The Structure Properties of Even-Even 216-226 Ra Isotopes

The properties of even-even 216-226 Ra isotopes were studied and its energy states calculated. To identify the properties of each isotope, the values of the first excited state and the ratio of the second to the first excited state were adopted. The phenomenon of back (or up)-bending, the E-GOS curve, the relationship between successive excited states to those preceding them and the odd-even staggering were studied. The appropriate limit in the interacting boson model IBM-1, the IVBM model and the Bohr and Mottelson (BM) model were used to calculate the energy states for each isotope and compared the results with the experimental values. Due to the inaccuracy of some calculations in the models IVBM and BM, an adjustment term was added for each model and recalculated the energy states and the results were improved.


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INTRODUCTION The even-even nuclei are characterized by certain energy states (Bohr and Mottelson, 1998). The first of these is the ground states band GSB, with even-angular momenta and positive parity states; ( ). These nuclei show other bands, such as the beta-vibrational band which exhibit similar energy states to the GSB and the gamma-vibrational-band which has positive parity and successive even and odd angular moment; ( ). Other states occur in these nuclei, such as negative parity states NPB with an odd-angular momentum; ).
The first excited state and the ratio of the second to the first excited state of the even-even nuclei provide an initial indication of the characteristics of the nucleus (Nomura, 2013), (Bonatsos and Skouras, 1991) . The vibrational nuclei are characterized by  and  , while the gamma-soft nuclei are characterized by   and  , and the rotational nuclei are characterized by and .
Several methods were introduced to identify the characteristics of even-even nuclei at higher excited states. The appearance of a back or up-bending is appropriate to determine the phase change of the nucleus (Wong, 1990). The relationship between the gamma-energy over spin ( ) as a function of the spin presented a method to identify the characteristics of the nucleus (Regan et al., 2003). Another method was introduced to identify the characteristics of the nucleus in its various excited states by the ratio between the energy of the various excited states to the one that proceeds it (Bonatsos and Skouras, 1991). By applying this relationship to a set of different nuclei, the characteristics of each nucleus were determined by the limits; for the vibrational nuclei, and for the transitional nuclei, while for the rotational nuclei.
Due to the absence of a uniform theory suitable for the study of different nuclei, different models were introduced to study these nuclei. Bohr and Mottelson introduced the collective model. The microscopic interacting bosons model IBM-1 calculation (Bonatsos et al., 2013), (Kumar and Gunye, 1985) is somehow in between the phenomenological and microscopic approaches. As a widely used model, the IBM-1 (Iachello and Arima, 1987) describes the quadrupole collective states of the medium and heavy nuclei and provides a unified description of collective nuclear states in terms of a system of interacting bosons. Different reductions of unitary group U(6) give three dynamical symmetry limits known as harmonic oscillator, deformed rotor and gamma-soft rotor, which are labeled as U(5), SU(3), and O(6), respectively (Arima and Iachello, 1976), , (Arima and Iachello, 1979).
In the present study different methods were used to investigate the properties of the Radium isotopes 216-226 Ra. These methods were the back (or up)-bending, the E-GOS, the ratios of the successive states to those proceeding them and the staggering phenomenon. The models, IBM-1, IVBM and BM, were used to calculate the GSB and NPB by fitting the parameters of the equations of each model to the experimental values which are then used to calculate many excited states of these isotopes and compared with the measured ones. Modified equations of IVBM and BM were suggested to improve the results for these two models.

CALCULATIONS
It was noted that for some nuclei, a dramatic change in the moment of inertia may occur by increasing the angular momentum , causing a drop in the rotational energy of the -transition from a state with to a state with . This behavior causes back or up-bending in the value of the energy which is given by (Sorenson, 1973): and the moment of inertia is given by (Wong, 1990): It is important to plot the ratio as a function of (E-GOS) to identify the characteristics of the nucleus along its excited states (Regan et al., 2003). The relation between and the angular momentum for the three limits are given by (Bohr and Mottelson, 1953), (Scharff-Goldhaber and Weneser, 1955) as: For a given GSB of a spin , the following ratio was constructed to define the characteristics of the even-even nucleus at its excited states (Bonatsos and Skouras, 1991): where is the measured energy ratio between the two states and .
Odd-even staggering patterns between GSB and NPB have been investigated (Bonatsos et al., 2000): where . All the levels are raised (or lowered) by an amount of energy with respect to the even levels. In other words, staggering takes alternatively positive and negative of zigzag values and may reach zero, followed by other increases again.
