Uncertainty evaluation and correlation analysis of single-particle energies in phenomenological nuclear mean field: An investigation of propagating uncertainties for independent model parameters
UUncertainty evaluation and correlation analysis of single-particle energies inphenomenological nuclear mean field: An investigation of propagating uncertaintiesfor independent model parameters
Zhen-Zhen Zhang, Hua-Lei Wang, ∗ Hai-Yan Meng, and Min-Liang Liu School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
Based on Monte Carlo approach and conventional error analysis theory, taking the heaviest dou-bly magic nucleus
Pb as an example, we firstly evaluate the propagated uncertainties of univer-sal potential parameters for three typical types of single-particle energies in the phenomenologicalWoods-Saxon mean field. Accepting the Woods-Saxon modeling with uncorrelated model parame-ters, we find that the standard deviations of single-particle energies obtained by the Monte Carlosimulation and the error propagation rules are in good agreement with each other. It seems that theenergy uncertaintis of the single-particle levels regularly evoluate with some quantum numbers to alarge extent for the given parameter uncertainties. Further, the correlation properties of the single-particle levels within the domain of input parameter uncertainties are analyzed using the method ofstatistical analysis, e.g., with the aid of Pearson correlation coefficients. It is found that the positive,negative or unrelated relationship may appear between two selected single-particle levels, which willbe very helpful for evaluating the theoretical uncertainty related to the single-particle levels (e.g., K isomer) in nuclear structural calculations. I. INTRODUCTION
The fundamental theory of the strong interactions is quantum chromodynamics [1]. As a final goal, all the phe-nomena in nuclear structure are expected to be derived from the interactions of quarks and gluons. To date, however,such a goal remains a daunting one to an extent though the density functional theory is attempting to approachit. In practice, in order to make the task tractable and more physically intuitive, numerous simplifications are usu-ally made in theoretical modeling for nuclei. As is well known, the first approximation is certainly the use of theconcept of nucleons and their interactions, which has been adopted in nearly all contemporary theories of nuclearstructure. Further, the mean-field approximations and nucleon effective interactions are respectively proposed due tothe difficulty of sloving the many-body problem and the complexity of the nucleon-nucleon interactions. Generallyspeaking, theoretical models for nuclear structure can be grouped into ab initio methods, mean-field theories, shellmodel theories, etc (cf. Ref. [2] and references therein).Nuclear mean-field theories include phenomenological or empirical [3–6] (e.g. the nuclear potentials of Woods-Saxonand Nilsson types) and self-consistent [7–9] (e.g. numerous variants related to the Hartree-Fock approximation) ones,which assume that all the nucleons independently move along their orbits. In this type of nuclear theories, theunderlying element contributing to good quality for theoretical calculations is the reliable mean-field single-particleenergies, which sensitively depend on the corresponding Hamiltonian modeling and model parameters. For a definedmathematical model, the sampling (selection) and the quality of the experimental data will determine the resultingoptimal parameter set and its quality. In principle, this can be done through the standard statistical fitting procedures,such as the least squares and χ fitting [10–13]. Then the physical quantity can be computed using the ‘optimalparemeters’. However, in the language of statistics, the overfitting (underfitting) phenomenon may appear if themodel contains more (less) parameters. For instance, it was pointed that the so-called realistic model-interactionsappear most of the time strongly over-parameterised [14]. Therefore, there will remain uncertainties orginating fromthe size of sample database, the errors of the experimental data, the limited reliability of the model and the numericalmethod [15]. In recent years, model prediction capacities and estimations of theoretical uncertainties are stonglyinterested in many subfields of physics and technological