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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 angleangular correlation''', '''PAC''' for short or '''PAC-Spectroscopy''', is a method of nuclear solid-state physics with which [[magnetic field|magnetic]] and [[electric fieldsfield]]s in [[crystal structuresstructure]]s can be measured. In doing so, electrical field gradients and the [[Larmor frequency]] in magnetic fields as well as dynamic effects are determined. With this very sensitive method, which requires only about 10-100010–1000 billion atoms of a radioactive [[isotope]] per measurement, material properties in the [[local structure]], phase transitions, magnetism and diffusion can be investigated. The PAC method is related to [[nuclear magnetic resonance]] and the [[Mössbauer effect]], but shows no signal attenuation at very high temperatures.
Today only the time-differential perturbed angular correlation ('''TDPAC''') is used.
== History and Developmentdevelopment ==
[[File:KoinzidenzdetectorCoincidenceDetector.pngsvg|thumb|right|Coincidence measurement in simplified depiction.]]
PAC goes back to a theoretical work by Donald R. Hamilton <ref>{{cite journal | last=Hamilton | first=Donald R. Hamilton:| title=On Directional Correlation of Successive Quanta. In:| journal=Physical Review. Band| publisher=American Physical Society (APS) | volume=58, Nr.| issue=2, 15. Juli| date=1940,-07-15 S.| 122–131,issn=0031-899X | doi:=10.1103/PhysRevphysrev.58.122 | pages=122–131| bibcode=1940PhRv...58..122H }}</ref> from 1940. The first successful experiment was carried out by Brady and Deutsch <ref>{{cite journal | last1=Brady | first1=Edward L. Brady,| last2=Deutsch | first2=Martin Deutsch:| title=Angular Correlation of Successive Gamma-Ray Quanta. In:| journal=Physical Review. Band| publisher=American Physical Society (APS) | volume=72, Nr.| issue=9, 1. November| date=1947,-11-01 S.| 870–871,issn=0031-899X | doi:=10.1103/PhysRevphysrev.72.870 | pages=870–871| bibcode=1947PhRv...72..870B }}</ref> in 1947. Essentially spin and parity of nuclear spins were investigated in these first PAC experiments. However, it was recognized early on that electric and magnetic fields interact with the nuclear moment,<ref>{{cite journal | last1=Aeppli | first1=H. Aeppli,| last2=Bishop | first2=A. S. Bishop,| last3=Frauenfelder | first3=H. Frauenfelder,| last4=Walter | first4=M. Walter,| last5=Zünti | first5=W. Zünti,| Phys.title=Influence Rev.of 82the Atomic Shell on Nuclear Angular Correlation in Cd<sup>111</sup> | journal=Physical Review | publisher=American Physical Society (1951APS) | volume=82 | issue=4 | date=1951-05-15 | issn=0031-899X | doi=10.1103/physrev.82.550 | pages=550| bibcode=1951PhRv...82..550A }}</ref>, providing the basis for a new form of material investigation: nuclear solid-state spectroscopy.
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>
Step by step the theory was developed.<ref>J.W. Gardner, Proc. Phys. Soc. (London) A62 (1949) 763.</ref><ref>D.S. Ling, D.L. Falkoff, Phys, Rev. 76 (1949) 1639.</ref><ref>M. Fierz, Helv. Phys. Acta 22 (1949) 489.</ref><ref>J.A. Spiers, Nat. Res. Council Canada, Publ. No. 1925 (1950) Phys. Rev. 80 (1950) 491.</ref><ref>D.L. Falkoff, G.E. Uhlenboeck, Phys. Rev. 79 (1950) 232.</ref><ref>G. Racah, Phys. Rev. 84 (1951) 910.</ref><ref>U. Fano, Nat'l. Bureau of Standards Report 1214; Phys. Rev. 90 (1953)577.</ref><ref>S.P. Lloyd, Phys. Rev. 85 (1952) 904.</ref><ref>K. Adler, Helv. Phys. Acta 25 (1952) 235.</ref><ref>S.R. de Groot, Physica 18 (1952) 1201.</ref><ref>F. Coester, J.M. Jauch, Helv. Phys. Acta 26 (1953) 3.</ref><ref>L.C. Biedenharn, M.E. Rose, Rev. Mod. Phys. 25 (1953) 729.</ref>
After Abragam and Pound <ref>{{cite journal | last1=Abragam | first1=A. Abragam,| last2=Pound | first2=R. V. Pound:| title=Influence of Electric and Magnetic Fields on Angular Correlations. In:| journal=Physical Review. Band| publisher=American Physical Society (APS) | volume=92, Nr.| issue=4, 15. November| date=1953,-11-15 S.| 943–962,issn=0031-899X | doi:=10.1103/PhysRevphysrev.92.943 | pages=943–962| bibcode=1953PhRv...92..943A }}</ref> published their work on the theory of PAC in 1953 inlcudingincluding 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>{{cite journal | last1=Collins | first1=Gary S. Collins,| last2=Shropshire | first2=Steven L. Shropshire,| last3=Fan | first3=Jiawen Fan:| title=Perturbed γ−γ angular correlations: A spectroscopy for point defects in metals and alloys. In:| journal=Hyperfine Interactions. Band| 62,publisher=Springer Nr.Science 1,and 1.Business AugustMedia LLC | volume=62 | issue=1–2 | year=1990, S.