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[[File:Pacshowoff.png|thumb|right|upright=2|Nuclear probe in a lattice.]]
[[File:PAC-Spectroscopy-Schema.png|thumb|right|upright=2|Schema of PAC-Spectroscopy]]
The '''perturbed γ-γ
Today only the time-differential perturbed angular correlation ('''TDPAC''') is used.
== History and
[[File:
PAC goes back to a theoretical work by Donald R. Hamilton
Step by step the theory was developed.<ref>{{cite journal | last=Gardner | first=J W | title=Directional Correlation between Successive Internal-Conversion Electrons | journal=Proceedings of the Physical Society. Section A | publisher=IOP Publishing | volume=62 | issue=12 | date=1949-12-01 | issn=0370-1298 | doi=10.1088/0370-1298/62/12/302 | pages=763–779| bibcode=1949PPSA...62..763G }}</ref><ref>{{cite journal | last1=Ling | first1=Daniel S. | last2=Falkoff | first2=David L. | title=Interference Effects in Gamma-Gamma Angular Correlations | journal=Physical Review | publisher=American Physical Society (APS) | volume=76 | issue=11 | date=1949-12-01 | issn=0031-899X | doi=10.1103/physrev.76.1639 | pages=1639–1648| bibcode=1949PhRv...76.1639L }}</ref><ref>{{cite journal|first=M. |last=Fierz|journal= Helvetica Physica Acta|volume=22|year=1949|issue=4|page=489|title=Zur Theorie der Multipolstrahlung|url=https://www.e-periodica.ch/digbib/view?pid=hpa-001:1949:22#499|language=de}}</ref><ref>J.A. Spiers, Nat. Res. Council Canada, Publ. No. 1925 (1950)</ref><ref>{{cite journal | last=Spiers | first=J. A. | title=On the Directional Correlation of Successive Nuclear Radiations | journal=Physical Review | publisher=American Physical Society (APS) | volume=80 | issue=3 | date=1950-11-01 | issn=0031-899X | doi=10.1103/physrev.80.491 | pages=491| bibcode=1950PhRv...80..491S }}</ref><ref>{{cite journal | last1=Falkoff | first1=David L. | last2=Uhlenbeck | first2=G. E. | title=On the Directional Correlation of Successive Nuclear Radiations | journal=Physical Review | publisher=American Physical Society (APS) | volume=79 | issue=2 | date=1950-07-15 | issn=0031-899X | doi=10.1103/physrev.79.323 | pages=323–333| bibcode=1950PhRv...79..323F }}</ref><ref>{{cite journal | last=Racah | first=Giulio | title=Directional Correlation of Successive Nuclear Radiations | journal=Physical Review | publisher=American Physical Society (APS) | volume=84 | issue=5 | date=1951-12-01 | issn=0031-899X | doi=10.1103/physrev.84.910 | pages=910–912| bibcode=1951PhRv...84..910R }}</ref><ref>U. Fano, Nat'l. Bureau of Standards Report 1214</ref><ref>{{cite journal | last=Fano | first=U. | title=Geometrical Characterization of Nuclear States and the Theory of Angular Correlations | journal=Physical Review | publisher=American Physical Society (APS) | volume=90 | issue=4 | date=1953-05-15 | issn=0031-899X | doi=10.1103/physrev.90.577 | pages=577–579| bibcode=1953PhRv...90..577F }}</ref><ref>{{cite journal | last=Lloyd | first=Stuart P. | title=The Angular Correlation of Two Successive Nuclear Radiations | journal=Physical Review | publisher=American Physical Society (APS) | volume=85 | issue=5 | date=1952-03-01 | issn=0031-899X | doi=10.1103/physrev.85.904 | pages=904–911| bibcode=1952PhRv...85..904L }}</ref><ref>{{cite journal|first=K. |last=Adler|journal=Helvetica Physica Acta|volume=25|year=1952|issue=3|page=235|title=Beiträge zur Theorie der Richtungskorrelation|url=https://www.e-periodica.ch/digbib/view?pid=hpa-001:1952:25#237|language=de}}</ref><ref>{{cite journal | last=De Groot | first=S.R. | title=On the theories of angular distribution and correlation of beta and gamma radiation | journal=Physica | publisher=Elsevier BV | volume=18 | issue=12 | year=1952 | issn=0031-8914 | doi=10.1016/s0031-8914(52)80196-x | pages=1201–1214| bibcode=1952Phy....18.1201D }}</ref><ref>F. Coester, J.M. Jauch, Helv. Phys. Acta 26 (1953) 3.</ref><ref>{{cite journal | last1=Biedenharn | first1=L. C. | last2=Rose | first2=M. E. | title=Theory of Angular Correlation of Nuclear Radiations | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=25 | issue=3 | date=1953-07-01 | issn=0034-6861 | doi=10.1103/revmodphys.25.729 | pages=729–777| bibcode=1953RvMP...25..729B }}</ref>
After Abragam and Pound
