3. Perturbed Angular Correlation Spectroscopy

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1 3. Perturbed Angular Correlation Spectroscopy Dileep Mampallil Augustine K.U.Leuven, Belgium Perturbed Angular Correlation Spectroscopy (PAC) is a gamma ray spectroscopy and can be used to investigate the hyperfine interactions at specific probe nuclei artificially induced into the crystal. They are based on the observation of the electric and magnetic hyperfine interaction between the nuclear moments of specific probe nuclei and the magnetic or electric fields experienced from the surroundings of the probes. PAC measures the time dependence of the -ray emission pattern. 3.1 Hyperfine interactions As a consequence of the Pauli exclusion principle, the electrons in an atom are located at different energy levels within the atom. These energy levels are separated by energies of the order of ev. The degeneracy of these levels are lifted due to the spin orbit coupling and the atomic levels are split into different energy levels separated by mev. This is the fine structure of the atomic levels. The atomic nucleus is not spherically symmetric. The charge-charge interactions between nucleus and electrons can bring nonzero quadrupole terms in their interaction Hamiltonian. This nonzero quadrupole terms produce an electric field gradient (EFG) in the atom. The dipole contributions of current-current interactions between nucleus and the electrons can produce magnetic hyperfine fields inside the crystal. The atomic energy levels are split into energy levels differing ev and is called the hyperfine structure of the atom Electric field gradient The electric field gradient (EFG) causes the nuclear energy levels to split up into non-equidistant levels. The EFG occurs if there is a non spherically symmetric charge distribution present in the vicinity of the nucleus. If a suitable coordinate system is chosen the EFG can be represented by three principal axes, V xx, V yy and V zz where direction of EFG is taken as z direction. Thus an asymmetry parameter is defined as = V xx V yy V zz (3.1)

2 and V zz V yy V xx (3.) The asymmetry parameter and the V zz determine the electric field gradient. If the EFG is axially symmetric, the asymmetry parameter becomes zero. And the quadrupole interaction energy is given by E m I = eqv zz 4II 1 [3m I I I 1] (3.3) where Q is the quadrupole moment, V zz the EFG and I the nuclear spin. The splitting between energy levels can be expressed as E m I m I ' =3 m I m I ' ħ Q (3.4) where Q = eqv zz 4II 1ħ (3.5) The energy level splitting for I=5/ level is illustrated in figure3.1 m I E ± 5 I=5/ 0 ± 3 0 ± 1 Figure 3.1: The energy level splitting for I=5/ in the presence of an axially symmetric electric field gradient. 0 =6 Q The electric field gradient is dependent on temperature. The EFG lowers when the temperature raises. regular spd compounds, V zz T =V zz 01 BT 1.5 (3.6) where the factor B has the order of magnitude of K 1.5. materials with f electrons, the temperature dependence is linear as, V zz T =V zz 01 BT 1 (3.7) 3.1. Magnetic hyperfine field Magnetic hyperfine field is the field present at the position of the nucleus. The origin of hyperfine

3 fields is the dipole contributions of the current-current interactions in the atom and the interaction energy is given by, E m I = g I N m I B hf (3.8) where g I is the g-factor and N the nuclear magneton. The nuclear spin I can be parallel or antiparallel to B hf, dependent on the sign of g I. the presence of a magnetic field, externally applied or a hyperfine field the nuclear level I splits into I+1 equidistant levels. The separation in energy is given by, E m I = g I N B hf =ħ L (3.9) The level splitting for I=5/ is illustrated in figure 3.. B I=5/ L m I Figure3.: The equidistant splitting of the nuclear spin I=5/ under a magnetic hyperfine field. 3. Angular correlation The angular correlation means that the two consecutive ray emissions are correlated with an angular dependence. An anisotropic angular distribution of radiation is obtained when the state from which the radiations are emitted is polarized. Polarization of a state means the unequal population of its substates. Decay scheme of a nucleus used in the PAC is shown below.

