Statické momenty jader –
metody jejich měření
Měření statických momentů jader
• Static moments of nuclei are measured via interaction of the nuclear charge
distribution and magnetism with electromagnetic fields in its immediate
surroundings. This can be the electromagnetic fields induced by the atomic
electrons or the fields induced by the bulk electrons and first neighboring
nuclei for nuclei implanted in a crystal, usually in combination with an
external magnetic field.
• Měření spinů a magnetických momentů jsou vzájemně „propletené“
• Nejstarší metoda – hyperjemné interakce atomových optických spekter
(dnes pomocí „colinear laser spectroscopy“)
• Funguje ale pouze pro stavy s nenulovým spinem!
Jádra s J=0 (základní stav sudo-sudých jader)
• Nelze měřit pomocí hyperjemné struktury
• Lze určit z měření
1.
hyperjemné struktury pro excitované stavy v daném rotačním pásu (za
předpokladu stejné deformace hladin v rotačním pásu)
měření
2.
vnitřní kvadrupólový moment
redukované pravděpodobnosti přechodu mezi prvním vzbuzeným stavem a
základním stavem
deformační parametr (ze změřených
Q momentů za použití vztahu):
typická hodnota pro deformovaná
jádra je
Hyperjemné interakce
• Energie stacionární soustavy nábojů a proudů ve vnějším poli
náboj;
elstat. potenciál
elektrický dipólový moment (= 0);
magnetický dipólový moment;
intenzita el. pole
intenzita magnet. pole
elektrický quadrupólový moment;
tenzor gradientu
el. pole
Liché elekrické momenty = 0 = sudé magnetické momenty (zákon zachování p)
Energie vyšších řádů lze zanedbat - jsou o několik řádů slabší
Zeemanův jev
Interakce magnetického dipólového momentu s vněším magnet. polem:
magnet. pole je součet pole od okolí a vnějšího pole
(pro Dm =1)  „úplné“ rozštěpení multipletu
Bohrův magneton,
magnetické kvantové č. m,
gyromagnet. poměr
57Fe
E
DE
hyperjemné
interakce
Ip
(2I +1) x
degenerovaná
hladina
0
posunutý
rozštěpený
multiplet
hladin
povolené pouze přechody
s DmI = -1,0,1
axiální symetrie = 0:
Elektrický kvadrupól v elstat. poli
(Starkův jev):
dochází jen k „částečnému“
rozštěpení multipletu
57Fe
kde eQ je elektrický kvadrupólový moment
(multipólový moment je obecně definován jako nultá
komponenta „momentu“ pro maximální projekci m = I ):
}
 u jádra s I = 0, ½ nelze takto „určit“ quadrupólový moment –
tento moment neexistuje
Table of nuclear magnetic dipole and
electric quadrupole moments
N.J. Stone
Atomic Data and Nuclear Data Tables 90 (2005) 75–176
seznam metod použitých při měření
(adoptovaných hodnot)
Metody
• Mößbauerův jev
– Omezeno jen na izotopy a hladiny měřitelné pomocí Mossbauera
• “PAC” (Time-Differential Perturbed Angular Distribution - TDPAD)
• NMR
– β-NMR pro hladiny s “krátkou dobou života“
• Nízkoteplotní orientace
• Rabiho experiment
• …
• Velikost hyperjemného
pole nezávisí pro daný
prvek na izotopu
• Lze pole změřit pole
pomocí jednoho izotopu
a pak měřit momenty u
dalších izotopů
Měření statických momentů jader
• Základní principy Mossbauerova jevu, NMR a porušených jaderných
korelací byly probrány v příslušných kapitolách
• Níže jsou jen některé metody rozebrány trošku podrobněji – spíše
příklady, jak se dají momenty měřit
TDPAD
• Spin-oriented isomeric states implanted into a suitable host will exhibit a
non-isotropic angular distribution pattern, provided the isomeric ensemble
orientation is maintained during its lifetime. If an electric field gradient
(EFG) is present at the implantation site of the nucleus, the nuclear
quadrupole interaction will reduce the spin orientation and thus the
measured anisotropy.
• If the implantation host is placed into a strong static magnetic field (order of
0.1–1 Tesla), the anisotropy is maintained. If the field is applied parallel to
the symmetry axis of the spin orientation, the reaction-induced spin
orientation can be measured.
• If a static magnetic field is placed perpendicular to the axial symmetry axis
of the spin orientation, the Larmor precession of the isomeric spins in the
applied field can be observed as a function of time, provided that the
precession period is of the same order as the isomeric lifetime (or shorter).
• Can also be used to measure the quadrupole moments of these isomeric
states, by implantation into a single crystal or a polycrystalline material with
a non-cubic lattice structure providing a static electric field gradient.
Příklady
•
TDPAD spectra for the γ-decay of the Iπ = 29/2−, t1/2 = 9 ns isomeric rotational
bandhead in 193Pb, implanted respectively in a lead foil to measure its magnetic
interaction (MI) and in cooled polycrystalline mercury to measure its quadrupole
interaction (QI).
• Detectors are placed in a plane perpendicular to the magnetic field direction
(θ = 90◦) and at nearly 90 ◦ with respect to each other (φ1 ≈ φ2 + 90), the
R(t) function in which the Larmor precession is reflected, is given by
Příklady
•
R(t) curves obtained in the study of g-factors of 9/2+ isomers in neutron-rich
isotopes of nickel and iron. The isomers, with lifetimes of 13.3 μs and 250 ns,
respectively, have been produced in a projectile fragmentation reaction at the
LISE high-resolution in-flight separator at GANIL.
β-NMR
• Time-differential measurements are only suited for short-lived nuclear states,
mainly because of relaxation effects causing a dephasing of the Larmor
precession frequencies with time (typically in less than 100 μs). To measure
nuclear moments of longer-lived isomeric states and also for ground states, a
time-integrated measurement is required. Time integration of R(t), taking
into account the nuclear decay time, will lead to a constant anisotropy.
• Therefore, a time-integrated measurement of the angular distribution of this
system will not allow one to deduce information on the nuclear moments.
Hence a second interaction, which breaks the axial symmetry of the
Hamiltonian, needs to be added to the system.
• One possibility to introduce a symmetry breaking in the system, is by adding
a radio-frequency (rf) magnetic field perpendicular to the static magnetic
field (and to the spin-orientation axis).
• If the nuclei are implanted into a crystal with a cubic lattice symmetry or
with a noncubic crystal structure inducing an electric field gradient,
respectively, one can deduce the nuclear g-factor or the quadrupole moment
from the resonances induced by the applied rf field between the nuclear
hyperfine levels.
β-NMR
• Consider an ensemble of nuclei submitted to a static magnetic field B0 and
an rf magnetic field with frequency ν and rf field strength B1. If the applied
rf frequency matches the Larmor frequency the orientation of an initially
spin-oriented ensemble will be resonantly destroyed by the rf field. For βdecaying nuclei that are initially polarized, this resonant destruction of the
polarization can be measured via the change in the asymmetry of the βdecay.
• For an ensemble of nuclei with the polarization axis parallel to the static
field direction, the angular distribution for allowed β-decay can be written as
with the NMR perturbation factor G1011 describing the NMR as a function of
the rf frequency or as a function of the static field strength. At resonance, the
initial asymmetry is fully destroyed if sufficient rf power is applied, which
corresponds to G1011 = 0. Out of resonance we observe the full initial
asymmetry and G1011 = 1.
β-NMR
• All forms of magnetic resonance require generation of nuclear spin
polarization out of equilibrium followed by a detection of how that
polarization evolves in time.
• In conventional NMR a relatively small nuclear polarization is generated by
applying a large magnetic field after which it is tilted with a small RF
