Perturbed angular correlation

The perturbed γ-γ angular correlation, PAC for short or PAC-Spectroscopy, is a method of nuclear solid-state physics with which magnetic and electric fields in crystal structures 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 10–1000 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.

The perturbed γ-γ angular correlation, PAC for short or PAC-Spectroscopy, is a method of nuclear solid-state physics with which magnetic and electric fields in crystal structures 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 10–1000 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. 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. R(t) , the count rate ratio, is obtained from the single spectra by using: R(t)=2\frac{W(180^\circ,t)-W(90^\circ,t)}{W(180^\circ,t)+2W(90^\circ,t)} 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 1 +ω 2 =ω 3 . As a rule, a different combination of 3 frequencies can be observed for each associated site in the unit cell. PAC is a statistical method: Each radioactive probe atom sits in its own environment. In crystals, due to the high regularity of the arrangement of the atoms or ions, the environments are identical or very similar, so that probes on identical lattice sites experience the same hyperfine field or magnetic field, which then becomes measurable in a PAC spectrum. On the other hand, for probes in very different environments, such as in amorphous materials, a broad frequency distribution or no is usually observed and the PAC spectrum appears flat, without frequency response. With single crystals, depending on the orientation of the crystal to the detectors, certain transition frequencies can be reduced or extinct, as can be seen in the example of the PAC spectrum of zinc oxide (ZnO). The amount of suitable PAC isotopes required for a measurement is between about 10 to 1000 billion atoms (10 10 -10 12 ). 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 6.22×10 23 particles. 10 12 atoms in one cubic centimeter of beryllium give a concentration of about 8 nmol/L (nanomol=10 −9 mol). The radioactive samples each have an activity of 0.1-5 MBq, which is in the order of the exemption limit for the respective isotope. 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: W(\Theta)=\sum^{k_{max}}_{k}A_{kk}P_{k}cos(\Theta) For a \gamma - \gamma -cascade, k is due to the preservation of parity: 0 Where I_S is the spin of the intermediate state and I_i with i=1;2 the multipolarity of the two transitions. For pure multipole transitions, is I_i=I'_i . A_ {kk} 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 \gamma -quanta upon decay. During the lifetime of the intermediate state, i.e. the time between \gamma_1 and \gamma_2 , the core experiences a disturbance due to the hyperfine interaction through its electrical and magnetic environment. This disturbance changes the angular correlation to: W(\Theta)=\sum^{k_{max}}_{k}A_{kk}G_{kk} G_{kk} is the perturbation factor. Due to the electrical and magnetic interaction, the angular momentum of the intermediate state I_i experiences a torque about its axis of symmetry. Quantum-mechanically, this means that the interaction leads to transitions between the M states. The second \gamma -quantum ( \gamma_2 ) is then sent from the intermediate level. This population change is the reason for the attenuation of the correlation. The interaction occurs between the magnetic core dipole moment \vec{\nu} and the intermediate state I_S or/and an external magnetic field \vec{B} . The interaction also takes place between nuclear quadrupole moment and the off-core electric field gradient V_{zz} . \omega_L=\frac{g\cdot u_N\cdot B}{\hbar} \Delta E=\hbar\cdot\omega_L=-g\cdot u_N\cdot B g is the Landé g-factor und u_N is the nuclear magneton. With N=M-M' follows: E_{magn}(M)-E_{magn}(M')=-(M-M')g\mu_NB_z=N\hbar\omega_L From the general theory we get: G_{k_1k_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: G_{k_1k_2}^{NN}=e^{\left({-iN\omega_Lt}\right)} E_Q=\sum_{ij}Q_{ij}V_{ij}     ij=1;2;3 This can be written as a product of the quadrupole moment Q_{ij} and the electric field gradient 