# γ-ray angular correlations, distributions and linear polarization in tracking arrays

T. Lauritsen, A. Korichi, C.M. Campbell, M.P. Carpenter, P.A. Copp, J.P. Greene, R.V.F. Janssens, T.L. Khoo, F.G. Kondev, C. Müller-Gatermann, et al.

► To cite this version:

T. Lauritsen, A. Korichi, C.M. Campbell, M.P. Carpenter, P.A. Copp, et al. γ-ray angular correlations, distributions and linear polarization in tracking arrays. The European physical journal. A, Hadrons and Nuclei, 2025, 61 (4), pp.76. 10.1140/epja/s10050-025-01533-5. hal-04964841

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# γ-Ray Angular correlations, distributions and linear polarization in tracking arrays.

T. Lauritsen¹ᵃ, A. Korichi¹,², C.M. Campbell³, M.P. Carpenter¹, P.A. Copp¹ᵇ, J.P Greene¹, R.V.F. Janssens⁴, T.L. Khoo¹, F.G. Kondev¹, C. Müller–Gatermann¹, S. Nandi¹ᶜ, W. Reviol¹, G. Savard¹, M. Siciliano¹, D. Seweryniak¹, and S. Zhu¹ᵈ

¹ Argonne National Laboratory, Argonne, Illinois 60439, USA

² IJClab IN2P3, bat 104-108, F-91405 Orsay Campus, France

³ Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

⁴ Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27559, USA and Triangle Universities Nuclear Laboratory, Duke University, Durham, North Carolina 27708, USA

Received: date / Revised version: date

**Abstract.** With γ-ray tracking arrays, the angular correlation and, especially, the linear polarization of γ rays can be measured with high precision. However, because of the complexity of these spectrometers, measuring angular distributions is more involved. The techniques and associated formalisms were developed and established in the 1950s. The experimental setups of that era were relatively simple when compared to the advanced γ-ray tracking arrays available today. This paper aims to review and refine these techniques, unify the nomenclatures and, specifically, adapt them for use with the newest tracking arrays. In addition, new methods for producing the required response functions for these arrays are introduced.

**PACS.** 07.85.Ne Gamma-ray tracking – 29.30.Kv Gamma-ray spectroscopy – 29.40.Gx Segmented germanium detectors

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>Angular correlations: sources</td><td>3</td></tr><tr><td>2.1</td><td>Formalism</td><td>3</td></tr><tr><td>2.2</td><td>Experimental extraction</td><td>3</td></tr><tr><td>3</td><td>Angular distributions</td><td>5</td></tr><tr><td>3.1</td><td>Formalism</td><td>5</td></tr><tr><td>3.2</td><td>Experimental extraction</td><td>7</td></tr><tr><td>4</td><td>Angular correlations: DCO</td><td>9</td></tr><tr><td>4.1</td><td>Formalism</td><td>9</td></tr><tr><td>4.2</td><td>Experimental extraction</td><td>12</td></tr><tr><td>5</td><td>Linear polarization</td><td>12</td></tr></table>

<table><tr><td>5.1</td><td>Linear polarization: radioactive source data</td><td>14</td></tr><tr><td>5.2</td><td>Linear polarization: in-beam data</td><td>15</td></tr><tr><td>5.3</td><td>Linear polarization: DCO in-beam</td><td>15</td></tr><tr><td>5.4</td><td>Experimental extraction</td><td>16</td></tr><tr><td>5.5</td><td>Linear polarization in-beam</td><td>18</td></tr><tr><td>6</td><td>Summary</td><td>19</td></tr><tr><td>A</td><td>Selected functions</td><td>19</td></tr><tr><td>A.1</td><td>Ordinary Legendre Polynomials</td><td>19</td></tr><tr><td>A.2</td><td>Associate Legendre Polynomials</td><td>19</td></tr><tr><td>A.3</td><td>The 3-j and Clebsch-Gordan coefficients</td><td>19</td></tr><tr><td>A.4</td><td>The 6-j and Racah coefficients</td><td>20</td></tr><tr><td>A.5</td><td>The 9-J coefficients</td><td>20</td></tr><tr><td>B</td><td>Datasets</td><td>20</td></tr><tr><td>B.1</td><td>In-beam</td><td>20</td></tr><tr><td>B.2</td><td>Source data</td><td>20</td></tr><tr><td>C</td><td>Other <sup>152</sup>Eu source examples</td><td>20</td></tr><tr><td>D</td><td>the DCO φ method for sources</td><td>21</td></tr></table>

<footnote>ᵃ Corresponding Author: Torben@anl.gov</footnote>

<footnote>ᵇ Current address: Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA</footnote>

<footnote>ᶜ Current address: Subatech (IMT Atlantique, CNRS IN2P3, Nantes Université) 4 rue Alfred Kastler 44307 Nantes cedex 3 France.</footnote>

<footnote>ᵈ Deceased</footnote>
E The *caDCO 2D superpulse method* for sources . . . . . 22

F Selected examples of DCO formulas . . . . . . . . . . . . 23

# 1 Introduction

As described in Ref. [1], the concept of angular correlations for γ-ray emission was introduced in 1940 by Hamilton [2]. However, it was not until 1947 that the first successful experiment was performed with Geiger counters [3]. At that time, the experimental detector systems for measuring the correlations were modest. Later, with improved detection capabilities, angular correlations came to play a major role in nuclear structure, especially since the 1950s. These techniques enabled crucial measurements of spins and parities of nuclear states, quantum numbers which remain essential in nuclear γ-ray spectroscopy today.

With the emergence of highly-efficient tracking arrays, where segmented High Purity Germanium (HPGe) crystals results in superior angular and position resolution, scientists are now much better equipped for such measurements, especially when taking advantage of efficient tracking algorithms (see Ref. [4] and reference therein). This paper reviews the classical methods, adapts them for data from tracking arrays, and proposes new approaches and normalization procedures which take advantage of the continuous angle measurements offered by arrays such as AGATA [4–9] and GRETINA [4, 10, 11].

In Sec. 2.1, for completeness, we reintroduce the classical formalism of angular correlations based on the extensive literature available from the 1950s and 1960s. This section also discusses the experimental procedures, including a technique for mixing events which are not in coincidence to produce a response function. In Sec. 3, we present the theory for angular distributions of γ rays emitted by nuclei produced in-beam. As discussed in Ref. [12], it has been realized that measuring angular distributions in tracking arrays is non-trivial because of the complexity of the tracked data. This issue is addressed in Sec. 3.1, where a new method to determine the required response function, using radioactive sources, is presented. Next, Sec. 4.1 covers the theory of angular correlations for nuclei produced in-beam, where the first state is aligned, significantly complicating the measurements. The full theory of Directional Correlation from Oriented nuclei (DCO) is necessary and is introduced accordingly, and we present novel experimental methods tailored to tracking arrays. In the final section, the techniques for measurements of linear polarization of γ rays using tracking arrays are elucidated. In Sec. 5.1, the linear polarization using sources, requiring two γ rays in coincidence, is covered. Subsequently, the extraction of linear polarization of γ rays from nuclei produced in-beam (and, therefore, polarized in the nuclear reaction) is presented in Sec. 5.2. The case where two γ rays, in coincidence, are observed is covered as well (Sec. 5.3). In all cases, we present new methods that do not require a normalization using radioactive sources.

This paper exclusively uses data from GRETINA that have been tracked [4]. Hereafter, we refer the reader to Ref. [4] for the description of the tracking, including the nomenclature used throughout the present paper. In most cases, a Figure Of Merit (FOM) cut of <0.8 is applied, and the tracking clustering angle is optimized to maximize the efficiency versus the peak-to-total ratio. For most in-beam data, a smaller clustering angle of 12°–16° (depending on the cascade multiplicity) is applied in order to minimize summing, whereas an angle of up to ~20° may be used for radioactive sources. In all cases, the first interaction point, as determined by the tracking algorithm, defines the direction of the emitted γ ray(s).

As discussed in Ref. [4], one of the significant advantages of γ-ray tracking arrays is their ability to determine the position of the photon interaction points with high precision, typically within a few millimeters. Consequently, the usual attenuation coefficients [13] correcting for the detector solid angle can generally be set to unity. This capability effectively simplifies the analysis of data from these arrays. However, one might have to account for the potential presence of a slight, energy-dependent halo caused by incomplete decomposition and/or tracking of the data [14].

For nuclei created in-beam, e.g., in heavy ion fusion-evaporation reactions, there will be an *alignment* with respect to the beam axis. For a state with spin J, the $2J + 1$ m-substates are distributed such that the population of the *m* state equals that of the *-m* one [15]. If this symmetry is broken, the level of interest is said to be *polarized* <fnref n="1" />. Conversely, if all m-substates are equally populated, the nucleus is not oriented at all. By analyzing the emitted γ rays using angular correlation or distribution techniques, one can determine the angular momentum removed by the γ rays, and by measuring the linear polarization, the assignment of parity to the states can be achieved as well. This makes tracking arrays a potent instrument for complete spectroscopy even of weakly populated states and bands of levels. It is interesting to note that, with a well populated tracking array, it should be possible to place gates on the entry distribution [17], i.e., the distribution in excitation energy and angular momentum from which γ-ray emission starts, thereby controlling and selecting how nuclear states and bands are populated [18].

This paper aims to provide *all* necessary algorithms, formulas, special functions and rules to extract the angular distribution, correlation and linear polarization, i.e., no consultation of external references should be necessary. The nomenclature of all the algorithms have been unified to make their application easier to use.

<footnote n="1">¹ Only the more common linear polarized γ rays are included in this work, circular polarized photons [16] are not considered.</footnote>
## 2 Angular correlations: sources

### 2.1 Formalism

The measured angular correlation between two γ rays can be written as [1, 15, 19–22]

$$ \omega(\theta) = a_0 + \alpha_2 a_2 P_2(\cos\theta) + \alpha_4 a_4 P_4(\cos\theta) + \dots \quad (1) $$

where θ is the angle between the two γ rays, and P_n represents the ordinary Legendre polynomial of order n (see App. A.1). This expression is valid if the initial state has a random distribution in m-substates, meaning that it is neither polarized nor aligned. The a_k coefficients depend on the spins of the states involved and the multiplicities of the γ rays in the decays. These coefficients will be discussed in the following sections. The α_k factors account for any attenuation of the signal due to the solid angle covered by the detectors [13]. However, as discussed in Ref. [4], these factors are close to unity in tracking arrays. It should be noted that, if parity is not conserved in the decays, the odd terms in eq. 1 must be included. Such cases are only seen in β decays and will not be considered here (we refer to the discussion in Ref. [23]). Thus, eq. 1, as used here for the case of alignment, only contains even terms and those with an index higher than four are often ignored. The a_k factors can be written as [1, 15, 21]

$$ a_k = A_k(L_1, L'_1, I_i, I) A_k(L_2, L'_2, I_f, I) \quad (2) $$

where

$$ A_k(L, L', I_1, I_2) = \frac{1}{1+\delta^2} (F_k(L, L, I_1, I_2) + 2\delta F_k(L, L', I_1, I_2) + \delta^2 F_k(L', L', I_1, I_2)) \quad (3) $$

The F_k coefficients are determined by 3-j “()” and 6-j “{}” symbols along with a phase factor as:

$$ F_k(L, L', I_1, I_2) = (-1)^{I_1+I_2-1} \times [(2L+1)(2L'+1)(2I_2+1)(2k+1)]^{\frac{1}{2}} \times \begin{pmatrix} L & L' & k \\ 1 & -1 & 0 \end{pmatrix} \begin{Bmatrix} L & L' & k \\ I_2 & I_2 & I_1 \end{Bmatrix} \quad (4) $$

Explicit formulas for the 3-j and 6-j symbols can be found in Ref. [24] which also contains tabulations of these symbols (see App. A.3 and App. A.4). The symbols are related to Clebsch–Gordan coefficients. In this context, I_i represents the spin of the initial state, which decays to the intermediate level with spin I via a γ ray of multipolarity (L_1, L'_1) and a mixing ratio δ_1. Similarly, the second γ ray, with a multipolarity of (L_2, L'_2) and a mixing ratio δ_2, completes the transition from the intermediate state to the final one, with spin I_f. Observing the first γ ray in a specific direction ensures that the magnetic substate distribution, P(m), within the intermediate level is non-uniform. Consequently, the second γ ray is not emitted uniformly relative to the direction of the first.