While until about 2008 PAC instruments used conventional high-frequency electronics of the 1970s, in 2008 Christian Herden and Jens Röder et al. developed the first fully
▲After Abragam and Pound <ref>A. Abragam, R. V. Pound: Influence of Electric and Magnetic Fields on Angular Correlations. In: Physical Review. Band 92, Nr. 4, 15. November 1953, S. 943–962, doi:10.1103/PhysRev.92.943</ref> published their work on the theory of PAC in 1953 inlcuding extra nuclear fields, many studies with PAC were carried out afterwards. In the 1960s and 1970s, interest in PAC experiments sharply increased, focusing mainly on magnetic and electric fields in crystals into which the probe nuclei were introduced. In the mid-1960s, ion implantation was discovered, providing new opportunities for sample preparation. The rapid electronic development of the 1970s brought significant improvements in signal processing. From the 1980s to the present, PAC has emerged as an important method for the study and characterization of materials.<ref>Th. Wichert, E. Recknagel: Perturbed Angular Correlation. In: Ulrich Gonser (Hrsg.): Microscopic Methods in Metals (= Topics in Current Physics. Band 40). Springer, Berlin/Heidelberg 1986, {{ISBN|978-3-642-46571-0}}, S. 317–364, doi:10.1007/978-3-642-46571-0_11</ref><ref>Gary S. Collins, Steven L. Shropshire, Jiawen Fan: Perturbed γ−γ angular correlations: A spectroscopy for point defects in metals and alloys. In: Hyperfine Interactions. Band 62, Nr. 1, 1. August 1990, S. 1–34, doi:10.1007/BF02407659</ref><ref>Th. Wichert, N. Achziger, H. Metzner, R. Sielemann: Perturbed angular correlation. In: G. Langouche (Hrsg.): Hyperfine Interactions of Defects in Semiconductors. Elsevier, Amsterdam 1992, {{ISBN|0-444-89134-X}}, S. 77</ref><ref>Jens Röder, Klaus-dieter Becker: Perturbed γ–γ Angular Correlation. In: Methods in Physical Chemistry. John Wiley & Sons, Ltd, 2012, {{ISBN|978-3-527-32745-4}}, S. 325–349, doi:10.1002/9783527636839.ch10</ref><ref>Günter Schatz, Alois Weidinger, Manfred Deicher: Nukleare Festkörperphysik: Kernphysikalische Messmethoden und ihre Anwendungen. 4. Auflage. Vieweg+Teubner Verlag, 2010, {{ISBN|978-3-8351-0228-6}}</ref> B. for the study of semiconductor materials, intermetallic compounds, surfaces and interfaces. Lars Hemmingsen et al. Recently, PAC also applied in biological systems.<ref>Lars Hemmingsen, Klára Nárcisz Sas, Eva Danielsen: Biological Applications of Perturbed Angular Correlations of γ-Ray Spectroscopy. In: Chemical Reviews. Band 104, Nr. 9, 1. September 2004, S. 4027–4062, doi:10.1021/cr030030v</ref>
▲While until about 2008 PAC instruments used conventional high-frequency electronics of the 1970s, in 2008 Christian Herden and Jens Röder et al. developed the first fully-digitized PAC instrument that enables extensive data analysis and parallel use of multiple probes.<ref>C. Herden, J. Röder, J. A. Gardner, K. D. Becker: Fully digital time differential perturbed angular correlation (TDPAC) spectrometer. In: Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. Band 594, Nr. 2, 1. September 2008, S. 155–161, doi:10.1016/j.nima.2008.05.001</ref> Replicas and further developments followed.<ref>Matthias Nagl, Ulrich Vetter, Michael Uhrmacher, Hans Hofsäss: A new all-digital time differential γ-γ angular correlation spectrometer. In: Review of Scientific Instruments. Band 81, Nr. 7, 1. Juli 2010, S. 073501, doi:10.1063/1.3455186</ref><ref>M. Jäger, K. Iwig, T. Butz: A user-friendly fully digital TDPAC-spectrometer. In: Hyperfine Interactions. Band 198, Nr. 1, 1. Juni 2010, S. 167–172, doi:10.1007/s10751-010-0201-8</ref>
==Measuring principle ==
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[[File:Complexpacspectrum.png|thumb|right|Bottom: A complex PAC-spectrum, top: its Fourier transformation.]]
According to the number n of detectors, the number of individual spectra (z) results after z=n²-n, for n=4 therefore 12 and for n=6 thus 30. In order to obtain a PAC spectrum, the 90° and 180° single spectra are calculated in such a way that the exponential functions cancel each other out and, in addition, the different detector properties shorten themselves. The pure perturbation function remains, as shown in the example of a complex PAC spectrum. Its Fourier transform gives the transition frequencies as peaks.
