Differential Equation Model for the Ground to First 2+ State Excitation Energy E2 of Even-Even Nuclei
aa r X i v : . [ nu c l - t h ] M a y DIFFERENTIAL EQUATION MODEL FOR THE FIRST + → + STATEEXCITATION ENERGY E2 OF EVEN-EVEN NUCLEI ∗ R. C. NayakDepartment of Physics, Berhampur University, Brahmapur-760007,India.S. PattnaikTaratarini College, Purusottampur, Ganjam, Odisha, India.
We propose here a new model termed as the Differential Equation Model for the first0 + → + state excitation energy E2 of a given even-even nucleus, according to which theenergy E2 is expressed in terms of its derivatives with respect to the neutron and protonnumbers. This is based on a similar derivative equation satisfied by its complementaryphysical quantity namely the Reduced Electric Quadrupole Transition Probability B(E2) ↑ in a recently developed model. Although the proposed differential equation for E2 has beenperceived on the basis of its close similarity to B(E2) ↑ , its theoretical foundation otherwisehas been clearly demonstrated. We further exploit the very definitions of the derivativesoccurring in the differential equation in the model to obtain two different recursion relationsfor E2, connecting in each case three neighboring even-even nuclei from lower to highermass numbers and vice-verse. We demonstrate their numerical validity using the availabledata throughout the nuclear chart and also explore their possible utility in predicting theunknown E2 values. ∗ This is a slightly modified version of the article submitted for publication in Int. Jou. of Mod.Phys. (2014).
I. INTRODUCTION
Reduced electric quadrupole transition probability B(E2) ↑ and its complimentary quantity,namely the first 0 + → + state excitation energy E2 for a even-even nucleus play crucial rolesfor the study of excited states of nuclei and more importantly the inherent nuclear structure. Suchstudies got a boost with the advent of isotope facilities providing a large amount of experimentaldata for several nuclides throughout the nuclear chart. The existence of a large volume of exper-imental data led Raman et al. [1] at the Oak Ridge Nuclear Data Project [1, 2] to make a com-prehensive analysis of all those data, in preparing the most sought-after experimentally adopteddata table for both the above two physical quantities. Of late, Pritychenko et al. [3] followed theprocess in compiling the newly emerging data sets for even-even nuclei near N ∼ Z ∼
28. Thesenew data including the old set obviously put a challenge for the nuclear theorists to understandthem.Theoretically, possible existence of symmetry in nuclear dynamics first explored in developingmass formulas such as the Garvey-Kelson [4] mass formula that connects masses of six neigh-boring nuclei, got into the domain of properties of the excited states. In this regard possibleexistence of such symmetry led Ross and Bhaduri[5] in developing difference equations involvingboth B(E2) ↑ and the E2 excitation energies of the neighboring even-even nuclei. Patnaik et al.[6] on the other hand have also succeeded in establishing even more simpler difference equationsconnecting these values of four neighboring even-even nuclei.Just recently we[7] have succeeded in developing a new model for the B(E2) ↑ , termed as theDifferential Equation Model (DEM) according to which, the B(E2) ↑ value of a given even-evennucleus is expressed in terms of its derivatives with respect to the neutron and proton numbers.Since these two quantities more or less complement each other, it is expected that the excitationenergy E2 should also satisfy a similar differential equation. Therefore in the present work withthis view in the background, we propose a similar model for E2 and explore its validity and utilityin predicting hitherto unknown data. It is needless to stress here that any relation in the formof a differential equation of any physical quantity is intrinsically sound enough to posses a goodpredictive ability. This philosophy has been well demonstrated in case of B(E2) ↑ predictions [7]just recently, and also over the recent years in the development [8–11] of the Infinite NuclearMatter (INM) model of atomic nuclei specifically for the prediction [11] of nuclear masses. Weshould also note here that the development of the Differential Equation Model for the B(E2) ↑ and presently for E2 is also based on the local energy relation of the INM model, which happensto be an important component of the ground-state energy of a nucleus signifying its individualcharacteristic nature.In Sec. II, we show how such a relation in the form of a differential equation for E2 can beformulated followed by its possible theoretical justification. Sec. III deals with how the samedifferential equation can be used to derive two recursion relations in E2, connecting in each casethree different neighboring even-even nuclei. Finally we present in Section IV, their numericalvalidity when subjected to the known [1] experimental data throughout the nuclear chart, and theirpossible utility in predicting its unknown values. II. DERIVATION OF THE DIFFERENTIAL EQUATION FOR THE FIRST + → + STATEEXCITATION ENERGY E2