The mixing ratios are usually expressed through reduced matrix elements [21] as

$$ \delta_1 = \frac{\langle I || L'_1 \pi' || I_i \rangle}{\langle I || L_1 \pi || I_i \rangle}, \quad \delta_2 = \frac{\langle I_f || L'_2 \pi' || I \rangle}{\langle I_f || L_2 \pi || I \rangle} \quad (5) $$

There are different conventions regarding the sign of the mixing ratios [25–27]. It should be noted that the F_0 term in eq. 4 fulfills the relation F_0(L, L', I_1, I_2) = δ_{LL'} (Kro-necker delta) which ensures that a_0 is always 1.0, inde-pendent of the values of the mixing ratios.

### 2.2 Experimental extraction

To extract the angular correlation, one must, of course, observe two γ rays in coincidence which may make statistics a challenge in many measurements, especially for nuclei produced with low cross sections such as is the case with neutron-rich isotopes or super-heavy elements. We have developed a dedicated technique which uses the data itself for the correction of the more complex response of the tracking array. It involves mixing events which are not in coincidence, herewith ensuring that there is no correlation between the γ rays. By mixing back a substantial number of events, the statistical dominance of the angular correlation is preserved over the reference spectrum’s statistics. The event mixing technique is also discussed in Ref. [28] and was originally introduced in Ref. [29].

Simulations [30] have revealed that, if the source is not perfectly centered, the angular correlation spectra may be slightly distorted. A model accounting for this effect suggests that adjustments to the gain and offset calibration of the angle-axis prior to fitting the data can compensate for such distortions. Therefore, in all subsequent fits to the high-statistics data, the gains and offsets were fine-tuned to maximize the a_2 value. These adjustments were found to be less than ~1%.

The angular correlation spectrum for the 1173- and 1333-keV lines from a ⁶⁰Co source is often shown as a classic example of correlations between E2 transitions in textbooks. This correlation, extracted using the GRETINA tracking array, is presented in Fig. 1. The fitted parameters are detailed in Table 1 along with the calculated values based on eq. 1 when the attenuation factors are assumed to be unity. The sequence 4⁺ → 2⁺ → 0⁺ through the 1173- and 1333-keV E2 transitions, is assumed to be pure<fnref n="2" /> and is commonly referred to as a 4(2)2(2)0 cascade [1]. Figure 2 provides the background subtracted and scaled spectra which produced the ⁶⁰Co angular correlation depicted in Fig. 1.

The angular correlation spectrum for the 898- and 1836-keV lines from an ⁸⁸Y source is also depicted in Fig. 1 and Table 1 provides the corresponding angular correlation coefficients. Here, the γ rays are from the 3⁻ → 2⁺ → 0⁺

<footnote n="2">² There may be a small mixing of the 1173-keV line with an M3 component, of the order of δ ~0.0025 [31].</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** The measured angular correlations for the 1173 – and 1333-keV lines from a $^{60}$Co source and the 1836 – and 898-keV ones from an $^{88}$Y source using tracked data from GRETINA. The data were fitted from ~34° and up and the fits are represented by the solid lines. The so-called clustering hole observed at small angles is discussed further in the text.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** The signal (in red)– and response (in blue) spectra that produced the angular correlation for the lines in $^{60}$Co displayed in Fig. 1. The spectra are plotted as a function of the relative angle between the 1173 – and 1333-keV lines. The spectra are background subtracted and scaled to have the same counts in the region from ~34° to ~170°.</figcaption></figure>

sequence and, thus, the sign of the $a_2$ coefficient is, as expected, negative for this pair of E1 and E2 transitions.

<table><caption>**Table 1.** Fitted angular correlation coefficients, as well as the calculated ones based on eq. 1, for the data shown in Fig. 1. The normalized $\chi^2$ values of the fits are given as well.</caption><thead><tr><th rowspan="2">source</th><th colspan="3">measured</th><th colspan="2">calculated</th></tr><tr><th>a<sub>2</sub></th><th>a<sub>4</sub></th><th>χ<sup>2</sup></th><th>a<sub>2</sub></th><th>a<sub>4</sub></th></tr></thead><tbody><tr><td><sup>60</sup>Co</td><td>0.100(2)</td><td>0.014(2)</td><td>1.08</td><td>0.1020</td><td>0.0091</td></tr><tr><td><sup>88</sup>Y</td><td>-0.069(2)</td><td>0.002(3)</td><td>1.64</td><td>-0.0714</td><td>0</td></tr></tbody></table>

Both of the $\gamma$ rays are reported to be pure [32], and the cascade may be referred to as a 3(1)2(2)0 one.

For both of these cases, the current event was mixed with twenty previous coincidence events to construct the reference (response function). In the mixing, four combinations: $(\gamma_1, \gamma_1^{old})$, $(\gamma_1, \gamma_2^{old})$, $(\gamma_2, \gamma_1^{old})$ and $(\gamma_2, \gamma_2^{old})$ were used. The $\gamma$ rays marked *old* are from the queue of previous events. This procedure results in the event mixed spectrum having 80 times the counts present in the raw angular correlation spectrum. Consequently, the statistical error contribution from the reference spectrum to the final one is estimated to be approximately ten percent. Each time a new coincidence event is found, the reference spectrum is updated as outlined above, and the current event is committed to the (in this case twenty events long) mixing queue in a first-in-first-out approach. This ensures that the reference spectrum always reflects the current experimental conditions, even if the latter change during the time the data is acquired.

A prominent feature of the curves depicted in Fig. 1 is the so-called clustering hole observed at small angles [4,33]. In this case, the tracking was performed with a clustering angle of 15°. Consequently, up to about 30°, one finds a reduced number of $\gamma$ rays in coincidence because the tracking algorithm tends to add $\gamma$ rays within about two times the clustering angle. This characteristic is described in Ref. [4] and references therein. A possible remedy for this effect, at least for sources, is presented in Ref. [34].

Beta decay and electron-capture from an $^{152}$Eu source produces transitions in two daughters: $^{152}$Gd and $^{152}$Sm, in each of these have several $\gamma$-ray transitions occur in coincidence. This provides further opportunities to evaluate the techniques discussed above. One example is the correlation of the 344 – and 779-keV transitions, and their angular correlation spectrum presented Fig. 3. The fitted $a_2$ coefficient of -0.067(2) compares well to the calculated value of -0.0714 for this 2(1)2(2)0 (pure) cascade.

For the well-known coincidence of the 122 – and 1408-keV lines in $^{152}$Sm, the angular correlation spectrum is extracted and presented in Fig. 4. The fitted coefficients are $a_2 = 0.175(3)$ and $a_4 = -0.008(4)$, while the calculated values are $a_2 = 0.25$ and $a_4 = 0$ for this 2(1)2(2)0 cascade of $\gamma$ rays if one assumes pure transitions. However, according to Ref. [35], the 1408-keV transition is characterized by a small M1/E2 mixing. Thus, a fit was performed in order to determine the mixing ratio, based on the measured $a_2$ and $a_4$ coefficients. The resulting ra-
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** The measured angular correlations of the 344 – and 779-keV lines in <sup>152</sup>Gd (daughter of <sup>152</sup>Eu). The data is presented in black, the data used for the fit in red. The fitted curve is given in blue, while the calculated angular correlation function, obtained using eq. 1 for a 3(1)2(2)0 pure cascade, is identified by the dashed blue curve. The χ<sup>2</sup> of the fit is 1.7.</figcaption></figure>

tio was determined to be -0.101(11). The χ² values for
the fit to the measured a₂ and a₄ coefficients are given
in Fig 5. In Fig. 4, the calculated curve (in dashed blue)
is produced using this mixing ratio and good agreement
with the data is observed. In the appendix, Sec. C, another
example from <sup>152</sup>Sm, where a very large mixing ratio is in-
volved, is provided as well. Different methods to determine
δ mixing ratios are discussed in Ref. [27].

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** The measured angular correlations of the 122 – and 1408-keV lines in <sup>152</sup>Sm (daughter of <sup>152</sup>Eu). The data is given in black, the data used for the fit in red. The fitted curve is shown in blue while the calculated angular correlation function, obtained using eq. 1, with the fitted mixing ratio of –0.101(11), is identified by the dashed blue curve. The mixing ratio fit is presented in Fig. 5. The χ<sup>2</sup> of the fit is 1.4.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** χ² values for the fit of the mixing ratio to the measured a₂ and a₄ coefficients for the 122 – and 1408-keV transitions in <sup>152</sup>Sm. See text for discussion.</figcaption></figure>

**3 Angular distributions**

**3.1 Formalism**

The formalism for angular distributions is closely tied to
the one describing angular correlations, but here, instead
of observing another γ ray in coincidence to determine the quantization axis, a beam striking a target polarizes the nucleus of interest and provides the required anisotropy. Thus, when γ rays are emitted from the excited nucleus, their distribution with respect to the beam axis<fnref n="3" /> becomes non-uniform, a phenomenon known as alignment. Given that the emission of the coincident γ rays is no longer required, one might argue that measuring an angular distribution spectrum is easier than obtaining the information from angular correlation data. However, it is worth noting that a coincidence requirement may be needed to obtain cleaner data.

The general formula for angular distribution of a γ ray decaying from a state with spin $I_i$ to one with spin $I_f$ can be expressed as [21, 36, 37]:

$$ \omega(\theta) = A_0 + \alpha_2 A_2 P_2(\cos\theta) + \alpha_4 A_4 P_4(\cos\theta) + \dots \quad (6) $$

where the $\alpha_k$ attenuation coefficients take into account the actual alignment of the state with spin $I_i$ from which the decay proceeds after (possibly many) preceding transitions have been emitted and have, possibly, reduced the original alignment. It is essential to differentiate these alignment coefficients from the attenuation factors accounting for any finite detector solid angle coverage [13]. As discussed, in tracking arrays, such attenuations can to first order be disregarded [4].

The $A_k$ factors can be calculated as

$$ A_k(I_i, L_1, L'_1, I_f) = \alpha_k A_k^{max}(I_i, L_1, L'_1, I_f) \quad (7) $$

with

$$ A_k^{max}(I_i, L_1, L'_1, I_f) = \frac{1}{1+\delta_1^2} \left( \begin{aligned} & f_k(I_f, L_1, L_1, I_i) \\ & +2\delta_1 f_k(I_f, L_1, L'_1, I_i) \\ & +\delta_1^2 f_k(I_f, L'_1, L'_1, I_i) \end{aligned} \right) \quad (8) $$

where $\delta_1$ is the mixing ratio, see eq. 5, and the $f_k$ factors are

$$ f_k(I_i, L_1, L'_1, I_f) = B_k(I_i) F_k(I_f, L_1, L'_1, I_i) \quad (9) $$

Under the assumption of complete alignment, the $B_k(J)$ function can be expressed as

$$ \begin{aligned} B_k(J) &= (2J + 1)^{1/2}(-1)^J C(J, 0, J, 0|k, 0) \\ B_k(J) &= (2J + 1)^{1/2}(-1)^{J-\frac{1}{2}} C(J, \frac{1}{2}, J, -\frac{1}{2}|k, 0), \quad (10) \end{aligned} $$

where the first expression refers to integer values of J, while the second is used for half-integer ones. In Ref. [21], the $F_k$ coefficients are calculated as

$$ \begin{aligned} F_k(I_f, L_1, L'_1, I_i) = (1)^{I_f - I_i - 1} & \times [(2L_1 + 1)(2L'_1 + 1)(2I_i + 1)]^{1/2} \\ & \times C(L_1, 1, L'_1, -1|k, 0) W(I_i, I_i, L_1, L'_1; k, I_f) \end{aligned} \quad (11) $$

where C is a Clebsch–Gordan coefficient and W a Racah coefficient, respectively. The latter is related to the 6–J symbol, see app. A.

