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Thermal properties and radiative strengths in 160,161,162Dy

The level densities and radiative strength functions (RSFs) in 160,161Dy have been extracted using the (3He,alpha gamma) and (3He,3He' gamma) reactions, respectively. The data are compared to previous measurements on 161,162Dy. The energy distribution in the canonical ensemble is discussed with respect to the nucleon Cooper pair breaking process. The gross properties of the RSF are described by the giant electric dipole resonance. The RSF at low gamma-ray energies is discussed with respect to temperature dependency. Resonance parameters of a soft dipole resonance at Egamma = 3 MeV are deduced.


I. INTRODUCTION
The well-deformed rare earth region appears to be ideal for studying statistical properties of nuclei as function of temperature. The single particle Nilsson scheme displays almost uniformly distributed single particle orbitals with both parities. However, the low-temperature thermal properties of these nuclei are only poorly known. The main reason for this is the lack of appropriate experimental methods.
The Oslo Cyclotron group has developed a method to extract first-generation (primary) γ-ray spectra at various initial excitation energies. From such a set of primary spectra, nuclear level density and radiative strength function (RSF) can be extracted [1,2]. These two functions reveal essential nuclear structure information such as pair correlations and thermal and electromagnetic properties. In the last couple of years, the Oslo group has demonstrated several fruitful applications of the method [3][4][5][6][7][8][9][10][11].
The subject of this work is to perform a systematic and consistent analysis of the three 160,161,162 Dy isotopes. Since the proton number (Z = 66) and the nuclear deformation (β ∼ 0.26) are equal for these cases, we expect to find the same electromagnetic properties. Furthermore, the underlying uniform distribution of single particle Nilsson states should from a statistical point of view give similar level densities for 160 Dy and 162 Dy. The present data set also allows us to check the results using the ( 3 He,αγ) and ( 3 He, 3 He ′ γ) reactions for one and the same residual nucleus.
In Sect. II an outline of the experimental procedure is given. The thermal aspects of the level density and RSF are discussed in Sects. III and IV, respectively. Finally, concluding remarks are given in Sect. V.

II. EXPERIMENTAL METHOD
The experiments were carried out with 45 MeV 3 Heions at the Oslo Cyclotron Laboratory. Particle-γ coincidences for 160,161,162 Dy were measured with the CACTUS multi-detector array. The charged ejectiles were detected with eight particle telescopes placed at an angle of 45 • relative to the beam direction. An array of 28 NaI γ-ray detectors with a total efficiency of ∼15% surrounded the target and particle detectors.  [3,4,7]. The reaction spin windows are typically I ∼ 2 − 6h. The self-supporting targets are enriched to ∼ 95% with thicknesses of ∼ 2 mg/cm 2 . The experiments were run with beam currents of ∼ 2 nA for 1-2 weeks.
The experimental extraction procedure and the assumptions made are described in Refs. [1,2] and references therein. For each initial excitation energy E, determined from the ejectile energy, γ-ray spectra are recorded. These spectra are the basis for making the first generation (or primary) γ-ray matrix [12], which is factorized according to the Brink-Axel hypothesis [13,14] as Here, ρ is the level density and T is the radiative transmission coefficient. The ρ and T functions can be determined by an iterative procedure [2] through the adjustment of each data point of these two functions until a global χ 2 minimum with the experimental P (E, E γ ) matrix is reached. It has been shown [2] that if one solution for the multiplicative functions ρ and T is known, one may construct an infinite number of other functions, which give identical fits to the P matrix bỹ Consequently, neither the slope (α) nor the absolute values of the two functions (A and B) can be obtained through the fitting procedure.
The parameters A and α can be determined by normalizing the level density to the number of known discrete levels at low excitation energy [15] and to the level density estimated from neutron-resonance spacing data at the neutron binding energy E = B n [16]. The procedure for extracting the total level density ρ from the resonance energy spacing D is described in Ref. [2]. Since our experimental level density data points only reach up to an excitation energy of E ∼ B n − 1 MeV, we extrapolate with the back-shifted Fermi gas model [17,18] where a constant η is introduced to fix ρ BS to the experimental level density at B n . The intrinsic excitation energy is estimated by U = E − C 1 − E pair , where C 1 = −6.6A −0.32 MeV and A are the back-shift parameter and mass number, respectively. The pairing energy E pair is based on pairing gap parameters ∆ p and ∆ n evaluated from even-odd mass differences [19] according to [20]. The level density parameter is given by a = 0.21A 0.87 MeV −1 . The spin-cutoff parameter σ I is given by σ 2 I = 0.0888aT A 2/3 , where the nuclear temperature is described by In cases where the intrinsic excitation energy U becomes negative, we put U = 0, T = 0 and σ I = 1. The spin distribution of levels (with equal energy) is given by [17] g(E, I) = 2I + 1 2σ 2 which is normalized to I g(E, I) ∼ 1. Figure 1 compares g(E, I) to the spin distributions of levels with known spin assignments [15] for nuclei along the βstability line with A = 150−170. Although these data are incomplete and include systematical errors 1 , the agreement is gratifying and supports the expressions adopted for σ I and g. Unfortunately, 159 Dy is unstable and no information exists on the level density at E = B n for 160 Dy. Therefore, we estimate the value from the systematics of other even-even dysprosium and gadolinium isotopes. In order to bring these data on the same footing, we plot the level densities as function of intrinsic energy U . From the systematics of Fig. 2, we estimate for 160 Dy a level density of ρ(B n ) = (9.7 ± 2.0)10 6 MeV −1 . Figure 3 demonstrates the level density normalization procedure for the 160 Dy case.