4 I i M i I M 1 l 1, m 1 l, m I f M f Figure 3.3: Decay of a nucleus used in the PAC spectroscopy. The angular distribution of the radiation depends on the spin of the nuclear level and on the population of the m levels and on the multipolarity of the radiation. The conservation of the angular momentum cause the nuclear spin to align in a particular direction after the first ray emission. Thus the alignment of the spin causes that the emission is angle dependent relative to the 1 emission Unperturbed angular correlation Consider a nuclear decay as shown in figure 3.3. The initial state I i M i decays by an emission of 1 into the intermediate state I M and then into the final state I f M f by emission of. Assuming equal M state occupation in the initial nuclear state, the population of an M substate in the intermediate state is given by P M = M i G M i M F l1 m 1 1 (3.10) where F l1 m 1 1 is the angular distribution function of gamma photon having multipolarity l 1 and m 1 =M i M. The factor G M i M is the integrated rate for the transition between I i M i and I M. The angular distribution function for the dipole radiation is illustrated in figure 3.4 The observation of 1 in a particular direction implies that the magnetic substates of I M are not equally populated. The alignment of the intermediate state cause the second gamma radiation to be anisotropic. Thus the emission of in coincidence with 1 has the following angular dependence.

5 W P M G M M f F l m M, M f (3.11) Figure 3.4: The angular radiation distribution of l=1 state. The principle of an angular correlation experiment is illustrated in figure 3.5. a fixed time number of coincidences are registered in the coincidence counter. The difference between the number of coincidence counts under 90 and 180 in equal time intervals is called the anisotropy. Detector Detector 1 1 Detector 3 Coincidence counter Figure 3.5: The principle of PAC. The angular distribution of radiation is shown. The detector or 3 detect. For detector, there is a nonzero probability for detecting.

6 3.. Perturbed angular correlation Consider a ray cascade for which the intermediate state has a certain lifetime. If the nucleus is subject to a hyperfine interaction, the interaction can cause repopulation or phase change of the magnetic substates of the intermediate state. The nuclear spin precesses in the extranuclear hyperfine field. Consequently, the anisotropy becomes time dependent. Textbooks [35] show that, one can obtain the following form for the time-dependent angular correlation, N W k 1, k, t= A k1 1 A k G 1, N t 1 k 1, k k 1, k, N 1, N k 1 1 k 1 Y N 1 N k 1 1, 1 Y k, --- (3.1) N where G 1, N t k 1, k is the perturbation factor. When the perturbation for the intermediate state vanishes, the above equation reduces to an equation for unperturbed angular correlation. When there is an axially symmetric magnetic dipole interaction, it can be derived that the perturbation factor becomes N, G N k, k t =exp i N L t (3.13) The angular correlation may precess at the fundamental frequency L as well as higher harmonics N L. The relative amplitude of the harmonics depends on the orientation of the hyperfine field relative to the detector geometry. Thus perturbed angular correlation spectroscopy is very sensitive to the orientation of the hyperfine field. case of a 5/ spin nuclear level the perturbation factor shows single and double frequencies and the anisotropy function is given by W t a 0 a 1 cos L ta cos L t (3.14) Where a 0, a 1 and a are the amplitude of the frequency harmonics. The direction of hyperfine field relative to the orientation of the detectors is shown in figure 3.6. It is straight forward to calculate the relative amplitudes a 0, a 1 and a for any orientation of the hyperfine field.

7 stop-180 B hff stop-90 start Figure 3.6: Showing the direction of hyperfine field relative to the detector setup and the angle and are defined For ( =0, =0 ) orientation, the hyperfine field is oriented along the start detector and there is no periodic contribution. For ( =90, ) the hyperfine field is perpendicular to the detector plane and there will be only a double harmonic frequency and all other components including the single harmonic are zero. For ( =0, =45 ) only single harmonic will be observed, all other components are zero including the constant term. And for ( =45,=45 ) orientation, both the single and the double harmonic frequencies are observed with equal amplitude. the presence of a static axially symmetric electric quadrupole interaction, the perturbation factor reduces to N, G N k k 1, k t = s 1, k 0 nn cosnq t (3.15) n where ħ Q 0 is the smallest non vanishing energy difference between M levels. The equation shows that the angular correlation rotates with frequencies Q 0, Q 0,..., n max Q 0 and orientation dependent amplitudes Overhauser distribution Cr the magnetic moment and the magnetic hyperfine field are spatially varying according to a sine. Therefore the probability P to find a specific hyperfine field H is Overhauser distributed. The hyperfine field as a function of the position x can be written as H x =H 0 sin x (3.16) The probability function for the hyperfine field is given by