The other component in eq. 7 is the attenuation coefficient (also referred to as the deorientation coefficient), $\alpha_k$, which is usually calculated as:

$$ \alpha_k(J) = \sum_m \alpha_k^{(m)}(J) P(m)(J), \quad (12) $$

where $P(m)(J)$ describes the distribution of the m sub-states of a level with spin J. This distribution is usually taken to be of a Gaussian form:

$$ P_m(J) = \frac{e^{-\frac{m^2}{2\sigma^2}}}{\sum_{m'=-J}^{+J} e^{-\frac{m'^2}{2\sigma^2}}} \quad (13) $$

and is defined with respect to the beam axis. The function is characterized by a $\sigma/J$ ratio, where a value of 0 indicates full alignment. This approach works well for nuclei produced in fusion–evaporation reactions but, in fragmentation reactions, for example, it may be necessary to explicitly specify $P_m(J)$ [38].

The $\alpha_k^{(m)}$ functions are written as [39]:

$$ \begin{aligned} \alpha_2^{(m)}(J) &= 1 - \frac{3m^2}{J(J+1)} \\ \alpha_4^{(m)}(J) &= 1 - 10 \frac{m^2}{J(J+1)} + \frac{35}{3} \frac{m^4}{J^2(J+1)^2} \end{aligned} \quad (14) $$

In reality, a nucleus produced in a fusion–evaporation reaction is characterized by a non zero $\sigma/J$-ratio, due to the evaporation of particles (neutrons, protons and $\alpha$ particles) from the compound nucleus. If the $\alpha_k(I_i)$ coefficient is known, the $\alpha_k(I_f)$ ones following $\gamma$ decay can be calculated as

$$ \alpha_k(I_f) = U_k(I_i L_1 L'_1 I_f) \frac{B_k(I_i)}{B_k(I_f)} \alpha_k(I_i) \quad (15) $$

where the $B_k(J)$ factors are given by eq. 10 and the $U_k$ functions can be calculated as

$$ \begin{aligned} U_k(I_i L_1 L'_1 I_f) &= \frac{1}{1+\delta_1^2} [u_k(I_i L_1 I_f) + \delta_1^2 u_k(I_i L'_1 I_f)] \\ u_k(I_1 L I_2) &= (-1)^{I_1+I_2-L} [(2I_1+1)(2I_2+1)]^{1/2} \\ & \qquad W(I_1 I_1 I_2 I_2; kL) \end{aligned} \quad (16) $$

Unless the mixing ratio $\delta_1$ from eq. 5 is zero, the first coefficient in eq. 6 is not one and this equation is usually written as

$$ \omega(\theta) = A_0(1 + \alpha_2 A_2 P_2(\cos\theta) + \alpha_4 A_4 P_4(\cos\theta) + \dots) \quad (17) $$

This form facilitates comparison between fits and measurements and one typically extracts and reports the ratios $A_2/A_0$ and $A_4/A_0$. For the angular correlations discussed in Section 2.1, the equivalent $A_0$ term is always one [1].

<footnote n="3">³ Another spin axis may be considered as well.</footnote>
If one can find a γ-ray sequence in the nucleus of interest where there is no mixing, a good example being a rotational band, then one can determine the $\sigma/J$ ratio by determining the value which reproduces the observed $A_2$ and $A_4$ values for the band members using the formulas above.

We note that, as pointed out in Ref. [40, 41], the $\theta$ angle in eq. 6 is expressed in the Center of Mass (CM) system. The relationship between the cosines in the laboratory and CM systems is given by:

$$ \cos(\theta) = \frac{\cos(\theta^{lab}) - \beta}{1 - \beta\cos(\theta^{lab})} \quad (18) $$

Furthermore, for fast-moving nuclei, the Lorentz transformation must be taken into account by multiplying $\omega(\theta)$ by the velocity-dependent factor [40]:

$$ \frac{1 - \beta^2}{[\beta\cos(\theta) - 1]^2} \quad (19) $$

where $\beta=v/c$, $v$ being the velocity of the nucleus and $c$ the speed of light.

If one requires other γ rays to be measured in coincidence (in order for example to obtain cleaner spectra), then eqs. 6 to 19 may require modification before extracting the angular distribution spectrum, because the observation of γ rays in coincidence in an array which does not cover $4\pi$ can alter the m-distribution. Sec 4.1 generally addresses this case, although the first γ ray is not explicitly observed. In principle, one could integrate out the non-explicitly observed first γ ray in the formalism of Sec 4.1 to derive the appropriate formula, replacing eqs. 6 to 19 in this context.

## 3.2 Experimental extraction

The key to extracting angular distributions from in-beam, tracked data lies in constructing a reliable response function, which represents the tracking array's response in the absence of angular distribution effects. However, compared to data from Compton-suppressed systems, such as Gammasphere [42], tracked data is notably more complex, posing a challenge in constructing this response. Achieving this goal requires a number of careful source measurements, each with sufficient statistical precision. Unfortunately, contrary to the angular correlation and linear polarization cases, the tracked data itself cannot be readily employed to generate the response function due to its inherent 'polarized' nature. Compared to Compton-suppressed arrays, which rely solely on central contact charge collection for energy measurements, tracked ones involve additional steps such as pulse shape analysis (decomposition) and tracking [4]. These steps significantly impact the extraction of angular distributions. Moreover, in Compton-suppressed systems, each detector contributes to a single angle only due to the ring-like construction of

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** The angular distribution response matrix constructed as described in the text representing the response of the GRETINA array in the absence of an angular distribution signal. Strong lines from the following radioactive sources were used: $^{152}$Eu, $^{207}$Bi, $^{60}$Co, $^{137}$Cs, $^{88}$Y, $^{243}$Am, $^{226}$Ra and $^{57}$Co. The y-axis is the polar angle [degrees] and the x-axis the γ-ray energy [keV].</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** As Fig. 6; but after interpolation and linear extrapolation to the energies in between and outside the measured responses. The response matrix is specific to the configuration of the array at the time of the $^{152}$Dy experiment (see App. B.1) which had a compact configuration with detectors preferentially mounted in the backward hemisphere.</figcaption></figure>

these arrays (e.g., Gammasphere comprises 17 rings of detectors, hence offering only 17 angles for angular distribution extraction). Conversely, tracking arrays feature a more involved tiling pattern. A single detector (crystal) typically covers a range of polar angles and overlaps with other detectors in a complex manner, complicating the angular distribution extraction process.

In general, it is not possible to use just one source line in the vicinity of the γ ray of interest as a suitable response.
Thus, the response determined from a source placed in, e.g. Gammasphere is not applicable for tracking arrays. The question may arise as to why smooth angular correlation spectra can be extracted when angular distributions present challenges. The reason lies in the integration of imperfections across polar angles, as we focus solely on relative angles between two γ rays across the full polar angle coverage.

We propose a novel approach for extracting an effective response function tailored to the case of tracking arrays. It consists in collecting the angular distribution (with respect to the beam axis) of strong γ rays from a number of radioactive sources, collecting as many different transition energies as possible. The resulting angular distributions, after a re-normalization, are placed into a response matrix where the distribution is on the y-axis (counts in the polar angles) and the energy of transition along the x-axis, see Fig. 6. As can be seen, one γ line's polar angle spectrum is placed in just one energy bin.

The remainder of the matrix from Fig. 6 is subsequently filled by interpolation values between the measured polar angle distributions, in terms of γ-ray energy, in order to produce the final response matrix exhibited in Fig. 7. Linear extrapolation is used below the first and after the last measured distribution to complete the response matrix. To account for error propagation, a variance matrix for the response function is also constructed and interpolated (and extrapolated as needed), while adjusting for the fact that the error in the response does not solely correspond to the scaled square root of the counts due to its construction. At low energies, γ rays around 90° are distorted due to attenuation by the target frame and target holder.

The procedure consists then in extracting the measured in-beam angular distribution from an energy vs. angular distribution matrix by placing a gate on the energy of interest. A response is extracted at the same energy from the response function matrix and the two spectra are divided by one another, properly propagating the errors. The data presented here are from an experiment with GRETINA at Argonne National Laboratory (ANL) where the array was in a compact configuration [4]. The polar angles covered were from ~70° to ~175° and a bin width of 3° was used in the analysis. As is often the case for heavy-ion in-beam measurements, we opt for a larger coverage of backward angles compared to forward ones.

An example of a measured angular distribution for a transition in ¹⁵²Dy, the 23⁻ → 21⁻ 991-keV line, is introduced in Fig. 8. For details on the reaction used, refer to App. B.1. Using eq. 6 to 19 with $\sigma/J = 0.3$ and $\delta_1 = 0$, one would expect the angular distribution coefficients to be $A_2 \sim 0.38$ and $A_4 \sim -0.13$. Using the same reaction, the angular distribution of the 34⁻ → 33⁺ 781-keV transition was measured and can be found in Fig. 9.

Despite careful measurement of the response function, non-statistical fluctuations in the data are clearly evident. The error estimates for the measured $A_2$ and $A_4$ values, from

<figure><img src="image_9.png" /></figure>

Tue Dec 17 16:11:41 2024

**Fig. 8.** The angular distribution of the 991-keV γ ray in ¹⁵²Dy. The usual $A_2$, $A_4$ Legendre polynomial fit to the data results in coefficients of $A_2 = +0.251(2)$ and $A_4 = -0.046(3)$. The dashed green line is calculated using eq. 6 to 19 with a $\sigma/J$ ratio of 0.3 while assuming a stretched transition. The $\chi^2$ of the fit is 35 and reflects systematic errors.

<figure><img src="image_10.png" /></figure>

**Fig. 9.** The angular distribution of the 781-keV line in ¹⁵²Dy. An $A_2$ coefficient value of -0.210(3) was extracted from the fit to the data. The dashed green line is calculated using eq. 6 to 19 with a $\sigma/J$ ratio of 0.3 and $\delta = 0$. The $\chi^2$ of the fit is 21.

the fits to the data, do not fully account for these systematic fluctuations.
<figure><img src="image_11.png" /><figcaption>Fig. 10. The angular distribution of the 432-keV line in $^{152}$Dy. An A$_{2}$ coefficient value of +0.363(2) and an A$_{4}$ coefficient value of +0.123(3) were extracted from the fit to the data. The dashed green line is calculated using eq. 6 to 19 with a $\sigma/J$ ratio of 0.3 and $\delta_{1} = 0.7$. Thus, this is an example of a $\gamma$ ray of mixed character. Because of interference with a stopped line just above the 432-keV line, only polar angles near or above 90° can be used. The $\chi^{2}$ of the fit is 142 and reflects systematic errors.</figcaption></figure>

The angular distribution of the 432-keV line in $^{152}$Dy is presented in Fig. 10. This transition is assigned a tentative M1 character in the literature [35]. However, from the distribution curvature (A2 is positive), it is clear that it does not correspond to a pure M1 transition; but rather to a mixed one. A mixing ratio of 0.7 fits the data well. From the angular distribution alone, one cannot infer if the transition is of mixed M1/E2 or a E1/M2 nature – nor if it might be of E2 character. However, a DCO measurement of this transition in presented in Sec. 4 below and in Sec. 5, the linear polarization is measured as well. Those measurements will clarify the nature of the $\gamma$ ray.

# 4 Angular correlations: DCO

## 4.1 Formalism

Measuring the angular distribution of $\gamma$ rays from excited, recoiling nuclei created in-beam is the standard technique for determining the angular momentum of the transition involved and thereby inferring the spins and possibly the parities of the states associated with the decay. However, as discussed in Ref. [43], the method does not necessarily determine all the parameters and, sometimes, yields ambiguous results. More information can, in general, be

extracted by additionally determining the angular correlation between two $\gamma$ rays measured in coincidence. Because of the coincidence requirement, the signal is usually cleaner; but more data is needed. Furthermore, if a cascade of $\gamma$ rays flows through a sufficiently long-lived isomer, it may not be possible to measure the angular distribution of the transitions below the latter state because the m-distribution is likely to be fully or partially randomized by the atomic fields. In such cases, angular correlations remain a powerful approach. The general method used when the initial state is not characterized by a random m-distribution is referred to as Directional Correlation from Oriented nuclei (DCO).

