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)

2. 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

měření vnitřní kvadrupólový moment

Hyperjemné interakce• Energie stacionární soustavy nábojů a proudů ve vnějším poli

Liché elekrické momenty = 0 = sudé magnetické momenty (zákon zachování )

Energie vyšších řádů lze zanedbat - jsou o několik řádů slabší

náboj;

elektrický dipólový moment (= 0);

magnetický dipólový moment;

elektrický quadrupólový moment;

elstat. potenciál

intenzita el. pole

intenzita magnet. pole

tenzor gradientu el. pole

Zeemanův jev

0

E

I

(2I +1) x degenerovaná hladina

hyperjemnéinterakce

posunutýrozštěpený multiplethladin

E

Interakce magnetického dipólového momentu s vněším magnet. polem:magnet. pole je součet pole od okolí a vnějšího pole

magnetické kvantové č. m,Bohrův magneton, gyromagnet. poměr

(pro m =1) „úplné“ rozštěpení multipletu

57Fe

povolené pouze přechodys mI = -1,0,1

Elektrický kvadrupól v elstat. poli(Starkův jev):

axiální symetrie = 0:

kde eQ je elektrický kvadrupólový moment(multipólový moment je obecně definován jako nultá komponenta „momentu“ pro maximální projekci m = I ):

dochází jen k „částečnému“rozštěpení multipletu

} u jádra s I = 0, ½ nelze takto „určit“ quadrupólový moment – tento moment neexistuje

57Fe

Table of nuclear magnetic dipole and electric quadrupole moments

N.J. StoneAtomic Data and Nuclear Data Tables 90 (2005) 75–176

seznam metod použitých při měření (adoptovaných hodnot)

Metody

• Velikost hyperjemného pole nezávisí pro daný prvek na izotopu

• Lze pole změřit pole pomocí jednoho izotopu a pak měřit momenty u dalších izotopů

• 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

• …

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 β-decaying nuclei that are initially polarized, this resonant destruction of the polarization can be measured via the change in the asymmetry of the β-decay.

• 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 G10

11 = 0. Out of resonance we observe the full initial asymmetry and G10

11 = 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.htmlhttp://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 Spin Quadrupole moment

(mb)

T1/2 (s) γ (MHz/T) beta-Decay asymmetry

(A)

production rate (s-1)

μ+ 1/2 2.2x10-6 135.5 0.33 75

8Li 2 +32 0.842 6.3018 0.33 108

11Be 1/2 13.8 22 0.33 107

15O 1/2 122 10.8 .7 108

17Ne 1/2 0.1 .33 106

• 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 β-

asymmetry 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.

• 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

• 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.

• 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 (-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 non-collinear 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 metodaTechnique developed for measuring the nuclear spin (can be used for measurement of )

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 K) 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:-mesic x-rays (measurements in 1960’s)

–Alpha particle decay (theory is needed)

–(cross section of fast neutrons) – not really used

Diffraction scattering

q = momentum transfer

ki

kf

ki

-kfq

k i

k f k q 2k sin( /2)

• Measure the scattering intensity as a function of to infer the distribution of charge in the nucleus,

r

F q 2

is the inverse Fourier transform of , which is known as the form factor for the scattering

e r

F q

rdrrrqq

qF

dvrVeqF

dvrVkkF

e

rqi

iffi

sin4

, *

Diffraction scattering

r2 3

5R2

• 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

e r constant

e r A

4 R3

4 R3 A

R Ro A1/3

Modulus squared of charge form factors (a) calculated by solving the Dirac equation with HF+BCS proton densities (b)

• For an electron in a given state, its energy depends on• Assume does not change appreciably if Vpoint Vsphere

• Then, E = Esphere - Epoint

• Assume can be E between sphere and point nucleus for

• Compare this E to measurement and we have R.

Atomic X-rays

• Assume the nucleus is uniform charged sphere.• Potential V is obtained in two regions:

– Inside the sphere

– Outside the sphere

V r Ze2

4oR

3

2

1

2

r

R

2

r R

V r Ze2

4or r R

V n*Vn dv

n

V n* V n dv

rR n

*Vn dvrR

1,1(1s), n1, 0

1,1(1s)

E1s

E1s(pt)

E1s(sphere)

n

E1s 2

5

Z4e2

4o

R2

ao3

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?

