Statické momenty jader – metody jejich měření

1. Statické momenty jader – metody jejich měření

2. 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

3. 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).

4. 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.

5. 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.

6. 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 • …

7. TDPAD • Spin-oriented isomeric states implanted into a suitable host will exhibit anon-isotropic angular distribution pattern, provided the isomeric ensembleorientation 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.

8. 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 itsmagnetic interaction (MI) and in cooled polycrystalline mercury to measure its quadrupole interaction (QI). • Detectors are placed in a plane perpendicular to the magnetic fielddirection (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

9. Příklady • R(t) curves obtained in the study of g-factors of9/2+ 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 projectilefragmentation reaction at the LISE high-resolution in-flight separator at GANIL.

10. β-NMR • Time-differential measurements areonly suited for short-lived nuclear states, mainly because of relaxation effectscausing a dephasing of the Larmor precession frequencies with time (typicallyin less than 100 μs). To measure nuclear moments of longer-livedisomeric states and also for ground states, a time-integrated measurement isrequired. 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.

11. 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 functionof the rf frequency or as a function of the static field strength. At resonance, the initial asymmetry is fully destroyed if sufficientrf power is applied, which corresponds to G1011 = 0. Out of resonancewe observe the full initial asymmetry and G1011 = 1.

12. 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.

13. Příklad • NMR curve for 11Be implanted in metallic Be at T = 50K. At thistemperature the spin-lattice relaxation time T1 is of the order of the nuclear lifetimeτ = 20 s. Larmorova frekvence:

14. 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). • 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.

15. 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

18. 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 interactionwith a rf field, but by misaligning the magnetic dipole and electricquadrupole interactions. This experimental technique does not need anadditional rf field to induce changes of the spin orientation. The change ofthe spin orientation is induced by the quantum mechanical “anti-crossing” ormixing of levels, which occurs in quantum ensembles where the axial symmetry is broken.

19. Nuclear HF levels of a nucleus with spin I = 3/2 submitted to acombined 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 occursat well-defined values for the ratio of the involved interactions frequencies, if • (d) At these positions, resonances are observed in the decayangular distribution of oriented radioactive nuclei, from which the nuclear spin and moments can be deduced

21. 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 forJ < I andJ > 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

22. 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.

23. Optical pumping • Polarization of a fast beam by optical pumping was introduced for theβ-asymmetry detection of optical resonance in collinear laser spectroscopy. • Most applications took advantage of theadditional option to perform nuclear magnetic resonance spectroscopy withβ-asymmetry detection (β-NMR) on a sample obtained by implantation ofthe polarized beam into a suitable crystal lattice. Whatever is the particulargoal of such an experiment, it is important to achieve a high degree of nuclear polarization. • Optical pumping within the hyperfine structure Zeeman levels for polarizationof 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 atomsin one of the extreme MF states for which the total angular momentumF = J +I, for an S state just composed of the electron spin and the nuclear spin, is polarized.

25. Optical pumping • If a weak magnetic field defines the quantization axis in the direction of the atomic and the laserbeam, each absorption of a circularly polarized photon introduces one unitof 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 polarizationof the light with respect to the direction of the magnetic field. • Repeated absorption and spontaneous emission of photons results in an accumulation of the atomsin one of the extreme MF states for which the total angular momentumF = J +I, for an S state just composed of the electron spin and the nuclear spin, is polarized.

26. Collinear Laser Spectroscopy

29. Rabiho metoda The experiment setup contains 3 parts: • an inhomogeneous magnetic field in front (A), • the weak rotating + 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 Larmorfrequency in the strong uniform field (at C), it will change the spin direction and neutrons do not focus on the detector (blue line).

30. Měření rozměrů (poloměrů) atomových jader

31. Measurement of nuclear radius • Distribution of charge can differ from distribution of matter • Methods outlined for charge matter radius: • Diffraction (electron) scattering (form factor) – measurement of charge distribution • Atomic x-rays • Muonic x-rays • Mirror Nuclides • Methods outlined for nuclear matter radius: • Rutherford scattering (via strong interaction) • Alpha particle decay • -mesic x-rays • (cross section of fast neutrons)

32. Diffraction scattering q = momentum transfer • Measure the scattering intensity as a function of  to infer the distribution of charge in the nucleus, is the inverse Fourier transform of , which is known as the form factor for the scattering

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

34. Atomic X-rays • Assumethe nucleus is uniform charged sphere. • Potential V is obtained in two regions: • Inside the sphere • Outside the sphere • For an electron in a given state, its energy depends on • Assume does not change appreciably if Vpt Vsphere • Then, E = Esphere - Ept • Assume can be • E between sphere and point nucleus for • Compare this E to measurement and we have R.

35. 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 transitions…

36. Use for short-lived nuclei • Let A, A and mA, mAbe the mass numbers and atomicmasses of the isotopes involved. Then for an atomic transition i the isotopeshift, 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 (secondterm) are factorized into an electronic and a nuclear part. The knowledgeof the electronic factors Fi(field shift constant) and Mi(mass shiftconstant) allows one to extract the quantity δr2of 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 approachto get precise information on the nuclear charge radii, because it issensitive enough to be performed on the minute quantities of (short-lived)radioactive atoms produced at accelerator facilities. • Other techniques aresuitable only for stable isotopes of which massive targets are available.

37. Use for short-lived nuclei • Elasticelectron scattering even gives details of the charge distribution, andX-ray spectroscopy on muonic atoms is dealing with systems for whichthe absolute shifts with respect to a point nucleus can be calculated. Thusboth methods give absolute values of r2and not only differences. Eventually,the combination of absolute radii for stable isotopes and differences ofradii for radioactive isotopes provides absolute radii for nuclei all over therange that is accessible to optical spectroscopy.

38. Muonic X-rays • Similar to “standard” X-rays measurement • Muons are heavier than electrons (106 MeV x 511 keV) which causes the difference in the radius and energy (energy difference) 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.

39. Coulomb Energy Differences • Coulomb energyof the charge distribution • Considermirror nuclides: • Can be determined from the b-decay of mirror nuclides (from maximum electron/positron energy) – the only difference in mirror nuclides is expected due to the Coulomb energy • Change in the Coulomb energy can be expected to depend as A2/3 (from A/R):

40. Coulomb Energy Differences Maximum energy of b-ray spectrum (MeV) • From experimental evidence analyzing mirror nuclei, we knowthat nuclear forces are symmetrical in neutrons and protons andthat 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 coulombenergywhich correspond to a constant-density modelRC=R0A1/3 • Dotted lines forR0=1.4 and 1.6·10-13 cm clearly constitute an interval for theCoulomb-energy unit radius. A2/3

41. Measurement of nuclear radius • Distribution of charge can differ from distribution of matter • Methods outlined for charge matter radius: • Diffraction (electron) scattering (form factor) – measurement of charge distribution • Atomic x-rays • Muonic x-rays • Mirror Nuclides • Methods outlined for nuclear matter radius: • Rutherford scattering (via strong interaction) • Alpha particle decay • -mesic x-rays • (cross section of fast neutrons)

42. 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. Example of influence of the radius on lifetime – simple calculations

43. 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 probably a bit more complicated

45. Měření hmot jader

46. 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 Formass 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

47. 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.

48. 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.

49. 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

50. 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.