When the nucleus of interest is produced in a fusion-evaporation reaction, it will be polarized with respect to the beam axis. For the case of angular correlations between two $\gamma$ rays emitted from such nuclei, a more general form of the $w(\theta)$ function in eq. 1 must be used to account for the fact that the initial emitting state does not have a random m-distribution. Not surprisingly, the theory contains elements from the regular correlation formalism covered in Sec. 2.1 and the angular distribution covered in Sec. 3.1. In Ref. [44], the general formula for the DCO is expressed as

$$w(\theta_1, \theta_2, \phi) = \sum_{\lambda_1, \lambda_2} B_{\lambda_1}(I) A_{\lambda}^{\lambda_2 \lambda_1}(X_1) A_{\lambda_2}(X_2) H_{\lambda_1 \lambda \lambda_2}(\theta_1, \theta_2, \phi) \quad (20)$$

where $\phi$ is the angle between the two planes defined by the beam axis and the direction in which the two coincident $\gamma$ rays are emitted, see Fig. 11. The angle is, therefore, also referred to as the relative azimuthal angle [43] between the $\gamma$ rays. In addition, the expression in eq. 20 also has a dependence on the direction of the emitted $\gamma$ rays with respect the beam axis through their $\theta_1, \theta_2$ polar angles (in the CM system). In Fig. 11, $\gamma_1$ and $\gamma_2$ denote the directions of the first and second $\gamma$ ray, respectively. The spin of the first state is (in this notation) $I_i$ and the $\gamma$ ray emitted has a multipolarity of $L_1, L'_1$ with a mixing ratio of $\delta_1$. The intermediate state then has a spin of $I$ and the transition in the decay to the final state, with a spin of $I_f$, has a multipolarity of $L_2, L'_2$ with a mixing ratio of $\delta_2$. For the special case where the first state has a random m-distribution, the expression in eq. 20 can be simplified to eq. 1 [44]. Since a random distribution of m-states of the first state is just a special case of polarization, eq. 20 in fact also describes this case.

The first term in eq. 20, known as the orientation parameter, is directly connected to the angular distribution described in Sec. 3.1. We make the connection as:

$$B_{\lambda}(I) = A_{k}(I_{i}, L_{1}, L'_{1}, I) \quad (21)$$

where the formalism is given in eqs. 7 to 16.

The next term in eq. 20, $A_{\lambda}^{\lambda_2 \lambda_1}(X_1)$, closely resembles eq. 3

$$A_{k}^{\lambda_2 \lambda_1}(L_2, L'_2, I_f, I) = \frac{1}{1+\delta^2} [F_{k}^{\lambda_2 \lambda_1}(L_2, L_2, I_f, I) + 2\delta F_{k}^{\lambda_2 \lambda_1}(L_2, L'_2, I_f, I) + \delta^2 F_{k}^{\lambda_2 \lambda_1}(L'_2, L_2, I_f, I)]; (22)$$
<figure><img src="image_12.png" /><figcaption>Fig. 11. The geometry used in the DCO formalism. The angle, $\phi$, is the relative azimuthal angle of the two planes defined by the z-axis (or beam axis) and the direction of the two coincident $\gamma$ rays. The two angles, $\theta_1$ and $\theta_2$, are the polar angles with respect to the z-axis of the two emitted photons, respectively.</figcaption></figure>

but the ordinary F-coefficients are replaced by Weidenmuller's generalized F-coefficients:

$$ F_{\lambda}^{\lambda_2 \lambda_1}(L_2, L'_2, I_f, I) = \left[ (2I + 1)(2I_f + 1)(2L_2 + 1)(2L'_2 + 1) \right. \\ \left. \qquad (2\lambda + 1)(2\lambda_1 + 1)(2\lambda_2 + 1) \right]^{1/2} (-1)^{L'_2 + \lambda + \lambda_2 + 1} \times \begin{pmatrix} L_2 & L'_2 & \lambda \\ 1 & -1 & 0 \end{pmatrix} \begin{Bmatrix} I_f & L_2 & I \\ I_f & L'_2 & I \\ \lambda_2 & \lambda & \lambda_1 \end{Bmatrix} \quad (23) $$

containing 3-j "()" and 9-j "{}" symbols which can be found in Ref. [24], where the 3-j symbols are also tabulated. Numerical formulas can furthermore be found in the App A.3 and A.5.

The third term in eq. 20, $A_{\lambda_2}(X_2)$, also resembles eq. 3 after a few substitutions:

$$ A_k(L_2, L'_2, I_f, I) = \frac{1}{1+\delta^2} (F_k(L_2, L_2, I_f, I) + 2\delta F_k(L_2, L'_2, I_f, I) + \delta^2 F_k(L'_2, L'_2, I_f, I)) \quad (24) $$

where $F_k$ is given in eq. 4. The last, and final term in eq. 20, $H_{\lambda_1 \lambda_2 \lambda_2}(\theta_1, \theta_2, \phi)$, can generally be expressed for most in-beam cases (see discussion in Ref. [44]) as

$$ H_{\lambda_1 \lambda_2 \lambda_2}(\theta_1, \theta_2, \phi) = \sum_q (2 - \delta_{q0}) < \lambda_1 0 \lambda q | \lambda_2 q > \frac{2^{\lambda_1+1}}{2^{\lambda_2+1}} \frac{(\lambda-q)!(\lambda_2-q)!}{(\lambda+q)!(\lambda_2-q)!} P_{\lambda}^{q}(\cos\theta_1) P_{\lambda}^{q}(\cos\theta_2) \cos(q\phi) \quad (25) $$

where $q$ takes the even and odd values from 0 to the smallest values of $\lambda$ and $\lambda_2$ – both of which are even. This formula applies when $\lambda1 + \lambda + \lambda_2 = \text{even}^4$. $P_{\lambda}^{q}$ are the as-

sociated Legendre polynomials which, in this general case, include odd q terms (see App. A.2).

As pointed out in Ref. [40], the $cos(\theta_1)$ and $cos(\theta_2)$ terms in eq. 25 are understood to be referring to the angles in the CM system. The angle, $\phi$, is the same in both the laboratory and CM systems, and the relationship between the cosines in the laboratory and CM system was introduced in eq. 18. Furthermore, in order to also take into account the Lorentz transformation, eq. 20 must be augmented with the following multiplicative terms (see also eq. 19):

$$ \times \frac{1 - \beta^2}{[\beta \cos(\theta_1) - 1]^2} \times \frac{1 - \beta^2}{[\beta \cos(\theta_2) - 1]^2} \quad (26) $$

With the modifications of eq. 18 and 26 to eq. 20, the angular correlation of $\gamma$ rays emitted from oriented nuclei can now be described in the laboratory system. The DCO formalism is clearly more complex than the angular correlation for the case covered in Sec. 2.1 (See App. F for some explicit examples).

In Refs. [43,45], the parameter-space is investigated. If one assumes that only dipole (L=1) and quadrupole (L=2) radiation is involved in the gamma decays, $\lambda$ and $\lambda_2$ can only take the even values 0, 2, and 4. Consequently, there are up to 19 non-vanishing terms in eq. 20.

With individual detectors, it is possible to select specific geometries, for which some terms in eq. 20 vanish, thus making the data analysis easier to perform [43–45]. In particular, the so-called 'normal geometries' are commonly employed. In these, one (or more) of the angles $\theta_1, \theta_2$ or $\phi$ are chosen to be $\pm \frac{\pi}{2}$ or 0 so that only the terms with even $\lambda$ and $q$ are non-zero. In such cases, the associated Legendre polynomials can be expressed as a sum of the ordinary ones enabling a fit of the resulting angular correlation spectra with a simple Legendre formula as given by eq.1. While selecting pseudo-normal geometries in tracking arrays is a possibility, it may result in a significant loss of statistics. In such cases, methods which exploit the continuous-angle measurements these arrays provide should be used.

In this work, we have explored two ways to practically extract DCO signals from tracking arrays. In both cases, from the experimental data, we construct DCO cubes with $180^3$ voxels of $(\theta_1, \theta_2, \phi)$, i.e., each voxel has a resolution of 1 degree.

Two gates are placed on the $\gamma$ rays of interest, which are in coincidence, and two gates are placed on nearby background areas. Three DCO cubes are now generated for peak–peak, peak–background and background–background coincidences, respectively. The latter two are used to background subtract the peak–peak DCO cube. Concurrently, a similar set of response DCO cubes are constructed by mixing $\gamma$ rays from the current coincidence event with $\gamma$

<footnote n="4">⁴ An equivalent equation can be written when $\lambda1 + \lambda + \lambda_2 = \text{odd}$. It contains $\sin(q\phi)$ rather than $\cos(q\phi)$ terms; but this</footnote>

<footnote>formula only applies under some special circumstances which are not covered in this work (see discussion in Ref. [44])</footnote>
<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Illustration of the sparseness of the DCO cubes described in the text.</figcaption></figure>

rays from previous ones. This ensures that the phase space is preserved while preventing the response cubes from having any DCO signal – as the γ rays selected in this way are *not* in coincidence. It follows that, contrary to the signal DCO cubes, the response DCO cubes do not suffer from the so-called clustering hole effect, see Ref. [4,46]). For source data, this clustering hole effect can be seen prominently in Fig 1. Since one cannot correct the signal DCO cubes for this effect, one can instead *impose* the clustering hole effect on the response DCO cubes. This is accomplished by *not* updating the response DCO cubes if the angle between the γ rays involved is below a minimum angle, ~1.6 × the clustering angle. This angle is adjusted slightly to match the clustering hole observed in the signal DCO cubes, where the clustering hole effect is in actuality observed as a 'clustering trench' at small φ angles in the cubes (best discovered through cross-diagonal projections).

Due to the complex coordinate system used (see Fig. 11), the DCO response function produced by mixing of uncorrelated events may not be a perfect representation. However, for in-beam data, the method provides a good approximation, and experimentally, the resulting response is sufficient for the extraction of a reliable DCO function.

It is important to recognize that, for the current γ-ray tracking spectrometers, the DCO cubes are sparse; only a fraction of voxels contain counts because the detector systems are not yet fully populated. This is illustrated in Fig. 12 with data from GRETINA when the array comprised 47 crystals in a compact configuration at ANL (see App. B.1).

In the first, and simplest, method of analyzing the data in the DCO cubes (dubbed the *DCO φ method*), a projection is made on the *φ* axis of the experimental data from the signal (peak-peak) DCO cubes, after background subtraction. A similar projection is produced from the response DCO cubes. Finally, a ratio of the two projections is generated. This ratio spectrum, with propagated errors, represents the measured DCO signal and examples will be presented below.

Simultaneously, a *calculated* spectrum is constructed using the information from the signal DCO cubes, with the parameters of eq. 20 where only $θ_1$, $θ_2$ and $φ$ from voxels that have counts are used. By design, the calculated spectrum imposes the experimental phase space on the data. This method produces a 1D experimental (ratio) data spectrum, with errors, and a DCO spectrum calculated consistently. These spectra can be compared, and a $χ^2$ value can be extracted. The experimental $φ$ spectrum has been extracted previously (e.g., see Ref. [12]). However, the DCO cubes discussed above preserve sufficient experimental information to accurately calculate the theoretical DCO signal based on the relevant parameters: spins, multipolarities, and mixing ratios; and this is done while conforming to the phase space imposed by the configuration of the continuous angle tracking array.

In the second method used to analyze the data in the DCO cubes (dubbed the *caDCO 2D superpulse method*), 2D projections on the $θ_1$ and $θ_2$ face of the DCO cubes are made, with slices on the $φ$ axis. Here, the *ca* symbol refers to the continuous angle aspect of the method.

Two DCO cubes (as defined above) are produced: one for the experimental signal and one holding the associated errors. A similar calculated DCO 2D matrix is constructed using only non-zero voxels in the signal cube. In principle, one would like to show the 2D matrices of the experimental signal with errors along with the calculated DCO signal surface. Since there is no straightforward graphics rendering, we construct a 1D super-pulse spectrum where the index is $[180 × θ_1 + θ_2]$ instead (referred to as a pseudo-polar angle in the following). This allows for the usual display of data points with errors and a calculated spectrum, albeit with many more channels than we are used to. If the index is further compressed, typically contracting to 10-15°, it is possible to render at least a fraction of the data and, after some normalization (adjustment of y offset), to generate a $χ^2$ as well using a few thousand data points. It is found that one needs good statistics to perform this analysis, but it may offer a high sensitivity for DCO fits to determine, e.g., mixing ratios (see discussion in App. E). One also observes that the $φ$ slices at 180° and 0° have the largest signal amplitude and the slices near 90° the smallest one, respectively. This can be understood by observing that the next leading terms in the DCO formula, see App. F, have $cos()$ terms which are thus reduced at 90°. The data for slices below $φ \sim 2$ times the cluster angle are still distorted, even though we tried to impose the clustering hole effect in the response DCO cubes. The superpulse spectrum with $150° < φ < 180°$
<figure><img src="image_14.png" /><figcaption>**Fig. 13.** The measured angular DCO correlations of the 967- and 541-keV lines in <sup>152</sup>Dy (from in-beam data). The 967-keV γ ray is an E1 transition (28<sup>+</sup> → 27<sup>-</sup>) and the 541-keV one is an E2 transition (25<sup>-</sup> → 23<sup>-</sup>). The 221-keV (27<sup>-</sup> → 25<sup>-</sup>) E2 transition is unobserved. See text for details.</figcaption></figure>

exhibits a particularly clear oscillatory behavior. An example is presented in App. E.