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 \Phi(\vec{r}) at the core: 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} V_{ij} becomes diagonalized, that: |V_{zz}|\ge|V_{yy}|\ge|V_{xx}| The matrix is free of traces in the main axis system (Laplace equation) V_{xx}+V_{yy}+V_{zz}=0 Typically, the electric field gradient is defined with the largest proportion V_{zz} and \eta : \eta=\frac{V_{yy}-V_{xx}}{V_{zz}} ,         0\le\eta\le 1 In cubic crystals, the axis parameters of the unit cell x, y, z are of the same length. Therefore: V_{zz}=V_{yy}=V_{xx} and \eta=0 In axisymmetric systems is \eta=0 . For axially symmetric electric field gradients, the energy of the substates has the values: E_Q=\frac{eQV_{zz}}{4I(2I-1)}\cdot (3m^2-I(I+1)) The energy difference between two substates, M and M' , is given by: \Delta E_Q=E_m-E_{m'}=\frac{eQV_{zz}}{4I(2I-1)}\cdot 3|M^2-M'^2| The quadrupole frequency \omega_Q is introduced. The formulas in the colored frames are important for the evaluation: \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)} \nu_Q=\frac{eQ}{h}V_{zz}=\frac{4I(2I-1)\omega_Q}{2\pi} The publications mostly list \nu_Q . e as elementary charge and h as Planck constant are well known or well defined. The nuclear quadrupole moment Q is often determined only very inaccurately (often only with 2-3 digits). Because \nu_Q can be determined much more accurately than Q , it is not useful to specify only V_{zz} because of the error propagation. In addition, \nu_Q is independent of spin! This means that measurements of two different isotopes of the same element can be compared, such as 199m Hg(5/2−), 197m Hg(5/2−) and 201m Hg(9/2−). Further, \nu_Q can be used as finger print method. For the energy difference then follows: \Delta E_Q=\hbar\omega_Q\cdot 3|m^2-m'^2| If \eta=0 , then: \omega^n=n\cdot\omega^{0}_{Q} with: \omega^{0}_{Q}=\textrm{min}\left(\frac{\Delta E_Q}{\hbar}\right) For integer spins applies: \omega^{0}_{Q}=3\cdot\omega_Q          und          n=|M^2-M'^2| For half integer spins applies: \omega^{0}_{Q}=6\cdot\omega_Q          und          n=\frac{1}{2}|M^2-M'^2| The perturbation factor is given by: G_{k_1k_2}^{NN}=\sum_ns_{nN}^{k_1k_2}\cos{(n\omega_Q^0t)} With the factor for the probabilities of the observed frequencies: s_{nN}^{k_1k_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 \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 \eta . For a probe with different probe nuclei, one needs a parameter that allows a direct comparison: Therefore, the quadrupole coupling constant \nu_Q independent of the nuclear spin \vec{I} is introduced. G_{22}^{dyn}(t)=e^{-\lambda_d t}G_{22}(t)             \lambda_d=(N_s-1)\omega_s Here \lambda_d is a constant to be determined, which should not be confused with the decay constant \lambda=\frac{1}{\tau} . For large values of \omega_s , only pure exponential decay can be observed: G_{22}^{dyn}(t)=e^{-\lambda_d t} The boundary case after Abragam-Pound is \lambda_d , if \omega_s>3\cdot\nu_Q , then: \lambda_d\approx\frac{2,5\nu_Q^2}{N_s\omega_s} \mathrm{d}N_a=-N_a\left(\Gamma_r+\frac{1}{\tau_{7/2}}\right)\mathrm{d}t mit: \tau_{7/2}=\frac{120\textrm{ps}}{\ln{2}} From this one obtains the exponential case: 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: 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) The initial occupation probabilities \rho are for static and dynamic environments: \rho_{stat}=\frac{\Gamma_r\tau_{7/2}}{\Gamma_r\tau_{7/2}+1} \rho_{dyn}=\frac{1}{\Gamma_r\tau_{7/2}+1} 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 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 W(\vec{k}_1,\vec{k}_2)=W(\Theta)=\sum_{k_{gerade}}^{k_{max}}A_k(1)A_k(2)P_k(\cos{\Theta}) 0\leq k\leq Minimum von (2I,l_1+l_1',l_2+l_2') W(\Theta,t)=\sum_{k=2,4}A_{kk}P_k(\cos{\Theta}) |M_a\rangle\rightarrow\Lambda(t)|M_a=\sum_{M_b}|M_b\rangle\langle M_b|\Lambda(t) |M_a\rangle 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 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_1k_2}^{N_1N_2}(t) with: G_{k_1k_2}^{N_1N_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} ==References==

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