<math>R(t)</math>, the count rate ratio, is obtained from the single spectra by using:
:<math>R(t)=2\frac{W(180^\circ,t)-W(90^\circ,t)}{W(180^\circ,t)+2W(90^\circ,t)}
</math>
Depending on the spin of the intermediate state, a different number of transition frequencies show up. For 5/2 spin, 3 transition frequencies can be observed with the ratio ω<sub>1</sub>+ω<sub>2</sub>=ω<sub>3</sub>. As a rule, a different combination of 3 frequencies can be observed for each associated site in the unit cell.
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[[File:Pacaufbau.png|thumb|right|Instrumental setup of detectors around the probe.]]
[[File:Energywiki.png|thumb|right|Energy spectrum of <sup>149</sup>Gd with energy windows for start and stop.]]
In the typical PAC spectrometer, a setup of four 90° and 180° planar arrayed detectors or six octahedral arrayed detectors are placed around the radioactive source sample. The detectors used are scintillation crystals of BaF<sub>2</sub> or NaI. For modern instruments today mainly LaBr<sub>3</sub>:Ce or CeBr<sub>3</sub> are used. Photomultipliers convert the weak flashes of light into electrical signals generated in the scintillator by gamma radiation. In classical instruments these signals are amplified and processed in logical AND/OR circuits in combination with time windows the different detector combinations (for 4 detectors: 12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, 43) assigned and counted. Modern digital spectrometers use digitizer cards that directly use the signal and convert it into energy and time values
== Sample materials ==
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=== Implantation ===
[[File:ISOLDE Schema.png|thumb|upright=1.5|Schema of '''Isotope Separator On Line DEvice'''' ([[ISOLDE]]) am [[CERN]]. The proton beam of the [[proton synchrotron|proton synchrotron booster]]s (PSB) creates by fission in targets radioactive nuclei. These are ionized in ion sources, accelerated and due to their different mases separated by magnetic mass sperarators either by GPS (''General Purpose Separator'') or HRS (''High Resolution Separator'').]]
During implantation, a radioactive ion beam is generated, which is directed onto the sample material. Due to the kinetic energy of the ions (1-500 keV) these fly into the crystal lattice and are slowed down by impacts. They either come to a stop at interstitial sites or push a lattice-atom out of its place and replace it. This leads to a disruption of the crystal structure. These disorders can be investigated with PAC. By tempering these disturbances can be healed. If, on the other hand, radiation defects in the crystal and their healing are to be examined, unperseived samples are measured, which are then annealed step by step.
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=== Neutron activation ===
In [[neutron activation]], the probe is prepared directly from the sample material by converting very small part of one of the elements of the sample material into the desired PAC probe or its parent isotope by neutron capture. As with implantation, radiation damage must be healed. This method is limited to sample materials containing elements from which neutron capture PAC probes can be made. Furthermore, samples can be intentionally contaminated with those elements that are to be activated. For example,
=== Nuclear reaction ===
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== Laboratories ==
The currently largest PAC laboratory in the world is located at [[ISOLDE]] in [[CERN]] with about 10 PAC instruments, that receives its major funding form [[BMBF]]. Radioactive ion beams are produced at the ISOLDE by bombarding protons from the booster onto target materials (uranium carbide, liquid tin, etc.) and evaporating the spallation products at high temperatures (
== Theory ==
[[File:Cascade3.png|thumb|right|General γ-γ-cascade with life-time <math>\tau_N</math> of the intermediate state.]]
The first <math>\gamma</math>-quantum (<math>\gamma_1, k_1</math>) will be emitted
:<math>W(\Theta)=\sum^{k_{max}}_{k}A_{kk}P_{k}cos(\Theta)
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</math>
Where <math>I_S</math> is the spin of the intermediate state and <math>I_i</math> with <math>i=1;2</math> the
<math>A_ {kk}</math> is the anisotropy coefficient that depends on the [[angular momentum]] of the intermediate state and the multipolarities of the transition.
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</math>
Typically, the electric field gradient is defined with the largest proportion <math>V_{zz}</
:<math>\eta=\frac{V_{yy}-V_{xx}}{V_{zz}}
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If <math>\eta=0</math>, then:
:<math>\
</math>
with:
:<math>\
</math>
For integer spins applies:
:<math>\
For half integer spins applies:
:<math>\
The perturbation factor is given by:
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==References==
{{Reflist}}
{{Authority control}}
[[Category:Nuclear physics]]
[[Category:Atomic physics]]
[[Category:Electromagnetism]]
[[Category:Spectroscopy]]
[[Category:Scientific techniques]]
[[Category:Laboratory techniques in condensed matter physics]]
[[Category:Solid-state chemistry]]
[[Category:Materials science]]
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