magnetic field. An inductive pickup coil is used to detect the resulting
precession of the nuclear magnetization. Typically one needs about 1018
nuclear spins to generate a good NMR signal with stable nuclei. Consequently
conventional NMR is mostly a bulk probe of matter. On the other hand, in
related nuclear methods such as muon spin rotation (μSR) or β-detected NMR
(β-NMR) a beam of highly polarized radioactive nuclei (or muons) is
generated and then implanted into the material. The polarization tends to be
much higher – between 10% and 100%. Most importantly, the time evolution
of the spin polarization is monitored through the anisotropic decay properties
of the nucleus or muon which requires about 10 orders of magnitude fewer
spins. For this reason nuclear methods are well suited to studies of dilute
impurities, small structures or interfaces where there are few nuclear spins.
Příklad
•
NMR curve for 11Be implanted in metallic Be at T = 50K. At this temperature the
spin-lattice relaxation time T1 is of the order of the nuclear lifetime τ = 20 s.
Larmorova frekvence:
https://groups.nscl.msu.edu/becola/bnmr.html
http://bnmr.triumf.ca/
…
β-NMR
• At radioactive ion beam facilities such as ISOLDE and ISAC it is possible
to generate intense (>108/s) highly polarized (80%) beams of low energy
radioactive nuclei.
• Furthermore one has the added possibility to control the depth of
implantation on an interesting length scale (6–400 nm) … important for
measurement of fields (not moments measurement)
• Although in principle any beta emitting isotope can be studied with β-NMR
the number of isotopes suitable for use as a probe in condensed matter is much
smaller. The most essential requirements are:
– (1) a high production efficiency
– (2) a method to efficiently polarize the nuclear spins and
– (3) a high β decay asymmetry.
• Other desirable features are:
– (4) small Z to reduce radiation damage on implantation,
– (5) a small value of spin so that the β-NMR spectra are relatively simple and
– (6) a radioactive lifetime that is not much longer than a few seconds.
Isotope
+
μ
Quadrupol
e moment
(mb)
-6
75
0.842
6.3018
0.33
10
1/2
13.8
22
0.33
10
O
1/2
122
10.8
.7
10
Ne
1/2
0.1
.33
10
11
Be
15
17
2
2.2x10
γ (MHz/T) beta-Decay production
-1
asymmetry rate (s )
(A)
0.33
Li
1/2
T1/2 (s)
135.5
8
•
Spin
+32
8
7
8
6
Table gives a short list of the isotopes we have identified as suitable for
development at ISAC. Production rates of 106/s are easily obtainable at ISAC.
8Li is the easiest to polarize and therefore was selected as the first one to
develop as a probe at ISAC
Atomic hyperfine structure
• For a particular atomic level characterized by the angular momentum J,
the coupling with the nuclear spin I gives a new total angular momentum
F, F = I + J, |I − J| ≤ F ≤ I + J. The HF interaction removes the
degeneracy of the different F levels and produces a splitting into 2J + 1 or
2I+1 hyperfine structure levels for J < I and J > I, respectively.
•
Example of the atomic fine
and hyperfine structure of
8Li. For free atoms the
electron angular momentum
J couples to the nuclear spin
I, giving rise to the HF
structure levels F. The
atomic transitions between
the 2S1/2 ground state to the
first excited 2P states of the
Li atom are called the D1
and D2 lines
Optical pumping
• Polarization of a fast beam by optical pumping was introduced for the βasymmetry detection of optical resonance in collinear laser spectroscopy.
• Most applications took advantage of the additional option to perform nuclear
magnetic resonance spectroscopy with β-asymmetry detection (β-NMR) on a
sample obtained by implantation of the polarized beam into a suitable crystal
lattice. Whatever is the particular goal of such an experiment, it is important to
achieve a high degree of nuclear polarization.
• Repeated absorption and
spontaneous emission of photons
results in an accumulation of the
atoms in one of the extreme MF
states for which the total angular
momentum F = J +I, for an S state
just composed of the electron spin
and the nuclear spin, is polarized.
•
Optical pumping within the hyperfine structure Zeeman levels for polarization
of the nuclear spin. The example shows the case of I = 1 for the case of 28Na
• Using vector coupling rules the HF structure energies of all F levels
• The determination of nuclear moments from hyperfine structure is
particularly appropriate for radioactive isotopes, because the electronic
parts Be(0) and Vzz(0) are usually known from independent measurements
of moments and hyperfine structure on the stable isotope(s) of the same
element.
LMR
• Another possibility to NMR is Beta-Ray Detected Level Mixing Resonance
(b-LMR)
• Here, the axial symmetry is broken via combining a quadrupole and a dipole
interaction with their symmetry axes non-collinear. This gives rise to
resonant changes in the angular distribution at the magnetic field values
where the nuclear hyperfine levels are mixing.
• The resonances observed in a LMR experiment are not induced by the
interaction with a rf field, but by misaligning the magnetic dipole and
electric quadrupole interactions. This experimental technique does not need
an additional rf field to induce changes of the spin orientation. The change of
the spin orientation is induced by the quantum mechanical “anti-crossing” or
mixing of levels, which occurs in quantum ensembles where the axial
symmetry is broken.
•
Nuclear HF levels of a
nucleus with spin I = 3/2
submitted to a combined
static magnetic
interaction and an
axially symmetric
quadrupole interaction:
(a) for collinear
interactions, β = 0◦;
(b) and (c) for noncollinear interactions
with β = 5◦ and β = 20◦,
respectively.
•
•
•
Crossing or mixing of hyperfine levels occurs at well-defined values for the ratio of
the involved interactions frequencies, if
•
(d) At these positions, resonances are observed in the decay angular distribution of
oriented radioactive nuclei, from which the nuclear spin and moments can be deduced
Rabiho metoda
Technique developed for measuring the nuclear spin (can be used for measurement of m)
The experiment setup contains 3 parts:
• an inhomogeneous magnetic field in front (A),
• the weak rotating (perpendicular to uniform) + strong uniform field at the middle (C),
• and another inhomogeneous magnetic field at the end (B).
Atoms after passed the first inhomogeneous field will split into 2 beams corresponding
the spin up and spin down state. If the gradient in (A) and (B) is the same in magnitude
but opposite in direction and there is no change in the spin direction, all the neutrons
enter the detector (red lines). If the weak rotating field has frequency equal to the Larmor
frequency in the strong uniform field (at C), it will change the spin direction and neutrons
do not focus on the detector (blue line).
Měření rozměrů (poloměrů)
atomových jader
Measurement of nuclear radius
• Distribution of charge can differ from distribution of matter
• Methods outlined for charge matter radius:
„přímé“
– Diffraction (electron) scattering (form factor) – measurement of charge
distribution
– Muonic x-rays
„relativní“
– Atomic x-rays (shift in Ka) or optical spectra
– Mirror Nuclides (not exactly used for determination of a radius, see below)
• Methods outlined for nuclear matter radius:
– Rutherford scattering (via strong interaction)
v principu se dají použít i následující metody:
– p-mesic x-rays (measurements in 1960’s)
– Alpha particle decay (theory is needed)
– (cross section of fast neutrons) – not really used
Diffraction scattering
ki  k f  k
ki
 q  2k sin( a /2)
a
q = momentum transfer
kf