To calculate the energy states of the used nuclei, the Bohr-Mottelson model (BM), interacting bosons model (IBM-1) and interacting vector bosons model (IVBM) were used. The BM model introduced an energy expansion of GSB and NPB in powers of for the deformed nuclei, which are given by (Bohr and Mottelson, 1998), (Bonatsos et al., 2000): where is the band head energy of the NPB, and the coefficients can be determined by fitting their equations with the available measured energy levels of GSB and NPB. In IBM-1, the general Hamiltonian is given by (Scholten et al., 1978): (10) where is the intrinsic boson energy and is the interaction strength between bosons i and j; the multipole form of the Hamiltonian is (Casten and Warner, 1988): where are the strengths of pairing, angular momentum, quadrupole, octupole and hexadecupole interactions of each terms in the equation (11).
The Hamiltonian in terms of multipole expansion tends to reduce three forms to meet the requirements of the three symmetry limits; the vibrational U(5), the rotational SU(3) and the soft O(6). In the U(5) limit, the effective parameter is , in the SU(3) limit, the dominating parameter is and in the O(6) limit, the predominate parameter is . The eigenvalues of these three limits are (Casten and Warner, 1988): where are other forms of the strength parameters and is the total boson number. Many nuclei have a transition property of two or three of the above limits.
The eigenvalues in IVBM model for the GSB and NPB states are given by (Georgieva et al., 1982): The values of β and γ can be determined from a fit to the positive GSB while η and ζ are estimated from the NPB.
RESULTS AND DISCUSSION Radium nucleus has 88 protons, with 6 protons above the magic number 82 that gives close properties to the vibrational characteristics. The isotopes studied in this paper are 216-226 Ra with 128 neutrons for 216 Ra which has 2 neutrons above the magic number (126) giving the magic properties for this isotope. The number of neutrons for 218 Ra is 130 which is still close to the magic number (126) and gives it the vibrational properties. The number of neutrons exceeds contiously by 2 for the rest 220-226 Ra isotopes which give them an alternate properties from gamma-soft to the rotational.
The number of protons and neutrons is not enough to indicate properly the properties of nuclei along their excited states, so several methods were used to ensure these properties. It was found that the values of the first measured excited states were equal to 688. The back or up-bending curves, the E-GOS curves, the ratios of successive excited states to that proceeding it and the staggering phenomena are used to confirm the properties of these isotopes along their GSB. The relation between the moment of inertia and the rotational frequency of the photons emitted during the transition between the different states of 216-226 Ra isotopes was plotted in Fig. (1) which shows that there is a back-bending curvature that occurred in the value of the both isotope 216 Ra and 218 Ra, while no bending appered in 220-226 Ra isotopes. Fig. (2) shows the E-GOS curves, where the 216 Ra curve shows a magic shape curve, and the shape of 218 Ra curve was very close to the standard vibrational curve, while the 220 Ra curve close to the standard gamm-soft nuclei, and the shape of the 222-226 Ra curves were close to the rotationalgamma-soft charecteristic. Fig. (3) shows the relationship between the ratio and the spin . Some negative values of occurred for 216 Ra which ensure the magic properties of this isotope. The relationship between and of the 218 Ra isotope showed values which coincide with the values of the vibration limit U(5) with a single negative value that perturbed this properties. The values for 220 Ra coincide with the r values of the vibrational nuclei, the values of r for 222 Ra are coincide with the gamma-soft nuclei and finally the values of r for 224,226 Ra are coincide with the r values of the gamma-softrotational nuclei. Fig. (4) shows the apparent staggering in the differences between the energies of GSB and NPB of 218-226 Ra, but not for 216 Ra due to the poor information of NPB for this isotope. The staggering approximated to zero at one point for 218 Ra but many points close to zero for the other isotopes 220-226 Ra. The IBM-1, IVBM, and BM models were used to calculate the energy states in the ground states band (GSB), while IVBM and BM models were used to calculate the energy states in the negative parity band (NPB) of the 216-226 Ra isotopes. The BM model is perfect for calculating the energy states of rotational nuclei but it fails to calculate the energy states in other nuclei. To make the BM model successful for not rotational nuclei, a vibrational limit term was added to improve the calculations. A significant improvement was observed in the calculations for non-rotational nuclei, and the modified equation of BM is given by: There are two terms in the IVBM model; one represents the rotational movement and the other represents the vibrational movement of the nuclei. Because of the perturbation which is caused by the effects of the gamma-soft characteristics, a third term was added for the isotopes which are affected by this characteristic, and the following equation illustrates this.
