Implantation

Schema of Isotope Separator On Line DEvice' (ISOLDE) am CERN. The proton beam of the proton synchrotron boosters (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.

The implantation is usually the method of choice, because it can be used to produce very well-defined samples.

Evaporation

In a vacuum, the PAC probe can be evaporated onto the sample. The radioactive probe is applied to a hot plate or filament, where it is brought to the evaporation temperature and condensed on the opposite sample material. With this method, e.g. surfaces are examined. Furthermore, by vapor deposition of other materials, interfaces can be produced. They can be studied during tempering with PAC and their changes can be observed. Similarly, the PAC probe can be transferred to sputtering using a plasma.

Diffusion

In the diffusion method, the radioactive probe is usually diluted in a solvent applied to the sample, dried and it is diffused into the material by tempering it. The solution with the radioactive probe should be as pure as possible, since all other substances can diffuse into the sample and affect thereby the measurement results. The sample should be sufficiently diluted in the sample. Therefore, the diffusion process should be planned so that a uniform distribution or sufficient penetration depth is achieved.

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 mm), micro-reactors can be used. Ideally, the probe is added to the liquid phase of the sol-gel process or one of the later precursor phases.

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, hafnium is excellently suited for activation because of its large capture cross section for neutrons.

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.

Magnetic dipole interaction

For the magnetic dipole interaction, the frequency of the precession of the nuclear spin around the axis of the magnetic field ${\displaystyle {\vec {B}}}$ is given by:

${\displaystyle \omega _{L}={\frac {g\cdot u_{N}\cdot B}{\hbar }}}$

${\displaystyle \Delta E=\hbar \cdot \omega _{L}=-g\cdot u_{N}\cdot B}$

${\displaystyle g}$ is the Landé g-factor und ${\displaystyle u_{N}}$ is the nuclear magneton.

With ${\displaystyle N=M-M'}$ follows:

${\displaystyle E_{magn}(M)-E_{magn}(M')=-(M-M')g\mu _{N}B_{z}=N\hbar \omega _{L}}$

From the general theory we get:

${\displaystyle G_{k_{1}k_{2}}^{NN}={\sqrt {(2k_{1}+1)(2k_{2}+1)}}\cdot e^{-iN\omega _{L}t}\times \sum _{M}{\begin{pmatrix}I&I&k_{1}\\M'&-M&N\\\end{pmatrix}}{\begin{pmatrix}I&I&k_{2}\\M'&-M&N\\\end{pmatrix}}}$

For the magnetic interaction follows:

${\displaystyle G_{k_{1}k_{2}}^{NN}=e^{\left({-iN\omega _{L}t}\right)}}$

Static electric quadrupole interaction

The energy of the hyperfine electrical interaction between the charge distribution of the core and the extranuclear static electric field can be extended to multipoles. The monopole term only causes an energy shift and the dipole term disappears, so that the first relevant expansion term is the quadrupole term:

${\displaystyle E_{Q}=\sum _{ij}Q_{ij}V_{ij}}$     ij=1;2;3

This can be written as a product of the quadrupole moment ${\displaystyle Q_{ij}}$ and the electric field gradient ${\displaystyle V_{ij}}$. Both [tensor]s are of second order. Higher orders have too small effect to be measured with PAC.

The electric field gradient is the second derivative of the electric potential ${\displaystyle \Phi ({\vec {r}})}$ at the core:

${\displaystyle V_{ij}={\frac {\partial ^{2}\Phi ({\vec {r}})}{\partial x_{i}\partial x_{j}}}={\begin{pmatrix}V_{xx}&0&0\\0&V_{yy}&0\\0&0&V_{zz}\\\end{pmatrix}}}$

${\displaystyle V_{ij}}$ becomes diagonalized, that:

${\displaystyle |V_{zz}|\geq |V_{yy}|\geq |V_{xx}|}$

The matrix is free of traces in the main axis system (Laplace equation)

${\displaystyle V_{xx}+V_{yy}+V_{zz}=0}$

Typically, the electric field gradient is defined with the largest proportion ${\displaystyle V_{zz}}$ and ${\displaystyle \eta }$:

${\displaystyle \eta ={\frac {V_{yy}-V_{xx}}{V_{zz}}}}$,        ${\displaystyle 0\leq \eta \leq 1}$

In cubic crystals, the axis parameters of the unit cell x, y, z are of the same length. Therefore:

${\displaystyle V_{zz}=V_{yy}=V_{xx}}$ and ${\displaystyle \eta =0}$

In axisymmetric systems is ${\displaystyle \eta =0}$.

