
FWHM at longer waiting times. If the cyclotron fre-
quency uncertainty would only depend on the FWHM, 5
a similar behavior as observed in Fig. 12 would be ex-
0.0 0.2 0.4 0.6 0.8
pected. However,as mentioned before, it is obvious that
u
there is also aneffect of the overallline-shape, especially
of the steepness of the curve and the pronounced side-
bands,ontheuncertaintyinthefrequencydetermination
of νc. FIG. 13: In (a) the uncertainty of the measured cyclotron
The excitation scheme used for the data points given in frequency is given as a function of the waiting time for the
Fig. 13 (a) consists of two, three, and four pulses, where two-pulse, three-pulse, and four-pulse excitation scheme. In
the total excitation cycle is fixed to 300 ms. The un- (b) the uncertainty of the measured cyclotron frequency is
certainty is obviously decreasing for shorter excitation given for the two-pulse scheme with a total cycle of 300 ms,
pulses, i.e. longer waiting times τ . In case of the two- 600 ms, and 900 ms, where u=(n 1)τ0/τtot is the fraction
0 −
of the total cycle time spent duringwaiting periods.
pulse excitation it can be reduced by more than a factor
of three (from δν 27 mHz down to δν 8 mHz) as
c c
≈ ≈
compared to the conventional procedure just by chang-
longer total cycle times must be assigned to the relative
ing the excitation scheme to the two-pulse method. The
decrease of the statistical uncertainty in comparison to
uncertainty using the three- and four-pulse excitation
the constant systematic uncertainty. Unsymmetric exci-
scheme decreases to δν 13 mHz and δν 15 mHz,
c c
≈ ≈ tationschemes where the individual Ramsey pulses have
respectively.
different lengths have also been investigated. However,
The result is summarized in Tab. II, where the max-
the symmetric two-pulse excitation scheme remains the
imal and the minimal uncertainty for all investigated
best one in respect to line-width reduction and uncer-
excitation schemes under identical experimental condi-
tainty gain (see Tab. II).
tions are listed. As for the investigation of the FWHM,
the scheme where the largest reduction in the cyclotron
frequency uncertainty can be achieved is the two-pulse
Ramsey scheme. Comparing the two-pulse excitation V. FIRST ONLINE MASS SPECTROMETRY
APPLICATION OF THE RAMSEY METHOD
scheme with 300 ms, 600 ms, and 900 ms total cycle
time, the tendency is similar to the results obtained for
the FWHM of the resonances (see Fig. 13 (b)). The The first online mass measurement by the Ramsey
slightlydecreasinggainfactorfortwo-pulseschemeswith excitation method was carried out for the short-lived