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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
==Measuring principle ==
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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:
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 ω1+ω2=ω3. As a rule, a different combination of 3 frequencies can be observed for each associated site in the unit cell.▼
:<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
[[File:Zno200NEU.png|thumb|right|PAC-spectrum of single crystal ZnO with fit.]]
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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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== Sample preparation ==
The amount of suitable PAC isotopes required for a measurement is between about 10 to 1000 billion atoms (10<sup>10</sup>-10<sup>12</sup>). The right amount depends on the particular properties of the isotope. 10 billion atoms are a very small amount of substance. For comparison, one mol contains about 6.22x10<sup>23</sup> particles. 10<sup>12</sup> atoms in one cubic centimeter of beryllium give a concentration of about 8 nmol/L (nanomol=10<sup>
How the PAC isotopes are brought into the sample to be examined is up to the experimenter and the technical possibilities. The following methods are usual:
=== 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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=== Added during synthesis ===
PAC probes may also be added during the synthesis of sample materials to achieve the most uniform distribution in the sample. This method is particularly well suited if, for example, the PAC probe diffuses only poorly in the material and a higher concentration in grain boundaries is to be expected. Since only very small samples are necessary with PAC (about 5
=== 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 ===
Rarely used are direct nuclear reactions in which nuclei are converted into PAC probes by bombardment by high-energy elementary particles or protons. This causes major radiation damage, which must be healed. This method is used with PAD, which belongs to the PAC methods.
== 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
:<math>W(\Theta)=\sum^{k_{max}}_{k}A_{kk}P_{k}cos(\Theta)
</math>
For a <math>\gamma</math>-<math>\gamma</math>-cascade, <math>k</math> is due to the preservation of [[
▲For a <math>\gamma</math>-<math>\gamma</math>-cascade, <math>k</math> is due to the preservation of [[parity_(physics)|parity]]:
:<math>0<k<\textrm{min}(2I_S, I_i+I'_i)
</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.
The radioactive nucleus is built into the sample material and emits two <math>\gamma</math>-quanta upon decay. During the lifetime of the intermediate state,
:<math>W(\Theta)=\sum^{k_{max}}_{k}A_{kk}G_{kk}
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</math>
<math>g</math> is the [[
With <math>N=M-M'</math> follows:
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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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In cubic crystals, the axis parameters of the unit cell x, y, z are of the same length. Therefore:
:<math>V_{zz}=V_{yy}=V_{xx}</math> and <math>\eta=0</math>
In axisymmetric systems is <math>\eta=0</
For axially symmetric electric field gradients, the energy of the substates has the values:
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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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As far as the magnetic dipole interaction is concerned, the electrical quadrupole interaction also induces a precision of the angular correlation in time and this modulates the quadrupole interaction frequency. This frequency is an overlap of the different transition frequencies <math>\omega_n</math>. The relative amplitudes of the various components depend on the orientation of the electric field gradient relative to the detectors (symmetry axis) and the asymmetry parameter <math>\eta</math>. For a probe with different probe nuclei, one needs a parameter that allows a direct comparison: Therefore, the quadrupole coupling constant <math>\nu_Q</math> independent of the nuclear spin <math>\vec{I}</math> is introduced.
=== Combined interactions ===
If there is a magnetic and electrical interaction at the same time on the radioactive nucleus as described above, combined interactions result. This leads to the splitting of the respectively observed frequencies. The analysis may not be trivial due to the higher number of frequencies that must be allocated. These then depend in each case on the direction of the electric and magnetic field to each other in the crystal. PAC is one of the few ways in which these directions can be determined.
=== Dynamic interactions ===
If the hyperfine field fluctuates during the lifetime <math>\tau_n</math> of the intermediate level due to jumps of the probe into another lattice position or from jumps of a near atom into another lattice position, the correlation is lost. For the simple case with an undistorted lattice of cubic symmetry, for a jump rate of <math>\omega_s<0.2\cdot \nu_Q</math> for equivalent places <math>N_s</math>, an exponential damping of the static <math>G_{22}(t)</math>-terms is observed:
:<math>G_{22}^{dyn}(t)=e^{-\lambda_d t}G_{22}(t)</math> <math>\lambda_d=(N_s-1)\omega_s
</math>
Here <math>\lambda_d</math> is a constant to be determined, which should not be confused with the decay constant <math>\lambda=\frac{1}{\tau}</math>. For large values of <math>\omega_s</math>, only pure exponential decay can be observed:
:<math>G_{22}^{dyn}(t)=e^{-\lambda_d t}
</math>
The boundary case after Abragam-Pound is <math>\lambda_d</math>, if <math>\omega_s>3\cdot\nu_Q</math>, then:
:<math>\lambda_d\approx\frac{2,5\nu_Q^2}{N_s\omega_s}
</math>
=== After effects ===
[[File:PAC-Spectroscopy-after-effects.png|thumb|right|Decay scheme of <sup>111</sup> In after <sup>111</sup>Cd, illustrating the initial occupation probabilities between a static Cd<sup>2+</sup> and a dynamic highly ionized state Cd<sup>x+</sup>.]]
Cores that transmute beforehand of the <math>\gamma</math>-<math>\gamma</math>-cascade usually cause a charge change in ionic crystals (In<sup>3+</sup>) to Cd<sup>2+</sup>). As a result, the lattice must respond to these changes. Defects or neighboring ions can also migrate. Likewise, the high-energy transition process may cause the [[Auger effect]], that can bring the core into higher ionization states. The normalization of the state of charge then depends on the conductivity of the material. In metals, the process takes place very quickly. This takes considerably longer in semiconductors and insulators. In all these processes, the hyperfine field changes. If this change falls within the <math>\gamma</math>-<math>\gamma</math>-cascade, it may be observed as an after effect.
The number of nuclei in state (a) in the image on the right is depopulated both by the decay after state (b) and after state (c):
:<math>\mathrm{d}N_a=-N_a\left(\Gamma_r+\frac{1}{\tau_{7/2}}\right)\mathrm{d}t</math>
mit: <math>\tau_{7/2}=\frac{120\textrm{ps}}{\ln{2}}</math>
From this one obtains the exponential case:
:<math>N_a(t)=N_{a_0}\cdot e^\left({-(\Gamma_r +\frac{1}{\tau_{7/2}})t}\right)
</math>
For the total number of nuclei in the static state (c) follows:
:<math>N_{c}(t)=\Gamma_r\int\limits_0^tN_a(t)\mathrm{d}t=N_0\frac{\Gamma_r\tau_{7/2}}{\Gamma_r\tau_{7/2}+1}\left(1-e^{-(\Gamma_r+\frac{1}{\tau_{7/2}})t}\right)
</math>
The initial occupation probabilities <math>\rho</math> are for static and dynamic environments:
:<math>\rho_{stat}=\frac{\Gamma_r\tau_{7/2}}{\Gamma_r\tau_{7/2}+1}
</math>
:<math>\rho_{dyn}=\frac{1}{\Gamma_r\tau_{7/2}+1}
</math>
=== General theory ===
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</math>
==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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