A numerical study of the anisotropic distribution of γ-ray emission from oriented 129m,131m,133mXe

Spin-polarized radioactive nuclei emit radioactive decay products in an anisotropic manner that is characteristic of their degree of nuclear orientation. This property can be utilized for nuclear magnetic resonance spectroscopy and imaging. In this manuscript, we present Python and MATLAB numerical simulations that investigate the angular distribution and measurable asymmetry of γ-ray emission from spin-oriented nuclei of the metastable isotopes 129m,131m,133m Xe. We examine several cases that represent different degrees of spin alignment and detection geometries. The results of our simulations provide a benchmark for experiments that use γ-decay anisotropy, such as the GAMMA-MRI project, which aims to develop a novel medical-imaging technique combining the strengths of magnetic resonance and nuclear medicine imaging.


Introduction
Over the past few decades, several authors have presented detailed discussions on the topic of nuclear spin * Authors to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.orientation 5 and the resulting angular correlations and angular distribution of nuclear radiation [2][3][4][5][6][7].Some of these studies were specifically focused on γ-emitting nuclei.In nuclearstructure research, γ-emitting nuclei were utilized to determine the spins and parities of excited states in nuclei [2,3], or to measure magnetic moments of unstable nuclei through γ-detected nuclear magnetic resonance (gamma-NMR) [8,9].More recently, γ-ray-detected magnetic resonance imaging (gamma-MRI) has even been proposed as a potential medical diagnostic application [10], which is now the focus of the European GAMMA-MRI project, as described at https:// gamma-mri.eu.
The GAMMA-MRI project aims to combine the principles of nuclear spin manipulation used in magnetic resonance imaging (MRI), with the highly efficient detection of γ-rays used in single photon emission tomography.Specifically, the project relies on: (1) external spin orientation of unstable γemitting nuclei inside a magnetic field; (2) measurement of the asymmetry in the angular distribution of the emitted γ radiation, which depends on the degree of orientation; and (3) application of MRI sequences to manipulate the spins in a selected volume of space and thus to modify the angular distribution of radiation for image reconstruction [11].
Here, we present analytical calculations and numerical simulations of the anisotropic distribution of γ-rays emitted by metastable xenon isotopes 129m,131m,133m Xe, which are feasible candidates for the GAMMA-MRI technique.The calculations are based on the additional assumption that the spin orientation can be described by the so-called spin temperature.Our results aim to support the optimization of spin orientation and detector asymmetry in the GAMMA-MRI project.
Conventional MRI works by manipulating the net magnetization of protons, i.e. the population difference between the spin-up and spin-down levels.The Boltzmann distribution predicts that in the static magnetic field of a typical MRI scanner with a field strength of B = 1.4 T the relative difference in these two populations is small, on the order of 10 −5 .Therefore, conventional MRI is heavily dependent on the abundance of hydrogen atoms in biological tissues to achieve feasible signals, and necessitates strong magnetic fields.
However, it is possible to engineer ex situ a larger difference between spin populations, by means other than increasing the magnetic field.These approaches are known as hyperpolarization (HP).One such technique is spin-exchange optical pumping (SEOP), which has been extensively described for non-radioactive noble gases by Walker and Happer [12], and others [13,14].SEOP uses circularly polarized light from a laser to polarize alkali atoms in a gas phase (rubidium or cesium), within a glass cell placed in a weak magnetic field (several mT).A pair of Helmholtz coils generates a homogeneous magnetic field in the cell's region, providing an orientation axis for the polarized atoms.During binary collisions and three-body interactions in the formation of van der Waals molecules with nitrogen, the hyperfine interaction between the alkali-metal electron and the noble-gas nucleus makes them partially exchange angular momentum.Repeated interactions lead to a build-up of nuclear spin orientation of mXe up to several tens of percent.The build-up of nuclear orientation can be monitored conveniently by continuous or pulsed NMR excitations.
In comparison to stable He and Xe, SEOP of radioactive noble gases (namely Xe) was performed only by Calaprice et al [8], and much more recently by Zheng et al [10].In the above cases, the SEOP process was followed by NMR or MRI rf pulse sequences, respectively.
In the GAMMA-MRI approach, in addition to exploiting HP, one utilizes a different detection scheme compared to conventional MRI.Instead of detecting the weak electromagnetic signal induced by rotating spins using inductive pickup coils, one measures the anisotropic emission of γ-rays, as demonstrated in [8,10].This method offers an advantage over conventional MRI as, unlike radio-frequency (rf) photons, even individual γ-ray photons can be easily discerned and detected with high probability.In principle, this detection scheme allows us to reach excellent sensitivity to small amounts of tracer nuclei in the sample volume.
The paper is organized as follows: in section 2, we summarize the mathematical description of nuclear orientation and γ decay characteristics, and lay out the idealized measurement geometry that was used for the computations presented in the section 3, and discussed in section 4. Here, a number of test cases are introduced to illustrate the effect of various parameters on the detected asymmetry.Specifically, we consider the impact of polarization and alignment, the relevance of higherorder multipolar emission contributions, and differences in the experimental geometry of the sample and detector.We conclude our findings with respect to the GAMMA-MRI project in section 4.