For axially symmetric electric field gradients, the energy of the substates has the values:

${\displaystyle E_{Q}={\frac {eQV_{zz}}{4I(2I-1)}}\cdot (3m^{2}-I(I+1))}$

The energy difference between two substates, ${\displaystyle M}$ and ${\displaystyle M'}$, is given by:

${\displaystyle \Delta E_{Q}=E_{m}-E_{m'}={\frac {eQV_{zz}}{4I(2I-1)}}\cdot 3|M^{2}-M'^{2}|}$

The quadrupole frequency ${\displaystyle \omega _{Q}}$ is introduced. The formulas in the colored frames are important for the evaluation:

${\displaystyle \omega _{Q}={\frac {eQV_{zz}}{4I(2I-1)\hbar }}={\frac {2\pi eQV_{zz}}{4I(2I-1)h}}={\frac {2\pi \nu _{Q}}{4I(2I-1)}}}$

${\displaystyle \nu _{Q}={\frac {eQ}{h}}V_{zz}={\frac {4I(2I-1)\omega _{Q}}{2\pi }}}$

The publications mostly list ${\displaystyle \nu _{Q}}$. ${\displaystyle e}$ as elementary charge and ${\displaystyle h}$ as Planck constant are well known or well defined. The nuclear quadrupole moment ${\displaystyle Q}$ is often determined only very inaccurately (often only with 2-3 digits). Because ${\displaystyle \nu _{Q}}$ can be determined much more accurately than ${\displaystyle Q}$, it is not useful to specify only ${\displaystyle V_{zz}}$ because of the error propagation. In addition, ${\displaystyle \nu _{Q}}$ is independent of spin! This means that measurements of two different isotopes of the same element can be compared, such as ^199mHg(5/2−), ^197mHg(5/2−) and ^201mHg(9/2−). Further, ${\displaystyle \nu _{Q}}$ can be used as finger print method.

For the energy difference then follows:

${\displaystyle \Delta E_{Q}=\hbar \omega _{Q}\cdot 3|m^{2}-m'^{2}|}$

If ${\displaystyle \eta =0}$, then:

${\displaystyle \omega _{n}=n\cdot \omega _{0}}$

with:

${\displaystyle \omega _{0}={\textrm {min}}\left({\frac {\Delta E_{Q}}{\hbar }}\right)}$

For integer spins applies:

${\displaystyle \omega _{0}=3\cdot \omega _{Q}}$         und          ${\displaystyle n=|M^{2}-M'^{2}|}$

For half integer spins applies:

${\displaystyle \omega _{0}=6\cdot \omega _{Q}}$          und          ${\displaystyle n={\frac {1}{2}}|M^{2}-M'^{2}|}$

The perturbation factor is given by:

${\displaystyle G_{k_{1}k_{2}}^{NN}=\sum _{n}s_{nN}^{k_{1}k_{2}}\cos {(n\omega _{Q}^{0}t)}}$

With the factor for the probabilities of the observed frequencies:

${\displaystyle s_{nN}^{k_{1}k_{2}}={\sqrt {(2k_{1}+1)(2k_{2}+1)}}\cdot \sum _{M,M'}{\begin{pmatrix}I&I&k_{1}\\M'&-M&N\\\end{pmatrix}}{\begin{pmatrix}I&I&k_{2}\\M'&-M&N\\\end{pmatrix}}}$

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 ${\displaystyle \omega _{n}}$. 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 ${\displaystyle \eta }$. For a probe with different probe nuclei, one needs a parameter that allows a direct comparison: Therefore, the quadrupole coupling constant ${\displaystyle \nu _{Q}}$ independent of the nuclear spin ${\displaystyle {\vec {I}}}$ 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 ${\displaystyle \tau _{n}}$ 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 ${\displaystyle \omega _{s}<0.2\cdot \nu _{Q}}$ for equivalent places ${\displaystyle N_{s}}$, an exponential damping of the static ${\displaystyle G_{22}(t)}$-terms is observed:

${\displaystyle G_{22}^{dyn}(t)=e^{-\lambda _{d}t}G_{22}(t)}$            ${\displaystyle \lambda _{d}=(N_{s}-1)\omega _{s}}$

Here ${\displaystyle \lambda _{d}}$ is a constant to be determined, which should not be confused with the decay constant ${\displaystyle \lambda ={\frac {1}{\tau }}}$. For large values of ${\displaystyle \omega _{s}}$, only pure exponential decay can be observed:

${\displaystyle G_{22}^{dyn}(t)=e^{-\lambda _{d}t}}$

The boundary case after Abragam-Pound is ${\displaystyle \lambda _{d}}$, if ${\displaystyle \omega _{s}>3\cdot \nu _{Q}}$, then:

${\displaystyle \lambda _{d}\approx {\frac {2,5\nu _{Q}^{2}}{N_{s}\omega _{s}}}}$

After effects

Decay scheme of ¹¹¹ In after ¹¹¹Cd, illustrating the initial occupation probabilities between a static Cd²⁺ and a dynamic highly ionized state Cd^x+.