applications[16–20]. Even, it was pointed out that modelpredictions without properly quantified theoretical errors will be of very limited utility [23].The phenomenological mean field, e.g., the realistic Woods-Saxon potential, has been used for many decades innuclear physics and is considered as of very high predictive power for single-nucleon energies whereas related computingalgorithms remain relatively simple. The model uncertainties and predictive power of spherically symmetric Woods- ∗ Corresponding author, [email protected] a r X i v : . [ nu c l - t h ] F e b Saxon mean field have been investigated [14], paying particular attention to issues of the parameter adjustment andparametric correlations. Prior to this work, we, based on the one-body Hamiltonian with a phenomenological meanfield of deformed Woods-Saxon type, have performed some studies [24–27] for different isotopes within the frameworkof macroscopic-microscopic model [28, 29] and cranking approximation [30, 31], focusing on different ground-stateand high-spin nuclear properties. The main interest of our present work is not the fitting of the new parameters,the parameter uncertainties and the investigation of parameter correlations but rather studying the propagation ofthe reasonably given parameter uncertainties and the statistical correlation properties of the calculated single-particlelevels within the domain of input parameter uncertainties using the same Woods-Saxon Hamiltonian. So far, sucha systematical study is scarce and meaningful, especially for the theoretical calculations (e.g., K isomer predictions)depended strongly on single-particle levels. As is well known, the single-particle levels are independent (which meansthe eigenfunctions of the Hamiltonian operator are orthogonal for different levels) in the mean-field approximationwithout the inclusion of the residual interaction. The wording of the ‘correlation properties’ for the levels may beconsidered to be not suitable, even be seriously misunderstood by a general reader. Therefore, it should be particularlynoted that the correlation property mentioned here means the statistical correlation (rather than something else, e.g.,the correlation between the spin partners j = l ± /
2) used for revealing the linear relationship of any two levels withinthe small domains related to their energy uncertainties. The calculated single-particle levels have the nature of theprobability distribution (namely, the property of a stochastic quantity) after considering the uncertainty propagationof model parameters. That is to say, each calculated single-particle level will have the fixed value when calculating ata fixed mean field without the consideration of the model parameter uncertainties, whereas it will possess a stochasticvalue near its ‘fixed’ one once the model parameter uncertainties are considered. In present work, as one of the aims,we will investigate the correlations between these stochastic values rather than the ‘relationships’ of those ‘fixed’ones. It should also be noted that we will, first of all, accept the Woods-Saxon modeling with independent modelparameters and then take the doubly magic nucleus
Pb (which has always been regarded as a benchmark in thestudy of nuclear structure) as an example to perform the present investigation. The parameter uncertainties for theWoods-Saxon potential, even parameter correlations, have been estimated based on the maximum likelihood and theMonte Carlo methods [14, 19].The paper is organized in the following way. In Sec. II, by three subsections, we briefly introduce our theoreticalframework on single-particle Hamiltonian, Monte Carlo method, propagation of the uncertainty and pearson productmoment correlation. Four subsections of Sec. III present our results and discussion on the evaluation of universalpotential parameters, generating pseudo data, uncertainties of single-particle energies and correlation effects betweenthem. Finally, we give a summary in Sec. IV.
II. THEORETICAL FRAMEWORK
Given that our main goal are the uncertainty evolution of single-particle levels and the assessment of correlationsamong them due to the error propagation of model paremeters rather than the Hamiltonian modeling, the fitting ofparameters or other physics issues. Here we will review some related points which are helpful for the general readersthough there are numerous related references for each part.