The level densities extracted from the five reactions are displayed in Fig. 4. The data have been normalized as prescribed above, and the parameters used for 160,161,162 Dy in Eq. (4) are listed in Table I. The level densities for the three reactions previously published [3,4,7] deviate slightly since we here have used updated and newly recommended data [15,16]. The results obtained with the very different reactions ( 3 He,α) and ( 3 He, 3 He ′ ), are almost identical, except for the level density of the ground state band in 162 Dy. Here, the ( 3 He, 3 He ′ ) reaction overestimates the level density, as has been discussed previously [4].

III. LEVEL DENSITY AND THERMAL PROPERTIES
The level densities of 160 Dy and 162 Dy are very similar, however, 161 Dy reveals several times higher level densities. In a previous work [6], it was claimed that the entropy for the excited quasi-particles is approximately extensive. To investigate this assumption further, we express the entropy as where Boltzmann's constant is set to unity (k B = 1). The multiplicity Ω is directly proportional to the level density by Ω(E) = ρ(E)/ρ 0 . The normalization denominator ρ 0 is adjusted to give S = ln Ω ∼ 0 in the ground state bands of the even-even nuclei. Here, we assume temperatures close to zero, thus fulfilling the third law of thermodynamics. The same ρ 0 is used for the odd-mass neighboring nuclei. Figure 5 shows the entropies S for the two new reactions reported in this work, i.e., the ( 3 He,αγ) 160 Dy and ( 3 He, 3 He ′ γ) 161 Dy reactions. The results for the other reactions are very similar and are therefore not discussed here. The entropy of the 161 Dy nucleus is seen to display an almost constant entropy excess compared to 160 Dy. The difference, ∆S ∼ 2, represents the entropy carried by the valence neutron outside the even-even 160 Dy core (or hole coupled to the 162 Dy core). It is an interesting feature that this difference is almost independent of excitation energy and therefore, of the number of quasiparticles excited in dysprosium, thus manifesting an entropy of S qp ∼ 2 assigned to each quasi particle.
The probability that a system at fixed temperature T has an excitation energy E, is described by the probability density function 2 where the canonical partition function is given by The experimental excitation energies E i have energy bins of ∆E. In principle, the sum runs over all energies from zero to infinity, and we therefore use Eq. (4) to extrapolate to the higher energies. The energy distribution function p T (E) has a moment of the order n about the origin given by It is easy to show that the various moments also may be evaluated by the differentiation of Z(T ): The moments µ n of E about its mean value E is defined by µ n = (E − E ) n . Thus, the second and third moments become These two moments are connected to the heat capacity and skewness of p T (E) according to respectively. We also identify the standard deviation of the energy distribution as σ E = √ µ 2 . Figure 6 shows the probability density functions for 160 Dy and 161 Dy. Below T ∼ 0.6 MeV, the distribution is mainly based on experimental data, but at higher temperatures the influence of the somewhat arbitrary extrapolation of the level density by Eq. (4) will be increasingly important. The most interesting temperature region is around T = 0.5 − 0.6 MeV, where the Cooper pair breaking process is at the strongest. At this point, the eveneven and odd-even nuclei behave differently; 160 Dy shows a broader distribution than 161 Dy. This is due to the explosive behavior of ρ for E > E pair = 1.5 − 2 MeV in even-even nuclei. Roughly, the number of levels for the breaking of neutron or proton pairs increases by a factor of exp(2S qp ) ∼ 55 giving totally ∼ 110 times more levels. For the odd-even nuclei, Fig. 4 shows that the level density is a monotonically increasing function from the ground state to higher excitation energies.
The various experimental moments are best evaluated from p T (E), since the multiplicity Ω is directly known from the measured level densities. The left panels of Fig. 7 show the corresponding values of average excitation energy E , heat capacity C V and the skewness γ of p T as function of temperature T . These key quantities characterize p T (E), and thereby reveal the thermodynamic properties of the systems studied. In the right panels these functions are compared to predictions evaluated in the canonical ensemble. The model [21] applied here treats the excitation of protons, neutrons, rotation, and vibration adiabatically with a multiplicative partition function where the various energy moments E n are evaluated from Eqs. (11)(12)(13).