Description of the model
In the simulations presented here, we consider the instantaneous nuclear alignment of radioactive xenon nuclei and the resulting angular distribution of the emitted γ radiation.Notably, we do not include considerations on the process of SEOP, such as the achievable degree of Rb atom polarization, the rates of spin orientation transfer from rubidium atoms to xenon nuclei, as well as losses of xenon orientation over time.This has been treated in detail elsewhere [12,15].Spin temperature T s and spin temperature parameter β are useful parameters to describe an ensemble of identical interacting spins that have been brought out of thermal equilibrium and have not yet returned back to it (e.g.mXe after spin orientation via spin exchange optical pumping) [16].In that case, the population p m of the magnetic sublevels of the spins is characterized by a Boltzmann distribution [16] with p m ∝ e −m/(kTs) (k is the Boltzmann constant) and therefore can be described with only one parameter T s , analogous to a 'real' temperature for a system in thermal equilibrium.The dimensionless β which we use in the manuscript is defined as β = −1/(kT s ).
Three isomers (i.e.long-lived excited nuclear states) are of interest for the GAMMA-MRI project: 129m Xe, 131m Xe, and 133m Xe, to which we refer further as mXe.All three have spins I = 11 / 2 and decay via a pure M4 γ-ray transition to I = 3 / 2 state, which is either the ground state or first excited state.Their decay schemes are shown in figure 1 [17][18][19].In the following, we describe the angular distribution of γ radiation for these nuclear spins and the type of γ radiation.

Anisotropic distribution of γ radiation from oriented nuclei
In order to describe the angular distribution W(θ) for a certain type of decay radiation emitted by an ensemble of oriented nuclei (at angle θ) to the quantization axis), several coefficients have to be known.Figure 2 illustrates the nature of these coefficients.
I 0 is the spin of the original oriented state, the parent state of the radioactive decay, or the initial state of a nuclear reaction.I f is the final state, and I m the intermediate state, for cases with two or more intermediate radiations.B λ are the nuclear orientation parameters describing the initial oriented state.λ is connected to the type of radiation and the multipolarity L of the radiation emitted between the oriented I 0 state and the final state I f .For example, for β emission only odd λ's are relevant, compared to even λ's for γ-ray emission.The maximum λ is given by the multipolarity L of the emitted radiation λ max = 2L (with L = 2 for dipole transitions, L = 4 for quadrupole transitions, etc).For M4 γ-ray transitions studied here, L = 4 and thus λ ∈ {2, 4, 6, 8}.
In the most general case, the B λ (I) are calculated using statistical tensors in the density matrix formalism [20].In the particular case when the orienting interaction has an axis of symmetry (such as is the case in here), the diagonal elements of the density matrix are used as population parameters p m , with m ′ = m, and the equation for B λ (I = I 0 ) simplifies to: where the weights are given by the Wigner 3-j symbol.U λ are the orientation coefficients that depend on the properties of the intervening intermediate radiations.U λ determines the orientation of I i by modifying the orientation of I 0 .If there is no intervening radiation, I i is identical to I 0 , and U λ are equal to 1.This is the case for the 11 / 2 state of metastable xenon.
G λ is the perturbation coefficients describing the direct interaction of the electromagnetic moments of a long-lived intermediate state I i with the nuclear environment (if such long-lived states exist).Coefficients G λ refer to the same phenomenon as relaxation in magnetic resonance terms.In the numerical simulations, they were assumed to be 1, and thus the initial state was the same as the originally oriented state, and there were no time-varying relaxation processes.Finally, A λ are the angular distribution coefficients that depend on the radiation type and transition multipolarity for the observed radiation.In the case of multiple radiations, where it might be advantageous to measure two emissions in coincidence, we would differentiate between A λ1 and A λ2 (generalized angular distribution coefficients in figure 2).In our case, where we have a single transition, i.e. for I i → I f transition, A λ is: where L and L ′ are the previously-mentioned angular momenta of the γ radiation and L ′ = 0, as the mixing ratio σ = 0 for the pure multipole transitions.F λ (L, L ′ , I f , I i ) are multipole functions with Wigner 3-j symbol and 6-j symbol.F λ (L, L ′ , I f , I i ) are tabulated, e.g. in [2,21]: For the case studied here (σ = 0), coefficients For the axially symmetric oriented states we discuss here, the general expression for the normalized directional distribution of γ-ray transition given in [2] (chapter 2.6) can be simplified to a form of a multiple expansion: with the solid angle correction factor Q λ that compensates for the finite solid angles subtended by the source and the detector.If we assume that the source and the detector are each geometrical points, and thus that the direction of the radiation emission can be uniquely specified, then Q λ = 1.Finally, P λ (θ) is the Legendre polynomial of order λ, and θ is the azimuthal angle between radiation and z-axis, around which the radiation is axially symmetric.Rodrigues' formula gives a compact expression for the Legendre polynomials, namely: dx n (x 2 − 1) n .In the simulations, we assume a negligible size for the source and detectors, and no intervening intermediate radiations for the 11 / 2 state.Thus, we take U λj , G λ , and Q λ as each equal to 1. Coefficients A λ and B λj are explicitly calculated and given as an output of the simulations, and so is the resulting angular distribution W(θ) plotted as a two-dimensional figure.