A different technique has been presented in Ref. [47], called SpeeDCO, which makes use of Fourier transformations. The technique further uses the γγ spectra channel by channel rather than on areas of peaks. This may not work so well for tracked data because the effective FOM cuts can be different at the center of peaks as opposed to their flanks.

## 4.2 Experimental extraction

The DCO signal presented in Fig. 13 is from a pair of transitions of E1-E2 character with an unobserved E2 transition between them [48,35]. The *DCO* *ϕ* method described above was used to extract the signal. As can be seen in Fig. 13, the stretched nature of the lines are consistent with the calculated curve. The difference between the DCO signal from the actual GRETINA array with 47 crystals in the setup (blue solid line) and the signal one for a 4π array (black dashed line) suggests that the experimental signal can be enhanced to some extent by configuring the array carefully.

In Fig. 14, the DCO signal from the 432-keV transition of mixed E1/M2 nature (32⁻ → 31⁺) and the stretched M1 712-keV γ ray (31⁺ → 30⁺) in <sup>152</sup>Dy is presented. As was discussed in Sec. 3.2 (Fig. 10), the 432-keV lines is probably not associated with an M1 transition as described in Refs [35,48]. The angular distribution of the line is

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** The measured angular correlations of the 432- and 712-keV γ rays in <sup>152</sup>Dy (from in-beam tracked data). See text for discussion.</figcaption></figure>

consistent with a transition of either M1/E2 or E1/M2 nature with a mixing ratio of ~0.7 (see Fig. 10). Figure 14 clearly rules out the 432-keV transition as being of a stretched E2 nature – and a E1/M2 assignment is slightly favored.

For the *caDCO* 2D *superpulse method*, significant statistics are required and it has been difficult to apply it to the limited, in-beam dataset described in Appendix B.1. However, some statistically substantial source datasets (see App. B.2) are available and this second method can be applied with $\frac{v}{c} = 0$ and when $\sigma_J$ is given a large value to effectively randomize the initial m-state population of the first state. Such a DCO extraction for sources is pursued in App. E and overcomes some of the limitations on statistics encountered for the in-beam data. Note that the *DCO* *ϕ* method using the same very high statistics source data is explored in App. D.

## 5 Linear polarization

Before large tracking arrays became available, the linear polarization of γ rays were, in general, measured as a ratio of signals from detectors near 90° versus a signal from those around 0° with respect to a Compton scattering plane. Such a ratio establishes the probability of Compton scattering into and out of the scattering plane. For such cases, simple explicit formulas can be produced, especially when only transitions of stretched character are considered [22, 39, 49–62]. For modern tracking arrays, improved techniques are presented here which take full advantage of continuous angle measurements. Furthermore,
a new technique not requiring source data for normalization is introduced as well. The data itself is used to generate a response function. In contrast to many presentations covering linear polarization, the cases of mixed transitions are included in the formalism (and calculated spectra) presented here as well.

An example of the resolution of an ambiguity for a mixing ratio determined from angular distribution measurements complemented by linear polarization data is presented in Ref. [52]; see also the discussion in Ref. [53,63]. Clearly, using tracking arrays to simultaneously measure angular distributions and/or correlations as well as linear polarizations without any dedicated setup is a significant advantage.

For the case of radioactive sources, two γ rays must be observed in order to determine the linear polarization of a γ ray. The observation of a first photon causes a polarization of the intermediate state which enables the determination of polarization for the second γ ray. This case is covered in Sec. 5.1. For in-beam spectroscopy, the nuclear reaction will (in general) result in an alignment so that the polarization of individual γ rays can be measured. This case is presented in Sec. 5.2. Two γ rays can also be measured in coincidence especially for nuclei produced in fusion-evaporation reactions and the method of Sec. 5.3 covers such cases.

The overall linear polarization signal to be expected can be elucidated by following the procedures in Ref. [15]. The Klein–Nishina scattering formula, with the linear polarization term included, can be written as:

$$d\sigma_{\phi} = \frac{r_0^2 E \gamma'}{2E\gamma} \left[ \frac{E\gamma}{E\gamma'} + \frac{E\gamma'}{E\gamma} - 2\sin^2\theta \cos^2\phi \right] d\Omega \quad (27)$$

where $E\gamma$ is the incident γ-ray energy, $E\gamma'$ the energy of the γ ray after the Compton scattering through the angle $\theta$ and $r_0$ is the electron radius. The angle $\phi$ is the azimuth angle of the scattered photon with respect to the reaction plane, see Fig. 16, and the discussion below. The energy and scattering angle are related through the well-known Compton formula [64]

$$E'_{\gamma} = \frac{E_{\gamma}}{1 + \frac{E_{\gamma}}{m_{0}c^{2}}(1 - \cos(\theta))} \quad (28)$$

where $m_0c^2$ is the electron rest mass. For the present purpose, this formula can be written as:

$$\frac{E\gamma}{E\gamma'} = 1 + \frac{E\gamma}{511}(1 - \cos\theta) \quad (29)$$

with energies in units of keV.

The linear polarization sensitivity can now be evaluated by forming the ratio:

$$R = \frac{d\sigma_{90}}{d\sigma_0} = \frac{\frac{E\gamma}{E\gamma'} + \frac{E\gamma'}{E\gamma}}{\frac{E\gamma}{E\gamma'} + \frac{E\gamma'}{E\gamma} - 2\sin^2\theta}, \quad (30)$$

<figure><img src="image_16.png" /><figcaption>Fig. 15. The linear polarization sensitivity, R, as a function of the Compton scattering angle for the incident γ-ray energies indicated. The curves are similar to those presented in Ref. [58]; but the R sensitivity used here is the one suggested in Ref. [15].</figcaption></figure>

which is a measure of the probability for scattering 'out of plane' versus the 'in plane' one. For a given energy, R will have a maximum at a (slightly) energy dependent scattering angle, $\theta^{max}(E\gamma)$.

In Fig. 15, R is plotted as a function of the Compton scattering angle for a number of incident γ-ray energies. As can be seen, the sensitivity is larger at low energies where $\theta^{max}$ is close to ~ 90°. For higher energies $\theta^{max}$ occurs at a lower angle (~ 69° for $E\gamma = 1500$ keV). However, the curves in Fig. 15 must be modified to include the actual probability of observing at least two interactions in the tracking array for the γ ray of interest. This probability becomes small for energies below ~300-keV as photo-absorption starts to dominate and, consequently, there is no second scattering.

Furthermore, the measured polarization signal depends on the actual sensitivity of the detector system in use. The ratio of the measured signal over the calculated one is often referred to as Q the polarization sensitivity parameter (sometimes also referred to as the analyzing power [61, 65]). This Q value depends on the number of crystals in the detector array and the configuration in use as well as on the γ-ray energy [51] and the gating conditions used to extract a clean signal. In this work, the Q value can be extracted for the examples presented in Sec. 5.4 through the ratio of the amplitude of fitted curves (in blue) to that of the corresponding calculated ones (in black), see Figs. 17 to 20. In general, tracking arrays have larger Q values whereas Compton suppressed ones (because of their construction) have smaller values [66,67]. In principle, one can also extract linear polarization information from any
Compton scattering after the second interaction [68]; but this is not pursued in this work.

The latest versions of the GRETINA tracking code [4] attempts to identify and process pair-production interactions [69–71]. If identified as such, the sub-event is marked as a single interaction event and the part of the code that addresses Compton scatterings is not engaged. Thus, even for high-energy γ rays, with mixed pair production and Compton scatterings, a polarization signal can be extracted from the latter when properly identified, but pair production, as reported by the tracking code, is ignored. For the latter, there is indeed information about the linear polarization of the γ ray [71–73], but the limited information extracted by the current decomposition task does not allow for the use of it.

## 5.1 Linear polarization: radioactive source data

The full version of the angular correlation formula (eq. 1), which also includes the linear polarization of the second γ ray, can be written as [15]:

$$ \omega(\theta, \zeta) = w(\theta) + P_{\pm} \cos(2\zeta) \\ (A_2(L_1, L'_1, I_i, I) \quad \Re_2(L_2, L'_2, I_f, I) P_2^2(\cos\theta) + \\ A_4(L_1, L'_1, I_i, I) \quad \Re_4(L_2, L'_2, I_f, I) P_4^2(\cos\theta) + \dots) \quad (31) $$

where θ is the angle between the two photons and w(θ) is given in eq. 1 through eq. 5. This formula is only valid if the initial state has a random orientation, which is normally the case for sources which are not cryogenically cooled. The A_k coefficients are given in eq. 3 and the P_k^2 functions are the normalized associated Legendre polynomials (see App. A.2). The \Re_k coefficients are 'modified' from the A_k ones by the K_k coefficients as:

$$ \Re_k(L, L', I_1, I_2) = \frac{1}{1+\delta^2} \left( \begin{aligned} & K_k(L, L)F_k(L, L, I_1, I_2) \\ & +2\delta K_k(L, L')F_k(L, L', I_1, I_2) \\ & +\delta^2 K_k(L', L')F_k(L', L', I_1, I_2) \end{aligned} \right) \quad (32) $$

where the latter K_k coefficients are evaluated as [15]:

$$ K_k(L, L') = \left[ \frac{(k-2)!}{(k+2)!} \right]^{1/2} \frac{C(L1L'1|k2)}{C(L1L' - 1|k0)} \quad (33) $$

with C being a Clebsch–Gordan coefficient (see app. A.3). The angle ζ is the azimuthal angle of the second scattered γ ray from the first interaction point in the coordinate system which will be described next. P_± takes on the value of +1 or −1 for Electric – and Magnetic transitions, respectively (as determined from known γ rays using this coordinate system).

In order to experimentally extract the linear polarization from experimental, tracked data, the following procedures are applied: a reaction plane is first constructed from the direction of the polarizing (first) photon and that of the

<figure><img src="image_17.png" /><figcaption>Fig. 16. The coordinate system used to extract the linear polarization signal. The reaction plane coincides with that of the figure. The X, Y and Z vectors are normalized to be unit vectors, and the Y vector points out of the plane. See the text for a full description of the coordinate system.</figcaption></figure>

(second) γ ray for which the polarization will be determined. This reaction plane defines a coordinate system where the y-axis is the normal vector to the reaction plane, the z-axis is in the direction of the second photon, while the x-axis is produced as a dot product between the former two normal vectors, see Fig. 16. The polarization spectrum is then formed by binning the azimuthal angle (with respect to the X and Y axis) of the scatter of the second γ ray in this coordinate system.

In the analysis, limits are placed on angles between the first and second photons, around 90° (or slightly smaller, see Fig. 15), in order to achieve a well-defined reaction plane and, furthermore, optimize the amplitude of the polarization signal [74,75]. Limits are placed on the scattering range of the second scatter of the second γ ray, as well as the polar angle of the latter with respect to the first one, in order to also obtain a well-defined scattering plane. This ensures that the extraction of the azimuthal angle as described above is meaningful.

Note that, in this formalism (eq. 31), one is always obtaining the linear polarization of the second γ ray. However, because of symmetry, one can equally well calculate the cascade 'the other way', e.g., a 3(1)2(2)0 cascade can be calculated as a 0(2)2(1)3 one (as is done in the $^{207}$Pb example below).