-k f
q
a
ki
e r is the inverse Fourier transform of
F q , which is known as the form factor
for the scattering


 
F ki , k f   *f V r  i dv
F q    eiq rV r dv
 
F q  
4p
sin qr  e r r dr 
q 
•
Measure the scattering intensity F q
as a function of a to infer the
distribution of charge in the
nucleus,  r 


2
Diffraction scattering
• Density of electric charge in the nucleus is almost constant
• The charge distribution does not have a sharp boundary
– Edge of nucleus is diffuse - “skin”
– Depth of the skin ≈ 2.3 fm
– RMS radius is calculated from the charge distribution and, neglecting the
skin, it can be shown
3 2
2
r
 R
5
e r  constant
e r 

A
4p R 3
4p R 3  A
R  Ro A1 / 3

Modulus squared of charge form factors (a)
calculated by solving the Dirac equation with
HF+BCS proton densities (b)
Atomic X-rays
• Assume the nucleus is uniform charged sphere.
• Potential V is obtained in two regions:
3 1  r 2

Ze 2 
– Inside the sphere
V r  
     r  R
4 p oR 
2
2 R  

Ze 2
V r  
rR
4 po r
– Outside the sphere

V    n* V  n dv
• For an electron in a given state, its energy depends on
• Assume n does not change
appreciably if Vpoint Vsphere

V     n* V  n dv    n* V  n dv
rR

rR
• 
Then, DE = Esphere - Epoint
• Assume n can be 1,1(1s), n1, 0
• D
E between sphere and point nucleus for
2 Z 4e 2
DE1s 
5 4po

•
1,1(1s)
E1s(sphere)
R2
E1s( pt)
ao3


Compare this DE to measurement and we have R.


DE1s


Atomic X-rays
• In reality, we will need two measurements (on two neighbor isotopes)
to get R
• Consider a 2p 1s transition for (Z,A) and (Z,A’) where
A’ = (A-1) or (A+1) ; what x-ray does this give?

 
 

EKa  A  EKa  A  E2 p  A  E1s  A  E2 p  A  E1s  A  E2 p  A  E2 p  A  E1s  A  E1s  A
• Assume that the first term will be ≈ 0 – larger radius (smaller influence)
• Then, use DE1s from previous slide for each E1s term:

2 Z 4e 2 1 2 2 / 3
2/3

EKa  A  EKa  A  DE1s  A  DE1s  A 
R
A

A
o
5 4p  o ao3

• This X-ray energy difference is called the “isotope shift”
• One can use optical transitions instead of X-ray (Ka) transitions…
Muonic X-rays
• Similar to “standard” X-rays
• Muons are heavier than electrons (106 MeV x
511 keV) which causes the difference in the
radius and energy (energy difference)
• Muonic orbitals feel the nucleus „as a sphere“ –
energy of X-rays is sensitive to nuclear radius
4po  2
mZ 2e 4
ao 
, En  
2
me
32p 2 o2  2 n 2
Prompt X-ray spectra from deuteron:
The curves are the results of the fitting
and the components of pμ X-rays
and dμ X-rays are also shown respectively.
Use for short-lived nuclei
• Let A, A and mA, mA be the mass numbers and atomic masses of the isotopes
involved. Then for an atomic transition i the isotope shift, i.e. the difference
between the optical transition frequencies of both isotopes, is given by
• This means that both the field shift (first term) and the mass shift (second
term) are factorized into an electronic and a nuclear part. The knowledge of
the electronic factors Fi (field shift constant) and Mi (mass shift constant)
allows one to extract the quantity δr2 of the nuclear charge distribution.
These atomic parameters have to be calculated theoretically or semiempirically.
• For unstable isotopes high-resolution optical spectroscopy is a unique
approach to get precise information on the nuclear charge radii, because it
is sensitive enough to be performed on the minute quantities of (shortlived) radioactive atoms produced at accelerator facilities.
• Other techniques are suitable only for stable isotopes of which massive
targets are available.
Use for short-lived nuclei
• Elastic electron scattering even gives details of the charge distribution, and
X-ray spectroscopy on muonic atoms is dealing with systems for which
the absolute shifts with respect to a point nucleus can be calculated. Thus
both methods give absolute values of r2 and not only differences.
Eventually, the combination of absolute radii for stable isotopes and
differences of radii for radioactive isotopes provides absolute radii for
nuclei all over the range that is accessible to optical spectroscopy.
• 799 ground state nuclear charge radii are presented in a survey from 2004
– I. Angeli, At. Data and Nucl. Data Tables 87 (2004), 185
• They are obtained from electron scattering, muonic atom X-rays, Ka
isotope shifts and optical isotope shifts
Coulomb Energy Differences
• Coulomb energy of the charge distribution
• Consider mirror nuclides:

A 1
A 1
;N 
2
2
A 1
A 1
Z
;N 
2
2
3 Q2
EC 
5 4po R
Z

3 e2
3 e2
2
2
2Z  1
DEC 
Z  Z  1 
5 4po R
5 4po R
• Assuming the only difference in mirror nuclides is only due to
Coulomb (charge independence of strong forces) – we can get
information on radius (and check the assumption)
• DEc can be determined from the b-decay of mirror nuclides (from
maximum electron/positron energy) Change in the Coulomb energy
can be expected to depend as A2/3 (from A/R):
3 e2
DE C 
A2 /3
5 4 po R o
Coulomb Energy Differences
• From experimental evidence analyzing
mirror nuclei, we know that nuclear
forces are symmetrical in neutrons and
protons and that nuclear binding between
two neutrons is the same as that between
two protons.
• In the figure the fact that the
experimental values tend to lie on a
straight line indicates that these nuclei
have coulomb energy which correspond
to a constant-density model RC=R0A1/3
• Dotted lines for R0=1.4 and 1.6·10-13 cm
clearly constitute an interval for the
Coulomb-energy unit radius.
Maximum energy of b-ray spectrum (MeV)
A2/3
Measurement of nuclear radius
• Distribution of charge can differ from distribution of matter
• Methods outlined for charge matter radius:
„přímé“
– Diffraction (electron) scattering (form factor) – measurement of charge
distribution
– Muonic x-rays
„relativní“
– Atomic x-rays (shift in Ka) or optical spectra
– Mirror Nuclides (not exactly used for determination of a radius)
• Methods outlined for nuclear matter radius:
– Rutherford scattering (via strong interaction)
v principu se dají použít i následující metody:
– p-mesic x-rays (measurements in 1960’s)
– Alpha particle decay (theory is needed)
– (cross section of fast neutrons)
a-decay lifetime
• The penetration of a depends very critically on the shape and the height of
the of the potential energy barrier and on the kinetic energy of a after
penetration. The height of the barrier is given by the nuclear radius, since
the particle is under the influence of the Colomb repulsion without any
compensating nuclear attraction when its distance from the center is larger
than R. The probability of penetration is closely connected with the decay
lifetime.
• In principle, the theory
of a-decay allows
determination of
the nuclear radius R
from the decay
lifetime and energy
of a particle.
• But requires „perfect“
theory … problems
Example of influence of the radius
on lifetime – simple calculations
Cross section of fast neutrons
• In principle could be used, in reality it is rather problematic
• According to the elemental theory of scattering (QM)
the total cross section of a particle s = sel + sreaction = 2p(R + l)2 ,
where l is “an uncertainty in the position of the incident particle”
(probably “equivalent” to the wavelength of the the particle)
• In the case of fast neutrons, l is very small and there is no Coulomb
interaction
… but reality is a bit more complicated
Měření hmot jader
Quantities which can be measured:
• Maximum energy of a decay (Q-value) … (n,g), b decay
• Frequency measurement … determination of q/m
– storage rings
– mass spectrometer (ISOLTRAP) … ISOL = isotope separator on line
For mass measurements on radioactive nuclides, the two
world’s most prominent instruments today, both in terms of
the final mass uncertainty reached and its sensitivity and the
number of measurements performed, are the
• experimental storage ring (ESR) at GSI (Darmstadt) and
• Penning trap mass spectrometer ISOLTRAP at
ISOLDE/CERN.
Based on:
H.-J. Kluge et al. / Nuclear Instruments and Methods in Physics
Research A 532 (2004) 48–55
Klaus Blaum / Physics Reports 425 (2006) 1-78
Přesnost změřených hmot
Nuclear chart with the relative mass uncertainties dm/m of all known nuclides shown in a
color code (stable nuclides are marked in black). Masses of gray-shaded nuclides are
estimated from systematic trends. Precission of 10-10 – 10-11 can be reached for stable nuclei.
ESR
• When relativistic ions (from heavy ion synchrotron - SIS), accelerated to
almost the velocity of light, collide with a thick target, a broad spectrum of
nuclei with mass and charge numbers below those of the projectile nucleus
fly onward, close to the velocity of the primary beam. An exotic nucleus can
be separated from this mixture almost free of background. This is
accomplished by deflecting the ions in electromagnetic fields and, in
addition, slowing them down in thick layers of matter. This is the basic
principle of the FRS fragment separator at GSI.
FRS–ESR mass measurements
•
Schematic view of the principle of mass measurement in the ESR. The motion of up to four
different species labeled by (m/q)1...4, is indicated. For SMS (left) ions are cooled and have
the same mean velocity v whereas for IMS (right) the ions are ‘‘hot’’ and have different
velocities. gt is an ion-optical parameter, which characterizes the transition point of the ESR
ESR
•
•
•
At the ESR, two new, complementary techniques, Schottky-Mass-Spectrometry (SMS)
and Isochronous-Mass-Spectrometry (IMS), have been developed during the last years
and were used in several experimental runs for mapping large areas of the nuclidic mass
surface.
The target is located at the entrance of the FRagment Separator (FRS), a magnetic high
resolution spectrometer. Depending on the operation mode, the FRS can provide cocktail
beams (a mixture of nuclei, which are characterized by similar mass-to-charge ratio) or
monoisotopic beams. At relativistic velocities the reaction products leave the production
target as highly-charged ions and mainly bare ions occur. The ions are injected as a
bunch of about 400 ns pulse length into the ESR. After injection, the ESR is used as
high-resolution mass analyzer, and the masses are determined from the precise
measurement of their revolution frequencies.
For an unambiguous relation between frequency and mass, the second (velocity
dependent) term on the rhs of the equation on the previous slide must be canceled and
two methods apply. For SMS, the ESR is operated with gt = 2.4, electron cooling is
applied so that Dv/v → 0; and the revolution frequency is determined from a Schottkynoise analysis. For IMS, the ESR is operated in the isochronous mode at gt = 1.4: Ions
are injected with a suitable velocity so that their Lorentz factor g = gt; and their
revolution frequency is determined from their time-of-flight (TOF) for each turn.
Detection in IMS
In the IMS mode of the storage ring the revolution times of each individual
stored ion are measured by a destructive time-of-flight technique. To this end
the ions cross a very thin, metallized carbon foil, being typically a few mg.cm−2
thick, mounted in the ring aperture, and eject at each passage electrons which