A. Woods-Saxon single-particle Hamiltonian
The single-particle levels and wave functions are calculated by solving numerically the stationary Schr¨odingerequation with an average nuclear field of Woods-Saxon type. The single-particle Hamiltonian for this equation isgiven by [5, 6] H WS = − (cid:126) m ∇ + V cent ( (cid:126)r ; ˆ β ) + V so ( (cid:126)r, (cid:126)p, (cid:126)s ; ˆ β )+ 12 (1 + τ ) V Coul ( (cid:126)r, ˆ β ) , (1)where the Coulomb potential V Coul ( (cid:126)r, ˆ β ) defined as a classical electrostatic potential of a uniformly charged drop isadded for protons. The first part in the right side of Eq. (1) is the kinetic energy term. The central part of theWoods-Saxon potential which control mainly the number of levels in the potential well is [6] V cent ( (cid:126)r, ˆ β ) = V [1 ± κ ( N − Z ) / ( N + Z )]1 + exp[dist Σ ( (cid:126)r, ˆ β ) /a ] , (2)where the plus and minus signs hold for protons and neutrons, respectively and the a is the diffuseness parameter ofthe nuclear surface. The spin-orbit potential, which can strongly affects the level order, is defined by V so ( (cid:126)r, (cid:126)p, (cid:126)s ; ˆ β ) = − λ (cid:104) (cid:126) mc (cid:105) (cid:26) ∇ V [1 ± κ ( N − Z ) / ( N + Z )]1 + exp[dist Σ so ( (cid:126)r, ˆ β ) /a so ] (cid:27) × (cid:126)p · (cid:126)s, (3)where λ denotes the strength parameter of the effective spin-orbit force acting on the individual nucleons. In Eq. (2),the term dis Σ ( (cid:126)r, ˆ β ) indicates the distance of a point (cid:126)r from the nuclear surface Σ. The nuclear surface is parametrizedin terms of the multipole expansion of spherical harmonics Y λµ ( θ, φ ), namely,Σ : R ( θ, φ ) = r A / c ( ˆ β ) (cid:104) (cid:88) λ + λ (cid:88) µ = − λ α λµ Y ∗ λµ ( θ, φ ) (cid:105) , (4)where the function c ( ˆ β ) ensures the conservation of the nuclear volume with a change in the nuclear shape and ˆ β denotes the set of all the considered deformation parameters. It is similar in Eq. 3 but the new surface Σ so needs tocalculate using the different radius parameter.Based on the Woods-Saxon Hamiltonian as mentioned above, the Hamiltonian matrix is calculated by using theaxially deformed harmonic-oscillator basis in the cylindrical coordinate system with the principal quantum number N (cid:54)
12 and 14 or protons and neutrons, respectively. Then, after a diagonalization procedure, the single-particlelevels and their wave functions can be obtained. It is shown that the calculated single-particle levels with such a basiscutoff will be sufficiently stable with respect to a possibly basis enlargement in present work. Of course, one can seethat for a given (
Z, N ) nucleus, the calculated energy levels { e } π,ν depend on two sets of six free parameters, { V c , r c , a c , λ so , r so , a so } π,ν , (5)one set with the symbol π for protons, and the other set with ν for neutrons; the superscripts ‘c’ and ‘so’ denote theabbreviations for ‘central’ and ‘spin-orbit’, respectively.For convenience, we define the parameter set { p } ≡ { p , p , p , p , p , p } , which is associated to the original one asfollows, { p , p , p , p , p , p } π,ν → { V c , r c , a c , λ so , r so , a so } π,ν . (6)Further, following the notation of Ref. [32, 33], we can denote a point in such a parameter space by p =( p , p , p , p , p , p ). According to the inverse problem theory, the model parameters are usually determined by fittingto a set of observables within a selected sample (e.g., the available sample database of the experimental single-particlelevels). For a given mathematical modeling, e.g. accepting the Woods-Saxon Hamiltonian with free parameters, theoptimum parametrization p o can usually be obtained by a least-squares fit with the global quality measure [33–35], χ (p) = N (cid:88) n =1 (cid:32) O (th) n (p) − O (exp) n ∆ O n (cid:33) (7)where ‘th’ stands for the calculated values, ‘exp’ for experimental data and ∆ O for adopted errors, which generallycontain the contributions from both experimental and theoretical aspects. Note that the definition of the objectivefunction χ is standard and several powerful techniques for finding its minimum value have already been developed.The universal parameter set used in present investigation is, indeed, such one ‘optimal’ parameter set. Havingdetermined p o , in principle, any physics quantity, e.g, the single-particle level e i , can be computed at e i (p o ). Fromthis point, we can to an extent regard the calculated energy level e i as the function of the corresponding parameterset { p } , namely, e i = e i ( p , p , p , p , p , p ) . (8)There is no doubt, the p value depends on the size and quaility of the selected sample database. In fact, thefunctional relationship of Eq. (8) expresses not only a physcial law but also the measured and calculated processes.All the uncertainties during the physical modeling, the experimental measurement and the theoretical calculation maylead to the uncertainty of the p value. On the contrary, the uncertainty of the p value will propagate to the resultsduring the calculations. B. Uncertainty estimation of single-particle levels