As mentioned above that the development of the DEM model both for the B(E2) ↑ and presentlyfor E2 owes its origin to the local energy differential equation in the INM model of atomic nuclei.Physically the local energy h embodies all the characteristic properties of a given nucleus, mainlythe shell and deformation, and has been explicitly shown [12] to carry the shell-structure. There-fore it is likely to have some characteristic correspondence with the properties of excited statesof a given nucleus in general and in particular, the reduced transition probability B ( E ) ↑ and itscomplementary quantity E2. Accordingly the h -equation as well as the B(E2) ↑ -equation [see forinstance the Eqs. (1 and 4) of the DEM model [7]] can be used as an ansatz to satisfy a similarrelation involving the E2 value of a given even-even nucleus. As a result we write on analogy, asimilar equation for E2 as E [ N , Z ] / A = h ( + b ) (cid:16) ¶ E / ¶ N (cid:17) Z + ( − b ) (cid:16) ¶ E / ¶ Z (cid:17) N i . (1)Thus we see that we have a relation (1) that connects the E2 value of a given nucleus (N,Z) withits partial derivatives with respect to neutron and proton numbers N and Z. It is true that ourproposition of this differential equation for E2 is purely on the basis of intuition and on closeanalogy with that of B(E2) ↑ . However its validity needs to be established, which we show in thefollowing.For a theoretical justification of the above equation, we use the approximation of expressing E2
16 24 32 40 48N0.01.02.03.04.0 E ( M e V ) Z=16Z=20Z=24Z=28
32 40 48 56 64 72 N0.01.02.0
Z=36Z=40Z=44
64 72 80 88 96 104N0.01.02.0
Z=56Z=60Z=64Z=68
24 32 40 48Z0.01.02.0 E ( M e V ) N=36N=38N=40N=42
40 48 56 64 72Z0.01.0
N=60N=64N=68N=72Z=76N=80
72 80 88Z0.00.51.0
N=106N=110N=114N=118N=122 (a) (b) (c) (f)(e)(d)
Figure 1: Known E2 values plotted as isolines for even-even nuclei. Isolines drawn in the graphs (a-c)connect E2 values of various isotopes for Z=16 on wards with varying neutron number N, while the isolinesdrawn in the graphs (d-f) show the same for isotones for N=36 on wards with varying proton number Z.Other possible isolines are not shown here to avoid clumsiness of the graphs. as the sum of two different functions E ( N ) and E ( Z ) as E [ N , Z ] = E ( N ) + E ( Z ) . (2)The goodness of this simplistic approximation can only be judged from numerical analysis of theresulting equations that follow using the experimental data. Secondly we use the empirical fact[see Fig. 1] that E2s are more or less slowly varying functions of N and Z locally. This assumptionhowever cannot be strictly true at the magic numbers and in regions where deformations dras-tically change. In fact known E2 values plotted as isolines for isotopes and isotones in Fig. 1,convincingly demonstrate the above aspects in most of the cases. The usual typical bending andkinks at magic numbers like 20, 50, 82 and semi-magic numbers 28 and 40 can be seen as a resultof sharply changing deformations. Consequently E and E can be written directly proportional toN and Z respectively as E ( N ) = l N , and E ( Z ) = n Z , (3)where l and n are arbitrary constants and vary from branch to branch across the kinks. Thenone can easily see that just by substitution of the above two Eqs. (2,3), the differential Eq. (1)gets directly satisfied. Thus the proposed differential equation for E2 analogous to the B(E2) ↑ relation in the DEM model gets theoretically justified. However, the differential Eq. (1) has itsown limitations, and need not be expected to remain strictly valid across the magic-number nucleibecause of the very approximations involved in proving it. III. DERIVATION OF THE RECURSION RELATIONS IN E2