Evaluating spin orientation
Nuclear orientation can be described using B λ (I 0 ) parameters up to the maximum allowed λ, as introduced above, as well as using the notation of Tolhoek and Cox [21].Among all parameters, the most frequently used are these with λ = 1, 2, namely the first-and second-order orientation parameters, also known as polarization and alignment, respectively.
The polarization (also known as vector polarization [1]), i.e. the first-order orientation parameter f 1 , is defined as an average value of the spin projection on the selected orientation axis (usually taken to be the z-axis) I Z , normalized with the absolute spin value I [2]: The sign of polarization changes if the direction of the spins gets inverted.Alignment (also known as tensor polarization [1]), i.e. the second-order orientation parameter f 2 , on the other hand, quantifies the deviation from the uniform spin distribution towards a distribution favoring the spin projections along the z-axis: where B 2 here is the nuclear orientation parameter B λj for λ j = 2. Alignment is a measure of the axially symmetric distribution of the spins, which means that the following is true: p m = p −m (unlike for polarization where p m ̸ = p −m ) [22].Alignment is positive if the majority of spins are oriented along or against the z-axis, and negative if they are oriented perpendicular to it.The second-order parameter (as well as higherorder parameters with even values of λ) does not change in value if the direction of the nuclear spins is reversed.The degree of polarization and alignment (i.e.f 1 and f 2 ) can be related to a given spin temperature β and be provided   as the output of the simulation together with the asymmetry in γ-ray emission.

Simulation of mXe γ-decay asymmetry detection for selected spin temperatures β
In the first step of the calculations outlined in previous subchapters, the coefficients A λ and B λ for mXe decay (with I 0 = 11 / 2 and λ = 2, 4, 6, 8) were calculated for selected spin temperatures that represent an increasing degree of orientation.Next, they were used as input for MATLAB simulations of experimental γ-ray asymmetry for several detector geometries.
Figures 3 and 4 illustrate the geometry used in these simulations with three γ-ray detectors placed in two planes (blue in the x − y plane; red and green in the y − z plane), around an ensemble of γ-ray emitting xenon nuclei in a single voxel (i.e. a volume of space).The two panels represent the case of no nuclear alignment (a) and the case of perfectly aligned spins (b) along the external static magnetic field ⃗ B (z-axis), and thus emit γ-rays preferentially in the plane orthogonal to ⃗ B. The size of the detectors was set to 9 × 9 mm 2 , while the source was assumed to be point-like, unless specified otherwise.The distance of the detectors to the source d sd (set to be the same for three detectors) was a variable examined in this study, also represented as a solid angle Ω of γ-ray detection.The spherical and elliptical shapes of non-zero dimensions in figures 3 and 4 are the 3-dimensional reconstructions of the angular distribution of γ-ray emission from the source.
Given an equidistant placement of the γ-ray detectors from the point-like or finite-size unoriented source, the recorded count rate should be equal for all three detectors.In the case of nuclear alignment however, the γ-ray count rate increases for a detector placed transversely to ⃗ B, and decreases for the detector placed longitudinally to ⃗ B. Let us consider an example for a point-like source of 30 MBq activity: with perfect detectors of the dimensions given above and placed at 20 mm from the source, one should expect to register 7.1 • 10 5 photon counts per second (cps) in all 3 detectors, if the sample is not oriented (β = 0), as shown in figure 3.For a (near) perfectly aligned sample (β = 10, shown in figure 4), one may expect 12.9 • 10 5 cps in the detector placed transversely to ⃗ B, and 3.7 • 10 5 cps in the detector placed longitudinally to ⃗ B. This is a significant change in the count rates of both detectors, which can be used as an efficient way of detecting the change in spin orientation, e.g.under rf excitations, as will be done in the GAMMA-MRI project.