are guided by electric and magnetic fields to a suitable detector. In this way,
every ion produces periodically at each passage a time-stamp. With a proper
data analysis software the fast-sampled sum signal can be assigned to
individual ions and their mass can be determined via the measured time of
flight. Due to energy loses in the foil only a few hundred to a few thousand
turns can be observed for one and the same ion.
Detection in SMS
The SMS method in a storage ring is based on the detection of image charges
and provides, as in the case of a Penning trap, single-ion sensitivity. The
revolution frequency of the highly charged ions is determined from a Schottkynoise analysis, i.e., at each turn the induced mirror charges of the circulating
ions on two electrostatic pick-up electrodes is monitored. Typically the 30–34th
harmonics of the signals are picked up by a resonant circuit. The signals of both
pick-up plates are amplified with low-noise amplifiers and then summed. The
Fourier transformed signal delivers the frequency and thus the mass spectrum.
At a charge state of q = 30+ the detection sensitivity is high enough to detect
single ions.
FRS–ESR mass measurements
• In the ESR. After cooling, the nuclides are ‘‘sorted’’ according to their
mass-to-charge ratio in the spectrum (increasing mass-to-charge ratio with
decreasing revolution frequency). The nuclides with known masses
(indicated by full letters in the Fig. on the next slide) are used as calibrants
of the spectrum and thus the so far unknown masses can be obtained. The
inset shows that low-lying isomeric states can be resolved and that the
measurement reaches ultimate sensitivity, i.e., even single ions can be
detected and their mass can be determined with a precision in the order of
50 keV. This is ideally adapted to the requirements of an experiment with
exotic nuclei, which are produced in tiniest amounts, some of them with
rates of the order of a few ions per day.
• Neutron deficient nuclei were produced by bismuth fragmentation.
• Neutron-rich nuclei are of special interest. These neutron-rich nuclei can be
produced at the FRS by fission of high-energy uranium projectiles. IMS is
used, which has the potential to investigate nuclides with half-lives down to
the microsecond range because no cooling is required.
FRS–ESR mass measurements
•
Frequency spectrum of cooled exotic nuclei. The inset, which shows ground and
isomeric excited state of fully stripped 143Sm, demonstrates the ultimate sensitivity
of SMS to detect single ions.
T1/2(143mSm) = 66 s, T1/2(143gSm) = 8.8 min
FRS–ESR mass measurements
• The performance of SMS depends strongly on the features of electron
cooling. Thus, a large cooling force is desired, but a high electron current
causes rapid beam loss due to charge exchange by the capture of electrons
from the electron cooler.
• Mass precission about 35 keV
• With IMS, where no cooling is required at all. There, the ions make only a
few thousand revolutions before they are lost due to the energy loss in the
foil of the TOF-detector
• Mass precision of typically 100 keV is achieved
• In general, precision of ESR is about 10-7 – 10-6 and nuclei with lifetimes
shorter than 1 ms can be measured (using IMS)
The ISOLTRAP experiment
•
•
•
ISOLTRAP is a triple trap mass spectrometer connected to the on-line mass
separator ISOLDE. There, the radionuclides are produced by bombarding a thick
target with 1.4 GeV proton. The produced nuclides diffuse out of the target and are
ionized either by surface, plasma or resonant laser ionization. The 60 keV ion beam
is mass separated in a magnetic spectrometer with a resolving power (m/Dm) of up
to 8000 and delivered to different experiments.
ISOLTRAP measures the mass m via the determination of the cyclotron frequency
nc = (1/2p)(q/m)B of ions with charge q stored in a homogeneous and stable
magnetic field B. The main components of the ISOLTRAP setup are shown in the
Fig. on next page. It consists of three traps that perform specific tasks:
(i) the radiofrequency quadrupole (RFQ) used as a beam conditioning trap in which
the 60-keV ISOLDE beam is decelerated, cooled, and bunched to adapt the beam to
the requirements of ISOLTRAP with respect to its time structure and emittance;
(ii) the preparation Penning trap, in which contaminant ions are removed by a massselective buffer gas cooling technique; and
(iii) the precision Penning trap for the actual mass measurement.
A stable alkali reference ion source located upstream of the RFQ trap allows testing
and preparation of the complete setup before radioactive-beam experiments.
The ISOLTRAP experiment
•
Sketch of the triple
trap mass
spectrometer
ISOLTRAP at
ISOLDE/CERN.
Micro-channel plate
(MCP) detectors are
used to monitor the
ion transfer as well as
to record the TOF
resonance (MCP5) for
the determination of
the cyclotron
frequency. The inset
shows the cyclotron
resonance of 33Ar+
with the fit of a
theoretically expected
curve
Micro-channel plate detector
• A micro-channel plate is a slab made from highly resistive
material of typically 2 mm thickness with a regular array of
tiny tubes or slots (microchannels) leading from one face to
the opposite, densely distributed over the whole surface. The
microchannels are typically approximately 10 mm in diameter
(6 mm in high resolution MCPs) and spaced apart by
approximately 15 mm; they are parallel to each other and
often enter the plate at a small angle to the surface (~8°).
• A single x-ray interacting in a channel of the
MCP produces a charge pulse of about 1000
electrons that emerge from the rear of the
plate. Since the individual tubes confine the
pulse, the spatial pattern of electron pulses at
the rear of the plate preserve the pattern
(image) of x-rays incident on the front surface.
When coupled to an additional MCP and an
electronic readout and display the MCP
becomes an x-ray image intensifier.
• “a small photomultiplier”
Traps and Nobel Prizes
• The Nobel Prize in Physics 1989 - one half awarded to Norman F. Ramsey "for
the invention of the separated oscillatory fields method and its use in the
hydrogen maser and other atomic clocks", the other half jointly to Hans G.
Dehmelt and Wolfgang Paul "for the development of the ion trap technique"
• … Hans Dehmelt's contributions are mainly connected with the development
and use of the Penning trap. He invented ingenious methods of cooling,
perturbing, storing (one single electron was trapped for more than 10 months),
and communicating with the trapped particles, thus forcing them to reveal their
properties.
• The g-factor, being a measure of the magnetism of the electron, has been