It is always desirable to solve the differential Eq. (1) in order to utilize it for practical applica-tions. Therefore it is necessary to obtain possible recursion relations in E2 for even-even nucleiin (N,Z) space from it. The partial derivatives occurring in this equation at mathematical level aredefined for continuous functions. However for finite nuclei, these derivatives are to be evaluatedtaking the difference of E2 values of neighboring nuclei. Since our interest is to obtain recur-sion relations for even-even nuclei, we use in the above equation the usual forward and backwarddefinitions for the partial derivatives. These are given by (cid:16) ¶ E / ¶ N (cid:17) Z ≃ h E [ N + , Z ] − E [ N , Z ] i , (cid:16) ¶ E / ¶ Z (cid:17) N ≃ h E [ N , Z + ] − E [ N , Z ] i , (4) and (cid:16) ¶ E / ¶ N (cid:17) Z ≃ h E [ N , Z ] − E [ N − , Z ] i , (cid:16) ¶ E / ¶ Z (cid:17) N ≃ h E [ N , Z ] − E [ N , Z − ] i . (5)Substitution of the above two pairs of definitions for the derivatives in the differential equation (1)enabled us to derive the following two recursion relations for E2, each connecting three neighbor-ing even-even nuclei. These are E [ N , Z ] = NA − E [ N − , Z ] + ZA − E [ N , Z − ] , (6) E [ N , Z ] = NA + E [ N + , Z ] + ZA + E [ N , Z + ] .. (7)The first recursion relation (6) connects three neighboring nuclei (N,Z), (N-2,Z) and (N,Z-2) whilethe second one (7) connects (N,Z), (N,Z+2) and (N+2,Z). The first one relates E2 of lower to highermass nuclei while the second one relates higher to lower mass, and hence they can be termed as theforward and backward recursion relations termed as E2-F and E2-B respectively. Thus dependingon the availability of E2 data, one can use either or both of these two relations to obtain thecorresponding unknown values of neighboring nuclei. IV. NUMERICAL TEST OF THE RECURSION RELATIONS IN E2
Having derived the recursion relations in E2 from the differential equation (4), it is desirable toestablish their numerical validity to see to what extent they satisfy the known experimental datathroughout the nuclear chart. This would also numerically support the differential equation (1)from which the recursion relations are derived. For this purpose we use the experimentally adoptedE2 data set of Raman et al. [1] in the above relations throughout mass range of A=10 to 240 , andcompute the same of all possible anchor nuclei that are characterized by the neutron and protonnumbers (N,Z) occurring in the left hand sides of the relations (6,7). For better visualization ofour results, we calculate the deviations of the computed E2 values from those of the experimentaldata in terms of the percentage errors following Raman et al. [16]. The percentage error of aparticular calculated quantity is as usual defined as the deviation of that quantity from that ofthe experiment divided by the average of the concerned data inputs, and then expressed as thepercentage of the average. Obviously the larger the percentage error larger is the deviation ofthe concerned computed value. These percentages so computed are plotted in the figures 2 and3 against the proton and neutron numbers respectively. This is intentionally done to ascertain towhat extent possible deviations occur at proton and neutron magic numbers. From the presentedresults we see, that in most of the cases both forward and backward recursion relations (6,7) givereasonably good agreement with experiment. Numerically the deviations in 284 out of 417 cases P e r ce n t a g e E rr o r E2-F0 20 40 60 80 100Proton Number Z-100-50050100 P e r ce n t a g e E rr o r E2-B(a)(b)
Figure 2: Numerical Test of the Recursion Relations connecting E2 values of neighboring nuclei. Thepercentage errors of the computed E2 values of all the anchor nuclei are plotted against Proton NumberZ of those nuclei. The graph (a) shown as E2-F corresponds to the results of the relation (6) while graph(b) marked as E2-B shows those of the relation (7). The vertical solid lines are drawn just to focus largerdeviations if any at the magic and semi-magic numbers. for the forward relation [E2-F] and 278 out of 416 cases for the backward relation [E2-B] liewithin ±
25% error [shown within broken lines in the figures]. In view of this, the agreement ofthe model recursion relations with those of experiment can be considered good. However one cansee from the figures 2 and 3, that the percentage errors (deviations) are relatively higher for somenuclei in the neighborhood of the magic numbers 20, 50, 82, 126 and semi-magic number 40.Such increase in the vicinity of the magic numbers is expected, as the differential Eq. (1) fromwhich the recursion relations are derived need not be strictly valid at the magic numbers.To bring out the contrasting features of our results in a better way, we also present our results inthe form of histograms in Fig. 4, which displays the total number of cases having different ranges P e r ce n t a g e E rr o r E2-F0 20 40 60 80 100 120 140Neutron Number N-100-50050100 P e r ce n t a g e E rr o r E2-B(a)(b)