Numerical results
Here, we present the magnetic substate populations for 129m,131m,133m Xe at selected spin temperatures, and the resulting γ-ray counts registered in the detectors placed longitudinally and transversely to ⃗ B (at 0 • and 90 • ).Although, unlike real temperature, T s and β can be both positive and negative, because the angular distribution of gamma radiation is a function of even powers of m, it does not depend on the sign of T s .Therefore, we have selected only positive β values.

Study cases
Selected cases comprise: • β = 0 resulting in isotropic γ-ray emission: p m = 1 12 = 0.083 and all B λ = 0; • β = 10 giving extremely anisotropic γ-ray emission: p(m = −11/2) = 0.999 95 (when β → ∞, all populations are concentrated in m min : p(m min ) = 1 and alignment is maximal: values, including an example with β = 0.38, as discussed in [8].As detailed in table 1 and in the plot of f i (β)/f i (β max = 10) (figure 6), analyzed β values span across the entire value range of the f 2 (and f 1 ) parameter.β = 10 is close to β = ∞ because in the latter case 99.95% of population is in the most extreme substate m min = −I and alignment value f 2 is identical to 5 significant digits.Thus, β = ∞ was approximated by β = 10 in simulations while input β = ∞ was used for normalization.
Note that f 1 and f 2 are not mutually exclusive, and certain distributions of magnetic sublevels can be described simultaneously by non-zero values of polarization and alignment.

Numerical simulation of γ-ray angular distribution W(θ)
For the study cases of mXe, the γ-ray multipolarity is L = 4 and thus λ max = 8.The angular distribution coefficients A λ of this transition and the corresponding nuclear orientation coefficients B λ for selected β values are listed in table 2.  As mentioned in section 2.2, we take U λj , G λ , and Q λ as each equal to 1. Thus, (4) takes the form: Legendre polynomials P λ , using Rodrigues' formula, take the representation listed in table 3 for λ = 2, 4, 6, 8. From table 2, one can see that as a general rule B 2 > B 4 > B 6 > B 8 .Figure 7 presents the angular distribution of γ radiation for different β scenarios for mXe ) .The presented distributions are 2D cross-sections through 3D angular distributions along the x − z plane (or y − z plane due to the axial symmetry with respect to the x and y axes).Points at angles θ = 0 • and θ = 180 • lie in the axis of the  magnetic induction vector ⃗ B, denoted in red in figure 7. Along this axis, γ-ray emission decreases with an increasing degree of nuclear alignment (and spin temperature β).Points at angles θ = 90 • and θ = 270 • correspond to the plane orthogonal to the magnetic induction vector ⃗ B. Along this plane, γ-ray emission increases with an increasing degree of nuclear alignment.