| 1–34,issn=0304-3843 | doi:=10.1007/BF02407659bf02407659 | pages=1–34| s2cid=94593348 }}</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> Be.g. for the study of semiconductor materials, intermetallic compounds, surfaces and interfaces., Larsand Hemmingsena etnumber al.of Recently,applications PAChave also appliedappeared in biological systemsbiochemistry.<ref>Lars{{cite journal | last1=Hemmingsen, | first1=Lars | last2=Sas | first2=Klára Nárcisz Sas,| last3=Danielsen | first3=Eva Danielsen:| title=Biological Applications of Perturbed Angular Correlations of γ-Ray Spectroscopy. In:| journal=Chemical Reviews. Band| publisher=American Chemical Society (ACS) | volume=104, Nr.| issue=9, 1. September| year=2004, S.| 4027–4062,issn=0009-2665 | doi:=10.1021/cr030030v | pages=4027–4062| pmid=15352785 }}</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.{{cite journal | last1=Herden, J| first1=C. | last2=Röder, | first2=J. | last3=Gardner | first3=J.A. Gardner,| last4=Becker | first4=K. D. Becker:| title=Fully digital time differential perturbed angular correlation (TDPAC) spectrometer. In:| journal=Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. Band| publisher=Elsevier BV | volume=594, Nr.| issue=2, 1. September| year=2008, S.| 155–161,issn=0168-9002 | doi:=10.1016/j.nima.2008.05.001 | pages=155–161| bibcode=2008NIMPA.594..155H }}</ref> Replicas and further developments followed.<ref>Matthias{{cite journal | last1=Nagl, Ulrich| first1=Matthias | last2=Vetter, Michael| first2=Ulrich | last3=Uhrmacher, Hans| first3=Michael | last4=Hofsäss: | first4=Hans | title=A new all-digital time differential γ-γ angular correlation spectrometer. In:| journal=Review of Scientific Instruments. Band| publisher=AIP Publishing | volume=81, Nr.| issue=7, 1. Juli| year=2010, S.| 073501,pages=073501–073501–9 | issn=0034-6748 | doi:=10.1063/1.3455186 | pmid=20687716 | bibcode=2010RScI...81g3501N }}</ref><ref>{{cite journal | last1=Jäger | first1=M. Jäger,| last2=Iwig | first2=K. Iwig,| last3=Butz | first3=T. Butz:| title=A user-friendly fully digital TDPAC-spectrometer. In:| journal=Hyperfine Interactions. Band| 198,publisher=Springer Nr.Science 1,and 1.Business Media LLC | volume=198 | issue=1–3 Juni| year=2010, S.| 167–172,issn=0304-3843 | doi:=10.1007/s10751-010-0201-8 | pages=167–172| bibcode=2010HyInt.198..167J | s2cid=17531166 | url=http://ul.qucosa.de/id/qucosa%3A33061 }}</ref>
==Measuring principle ==
[[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.
=== Implantation ===
[[File:ISOLDE Schema.png|thumb|hochkantupright=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.
=== 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, haffniumhafnium is excellently suited for activation because of its large capture cross section for neutrons.
=== Nuclear reaction ===
== 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 (up to 2000 °C), then ionizing them and then accelerating them. With the subsequent mass separation usually very pure isotope beams can be produced, which can be implanted in PAC samples. Of particular interest to the PAC are short-lived isomeric probes such as: <sup>111m</sup>Cd, <sup>199m</sup>Hg, <sup>204m</sup>Pb, and various rare earth probes.
== 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 isotopicallyisotropically. Detecting this quantum in a detector selects a subset with an orientation of the many possible directions that has a given. The second <math>\gamma</math>-quantum (<math>\gamma_2, k_2</math>) has an anisotropic emisisonemission and shows the effect of the angle correlation. The goal is to measure the relative probability <math>W(\Theta)\textrm{d}(\Omega)</math> with the detection of <math>\gamma_2</math> at the fixed angle <math>\Theta</math> in relation to <math>\gamma_1</math>. The probability is given with the angle correlation ([[Perturbation theory (quantum_mechanics)#Time-dependent_perturbation_theory|perturbation theory]]):
:<math>W(\Theta)=\sum^{k_{max}}_{k}A_{kk}P_{k}cos(\Theta)
</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 [[polarity (physics)|multipolarity]] of the two transitions. For pure multipole transitions, is <math>I_i=I'_i</math>.
<math>A_ {kk}</math> is the anisotropy coefficient that depends on the [[angular momentum]] of the intermediate state and the multipolarities of the transition.
If <math>\eta=0</math>, then:
:<math>\omega_nomega^n=n\cdot\omega_0omega^{0}_{Q}
</math>
with:
:<math>\omega_0omega^{0}_{Q}=\textrm{min}\left(\frac{\Delta E_Q}{\hbar}\right)
</math>
For integer spins applies:
:<math>\omega_0omega^{0}_{Q}=3\cdot\omega_Q</math> und <math>n=|M^2-M'^2|</math>
For half integer spins applies:
:<math>\omega_0omega^{0}_{Q}=6\cdot\omega_Q</math> und <math>n=\frac{1}{2}|M^2-M'^2|</math>
The perturbation factor is given by:
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