Figure 3: Same as Fig.2 plotted versus Neutron Number N. of percentage errors. As can be seen, the sharply decreasing heights of the vertical pillars with theincreasing range of errors are a clear testimony of the goodness of our recursion relations.For exact numerical comparison, we also present in Table I results obtained in our calculationalong with those of the experiment [1] for some of the nuclides randomly chosen all over thenuclear chart. One can easily see that the agreement of the predictions with the measured valuesis rather good in most of the cases. In few cases such as Si , S , Ca and Rn there exists littlebit of discrepancy in between the predictions of the relation (7) and those of the experiment. Thusone can fairly say, that the overall agreement of our model predictions with those of experiment isexceedingly good. Therefore such agreement is a clear testimony of the goodness of our recursionrelations.Once we establish the goodness of the two recursion relations, it is desirable to compare ourpredictions with the latest experimentally adopted data of Pritychenko et al. [3]. It must be made Percentage Error050100150200250300 N o . o f C a s e s E2-F Percentage Error050100150200250300 E2-B0-25 25-5050-75 75-100 0-25 25-50 50-75 75-100
Figure 4: Vertical pillars showing number of cases having different ranges of absolute percentage errors.Those marked as E2-F and E2-B correspond to the results of our relations (6 and 7) respectively. clear that none of the values of the new experimental data set has been used in our recursionrelations. Rather we use only the available data set of Raman et al. [1] to generate all possiblevalues of a given nucleus employing the two recursions relations (6) and (7). One should notehere that each of these relations can be rewritten in three different ways just by shifting the threeterms occurring in them from left to right and vice-verse. Thus altogether, these two relationsin principle can generate up to six alternate values for a given nucleus subject to availability ofthe corresponding data. Since each of the values is equally probable, the predicted value for agiven nucleus is then obtained by the arithmetic mean of all those generated values so obtained.Our predictions here are confined to those isotopes for which measured values were quoted byPritychenko et al. [3]. The predicted values so obtained termed as the DEM values are presentedin Table II for various isotopes of Z=24, 26, 28 and 30 along with those of the latest experimental0
Nucleus Experiment B(E2)-B B(E2)-F Nucleus Experiment B(E2)-B B(E2)-F(MeV) (MeV) (MeV) (MeV) (MeV) (MeV) Ne Mg Si S Ar Ca Ti Ge Se Kr Sr Zr Mo Ru Pd Cd Sn Te X e Ba Ce Nd Sm Gd Dy Er Y b
H f W Os Pt Hg Pb Po Rn Ra T h U Pu Cm C f Fm [3] data.We also present our DEM predictions in Fig. 5 to convey a better visualization of our results.One can easily see that in all the cases except for , , Ni and , Zn , the agreement betweenthe predictions with those of the experiment are remarkably good. For these few nuclei, the dis-1 Nucleus Experiment [3] DEM Nucleus Experiment [3] DEM(MeV) (MeV) (MeV) (MeV) Cr Cr Cr Cr Cr Cr Cr Fe Fe Fe Fe Fe Fe Ni Ni Ni Ni Ni Ni Ni Ni Ni Ni Ni Ni Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn crepancies may be attributed to the possible sub-shell effect as either proton or neutron numbers orboth are close to semi-magic numbers 28 and 40. For sake of comparison we have also presentedin Fig. 5, results obtained from two shell-model calculations [3, 14] marked here as SM1 andSM2. One should note here, that the first one obtained using the effective interactions GXPF1A[14] did not succeed in getting reliable values for nuclei having neutron number beyond N=36 be-cause of its own limitations. Hence the second shell-model with JUN45 effective interaction wasperformed by Pritychenko et al. [3] for the nuclei Fe , Ni and , Zn . One can easily see thatthe shell-model values SM1 almost agree with those of ours for almost all the isotopes while the2other shell-model values SM2 are rather away from ours as well as from the experimental values.