The qualitative agreement between model and experiments shown in Fig. 7 indicates that our model describes the essential thermodynamic properties of the heated systems. The heat capacity curves show clearly a local increase in the T = 0.5 − 0.6 MeV region, hinting the collective and massive breaking of nucleon Cooper pairs. This feature was recently discussed in Ref. [22], where two different critical temperatures were discovered in the microcanonical ensemble using the method of Lee and Kosterlitz [23,24]: (i) The lowest critical temperature is due to the zero to two quasi-particle transition, and (ii) the second transition is due to the continuous melting of Cooper pairs at higher excitation energies. The first contribution is strongest for the even-even system ( 160 Dy), since the first broken pair represents a large and abrupt step in level density and thus a large contribution to the heat capacity. In 161 Dy, the extra valence neutron washes out this step. The second contribution to C V is present in both nuclei signalizing the continuous melting of nucleon pairs at higher excitation energies. This second critical temperature appears at ∼ 0.1 MeV higher values.
The skewness γ reveals higher order effects in the p T (E) distribution. For a symmetric energy distribution, γ is zero. Figure 7 shows positive values indicating distributions with high energy tails, as is confirmed by Fig. 6. The 160 Dy system shows a strong signal in γ around T ∼ 0.2 MeV. This signal is connected with the high energy tail of the p T (E) distribution into the E > 2∆ excitation region with high level density.

IV. RADIATIVE STRENGTH FUNCTION AND ITS RESONANCES
The slope of the experimental radiative transmission coefficient T (E γ ) has been determined through the normalization of the level densities, as described in Sect. II. However, it remains to determine B of Eq. (3), giving the absolute normalization of T . For this purpose we utilize experimental data [16] on the average total radiative width Γ γ at E = B n .
We assume here that the γ-decay taking place in the continuum is dominated by E1 and M1 transitions and that the number of positive and negative parity states is equal. For initial spin I and parity π at E = B n , the expression of the width [25] reduces to where the summation and integration run over all final levels with spin I f which are accessible by dipole (L = 1) γ radiation with energy E γ . From this expression the normalization constant B can be determined as described in Ref. [10]. However, some considerations have to be made before normalizing according to Eq. (19). Methodical difficulties in the primary γ-ray extraction prevents determination of the functions T (E γ ) and ρ(E) in the interval E γ < 1 MeV and E > B n −1 MeV, respectively. In addition, the data at the highest γ-energies, above E γ ∼ B n − 1 MeV, suffer from poor statistics. For the extrapolation of ρ we apply the back-shifted Fermi gas level density of Eq. (4). For the extrapolation of T we use a pure exponential form, as demonstrated for 160 Dy in Fig. 8. The contribution of the extrapolation to the total radiative width given by Eq. (19) does not exceed 15%, thus the errors due to a possibly poor extrapolation are expected to be of minor importance [10].
The radiative strength function for L = 1 transitions can be calculated from the normalized transmission coefficient by The RSFs extracted from the five reactions are displayed in Fig. 9. The data have been normalized with parameters from Tables I and II. Also here, the present results deviate slightly from the three data sets previously published [3,4,10]. The present RSFs seem not to show any clear odd-even mass differences, and again the ( 3 He,α) and ( 3 He, 3 He ′ ) reactions reveal similar results. The γ decay probability is governed by the number and the character of available final states and by the RSF. A rough inspection of the experimental data of Fig. 9 indicates that the RSFs are increasing functions of γ-energy, generally following the tails of the giant electric (GEDR) and magnetic (GMDR) resonances in this region. In addition, a small resonance around E γ ∼ 3 MeV is found, the so-called pygmy resonance. These observations have been previously verified for several well deformed rare earth nuclei [3,10].
Also in the present work we adapt the Kadmenskiȋ, Markushev and Furman (KMF) model [26] for the giant electric dipole resonance: Since the nuclei studied here have axially deformed shapes, the GEDR is split into two components GEDR1 and GEDR2. Thus, we add two RSFs with different resonance parameters, i.e., strength σ E1 , width Γ E1 and centroid E E1 . The M1 radiation, which is supposed to be governed by the spin-flip M1 resonsance [10], is described by The GEDR and GMDR parameters are taken from the systematics of Ref. [16] and are listed in Table II. The pygmy resonance is described with a similar Lorentzian function f py as described in Eq. (22). Thus, we fit the total RSF given by to the experimental data using the pygmy-resonance parameters σ py , Γ py and E py and the normalization constant κ as free parameters. In previous works [3,10,11], the temperature T of Eq. (21) was also used as fitting parameter, assuming that a constant temperature could describe the data. The fitting to experimental data gave typically T ∼ 0.3 MeV which is about the average of what is expected in this energy region. The use of a constant temperature approach is consistent with the Brink-Axel hypothesis [13,14], which is utilized in order to separate ρ and T through Eq. (1).