Cores that transmute beforehand of the ${\displaystyle \gamma }$-${\displaystyle \gamma }$-cascade usually cause a charge change in ionic crystals (In³⁺) to Cd²⁺). 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 ${\displaystyle \gamma }$-${\displaystyle \gamma }$-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):

${\displaystyle \mathrm {d} N_{a}=-N_{a}\left(\Gamma _{r}+{\frac {1}{\tau _{7/2}}}\right)\mathrm {d} t}$

mit: ${\displaystyle \tau _{7/2}={\frac {120{\textrm {ps}}}{\ln {2}}}}$

From this one obtains the exponential case:

${\displaystyle N_{a}(t)=N_{a_{0}}\cdot e^{\left({-(\Gamma _{r}+{\frac {1}{\tau _{7/2}}})t}\right)}}$

For the total number of nuclei in the static state (c) follows:

${\displaystyle N_{c}(t)=\Gamma _{r}\int \limits _{0}^{t}N_{a}(t)\mathrm {d} t=N_{0}{\frac {\Gamma _{r}\tau _{7/2}}{\Gamma _{r}\tau _{7/2}+1}}\left(1-e^{-(\Gamma _{r}+{\frac {1}{\tau _{7/2}}})t}\right)}$

The initial occupation probabilities ${\displaystyle \rho }$ are for static and dynamic environments:

${\displaystyle \rho _{stat}={\frac {\Gamma _{r}\tau _{7/2}}{\Gamma _{r}\tau _{7/2}+1}}}$

${\displaystyle \rho _{dyn}={\frac {1}{\Gamma _{r}\tau _{7/2}+1}}}$

General theory

General γ-γ-cascade with life-time ${\displaystyle \tau _{N}}$ of the intermediate state.

In the general theory for a transition ${\displaystyle M_{i}\rightarrow M_{f}}$ is given:

${\displaystyle W(M_{i}\rightarrow M_{f})=\left|\sum _{M}\langle M_{f}|{\mathcal {H}}_{2}|M\rangle \langle M|{\mathcal {H}}_{1}|M_{i}\rangle \right|^{2}}$

${\displaystyle W({\vec {k}}_{1},{\vec {k}}_{2})=\sum _{M_{i},M_{f},\sigma _{1},\sigma _{2}}\left|\sum _{M}\langle M_{f}|{\mathcal {H}}_{2}|M\rangle \langle M|{\mathcal {H}}_{1}|M_{i}\rangle \right|^{2}}$

${\displaystyle W({\vec {k}}_{1},{\vec {k}}_{2})=W(\Theta )=\sum _{k_{gerade}}^{k_{max}}A_{k}(1)A_{k}(2)P_{k}(\cos {\Theta })}$

${\displaystyle 0\leq k\leq }$ Minimum von ${\displaystyle (2I,l_{1}+l_{1}',l_{2}+l_{2}')}$

Detector angles

${\displaystyle W(\Theta ,t)=\sum _{k=2,4}A_{kk}P_{k}(\cos {\Theta })}$

${\displaystyle |M_{a}\rangle \rightarrow \Lambda (t)|M_{a}=\sum _{M_{b}}|M_{b}\rangle \langle M_{b}|\Lambda (t)|M_{a}\rangle }$

${\displaystyle W({\vec {k}}_{1},{\vec {k}}_{2},t)=\sum _{M_{i},M_{f},\sigma _{1},\sigma _{2}}\left|\sum _{M_{a}}\langle M_{f}|{\mathcal {H}}_{2}\Lambda (t)|M_{a}\rangle \langle M_{a}|{\mathcal {H}}_{1}|M_{i}\rangle \right|^{2}=\langle \rho ({\vec {k}}_{2})\rangle _{t}}$

${\displaystyle W({\vec {k}}_{1},{\vec {k}}_{2},t)=\sum _{k_{1},k_{2},N_{1},N_{2}}A_{k_{1}}(1)A_{k_{2}}(2){\frac {1}{\sqrt {(2k_{1}+1)(2k_{2}+1)}}}\times Y_{k_{1}}^{N_{1}}(\Theta _{1},\Phi _{1})\cdot Y_{k_{2}}^{N_{2}}(\Theta _{2},\Phi _{2})G_{k_{1}k_{2}}^{N_{1}N_{2}}(t)}$

with:

${\displaystyle G_{k_{1}k_{2}}^{N_{1}N_{2}}=\sum _{M_{a},M_{b}}(-1)^{2I+M_{a}+M_{b}}{\sqrt {(2k_{1}+1)(2k_{2}+)}}\times \langle M_{b}|\Lambda (t)|M_{a}\rangle \langle M_{b}'|\Lambda (t)|M_{a}'\rangle ^{*}\times {\begin{pmatrix}I&I&k_{1}\\M_{a}'&-M_{a}&N_{1}\end{pmatrix}}{\begin{pmatrix}I&I&k_{2}\\M_{b}'&-M_{b}&N_{2}\end{pmatrix}}}$