• Assume that the first term will be ≈ 0 – larger radius (smaller influence)• Then, use E1s from previous slide for each E1s term:

• This X-ray energy difference is called the “isotope shift”

• One can use optical transitions instead of X-ray (K) transitions…

AEAEAEAEAEAEAEAEAEAE ssppspspKK 11221212

3/23/223

24

11

1

45

2 AAR

a

eZAEAEAEAE o

oossKK

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

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.

2222

42

2

2

32,

4

n

emZE

ema

on

oo

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 semi-empirically.

• 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 (short-lived) 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, K isotope shifts and optical isotope shifts

Coulomb Energy Differences

• Coulomb energy of the charge distribution

• Consider mirror nuclides:

2

1;

2

12

1;

2

1

AN

AZ

AN

AZ

EC 3

5

e2

4oRo

A2 /3

R

QE

oC 45

3 2

1245

31

45

3 222

2

ZR

eZZ

R

eE

ooC

• 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)

• Ec can be determined from the -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):

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.

A2/3

Maximum energy of -ray spectrum (MeV)

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 K) 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:-mesic x-rays (measurements in 1960’s)–Alpha particle decay (theory is needed)–(cross section of fast neutrons)

-decay lifetime

• The penetration of depends very critically on the shape and the height of the of the potential energy barrier and on the kinetic energy of 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 -decay allows determination of the nuclear radius R from the decay lifetime and energy of 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 = el + reaction = 2(R + )2 ,where 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, 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,), 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) 48–55Klaus Blaum / Physics Reports 425 (2006) 1-78

Přesnost změřených hmot

Nuclear chart with the relative mass uncertainties m/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. t 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 t = 2.4, electron cooling is applied so that v/v → 0; and the revolution frequency is determined from a Schottky-noise analysis. For IMS, the ESR is operated in the isochronous mode at t = 1.4: Ions are injected with a suitable velocity so that their Lorentz factor = t; 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 g.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 Schottky-noise 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/m) of up to 8000 and delivered to different experiments.

• ISOLTRAP measures the mass m via the determination of the cyclotron frequency c = (1/2)(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 mass-selective 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 m in diameter (6 m in high resolution MCPs) and spaced apart by approximately 15 m; 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 confinementand a weak static electric field for axial trapping are superposed. The latter is created by a voltage U0 (or Udc) applied betweenthe 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 c = (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 (z, r) = (U0/2d2)(z2 − r2/2) given in cylindrical coordinates. The meaning of d is: 2d2 = z0

2 + 02/2.

Penning trap

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 z), and a radial motion that is a superposition of the modified cyclotron motion with frequency + and the magnetron motion with frequency −

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

Eigenfrequencies i = i/2 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.

- 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 c 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:

•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

• 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 c is probed by excitation of the ions motion by a radiofrequency signal and measurement of the TOF to the micro-channel-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 c.

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 as well as the associated energy E = 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(x)/(x).

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 m/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 ENDzazvonil zvonec a

všech pohádek je konec…

Collinear Laser Spectroscopy

https://collaps.web.cern.ch/collaps/colinear/ClassicalCollinear.htm

resonant interaction between accelerated ion beam and a parallel laser beam

Collinear Laser Spectroscopywith 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 (-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 t = 2.4, electron cooling is applied so that v/v → 0; and the revolution frequency is determined from a Schottky-noise analysis. For IMS, the ESR is operated in the isochronous mode at t = 1.4: Ions are injected with a suitable velocity so that their Lorentz factor = t; 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 two-photon transitions. The five levels are non-equidistant, resulting in four equidistant one-photon resonances in the NMR spectrum

On-Line NMR/ONOn-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

68Cu ΔE=const=δ(1/2mv2)≈m

vδvResolution ~1 MHz, resulting from the velocity compression of the line shape through energy increase.

In Source, Doppler width resolution ~ 250 MHzCollinear Concept - add constant energy to ions

On-Line Laser spectroscopy Collinear and In-Source Methods: Atomic Hyperfine Structure splitting

With nuclear spin I these each form a doublet with F (= I + J) = I +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 dipole) splitting

- good resolution (Collinear) for both A and B (smaller electric quadrupole splitting)

In Cu+ ion, electron states involved are s1/2 and p1/2.

The NSCL Fragment Separator, MSU

Fragmentation -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 2R2

• THE YIELD OF NUCLEAR REACTIONS INITIATED BY PROTONS OR -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