determined with twelve significant digits and is now the most accurately
known fundamental constant.
• The Nobel Prize in Physics 2012 - was awarded jointly to Serge Haroche and
David J. Wineland "for ground-breaking experimental methods that enable
measuring and manipulation of individual quantum systems“ – measurements
using traps
• To obtain full spatial confinement requires a potential minimum in all three
dimensions. Moreover, the most desirable confining force is one that causes
simple harmonic motion of the confined particle, i.e., one that is
proportional to the distance of the particle from the center of confinement.
Since no simultaneous trapping in three dimensions is possible by
purely electrostatic potentials, three-dimensional confinement is achieved
in a Penning trap by the superposition of a homogeneous magnetic field
providing radial confinement and an axially symmetric electrostatic
quadrupole field providing axial confinement. For Paul traps, a
radiofrequency quadrupole (RFQ) field is employed for the confinement of
ions.
• “Standard quadrupole mass filters” are 2D
Left: Radiofrequency quadrupole mass filter
electrodes having hyperbolic cross-section. Right:
Equipotential lines for a quadrupole field
generated with the electrode structure shown left.
Paul trap
• In a Paul trap the trapping effect is achieved solely with electric fields. They
consist of a ring electrode and two endcap electrodes that in ideal case are
hyperboles of revolution. Confinement of ions is achieved by using both DC
and AC electric fields. Motion of ions is described with Mathieu equations
which in short describes the suitable combinations of frequency and
amplitude of the electric field for storing ions with certain m/q ratio.
Radio-frequency Paul trap
consisting of two end caps and a
ring electrode. (a) Cutaway view
(after G. Kamas, ed., Time and
Frequency Users's Manual,
National Bureau of Standards
Technical Note 695, 1977). (b)
Cross section, showing the
amplitude of the instantaneous
oscillations for several locations
in the trap.
• Resolution of Paul traps is worse (limited by stability of the electric field)
than that of Penning traps. But they are used in many applications.
Paul trap
From http://mathworld.wolfram.com
• In nuclear physics Paul traps are used mainly for storing and cooling ions.
Some trap structures are prepared so that the center of the trap is exposed for
example for lasers and particle detectors.
Penning trap
• An ideal Penning trap consists of a strong
homogenous magnetic field and a weak
quadrupolar electrostatic potential.
• As a Paul trap, a Penning trap also consists
of ring and endcap electrodes. Quite often
so-called guard or correction electrodes
are placed between endcaps and the ring to
compensate for the truncation of the
hyperbolical electrodes.
• Two types of geometry configurations are
commonly used: hyperbolic and
cylindrical. Both constructs have their own
benefits although in precision experiments
usually hyperbolical are favored due to
better production of quadrupolar electric
field. On the other hand, cylindrical
electrodes are easier to manufacture and
sometimes more open geometry offer
other benefits such as better conductance
of gas.
In contrast to a Paul trap, full
confinement is achieved with
static trapping fields (R ≈1cm).
Penning trap
For the storage of charged particles in a
Penning trap a strong homogeneous magnetic
field B for radial confinement
and a weak static electric field for axial
trapping are superposed. The latter is created
by a voltage U0 (or Udc) applied between
the ring electrode and the two end electrodes.
• An ion with a charge-to-mass ratio q/m stored in a pure magnetic field B =
B(z) in the z-direction and with a velocity component v perpendicular to the
direction of the magnetic field will experience a Lorentz force FL = qv × B.
This force confines the charged particle in the radial direction and the ion
performs a circular motion with angular frequency wc = (q/m)B.
• Since there is no binding in the direction of the magnetic field lines, i.e. in
the axial direction, a three-dimensional confinement is obtained in the
Penning trap by superposing a weak static electric quadrupole potential
F(z, r) = (U0/2d2)(z2 − r2/2) given in cylindrical coordinates. The meaning
of d is: 2d2 = z02 + 02/2.
Penning trap
For an ideal electric quadrupole field there are
three eigenfrequencies of the ion motion
In order that the
motion be bounded,
the roots in Eqs. must
be real, leading to the
trapping condition
Schematic trajectory (three-dimensional and
projection onto the x–y-plane) with ideally three
independent eigenmotions of an ion in a
Penning trap: a harmonic oscillation in the axial
direction (axial motion with frequency wz), and
a radial motion that is a superposition of the
modified cyclotron motion with frequency w+
and the magnetron motion with frequency w−
Eigenfrequencies ni = wi/2p of singly charged ions with different masses in a
hyperbolic Penning trap with operation parameters r0 = 6.38mm, z0 = 5.5mm,
Udc = 10 V, and B = 7T.
n- is almost independent of mass
Cooling of ions in the RFQ trap
•
•
The operating principle of a linear RFQ is based on the radial confinement of ions in
the quadrupolar field of a four-rod structure. The time-averaged radial centering
force can be described as a harmonic pseudo-potential well. The ISOLTRAP RFQ is
in addition filled with He as buffer gas, thus ions are not only radially confined but
also cooled by collisions with buffer gas atoms, and the four rods are 26-fold
segmented and an axial DC potential is applied in order to allow the accumulation of
a number of ions in cooled bunches.
The total length of the RFQ is about 1m and the trap is operated at gas pressures of
about 1 Pa, at a radiofrequency of typically 1 MHz, and at peak-to-peak RF
amplitudes of up to 250 V, depending on the ion mass. After an accumulation period
of about 5–10 ms the ions are ejected towards the preparation trap through a pulsed
drift tube in which their energy is adapted to ground potential.
electrodes of a linear
paul trap (RFQ)
Left: Radiofrequency quadrupole mass filter electrodes having
hyperbolic cross-section. Right: Equipotential lines for a quadrupole
field generated with the electrode structure shown left.
Cooling in a Penning trap
• In ISOLTRAPs preparation Penning trap a combination of He buffer gas
collisions and application of a resonant azimuthal quadrupole
radiofrequency excitation at the true cyclotron frequency nc is used. Both,
cyclotron and axial oscillations are damped by buffer gas collisions. Due to