As can be seen from eq. 31, the linear polarization signal has an offset which depends on the w(θ) values. Therefore, one should average over the polar angles to fit the experimental conditions. However, in the practical extractions of the linear polarization signal, the offset from this term is normalized out. The amplitude of the P_±cos(2ζ) term is a function of the setup or, more precisely, of the probability of observing the range of polar angles within the gates on these angles that are often applied in the experimental
extraction, as described above. In eq. 31, one may use the substitution

$$P_{\nu}^{2}(\cos\theta) \rightarrow \sum_{\theta} P(\theta) P_{\nu}^{2}(\cos\theta) \quad (34)$$

when the polarization signal is extracted. Here, $P(\theta)$ is the normalized distribution of angles between the $\gamma$ rays under the additional conditions applied in the experimental extraction. Using this averaging procedure, the expected amplitude for the $P_{\pm}\cos(2\zeta)$ term as well as the sign can be calculated.

To extract the linear polarization spectrum, the spectrum described above it must be normalized by dividing by a reference. In many treatments, the reference spectrum (or response function) is formed by using (single) $\gamma$ rays from a radioactive source, which, by definition, have no polarization signal (see e.g., Ref. [58,76]) unless coincidence conditions are applied. This means one has to perform two measurements, one to obtain the desired data, and another to generate the reference, or response, spectrum (for a similar energy). It follows that the statistics achieved for the reference spectrum can have a significant impact on the final statistics of the normalized linear polarization spectrum [74,75].

In this work, we propose to use the data itself to produce the reference spectrum. For each event, the direction of the first of the two emitted $\gamma$ rays is stored away in a queue of 'reference' polarization directions. Stored $\gamma$ rays versus the current $\gamma$ ray under consideration have no linear polarization because the photons are not in coincidence. The reference spectrum is now produced by analyzing the polarization of the second photon using the members of the stored queue of previous 'first' $\gamma$ rays which can be hundreds of members deep. Thus, one can easily generate overwhelming statistics for the reference spectrum and, therefore, only the statistics of the real signal ends up being a factor. For each event, the reference queue is moved one down and the current first direction takes the lead in the reference queue for the next one.

Even though, when using tracking arrays, the linear polarization signal can be extracted as an aesthetically pleasing full spectrum, in practice, it is still largely the sign of the $\cos(2\zeta)$ term which ultimately matters.

## 5.2 Linear polarization: in-beam data

For the case of the linear polarization measurements of (single) $\gamma$ rays from nuclei produced in-beam, one must expand on eq. 6. Based on Refs. [15,22], we may write

$$\omega(\theta, \zeta) = \omega(\theta) + P_{\pm}\cos(2\zeta) \times (A'_{2}P_{2}(\cos\theta) + A'_{4}P_{4}(\cos\theta)) \quad (35)$$

where $\omega(\theta)$ is taken from eq. 6 and

$$A'_{k}(I_{i}, L_{1}, L'_{1}, I_{f}) = \alpha_{k} A_{k}^{\text{max}}(I_{i}, L_{1}, L', I_{f}) \quad (36)$$

with $\alpha_k$ given in eq. 12. Note that $A_k^{\text{max}}$ is modified from $A_k^{\text{max}}$ by the $K_k$ coefficients (see eq. 33) as

$$A_k^{\text{max}}(I_i, L_1, L'_1, I_f) = \frac{1}{1+\delta^2} (K_k(L_1, L_1)f_k(I_f, L_1, L'_1, I_i) + 2\delta K_k(L_1, L'_1)f_k(I_f, L_1, L'_1, I_i) + \delta^2 K_k(L'_1, L'_1)f_k(I_f, L'_1, L'_1, I_i)) \quad (37)$$

This formula holds if the required polarization is caused by the nuclear reaction and the reaction plane (see Fig. 16) then includes the beam axis (z-axis). Obviously, terms with $\lambda = 6$ and higher can easily be added to cover cases where the $\gamma$ transitions have angular momentum changes $L > 2$.

For in-beam linear polarization, to the best of our knowledge, a source has always been used to generate the normalization spectrum (see e.g., Ref. [58]). However, once again the in-beam data itself can be used to normalize. Essentially, a queue of uncorrelated events is used as was the case for the angular correlation in Sec. 2.2, but in this case, 3D random directions [77] of the beam axis are employed rather than the direction of the first $\gamma$ ray. This effectively creates a random reaction plane. Other than this, the data is treated the same way. Because of the limitations on the polar angle generally applied, the full range of directions in 3D are not in use.

One observes that, for sources, the character of the 'polarizing' $\gamma$ ray can easily change the sign of the observed polarization spectrum. Thus, a proper calculation is required to determine if the $\gamma$ ray of interest was indeed of electric or magnetic character; simple rules do not apply. For in-beam cases, mostly all the $\gamma$ rays see the same polarization, although in some special cases, the sign of the attenuation coefficient in a cascade, $\alpha_k$, (see eq. 7 to 16) can actually change sign (for an example, consult Ref. [21]). Note that eq. 35 should be modified to take into account the Lorentz transformation. Thus, the corrections of eq. 18 and 19 should be applied to eq. 35 in cases where the velocity parameter $v/c$ is significant.

## 5.3 Linear polarization: DCO in-beam

It it also possible to measure the linear polarization for in-beam cases taking advantage of the correlation between two $\gamma$ rays; i.e., using one of them to create a polarization for the other. This can be useful in instances where a nucleus is formed in-beam, but has partially lost its polarization, e.g., because the deexitation process involved an isomer. If the nucleus, created in-beam, loses all of it's polarization, one may revert the formalism of Sec. 5.1 with the modifications of eq. 19 and 26 to take into account the Lorentz transformation. As a matter of fact, the extraction of the experimental data for the DCO case is identical to the description in Sec. 4.1, including the formation of the response data using the mixing of events
not in coincidence. However, if the initial state is even partially polarized, the full DCO formalism presented below is required.

To arrive at the general DCO formula which includes the linear polarization term, one must include this substitute in e.g. 20

$$P_{\lambda}^{q}(\cos\theta_{l}) \rightarrow P_{\lambda}^{q}(\cos\theta_{l}) + P_{\pm}\cos(2\zeta)K_{k}P_{\lambda}^{q}(\cos\theta_{l}) \quad (38)$$

One must use $l = 1$ if the linear polarization is observed for the first $\gamma$ ray and $l = 2$ for the second one. Thus, the DCO formula including the linear polarization term for the second $\gamma$ ray can be written as:

$$w(\theta_1, \theta_2, \phi, \zeta) = w(\theta_1, \theta_2, \phi) + P_{\pm}\cos(2\zeta) \times \left[ \sum_{\lambda_1\lambda\lambda_2} B_{\lambda_1}(I_i) A_{\lambda}^{\lambda_2\lambda_1}(X_1) A'_{\lambda_2}(X_2) H_{\lambda_1\lambda\lambda_2}(\theta_1, \theta_2, \phi) \right] \quad (39)$$

where

$$A'_{\lambda_2}(X_2) = \frac{1}{1+\delta^2} \left( K_k(L_2, L_2) F_k(L_2, L_2, I_f, I) + 2\delta K_k(L_2, L'_2) F_k(L_2, L'_2, I_f, I) + \delta^2 K_k(L'_2, L'_2) F_k(L'_2, L'_2, I_f, I) \right) \quad (40)$$

The other terms can be found in Sec. 4.1 after appropriate substitutions.

In order to calculate the amplitude of the linear polarization signal, one must take into account the phase space of the angles $\theta_1, \theta_2$ and $\phi$. This requires using the ideas of the (sparse) DCO cubes discussed in Sec. 4.1 and an averaging of

$$P_{\lambda}^{q}(\cos\theta_{1})P_{\lambda}^{q}(\cos\theta_{2})\cos(q\phi) \rightarrow \sum P(\theta_{1}, \theta_{2}, \phi)P_{\lambda}^{q}(\cos\theta_{1})P_{\lambda}^{q}(\cos\theta_{2})\cos(q\phi) \quad (41)$$

in the $H_{\lambda_1\lambda\lambda_2}$ terms (see eq. 25) must be used since those angles are not observed in a typical linear polarization measurement. Here, $P(\theta_1, \theta_2, \phi)$ is the (normalized) probability for using these particular angle voxels in the DCO cubes in the experimental setup under all gating conditions.

We have not attempted to extract the DCO linear polarization signal from recoiling nuclei produced in-beam, using two $\gamma$ rays in coincidence, as a dataset with sufficient statistics is at present unavailable to do so. However, in this work, for completeness, we do suggest the formalism.

## 5.4 Experimental extraction

Figure 17 presents the measured linear polarization spectrum for the 1333-keV line from a $^{60}$Co source, when the observance of the 1173-keV $\gamma$ ray is used to polarize the intermediate state. A fit to the tracked data yields a polarization amplitude of +0.016(2). This transition, as discussed above, is of almost pure E2 character in a 4(2)2(2)0

<figure><img src="image_18.png" /><figcaption>Fig. 17. Linear polarization spectrum for the 1333-keV E2 transition in $^{60}$Ni (daughter of $^{60}$Co). The observation of the 1173-keV E2 line in GRETINA was used to polarize the $\gamma$ ray. The extracted sensitivity value Q is 0.33(4) and if the transitions are reversed, one finds a Q value of 0.37(4). The $\chi^2$ for the fit is 1.2.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 18. Linear polarization spectrum for the 1064-keV M4/E5 line in $^{207}$Pb (daughter of $^{207}$Bi) polarized by the observation of the 570-keV E2 transition in GRETINA. The data points are in red, the calculated curve with $\delta_2 = 0.02$ in thin black and the dashed line is a calculation with $\delta_2=-0.96$. The $\chi^2$ for the fit is 0.9. See text for discussion.</figcaption></figure>

cascade, and the calculated amplitude using eq. 31, including the averaging over the polar angles of eq. 34, was found
to be +0.048. Thus, Q is determined to be 0.33(4). The calculation is shown as a thin black curve.

In Fig. 18, the linear polarization spectrum for the 1064-keV $\gamma$ ray from a $^{207}$Bi source is displayed under the condition that the 570-keV transition is observed in coincidence. The 1064 keV transition is reported to be an M4 one [78] in an 1/2(2)5/2(4)13/2 cascade (considering the cascade backwards in order to make the second transition the 1064-keV $\gamma$ ray) with a small admixture of E5 multipolarity ($\delta_2$ ~0.02). The fitted polarization amplitude is -0.022(1) and the calculated amplitude was found to be -0.064. Thus, the measured Q value is 0.34(3) using a $\delta_2$ value of 0.02. The angular correlation of the two lines were also measured (but not shown here) and $A_2$ and $A_4$ values of +0.194(2) and -0.024(2), respectively, were determined. A $\delta_2$ ~0.02 reproduce these values; but a $\delta_2$ fit revealed that a slightly deeper minimum was located at -0.96. Thus, in Fig 18, the calculated curve for $\delta_2$ = -0.96 is also presented (dashed curve). The polarization measurement clearly indicates that the $\delta_2$ ~0.02 value is the correct assignment and that, with this mixing ratio, the measured Q value is in line with that of the other measurements. This is another example of how measuring the linear polarization can resolve ambiguities in mixing ratios obtained from angular correlations only.

Clearly, the opposite electric and magnetic character of the two cases (both with even L) discussed above are confirmed and these transitions were used to determine the sign, $P_{\pm}$, of the polarization in eq. 31, assuming that the 1333-keV transition is indeed of electric character. Calculations, based on eq. 31, then yield the same signs of the measured polarization amplitudes using our conventions.

For the daughter of $^{152}$Eu via the $\beta^-$ decay, $^{152}$Gd, there are two $\gamma$ rays in coincidence with energies of 779 - and 344-keV. The first of these is an E1 transition with a small mixture of M2 ($\delta$ ~0.03) and the second is a pure E2 transition to the ground state [35]. The cascade is therefore $3^- \rightarrow 2^+ \rightarrow 0^+$ and the angular correlation of these $\gamma$ rays is presented in Fig. 3 above. In Fig. 19, the measured linear polarization spectrum for the 344-keV $\gamma$ ray is presented under the condition that the 779-keV line is used to polarize the $2^+$ state. The measured amplitude is -0.014(2), whereas the calculated one is -0.033.