However, experimental data indicate that the RSF may depend also on how the temperature changes for the various final states. Data from the 147 Sm(n,γα) 144 Nd reaction [27] indicate a finite value of f E1 in the limit E γ → 0. Furthermore, in our study of the weakly deformed 148 Sm, where no clear sign of the pygmy resonance is present, the RSF also flattens out at small γ energies [11]. In the 56,57 Fe isotopes it has been reported [28] that the RSF reveals an anomalous enhancement for γ energies below 4 MeV. Also the 27,28 Si isotopes show a similar increase in the RSF below 6 MeV [29]. We should also mention that the extracted caloric curve E(T ) of Fig. 7 indicates a clear variation in T for the excitation energy region investigated. Figure 10 shows indeed that the strength of the tail of the GEDR, using the model of Eq. (21), is strongly temperature dependent. Therefore, from these considerations, we find it interesting to test the consequences by including a temperature dependent RSF in the description of the experimental data.
However, there is an inconsistency between such an approach and our extraction of the RSF using the Brink-Axel hypothesis through Eq. (1). The consequences have been tested in the following way: We first construct a typical level density and a temperature dependent transmission coefficient and multiply these two functions with each other to simulate a primary γ-ray matrix. Then Eq. (1) is utilized in order to extract ρ and T . It turns out that the output ρ is almost identical with the input. Also T is correctly extracted, except small deviations of ∼ 15% for γ-energies below 1 MeV. Thus, the mentioned inconsistency should not cause severe problems.
If we assume that the RSF depends on the temperature of the final states, it also depends on the primary γ-ray spectra chosen. Usually these spectra are taken at initial excitation energies between E 1 ∼ 4 and E 2 ∼ 8 MeV. Thus, the average temperature of the final states E f populated by a γ transition with energy E γ is given by where T (E f ) = (E f − C 1 − E pair )/a is the schematic temperature dependency taken from Eq. (5). Figure 11 shows T and the standard deviation σ T = T 2 − T 2 for states populated with a γ transition of energy E γ in 160 Dy. The temperature goes almost linearly from 0.6 MeV to zero, giving an average of 0.3 MeV consistent with earlier constant temperature fits [3,10,11]. The standard deviation is relatively large, σ T ∼ 0.1 MeV, indicating that one should not replace T by T in Eq. (21) but rather calculate f E1 (E γ ) analog to the evaluation of T (E γ ) in Eq. (24). Figure 12 shows fits to the experimental RSFs obtained from the ( 3 He,α) 160 Dy and ( 3 He, 3 He ′ ) 162 Dy reactions. The approaches using a varying temperature, f E1 , and a fixed temperature, f E1 (T = 0.3 MeV), are displayed as solid and dash-dotted lines, respectively. The f E1 contribution to the total RSF is seen to give an increased strength for E γ < 1 MeV, which the 162 Dy case seems to support. However, the 160 Dy case supports the fixed temperature approach. Unfortunately, the RSFs in the E γ ∼ 1 MeV region are experimentally difficult to measure. Here, a strong γ-decay intensity from vibrational states may not have been properly subtracted in the primary γ-ray spectra. Thus, the present data are not conclusive regarding the existence of enhanced radiative strength at low γ energies.
In Table III, we have summarized the fitted parameters for the pygmy resonance and the normalization constant κ using varying temperatures. All the investigated dysprosium nuclei show similar pygmy resonance parameters and a mean average for all five reactions is given in the last row. Since the pygmy resonance is superimposed on the GEDR, the extraction of its resonance parameters is rather independent of how the GEDR temperature dependency is formulated.

V. SUMMARY AND CONCLUSIONS
The present comparison between level densities and RSFs obtained with various reactions gives confidence to the Oslo method. The entropies of 161 Dy follow very parallel the even-even 160,162 Dy systems, assigning an entropy of ∼ 2 to the valence neutron. The evolution of the probability density function with temperature was presented for 160,161 Dy. The widths of these distributions increase anomalously in the T = 0.5 − 0.6 MeV region. This feature of local increase in the canonical heat capacity is a fingerprint of the depairing process. Also the skewness of these distributions are studied showing the variation in the high energy tails as function of temperature. A simple canonical model is capable of describing qualitatively the various thermodynamic quantities.
The five RSFs studied reveal very similar structures for all isotopes studied, as is expected since they all are considered to have the same deformation. The RSFs show a pygmy resonance superimposed on the tail of the giant dipole resonance. We have tested the consequences of introducing an RSF with varying temperatures in the GEDR case, which gives an enhanced strength at lower γ energies. Our data are not conclusive in determining whether such effects are real or not.