the potential energy loss by collisions with the buffer gas atoms the
magnetron radius increases. A mass selective recentering of the ions by a
radiofrequency field that couples the modified cyclotron and the magnetron
motion avoids ion loses.
• This mass selective technique allows ions to be cooled to a temperature
equivalent to that of the buffer gas and to eliminate at the same time
contaminant ions of other masses present in the trap. Using this technique, a
mass resolving power of 105 could be demonstrated with 100 ms cooling
time.
Mass determination in Penning trap
Two methods are used for measuring cyclotron frequencies in high-accuracy
mass spectrometry with ion traps:
(1) manipulation of the ion motion by radiofrequency fields and measurement
of the time of flight (TOF) of the ions from the ion trap after ejection to an
ion detector placed outside the magnetic field and
(2) broad-/ narrow-band observation of the oscillating image currents
induced by the motion of the ion in the trap electrodes (detection by image
charges).
TOF measurement in a Penning trap
• The ions’ cyclotron frequency nc is probed by excitation of the ions motion
by a radiofrequency signal and measurement of the TOF to the microchannel-plate (MCP) detector. The cyclotron resonance is determined by
repetition of this sequence and measurement of the TOF as a function of the
frequency of the applied signal. The value of the magnetic field B is
measured by a determination of the cyclotron frequency of a reference ion
with well-known mass both before and after the measurements of the
cyclotron frequency of the ion of interest.
An example for 33Ar+ is shown. A fit of the resonance curve
to the theoretical function yields the cyclotron frequency nc.
TOF measurement in a Penning trap – from a different paper
•
•
•
•
In the time-of-flight ion–cyclotron resonance (TOF-ICR) detection technique the ions are
first prepared at a well-defined radius of the magnetron motion. Here, the orbital frequency
and, therefore, the orbital magnetic moment m as well as the associated energy E = m.B ,
are small. By application of a resonant quadrupolar excitation, with an appropriate choice
of amplitude and excitation time, the magnetron motion is completely converted into the
(modified) cyclotron motion while the radial radius remains constant.
When the ions are ejected from the trap after one full conversion (by lowering the trapping
potential of the downstream end electrode) at initially low axial velocity they drift along
the axis out of the magnetic field. In passing through the magnetic field gradient the ions
get accelerated due to the gradient force and thus the axial velocity of the ions increases.
In each of several experimental cycles, different excitation frequencies are applied. Since
the magnetic moment and the radial energy of the ions are larger in resonance due to the
higher frequency of the cyclotron motion as compared to the magnetron frequency, the
resonantly excited ions arrive earlier at the detector than those ions that have been excited
non-resonantly.
A variation of the quadrupole frequency rf results in a
characteristic time-of-flight cyclotron resonance curve. The
theoretically expected line shape for such a resonance is
mainly determined by the Fourier transformation of the
rectangular time excitation profile and is similar to the
absolute value of the so called sinc(x)-function
f(x)=sin(ax)/(ax).
Image charges detection
• With the detection of the image charges a full resonance spectrum after one
experimental cycle can be obtained instead of repeated probing of the
expected cyclotron frequency.
• The signal of the charged particle stored in a Penning trap is picked up by
means of an attached narrow-band electronic resonance circuit working
under cryogenic conditions (T = 4.2K). It enables the detection of a single
ion as well as further successive measurements with the same ion.
• Generally the axial oscillation is monitored.
Experimental setup for a sensitive, narrow-band detection of a single stored ion. Due to a tuned
resonance circuit with a high quality factor Q an improved detection sensitivity is reached.
The ISOLTRAP experiment
•
•
ISOLTRAP looks back on a highly successful physics program. In total the masses
of 271 radionuclides throughout the entire nuclear chart of the nuclides have been
determined since its installation at the original ISOLDE facility in 1992. The
relative uncertainty is typically dm/m ≈10-8 and even almost up to one order of
magnitude better in some special cases … for nuclei with t1/2 = tens of ms
THE END
zazvonil zvonec a
všech pohádek je konec…
Collinear Laser Spectroscopy
resonant interaction between accelerated ion beam and a parallel laser beam
https://collaps.web.cern.ch/collaps/colinear/ClassicalCollinear.htm
Collinear Laser Spectroscopy
with optical detection of the fluorescent decay on continuous ion beam
measure fluorescent photon decay
Měření hmot jader
• An ideal Penning trap consists of a strong homogenous magnetic field and a
weak quadrupolar electrostatic potential. In contrast to a Paul trap, full
confinement is achieved with static trapping fields. As a Paul trap, a Penning
trap also consists of ring and endcap electrodes. Quite often so-called guard
or correction electrodes are placed between endcaps and the ring to
compensate for the truncation of the hyperbolical electrodes. Two types of
geometry configurations are commonly used: hyperbolic and cylindrical.
Both constructs have their own benefits although in precision experiments
usually hyperbolical are favored due to better production of quadrupolar
electric field. On the other hand, cylindrical electrodes are easier to
manufacture and sometimes more open geometry offer other benefits such as
better conductance of gas.
Optical pumping
• If a weak magnetic field defines the quantization axis in the direction of the
atomic and the laser beam, each absorption of a circularly polarized photon
introduces one unit of angular momentum in the atomic system. This can be
expressed by the selection rule ΔMF = ±1 for σ± light, with σ+ and σ− being
the conventional notations for the circular polarization of the light with
respect to the direction of the magnetic field.
• Repeated absorption and
spontaneous emission of photons
results in an accumulation of the
atoms in one of the extreme MF
states for which the total angular
momentum F = J +I, for an S state
just composed of the electron spin
and the nuclear spin, is polarized.
Atomic hyperfine structure
• Not only the radial distribution of the nuclear charge (monopole
moment) but also the higher multipole electromagnetic moments of