Computing programs were written using MATLAB 6.5 (Imad and Al-Jubbori, 2009), to calculate the energy levels of GSB and NPB for 216-226 Ra isotopes using BM, IBM-1, and IVBM models. The number of bosons and the best values of the parameters which give the best fitting between theoretical and experimental energy levels of the above isotopes are shown in (Table 1). It is clear from ( Ra is equal 10.95 which is very close to which is equal 11.3 for this isotope, and this is because the increasing the rotational properties for these isotopes by increasing (n). (Table 1) shows that there are modification parameters for both the BM and IVBM, due to the lack of the vibration parameter in the BM model and there is a lack of gamma-soft parameter in IVBM model. It is noticed from (Table 1) the negative value of A * for 216 Ra and the small values of this parameter for 218,220 Ra due to the pure vibrational properties for these isotopes, while the A * for 222-224 Ra close from specially for 224 Ra which refers to the rotational properties for these isotopes. The BM.M not use for 226 Ra since this nucleus is a pure rotational one. The D * parameter of 216 Ra is equal 436.6 keV which is big, because the magic properties of this nucleus. The D * parameters for other isotopes decreas continuously by increasing neutron numbers. The IVBM.M was used for 216 Ra where the * parameter is deleted and a big value of * are proper for the magic properties, but the negative values of * for 222-226 Ra refer to the reduction of the vibrational properties for these isotopes. (Table 2) shows the measured (Wu, 2007;Askor and Balraj, 2006;Brown andTuli, 2011, Sukhjeet et al., 2011;Sukhjeet and Balraj, 2015;Akovali, 1996) and the calculated energy levels in the GSB for 216-226 Ra, which show that the calculations of IBM-1 are the best compared with IVBM and BM for 216-224 Ra. The error in the BM calculations are equal to 72.1%, 57.3%, 39.7%, 24.1% and 13.7% of the first excited states for 216-224 Ra respectively and less error for the other states. The calculations were improved well by using the modified BM.M, while the BM calculations of 226 Ra was the best since this isotope is a pure rotational nucleus. The error in IVBM calculations of 216,222-226 Ra equal to 28.4, -12.9, 17.1 and -27 of the first excited states respectively and less error for the other states. Howe ever these calculations were improved well using the IVBM.M, while the IVBM calculations are good and equal to IBM-1 calculations for 218,220 Ra.   (Wu, 2007;Askor and Balraj, 2006;Brown andTuli, 2011, Sukhjeet et al., 2011;Sukhjeet and Balraj, 2015;Akovali, 1996) (Table 3) shows the best fitting parameters of IVBM, IVBM.M, BM and BM.M equations with the measured values of NPB. New parameters of BM and BM.M were used for 218-226 Ra in this band, but the same GSB parameters of IVBM and IVBM.M were used since an additional two parameters were used to calculate the NPB states. (Table 4) shows the measured and calculated energy states of NPB for 218-226 Ra, using BM, BM.M, IVBM and IVBM.M. There are no calculations for 216 Ra due to the poor information of NPB for this isotope. Reasonable results were obtained with low errors for all calculated states of the isotopes.   (Wu, 2007;Askor and Balraj, 2006;Brown andTuli, 2011, Sukhjeet et al., 2011;Sukhjeet and Balraj, 2015;Akovali, 1996)

CONCLUSIONS
Knowing the number of protons and neutrons and their distribution on different shells and subshells as well as the positions of their first excited states and the ratio of the second to the first excited states of 216-226 Ra isotopes are not enough to determine their properties among all their excited states. The back or up-bending, the E-GOS, the ratio of different successive states to those states proceeding them and the staggering of the difference between GSB and NPB states are helpful to determine the properties of the nuclei among their many excited states. From Tables (1-4), one can recognize that 216 Ra has a magic properties while 218,220 Ra have vibrational properties and 222-224 Ra have the O(6)-SU(3) properties, where 226 Ra has a pure rotational properties. Using different models, the IBM-1, the IVBM and Bohr-Mottelson to calculate the GSB and NPB show different accuracy. The IBM-1 calculations were the best one, since using suitable limits of IBM-1 for the 216-226 Ra isotopes; where the general form of IBM-1 was suitable for 216 Ra. On the other hand the U(5) limit was found for 218,220 Ra, while the SU(3)-O(6) was convenient for 222,224 Ra and the pure SU(3) appropriate for 226 Ra. A modification term was added to improve the calculations of IVBM and BM, where a gamma-soft term was added to the IVBM and a vibrational term was added to the BM model. An obvious improvement was observed which supports the modification used for each model.