Simulation of γ-ray asymmetry observed with finite-size detectors
In this section, we look at the asymmetry of γ-ray detection, i.e. the difference in counts registered by finite-size detectors placed at θ = 0 • and θ = 90 • with respect to vector ⃗ B. The detectors are placed around a point-like mXe source, all at the same distance d sd .
Using (7), we first calculate the intensity of γ radiation for θ = 0 • and for θ = 90 • , for which the biggest difference in the intensity of the emitted radiation is expected (see also figure 7 for the graphical representation of these equations): and: The resulting γ-ray asymmetry, i.e. relative difference in counts Asm Eq at θ = 0 • and θ = 90 • angles, is given as: The values of asymmetry from (10) calculated including only selected B λ parameters are illustrated in figure 8 for the range of investigated β ∈ ⟨0, 10⟩.From the figure, it is clear that one needs to include at least λ = 2 and λ = 4 to obtain an asymmetry that is close to the value including all four orders of B λ .As the contribution from λ = 6 and λ = 8 is below 2% for β > 2, and even below 0.2 % for β ⩽ 1, these two terms can be neglected in practical applications.Given that B 2 alone is not enough to describe accurately the decay asymmetry of mXe states, when discussing and interpreting γ-ray asymmetry, one should use both B 2 (alignment) and B 4 or spin temperature β (when the latter assumption can be considered to hold).
Using MATLAB simulations, we then studied the value of experimental asymmetry at selected β values for a point-or finite-size source, and finite-size detectors placed at varying distances d sd representing different solid-angle coverage Ω, as summarized in table 4.
All datasets obtained from MATLAB simulations used detectors of size 9 × 9 mm 2 , but a different distance d sd from the source and various sizes of the source.One detector was located at θ = 0, i.e. in the plane longitudinal to ⃗ B (blue detector in figures 3 and 4), while the second-at θ = 90, in the plane transverse to ⃗ B (green or red detector in figures 3 and 4).All d sd tabulated in table 4 were studied with point-like and finite-size sources, but for larger distances, d sd = 100 mm and 1000 mm, no difference in Asm was observed.Thus, Asm values for finite-size sources are presented here only for smaller d sd .
The values of Asm sd for different d sd in the function of spin temperature parameter β are represented with points in figure 9.For comparison, Asm Eq for λ = 2, 4, 6, 8 is shown as a black line.All combinations of source and detector size, and distance, reach near-unity asymmetry at β = 10, except for d sd = 10 mm.This is because in the latter case, the solid angle covered by one detector is no more negligible.

Conclusions and outlook
The purpose of this manuscript is to link the degree of nuclear orientation in metastable states of 129m,131m,133m Xe isotopes of spin I = 11 / 2 with the experimental asymmetry of their γray emission.To this end, exact and approximate analytical equations, as well as numerical simulations were used to calculate the number of γ-rays reaching point-like and finite-size detectors placed at 0 • and 90 • to the spin direction.The study cases presented here correspond to the spin temperature parameter values β ∈ ⟨0, 10⟩, which is expected to describe the distribution of the magnetic sub-level population after Xe orientation with spin exchange optical pumping.
Nuclear orientation parameters f 1 and f 2 , also known respectively as polarization and alignment, were also derived for each spin temperature value.These two parameters are commonly used to describe the nature and degree of spin orientation (it should be noted that they are not mutually exclusive and rather describe different orders/types of spin orientation).Although both are non-zero for most distributions of magnetic sublevels studied here, it is only the alignment that is relevant for the angular distribution of γ radiation.Furthermore, we saw that when assuming spin temperature, a relatively high polarization is accompanied by significantly lower relative alignment, except for high spin temperatures β ⩾ 2, where both parameters go over 90% of their maximal values.For instance, for β = 0.38 reached in [8] the degree of alignment corresponds to only 29 % of its maximal value, while the polarization is already at 63 %.Similar values were reached by [10].
Parameters A λ , B λ and Legendre polynomials P λ (cos θ) were then calculated to find the distribution of emitted γ radiation W(θ) and asymmetry Asm (W(0), W(90)).B 1 (proportional to spin polarization) and higher B λ with odd λ do not appear in the equation for W(θ) in gamma decay, while even B 2 (spin alignment), B 4 , B 6 , and B 8 do.Their values for different β were calculated.Gamma decay asymmetry was then determined analytically for a point-like source and detectors, as well as using MATLAB simulations that included also finite sizes.
Next, the importance of higher-order orientation parameters B λ was then studied.It was shown in particular that it is necessary to include B 4 parameter, while higher orders B 6 and B 8 can be fully neglected for spin temperature parameter β < 1, while giving a small, 2% correction for β ⩾ 2.
Furthermore, as expected, the smaller the solid angle covered by the detectors, the closer the asymmetry is to the idealized case of point-like detectors at 0 • and 90 • to the spin direction.
In the GAMMA-MRI project, a high degree of γ-decay asymmetry is desired.We have illustrated here that to achieve this goal, one ideally needs to reach with SEOP such a high spin temperature value that it corresponds not only to a high degree of polarization, but also that of alignment.In addition, the solid angle covered by the detectors should be as small as possible.Since the physical size of the detectors is a constraint, to increase the asymmetry, one can increase the distance to the source and/or use a small source.However, in the former case, the intrinsic detection probability of the γ-ray detectors and the activity of the γ-ray source have to be high enough to provide sufficient count rate statistics.
Recording angular γ-ray asymmetry is the basis for signal acquisition in the γ-ray detected MRI.For this reason, it was important to calculate the expected asymmetry for a range of experimental parameters, such as detector size and distance, and to test different analytical approximations.Such an analysis, presented in this manuscript, allowed us to determine that a high degree of spin polarization does not guarantee high alignment and high gamma-ray asymmetry.Also, certain approximations in the formula describing asymmetric decay are allowed.Finally, detector solid angles below 1% provide optimal geometries.The above observations are of use for the GAMMA-MRI project, which aims to bring in a hybrid technique that combines the advantages of magnetic resonance imaging and nuclear medicine imaging.
The current proof-of-concept apparatus being built within that project provides the functionalities necessary for SEOP HP and γ-ray asymmetry acquisition.Funding was received from the European Union's Horizon 2020 research and innovation program under Grant Agreement No. 964644 (GAMMA-MRI) to finance the progress of the prospective setup.The next steps in the project cover, among others, realistic simulations, including biological modeling of the evolution of oriented mXe in vivo in the organ of interest, and the time dependence of the measured signal.