8 P H = 1 H 0 H (3.17) And the correlation function is given by Rt =a o a 1 J 0 H 0 ta J 0 H 0 t (3.18) where J 0 is the zeroth order Bessel function. 3.3 PAC experiments This section explains the PAC experimental setup. There are different sources that can be used for the PAC experiments. But we used as the source for all our experiments The PAC source The isotope source. The decay of which decays to Cd is illustrated in figure 3.7 by electron capture (EC) is the most often used PAC Cd 171 kev 45 kev EC t 1/ =85ns t 1/ =.83 days 5/= N Q5/=0.8313b A = 0.18 A 44 = A 4 = 0.06 A 4 = Figure 3.7: The decay scheme of The nuclear moments of the intermediate nuclear level and the anisotropy coefficients of the gamma ray cascade are shown on the right The data in figure 3.7 show that that can be used are 181 Hf 181 Ta Cd and is an excellent probe for the PAC experiment. Other probes 100 Pd 100 Rh.

9 3.4 Experimental apparatus N The measurement of the perturbation factor G 1, N t k 1, k gives the complete information about the interactions. The PAC setup should be capable of measuring this perturbation factor. Due to the perturbations the emission pattern rotates and the detectors at fixed angles record time dependent coincident counts rates. The coincidence counts can be converted into frequencies that reflects hyperfine interaction strengths. The detector setup is illustrated in figure 3.8. The figure shows a four detectors set ( BaF detectors) arranged in one plane. There are four more detectors arranged in a perpendicular plane and at 45 relative to the detection in the horizontal plane. Figure 3.9: The setup for PAC measurements. The MCA is connected to a computer system. Two signals are derived from each detector, an energy and a time signal. The anode signal is transformed into a digital timing pulse by a constant fraction discriminator (CFD). Constant fraction discriminator is used to remove the pulse height dependence of the signal. a constant fraction discriminator the input signal is split; one part is delayed and the other part is inverted, attenuated, and then added to the first part. The resulting signal is independent of pulse height and slope of the signal.

10 The signals from the CFD are fed into time to amplitude converter (TAC). The signals from stop detector are delayed between CFD and TAC. The TAC acts like a stop watch. When a signal is detected in the start detector, the clock starts and stops when a signal is detected at the stop detector. It converts the time difference between two digital input signals into an output signal whose pulse height is proportional to the time difference. TAC is calibrated for time against number of channels using 'ORTEC time calibrator'. The TAC signals are converted to a digital number using analog to digital converter (ADC). The dynode or the energy signal is amplified and sent to single channel analyzer (SCA) which selects signal with desired energies. The energy range for the start and the stop signals can be set in the SCA. The signals from SCA are sent to the coincidence unit. The coincidence unit selects signals of correct coincidence and also determines which detectors are involved in it. The signals from the coincidence unit and the digital signal from the ADC are sent to ADC -Routing and then to the multi channel analyzer (MCA). MCA produces separate spectrum for each detector combination. the multichannel analyzer one obtains a count rate of form N, t= N 0 exp t / N W, t B (3.19) where N is the lifetime of the intermediate state, W, t is the time dependent angular correlation and B is the background which occurs due to random accidental coincidences. If N t be the average of the normalized individual coincidence spectra after background correction, the time-dependent anisotropy ratio is defined as Rt = N t N t N 180 tn 90 t (3.0) where N 90 t= N AD t N BC t 1/ (3.1) and N 180 t= N AC t N BD t 1/ (3.) Here the subscripts AC, AD, BC, BD represent the detector combinations of A, B,C and D detectors. 3.5 The PAC spectra An example of a PAC anisotropy function is illustrated in figure (The spectra is of a cadmium single crystal activated by by diffusion. The crystal is annealed at 50 C for 0 minutes). The

11 cadmium has a hexagonal crystal structure (c/a=1.89) which produces an axially symmetric electric field gradient. The c-axis of the crystal and the detectors are in the same plane. Figure 3.10: The R(t) function of a PAC spectrum for metal. Cd in cadmium The PAC spectra reflect the electric field gradient inside the lattice of the crystal. For instance measuring the frequency Q from PAC, the electric field gradient V zz can be calculated using the following relation, Q = eqv zz 4II 1 ħ (3.3) where the quadrupole moment of the intermediate state of Cd is known. The analysis of the PAC spectra of a magnetic material can reveal the hyperfine fields and its orientations inside the material. The hyperfine field B z and the frequency L are related by the relation, ħ L = g N B z (3.4) where g is the Landé g-factor and N is the nuclear magneton.