nuclei with a spin I ≠ 0 influence the atomic energy levels. By
interacting with the multipole fields of the shell electrons they cause an
additional splitting called hyperfine structure. For all practical purposes
it is sufficient to consider only the magnetic dipole and the electric
quadrupole interaction of the nucleus with the shell electrons.
• The shell electrons in states with a total angular momentum J ≠ 0
produce a magnetic field at the site of the nucleus. This gives a dipole
interaction energy E = −μ · B. The spectroscopic quadrupole moment of
a nucleus with I ≥ 1 interacts with an electric field gradient produced by
the shell electrons in a state with J ≥ 1 according to E = eQ (∂2V/∂z2).
Externally applied EM fields
• When a nucleus with spin I is implanted into a solid (or liquid) material,
the interaction between the nuclear spin and its environment is no longer
governed by the atomic electrons. For an atom imbedded in a dense
medium, the interaction of the atomic nucleus with the electromagnetic
fields induced by the medium is much stronger than the interaction with
its atomic electrons.
• The lattice structure of the medium now plays a determining role. This
“hyperfine interaction” is observed in the response of the nuclear spin
system to the internal electromagnetic fields of the medium, often in
combination with externally applied (static or radio-frequency) magnetic
fields.
Interakce jádra s vnějšími aplikovanými poli
• Experimental techniques based on measuring the angular distribution of the
radioactive decay are often more sensitive than the atomic HF methods, and
in some cases also allow more precise measurements of the nuclear g factor
and quadrupole moment. This angular distribution is influenced by the
interaction of the nuclear moments with externally applied magnetic fields
and/or electric field gradients after implantation into a crystal
• The radioactive decay intensity is measured as a function of time (TDPAD)
or as a function of an external variable, e.g., a static magnetic field or the
frequency of an applied radio-frequency magnetic field (b-NMR). The
former are called “time differential” measurements and the latter “time
integrated” measurements.
ESR
•
•
•
At the ESR, two new, complementary techniques, Schottky-Mass-Spectrometry (SMS)
and Isochronous-Mass-Spectrometry (IMS), have been developed during the last years
and were used in several experimental runs for mapping large areas of the nuclidic mass
surface.
The target is located at the entrance of the FRagment Separator (FRS), a magnetic high
resolution spectrometer. Depending on the operation mode, the FRS can provide cocktail
beams (a mixture of nuclei, which are characterized by similar mass-to-charge ratio) or
monoisotopic beams. At relativistic velocities the reaction products leave the production
target as highly-charged ions and mainly bare ions occur. The ions are injected as a
bunch of about 400 ns pulse length into the ESR. After injection, the ESR is used as
high-resolution mass analyzer, and the masses are determined from the precise
measurement of their revolution frequencies.
For an unambiguous relation between frequency and mass, the second (velocity
dependent) term on the rhs of the equation on next slide must be canceled and two
methods apply. For SMS, the ESR is operated with gt = 2.4, electron cooling is applied
so that Dv/v → 0; and the revolution frequency is determined from a Schottky-noise
analysis. For IMS, the ESR is operated in the isochronous mode at gt = 1.4: Ions are
injected with a suitable velocity so that their Lorentz factor g = gt; and their revolution
frequency is determined from their time-of-flight (TOF) for each turn.
Příklad
•
Nuclear magnetic resonances
for 8Li (I = 2) implanted into
different non-cubic crystals.
This illustrates the influence
of the implantation host on
the quadrupole frequency as
well as on the resonance line
widths. The nuclear level
splitting for a nucleus with
spin I = 2, submitted to a
magnetic field and an EFG,
and the corresponding
transition frequencies are
shown for one- and twophoton transitions. The five
levels are non-equidistant,
resulting in four equidistant
one-photon resonances in the
NMR spectrum
On-Line NMR/ON
Nuclear Magnetic Resonance on Oriented Nuclei is done at ~10 mK temperatures.
Polarised radioactive nuclei are exposed to an RF field of variable frequency.
When the Zeeman splitting frequency is found
resonant absorption changes the
sublevel populations and hence also the observed anisotropy
a resonance in the
anisotropy versus frequency plot.
• COLlinear LAser
SPectroscopy
On-Line Laser spectroscopy Collinear and In-Source
Methods:
Atomic Hyperfine Structure splitting
68C
u
In Source, Doppler width resolution ~ 2
Collinear Concept - add constant energy
ΔE=const=δ(1/2mv2)≈mvδv
Resolution ~1 MHz, resulting from
the velocity compression of the line
shape through energy increase.
In Cu+ ion, electron states involved are s1/2
andeach
p1/2.form a doublet with F (= I + J) = I
With nuclear spin I these
+1/2 and I - 1/2.
Transitions between these doublets give four lines in two pairs
with related splittings.
- poor resolution (In Source) only for the A (large magnetic
The NSCL Fragment Separator, MSU
Fragmentation b-NMR
Fragments are polarised in their creation.
Implanted in cubic materials, their polarisation can be detected by
measurement of the asymmetry of their beta decay. Application of a
magnetic field creates a Zeeman splitting which is deduced from
resonant destruction of the asymmetry, yielding the nuclear g-factor.
…
• The spectroscopic quadrupole moment can be related to an
intrinsic quadrupole moment Q0 reflecting the nuclear
deformation β, only if certain assumptions about the nuclear
structure are made. An assumption that is often made (but is
not always valid!), is that the nuclear deformation is axially
symmetric with the nuclear spin having a well-defined
direction with respect to the symmetry axis of the deformation
(strong coupling). In this case, the intrinsic and the
spectroscopic quadrupole moment are related as follows:
•
•
•
•
SCATTERING OF HIGH-ENERGY
NEUTRONS BY NUCLEI:
cross section of the very fast neutrons
(usually 14 and 25 MeV neutrons
used) reaches the value 2pR2
THE YIELD OF NUCLEAR
REACTIONS INITIATED BY
PROTONS OR a-PARTICLES:
Comparison of excitation functions
with theory can give information
about nuclear radius
•
•
•
Scattering of e- of high energy (200
MeV)
Diffraction pattern is expected if the
charge is expected to be uniformly
distributed around the nucleus (not
point-like)
Assuming different values of R and b,
one can try to find the best fit
observed angular distribution