Figure 2 .
Figure 2. Schematic diagram showing the nuclear states and parameters involved in the parametrization of angular distribution measurements from oriented nuclei.Drawing based on [2], chapter 2: Nuclear Orientation Formalism.

Figure 3 .
Figure 3.The first of two exemplary MATLAB simulation outputs of an ensemble of γ-ray emitting Xe nuclei emitted by a point-like source.Three detectors (green, blue, and red) with 10 × 10 sensors each are located around a sample of one voxel visualized with a red vector indicating its spin orientation and with a 3D surface plot indicating the directional distribution of emitted γ radiation.γ radiation is emitted isotropically from randomly oriented nuclei (no nuclear orientation: β = 0, f 1 = 0, and f 2 = 0), which is represented as a spherical 3D surface plot.

Figure 4 .
Figure 4.The second of two exemplary MATLAB simulation outputs of an ensemble of γ-ray emitting Xe nuclei emitted by a point-like source.Three detectors (green, blue, and red) with 10x10 sensors each are located around a sample of one voxel visualized with a red vector indicating its spin orientation and with a 3D surface plot indicating the directional distribution of emitted γ radiation.γ radiation is emitted anisotropically from (nearly) perfectly aligned spins (β = 10, f 1 = 1, and f 2 = 0.606), which is represented as an ellipsoidal 3D surface plot.

Figure 5 .
Figure 5. Population probability distributions pm of magnetic sublevels m for mXe with I 0 = 11 / 2 for selected values of spin temperature parameter β.Values of β are shown in the legend in the top-left corner.

Figure 5
Figure 5 below presents magnetic sublevel population probabilities p m for 129m,131m,133m Xe corresponding to selected positive β values.Although, unlike real temperature, T s and β can be both positive and negative, because the angular distribution of gamma radiation is a function of even powers of m, it does not depend on the sign of T s .Therefore, we have selected only positive β values.Selected cases comprise:

Figure 7 .
Figure 7.The angular distribution W(θ) of M4 γ radiation from mXe for the simulated cases of magnetic sublevel populations calculated using selected values of spin temperature parameter β.The direction of the external static magnetic field vector ⃗ B is marked in red.The extreme cases are marked in: purple (isotropic emission of γ radiation with f 2 = 0 for β = 0) and yellow (anisotropic emission of γ radiation with f 2max = 0.606 for β = 10).The remaining plots represent W(θ) for the intermediate values of f 2 and β.

Figure 9 .
Figure 9. Simulated γ-decay asymmetry Asm for different source and detector sizes and distances d sd as a function of spin temperature parameter β (top x-axis) and the corresponding relative alignment (bottom x-axis, f2 f2max).For two datasets finite-size source was considered: 3 mm for Asm 10s3 and 20 mm for Asm 50s20 .For other datasets, a point-like source was assumed.

Table 4 .
Size of γ-ray source, distance and size of detectors, and observed asymmetry obtained with a MATLAB code relative to asymmetry for point-like source and detector.