In Fig. 20, the related linear polarization is presented where now the observance of the 344-keV $\gamma$ ray provides the polarization of the $2^+$ state required to determine the linear polarization signal for the 779-keV $\gamma$ ray. The measured amplitude is +0.014(2) and the calculated one is +0.033. This is a case where it is necessary to calculate the sign of the polarization signal because the polarizing $\gamma$ ray can effectively change the observed sign. Using the formalism of eq. 31, one does indeed confirm that both $\gamma$ rays are of electric character, even though the signs of the measured polarization spectra are observed to be opposite in Figs. 19 and 20. This tends to happen when the two transitions have mixed odd and even L quantum numbers or whether there is a parity change or not [79]. To

<figure><img src="image_20.png" /><figcaption>**Fig. 19.** Linear polarization spectrum for the 344-keV E2 transition in $^{152}$Gd (daughter of $^{152}$Eu) after the observation of the 779-keV line in GRETINA. The extracted Q value is: 0.42(6) and the $\chi^2$ for the fit is 1.2.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 20.** Linear polarization spectrum for the E1 779-keV line in $^{152}$Gd (daughter of $^{152}$Eu) after the observation of the 344-keV $\gamma$ ray of M2 character in GRETINA. The extracted Q value is: 0.42(6). The $\chi^2$ for the fit is 0.9.</figcaption></figure>

calculate the linear polarization of the 779-keV transition, one considers the cascade as $0^+ \rightarrow 2^+ \rightarrow 3^-$ in order to make this $\gamma$ ray the 'second' one emitted.
<figure><img src="image_22.png" /><figcaption>**Fig. 21.** Linear polarization spectrum for the 968-keV E1 line (28⁺ → 27⁻) in ¹⁵²Dy produced in-beam, from tracked data. The calculated spectrum was produced using a σ/J = 0.3. The χ² of the fit is 3.1. See text for details.</figcaption></figure>

## 5.5 Linear polarization in-beam

For the nucleus ¹⁵²Dy produced in-beam (see App. B.1), the linear polarization of the 968-keV γ ray which corresponds to the 28⁺ → 27⁻ E1 transition, see Fig. 21 [35]. Measuring the $P_{\pm}cos(2\zeta)$ polarization amplitude yields a value of +0.127(2). Calculations, using eq. 35, confirm that a positive sign is expected for this electric transition and, with σ/J = 0.3, like in the angular distribution measurement (see Figs. 9 and 10), one calculates an amplitude of +0.086.

Another example from ¹⁵²Dy produced in-beam is the 432-keV γ ray which is reported as a tentative M1 transition in Ref. [35]. However, from the angular distribution analysis in Sec. 3.2 (Fig. 10) it becomes clear that the transition is either of mixed M1/E2 or E1/M2 character. The DCO analysis in Sec. 4.2 (Fig. 14) rules out the possibility of an E2 transition. The measured linear polarization is presented in Fig. 22 along with the fit and calculated M1/E2, E1/M2 and stretched M1 linear polarizations. The calculations were carried out with σ/J = 0.3 and, for the mixed transition, a δ mixing value of 0.7. Interestingly, a stretched M1 transition fits that data well (the dashed line). However, as just discussed above, this is inconsistent with the angular distribution. Ultimately, the linear polarization measurement resolves the ambiguity of M1/E2 or E1/M2 as only the latter solution is consistent with the data.

In Tab. 2 the polarization sensitivity, Q, is listed for the source examples discussed in this work. In Ref. [76], the Q sensitivity for GRETINA is reported to be 0.196(2)

<table><caption>**Table 2.** The measured polarization sensitivity, Q, as extracted in this work using sources.</caption><thead><tr><td>γ energy [keV]</td><td>Q</td></tr></thead><tbody><tr><td>344</td><td>0.42(6)</td></tr><tr><td>659</td><td>0.49(3)</td></tr><tr><td>779</td><td>0.42(6)</td></tr><tr><td>1064</td><td>0.34(3)</td></tr><tr><td>1173</td><td>0.37(4)</td></tr><tr><td>1333</td><td>0.33(4)</td></tr><tr><td>1442</td><td>0.31(5)</td></tr></tbody></table>

at 1368-keV, somewhat lower than what was found here. This reference also lists the polarization sensitivity of a number of other γ-ray detector arrays (as does Ref. [80]). In particular, Ref. [66] reports that Gammasphere has a Q sensitivity of 0.04, with only a small energy dependence. Thus, the polarization sensitivity of GRETINA is about ten times larger. Compton suppressed arrays are expected to have a low Q sensitivity because the Compton shields suppress the observation of most second interactions. However, for two fold segmented detectors in e.g., Gammasphere, it is possible to extract a linear polarization signal; but because of a poor definition of scattering direction, the fidelity of the signal is low. Besides, the fraction of segmented detectors in the array is limited.

<figure><img src="image_23.png" /><figcaption>**Fig. 22.** Linear polarization spectrum for the 432-keV E1/M2 mixed γ ray (32⁻ → 31⁺) in ¹⁵²Dy produced in-beam, from tracked data. In this case, the polar angle range was selected to be from 90° to 120° to avoid the stopped line discussed in Fig. 10. The χ² of the fit is 3.0. For the calculated spectra, see discussion in the text.</figcaption></figure>
# 6 Summary

We have re-introduced the classical formalism for angular correlations, but customized it for use with tracking arrays. We have described how the data itself can be used to obtain a response with overwhelming statistics (Sec. 2). The so-called clustering hole is evident in the angular correlation data; a deficiency that is shared with Compton suppressed arrays.

A new method to measure angular distributions from tracked data using a response function has been introduced (Sec. 3). This approach makes it easier to measure such distributions in tracking arrays – where otherwise large fluctuations are observed as a function of polar angle because the segmented germanium crystals are probed in a more complex way when compared to Compton suppressed spectrometers. This is a case where it is not possible to generate a response from the data itself, rather, radioactive sources must be used.

We introduced the DCO function which takes full advantage of the continuous angle measurements possible with tracking arrays (Sec. 4). Usually, only a ratio is measured. Two different approaches are offered to extract the desired information; i.e., the DCO φ method and the caDCO 2D superpulse method.

We have developed a formalism and the required software for calculating the linear polarization function for source data when two γ rays are emitted in coincidence, with one γ ray causing a polarization of the intermediate state (Sec. 5). We have also developed the formalism for calculating the linear polarization for in-beam data, where the reaction provides an initial alignment enabling the observation of the linear polarization of a γ ray. The formalism handles transitions of mixed character. For both cases, techniques have been developed where the data itself can be used for the response, – no source normalization is required. Typical examples have been provided. We also briefly touched on the possible formalism for observing linear polarization for two γ rays emitted in coincidence from oriented nuclei.

The nomenclature of all the algorithms have been unified and the appendix provides all the supplementary mathematical functions and rules used in this work (Sec. A). More examples of the analysis of data from sources, as well as measurements in-beam, are presented here as well (Sec. C to E).

# Appendix

# A Selected functions

## A.1 Ordinary Legendre Polynomials

The ordinary Legendre Polynomials used in this work are [22]:

$$ 
\begin{aligned}
P_0(x) &= 1 \\
P_2(x) &= \frac{1}{2}(3x^2 - 1) \\
P_4(x) &= \frac{1}{8}(35x^4 - 30x^2 + 3) \\
P_6(x) &= \frac{1}{16}(231x^6 - 315x^4 + 105x^2 - 5)
\end{aligned} 
\quad (42) $$

## A.2 Associate Legendre Polynomials

The Selected unnormalized associated Legendre Polynomials, $P_n^m$, used in this work, are:

$$ 
\begin{aligned}
P_0^0(x) &= 1 \\
P_2^0(x) &= \frac{1}{2}(3x^2 - 1) \\
P_2^1(x) &= -3x(1 - x^2)^{1/2} \\
P_2^2(x) &= 3(1 - x^2) \\
P_4^0(x) &= \frac{1}{8}(35x^4 - 30x^2 + 3) \\
P_4^1(x) &= -\frac{5}{2}(7x^3 - 3x)(1 - x^2)^{1/2} \\
P_4^2(x) &= \frac{15}{2}(7x^2 - 1)(1 - x^2) \\
P_4^3(x) &= -105x(1 - x^2)^{3/2} \\
P_4^4(x) &= 105(1 - x^2)^2
\end{aligned} 
\quad (43) $$

To normalize these polynomials, they must be multiplied by [81,82]:

$$ \sqrt{\left(\frac{2n+1}{2}\right) \frac{(n-m)!}{(n+m)!}} \quad (44) $$

## A.3 The 3-j and Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are related to the 3-J symbol as:

$$ \langle j_1 m_1 j_2 m_2 | [j_1 j_2] j_3 m_3 \rangle = (-1)^{j_2 - j_1 + m_3} \sqrt{2j_3 + 1} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & -m_3 \end{pmatrix} \quad (45) $$

This formula, and a list of other formulas using alternative phase factors, can be found in Ref. [24] which also provides the numerical formula for the fundamental 3-J symbol (see eq. 1.5 on page 2). This formula for the 3-J symbol is used throughout the software developed in this work. For completeness, this formula can be written as

$$ \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} = v1 \times \sqrt{\frac{v2}{(j_1+j_2+j_3+1)}} \times \sum_k \frac{(-1)^k}{k!v3} \quad (46) $$
with

$$
\begin{align}
v1 &= (-1)^{j_1-j_2-m_3} \nonumber \\
v2 &= (j_1 + j_2 - j_3)!(j_1 - j_2 + j_3)!(-j_1 + j_2 + j_3)! \nonumber \\
&\quad \times (j_1 + m_1)!(j_1 - m_1)!(j_2 + m_2)! \nonumber \\
&\quad \times (j_2 - m_2)!(j_3 + m_3)!(j_3 - m_3)! \nonumber \\
v3 &= (j_1 + j_2 - j_3 - k)!(j_1 - m_1 - k)!(j_2 + m_2 - k)! \nonumber \\
&\quad \times (j_3 - j_2 + m_1 + k)!(j_3 - j_1 - m_2 + k)! \tag{47}
\end{align}
$$

A.4 The 6-j and Racah coefficients

The 6-j coefficients are related to the Racah coefficients [24]
by

$$
\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1+j_2+j_4+j_5} w(j_1,j_2,j_5,j_4; j_3,j_6) \quad (48)
$$

The numerical formula for the 6–J symbol is used in this
work can be found in Ref. [24], eq. 2.3 and 2.4. The 6–j
coefficient numerically can be expressed as

$$
\begin{Bmatrix} j_1 & j_2 & j_3 \\ l_1 & l_2 & l_3 \end{Bmatrix} = v1v2 \sum_k \frac{(-1)^k (j_1 + j_2 + l_1 + l_2 + 1 - k)!}{k!v3} \quad (49)
$$

with

$$
\begin{align}
\Delta(a, b, c) &= \sqrt{\frac{(a+b-c)!(a-b+c)!(-a+b+c)!}{(a+b+c+1)!}} \nonumber \\
v1 &= (-1)^{j_1+j_2+l_1+l_2} \nonumber \\
v2 &= \Delta(j_1, j_2, j_3)\Delta(l_1, l_2, j_3)\Delta(j_1, j_2, l_3) \nonumber \\
v3 &= (j_1 + j_2 - j_3 - k)!(l_1 + l_2 - j_3 - k)! \nonumber \\
& \quad \times (j_1 + l_2 - l_3 - k)!(l_1 + j_2 - l_3 - k)! \nonumber \\
& \quad \times (-j_1 - l_1 + j_3 + l_3 + k)! \nonumber \\
& \quad \times (-j_2 - l_2 + j_3 + l_3 + k)! \tag{50}
\end{align}
$$

A.5 The 9–J coefficients

The 9–J coefficients can be calculated from a sum of 6–J
symbols as

$$
\begin{Bmatrix} j_{11} & j_{12} & j_{13} \\ j_{21} & j_{22} & j_{23} \\ j_{31} & j_{32} & j_{33} \end{Bmatrix} = \sum_j (1)^{2j} (2j + 1) \times \begin{Bmatrix} j_{11} & j_{21} & j_{31} \\ j_{32} & j_{33} & j \end{Bmatrix} \begin{Bmatrix} j_{12} & j_{22} & j_{32} \\ j_{21} & j & j_{23} \end{Bmatrix} \begin{Bmatrix} j_{13} & j_{23} & j_{33} \\ j & j_{11} & j_{12} \end{Bmatrix} \quad (51)
$$

where j takes all values that the triangle rules for (j₃₂, j₃₃, j),
(j₂₁, j, j₂₃) and (j, j₁₁, j₁₂) allow [24]. The triangle rule
may be expressed as the requirements that:

$$
\begin{align*}
(j_1 + j_2 - j_3) &\ge 0 \\
(j_1 - j_2 + j_3) &\ge 0 \\
(-j_1 + j_2 + j_3) &\ge 0
\end{align*}
$$

and that $(j_1 + j_2 + j_3)$ is an integer.