20 24 28 32 360.01.02.0 E ( M e V ) DEMEXPSM1
20 24 28 320.01.02.0 E ( M e V ) DEMEXPSM1
24 28 32 36 40 44 48 520.01.02.03.0 E ( M e V ) DEMEXPSM1SM2
32 36 40 44 48 52Neutron Number N0.51.01.5 E ( M e V ) DEMEXPSM1SM2
Z=24, CrZ=26, FeZ=28, NiZ=30, Zn
Figure 5: Both calculated (DEM) and the latest experimental (EXP) [3] E2 values [see text for details] arepresented for various isotopes of Z=24, 26, 28 and 30 versus Neutron Numbers N. DEM Values for differentisotopes are connected by solid lines just to guide the eye. Recent shell-model calculated values [SM1 andSM2] are also presented for sake of comparison
We have just demonstrated as shown above the utility of the recursion relations for predictingE2 values for some of the even-even isotopes of Cr, Fe, Ni and Zn in agreement with the latestexperimental [3] data. Therefore it is desirable to find out whether the model is good enough forsuch predictions in the higher mass regions of the nuclear chart. However in these regions thereis no new data to compare with and hence we can only compare with the data set of Raman etal. [1]. With this view, we repeated our calculations for higher isotope series following the same3
36 40 44 48 52 56 60 64 680.00.51.01.52.02.53.0 E ( M e V ) DEMEXP
48 52 56 60 64 68 72 76 800.51.01.52.0
DEMEXP
72 76 80 84 88 92 96 100Neutron Number N E ( M e V ) DEMEXP
120 128 136 144Neutron Number0.00.51.01.52.0
DEMEXP (a) Z=40, Zr (b) Z=48, Cd(d) Z=88, Ra(c) Z=64, Gd
Figure 6: Similar to Fig. 5 but for Z=40, 48, 64 and 88. The experimental data points marked as EXPcorrespond to those of Raman et al. [1]. methodology outlined above. Since the main aim of our present investigation is just to establishthe goodness of our model, we present here results of only few such series for which experimentaldata exist for a relatively large number of isotopes. Accordingly we have chosen four isotope seriesZ=40, 48, 64 and 88 covering nuclei both in mid-mass and heavy-mass regions. Our choice of thefirst two series namely Z=40 and 48 is again to see to what extent our model works across the4semi-magic number 40 and magic number 50. Our predictions along with those of experimentaldata of Raman et al. [1] are presented in Fig. 6. From the presented results we see that theagreement of the model values with experiment is remarkably good. All isotopic variations of ourmodel predictions clearly follow those of the experiment. However small discrepancies exist atthe magic numbers N=50, 82 and 126 as the DEM model need not be expected to hold good.Now taking stock of all the results discussed so far, one can fairly say that the recursion rela-tions for E2 work exceedingly well almost throughout the nuclear chart. Even across the magicnumbers and sharply changing deformations, these relations have succeeded in reproducing theexperimental data to a large extent with a little bit of deviation here and there. In a nutshell, therecursion relations for the excitation energies E2 derived here can be termed sound enough asto have passed the numerical test both in reproducing and predicting the experimental data , andthereby establish the goodness of the differential equation (1) from which they originate.
V. CONCLUDING REMARKS
In conclusion, we would like to say that we have succeeded in obtaining for the first time, anovel relation for the first 0 + → + state excitation energy E2 of a given even-even nucleus in termsof its derivatives with respect to neutron and proton numbers. We could establish such a differentialequation on the basis of one-to-one correspondence with the local energy of the Infinite NuclearMatter model of atomic nuclei and the recently developed Differential Equation Model for thecomplementary physical quantity B(E2) ↑ . We have also succeeded in establishing its theoreticalfoundation on the basis of the empirical fact that E2’s are more or less slowly varying functionsof neutron and proton numbers except across the magic numbers. We further used the standarddefinitions of the derivatives with respect to neutron and proton numbers occurring in the equation,to derive two recursion relations in E2. Both these relations are found to connect three differentneighboring even-even nuclei from lower to higher mass and vice-verse. The numerical validityof these two relations was further established using the known experimental data set compiledby Raman et al. [1] throughout the mass range of A=10 to 240. More importantly their utilitywas further demonstrated by comparing our predictions with the latest experimental data set ofPritychenko et al. [3] for the isotopes of Cr, Fe, Ni and Zn. The results so obtained convincingly5show the goodness of the recursion relations in E2 and thereby their parent differential equation. [1] S. Raman, C. W. Nestor, Jr, P. Tikkanen, At. Data and Nucl. Data Tables 78(2001)1-128[2] S. Raman, C. H. Malarkey, W. T. Milner, C. W. Nestor Jr., At. Data and Nucl. Data Tables 36(1987)1.[3] B. Pritychenko, J. Choquette, M. Horoi, B. Karamy and B. Singh , At. Data and Nucl. Data Tables98(2012)798-811[4] G. T. Garvey and I. Kelson, Phys. Rev. Lett. (1966) 197[5] C. K. Ross and R. K. Bhaduri, Nucl. Phys. A 196 (1972)369[6] R. Patnaik, R. Patra and L. Satpathy, Phys. Rev.
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