B Datasets

B.1 In-beam

The in-beam dataset from GRETINA used in this work
was obtained with the ¹⁰⁸Pd(⁴⁸Ca[191 MeV],4n)¹⁵²Dy re-
action (ATLAS exp1909). The data was acquired over 2
days and 10 hours in 2022, and had an isomer trigger
on the γ rays under the 17+, T₁/₂=60 ns isomer [35]. The
¹⁰⁸Pd target thickness was 0.789 mg/cm². The isomer was
stopped in a 24 mg/cm² Pb catcher foil located ~35 cm
behind the ¹⁰⁸Pd target, and photons were detected in a
LaBr₂ detector array which was shielded from the placed
target at the center of GRETINA. With 47 crystals, the
occupancy was 47/120 or ~39%. The v/c value of the ¹⁵²Dy
recoiling residues was 0.027. The dataset is small com-
pared to those produced in typical experiments. There
are 1.55×10⁹ good (FOM below 0.8) γ rays in the ¹⁵²Dy
dataset.

B.2 Source data

The following sources were used in GRETINA to obtain
the angular distribution response function: ¹⁵²Eu, ²⁰⁷Bi,
⁶⁰Co, ¹³⁷Cs, ⁸⁸Y, ²⁴³Am, ²²⁶Ra and ⁵⁷Co. Some of the
datasets are large and, therefore, also used to verify some
of the extraction methods discussed (e.g., for the DCO
angular correlation). Some source datasets originate from
GRETINA configurations that were different from the in-
beam ¹⁵²Dy data one. Thus, the polar-angle ranges may
differ.

The ¹⁵²Eu dataset was acquired in GRETINA over 7 hours and 55 minutes and has 3.77×10⁸ tracked γ rays with a FOM less than 0.8 (therefore considered 'good' γ rays). For the other sources with a multiplicity of ~2 the number of good tracked γ rays are 3.75×10⁸, 1.42×10⁸, 2.89×10⁸, and 3.77×10⁸, respectively for the ⁶⁰Co, ⁸⁸Y, ²⁰⁷Bi and ¹⁵²Eu sources. The other sources, used for other purposes, were typically measured in an hours or less.

C Other ¹⁵²Eu source examples

In the daughter of ¹⁵²Eu, ¹⁵²Sm, the 122 – and 964–
keV γ rays are in coincidence; the latter being strongly
mixed [35]. In Fig. 23 the angular correlation spectrum
(from GRETINA) is presented and the coefficients are
measured to be a₂ = −0.006(3) and a₄ = 0.228(4). This
results in this angular correlation spectrum being domi-
nated by the a₄ term. In Fig. 24, a fit to the measured
angular coefficients yields a value of the mixing ratio δ =
+10(3) which was used for the calculated curve (dashed
blue) in Fig. 23. Note that a mixing ratio of −9.4 is re-
ported in Ref. [35]; which would correspond to +9.4 with
the sign convention used in this paper.
<figure><img src="image_24.png" /><figcaption>**Fig. 23.** The measured angular correlations of the 122 − and 964-keV lines in <sup>152</sup>Sm (daughter of <sup>152</sup>Eu). The data is presented in black, the data used for the fit is shown in red. The fitted curve is given in blue while the calculated angular correlation, obtained using eq. 1, with a mixing ratio of +10(3) from a fit, is identified by the dashed blue curve. The χ<sup>2</sup> of the fit is 2.3. See text for details.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. 24.** The χ² as a function of δ for the fit to the measured a₂ and a₄ coefficients of the data presented in Fig. 23. The best fit is for a mixing ratio of 10(3). The local minimum at a small negative value does not provide a good fit, rather, one finds that the best fit corresponds to a large positive value. Since the minimum there is very shallow, the error of the fitted χ² is, as a result, large.</figcaption></figure>

D the DCO *φ* method for sources

The angular distribution of γ rays from radioactive sources
can be described using the formalism in Sec. 3.1. The for-
malism here is valid when the first state has a random
m-distribution and, in this case, just the relative angle
between the γ rays is used. However, the full DCO for-
malism described in Sec. 4.1 can also describe source data,
provided that one uses *v/c* = 0 and the first state is pre-
pared with a random m-state distribution. In this work,
the latter is accomplished by using a very large value of
*σ/J*, effectively randomizing the m-states. For sources,
in general, it is easier to obtain superior statistics with-
out having much background under the peaks. Thus, here
we explore the full DCO treatment, described in Sec. 4.1,
for the two sources, <sup>60</sup>Co and <sup>88</sup>Y, using the analysis of
just the *φ* angle as described in Sec. 4. The results are pre-
sented in Figs. 25 and 26 for the <sup>60</sup>Co and <sup>88</sup>Y radioactive
source, respectively.

<figure><img src="image_26.png" /><figcaption>**Fig. 25.** The DCO *φ* spectrum for a <sup>60</sup>Co source. The analysis follows the method outlined in Sec. 4. This spectrum may qualitatively be compared to Fig. 13.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Fig. 26.** The DCO *φ* spectrum for a <sup>88</sup>Y source. The analysis follows the method outlined in Sec. 4. This spectrum may qualitatively be compared to Fig. 14.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>**Fig. 28.** As Fig. 27, but for <sup>88</sup>Y source data. See text for details.</figcaption></figure>

## E The *caDCO 2D superpulse method* for sources

The *caDCO 2D superpulse method* described in Sec. 4.1 can be tested using radioactive source data with good statistics. The recoil velocity, *v/c*, is set to zero in the

<figure><img src="image_29.png" /><figcaption>**Fig. 27.** The caDCO superpulse spectrum for <sup>60</sup>Co. On the y-axis is the DCO signal and the x-axis is the pseudo polar angle defined in Sec. 4.1. Only a small fraction of the superpulse data is covered here. See text for discussion.</figcaption></figure>

formalism and $\sigma/J$ is chosen to be very large, effectively randomizing the m-substate population of the first state.

For the case of the <sup>60</sup>Co data described in Appendix B.2, the superpulse has 1288 pseudo-polar data points (with a range from 473 to 2098). A *φ* cut of 150° < *φ* < 180° was imposed on the DCO cube projections. For a calculated spectrum using a 4(2)2(2)0 (stretched) cascade, one finds a $\chi^2$ of 1.085 when the data is compared to this spectrum. Only a small fraction of the total dataset is presented in Fig. 27 where the data is in red and the calculated DCO signal is in blue.

In Fig. 28, the equivalent *caDCO 2D superpulse* for the source <sup>88</sup>Y is presented. If the calculated spectrum is generated for a 3(1)2(2)0 (stretched) cascade, one finds a reduced $\chi^2$ of 1.039 based on 1276 data points.

If, wrongfully, a 4(2)2(2)0 (stretched) cascade was specified for the <sup>88</sup>Y data, the $\chi^2$ increases to 1.64. Likewise, if a 3(1)2(2)0 (stretched) cascade was used for the <sup>60</sup>Co case, the $\chi^2$ increase to 2.425. This might indicate a good sensitivity to the parameters used to fit the DCO signal, possibly better than for the *DCO φ method*. However, as was already mentioned, only a fraction the data in the DCO cube are used here. As discussed in Sec. 4.1, if other *φ* slices are used, the signal is reduced, but these slices can nevertheless be used to aid in a fit of the calculated DCO signal to the data. The slices at small *φ* angles are, in general, too influenced by the clustering trench be useful despite the attempt to impose this tracking deficiency on the response function. It should be said that, for the source datasets used here, with the GRETINA configuration that was used, there is relatively more data for *φ*.
angles close to 180° than at other angles, thus, excluding the smaller φ angles from the analysis may not reduce the statistics significantly.

## F Selected examples of DCO formulas

To illustrate the complexity, the explicit DCO formula for a 4 → 2 → 0 sequence is presented below:

$$ 
\begin{aligned}
+1.000000000000 & \\
-0.000091594667 & \times P_2^0(\cos(\theta_2)) \\
-0.000091594667 & \times P_2^0(\cos(\theta_1)) \\
+0.102059509116 & \times P_2^0(\cos(\theta_1))P_2^0(\cos(\theta_2)) \\
+0.034016720907 & \times P_2^1(\cos(\theta_1))P_2^1(\cos(\theta_2))\cos(1\phi) \\
+0.008501843628 & \times P_2^2(\cos(\theta_1))P_2^2(\cos(\theta_2))\cos(2\phi) \\
+0.000007477116 & \times P_2^0(\cos(\theta_1))P_4^0(\cos(\theta_2)) \\
+0.000001246186 & \times P_2^1(\cos(\theta_1))P_4^1(\cos(\theta_2))\cos(1\phi) \\
+0.000000103849 & \times P_2^2(\cos(\theta_1))P_4^2(\cos(\theta_2))\cos(2\phi) \\
+0.000007477116 & \times P_4^0(\cos(\theta_1))P_2^0(\cos(\theta_2)) \\
+0.000001246186 & \times P_4^1(\cos(\theta_1))P_2^1(\cos(\theta_2))\cos(1\phi) \\
+0.000000103849 & \times P_4^2(\cos(\theta_1))P_2^2(\cos(\theta_2))\cos(2\phi) \\
+0.009071805313 & \times P_4^0(\cos(\theta_1))P_4^0(\cos(\theta_2)) \\
+0.000907157873 & \times P_4^1(\cos(\theta_1))P_4^1(\cos(\theta_2))\cos(1\phi) \\
+0.000050393883 & \times P_4^2(\cos(\theta_1))P_4^2(\cos(\theta_2))\cos(2\phi) \\
+0.000003599114 & \times P_4^3(\cos(\theta_1))P_4^3(\cos(\theta_2))\cos(3\phi) \\
+0.000000449811 & \times P_4^4(\cos(\theta_1))P_4^4(\cos(\theta_2))\cos(4\phi)
\end{aligned}
\quad (52) $$

For higher spins, there are in general 19, rather than 17, non-zero terms when only L=1,2 transitions are considered. Likewise, the explicit DCO formula for a 3 → 2 → 0 sequence is:

$$ 
\begin{aligned}
+1.000000000000 & \\
+0.000082469605 & \times P_2^0(\cos(\theta_2)) \\
-0.000057728723 & \times P_2^0(\cos(\theta_1)) \\
-0.071416790056 & \times P_2^0(\cos(\theta_1))P_2^0(\cos(\theta_2)) \\
-0.023807560248 & \times P_2^1(\cos(\theta_1))P_2^1(\cos(\theta_2))\cos(1\phi) \\
-0.005953362733 & \times P_2^2(\cos(\theta_1))P_2^2(\cos(\theta_2))\cos(2\phi) \\
+0.000004712549 & \times P_2^0(\cos(\theta_1))P_4^0(\cos(\theta_2)) \\
+0.000000785425 & \times P_2^1(\cos(\theta_1))P_4^1(\cos(\theta_2))\cos(1\phi) \\
+0.000000065452 & \times P_2^2(\cos(\theta_1))P_4^2(\cos(\theta_2))\cos(2\phi)
\end{aligned}
\quad (53) $$

In both cases, the Lorentz transformation factors are omitted. A σ/J value of 20 was used, effectively creating an un-oriented initial state, and v/c was set to zero. Thus, these particular formulas are appropriate for sources rather than for in-beam data. As can be seen, the DCO formulas are significantly more complex as compared to the formulas governing the angular distributions and angular correlations for sources.

## Acknowledgment

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-AC02-06CH11357 (ANL) and grant numbers DE-FG02-97ER41041 (UNC) and DE-FG02-97ER41033 (TUNL). This research used resources of the ANL's ATLAS facility, which is a DOE Office of Science User Facility. This work was supported by the French National Center of Research, CNRS. We would like to thank S. Zhu and D. Weisshaar for collecting some of the large radioactive-source datasets used in this work. We thank G. Savard for discretionary beam-time for the $^{152}$Dy experiment.

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