By reasonably assuming that input parameters { p } are Gaussian random variables, we will be able to estimatethe uncertainties of the single-particle levels due to input uncertainties of potential parameters using conventionalanalysis method [36–38] (e.g., the formula of uncertainty propagation) and Monte Carlo method [39–43]. Based on thefunctional relationship of Eq. (8) and the uncertainty propagation formula, the uncertainty of the i th single-particlelevel e i with random and uncorrelated inputs can be given analytically by σ e i = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) j =1 (cid:18) ∂e i ∂p j (cid:19) · σ p j , (9)where σ p j is the standard deviation of the input parameter p j ; the partial derivative ∂e i /∂p j is usually called sensitivitycoefficient, which gives the effect of the corresponding input parameter on the final result. Note that both linearityof the function (at least, near the calculated point p j ) and ‘small’ uncertainty of input parameter are prerequisites ofthe conventional method of uncertainty estimation. However, there is no such limitation for Monte Carlo simulationmethod, which can handle both small and large uncertainties in the input quantities. Moreover, the Monte Carlosimulation, which can be generally defined as the process of replication of the ‘real’ world, has the ability to takeaccount of partial correlation effects for input parameters. It is also convenient to study the correlation effect e.g.between two Gaussian-distributed variables whereas the conventional method cannot do this.As known in such a simulation, the availability of high quality Gaussian random numbers is of importance. Generallyspeaking, the realization of Gaussian-random-number generator can adopt the software and the hardware methods.The former has the limited speed and poor real-time characteristic while the latter (which is based on digital devices)is not only fast, real-time, but also has good flexibility and accuracy. At present, the majority of the frequentlyused digital methods for generating Gaussian random variables are based on transformations from uniform randomvariables. Popular methods, for instance, include the Ziggurat method [44], the inversion method [45], the Wallacemethod [46], the Box-Muller method [47–50] and so on. In present work, we realize the hardware Gaussian randomnumber generators using the Box-Muller algorithm. Namely, taking each value p oj of the universal parameter set { p o , p o , p o , p o , p o , p o } as the corresponding mean value, one can generate the random and uncorrelated input parameter p j following a normal distribution N ( p oj , σ p j ). With a large sample of input parameters, the uncertainties of single-particle levels can be estimated. For instance, considering the uncertainty of one input parameter p j and keepingother universal values unchanged, the variance of the calculated e i can be given by σ e i = 1 N − N (cid:88) k =1 [ e i ( p jk ) − e i ( p oj )] , (10)where the sampling number N should be chosen to be sufficiently large (e.g., 10 000 or more). Similar calculationscan be performed when the uncertainties of two or more input parameters are opened. Therefore, we will be able toinvestigate the effects of the uncertainties of different input parameters and their combinations on the uncertaintiesof single-particle levels. C. Pearson Product Moment Correlation
The single-particle levels with certain uncertainties can usually be regarded as the input variables in further nuclearmean-field calculations, e.g., the K isomeric calculations. In this case, for evaluating the further theoretical predictions,it will be very useful to know both the uncertainties and correlation properties of the single-particle levels. As a simpleexample, the energy uncertainty of one-particle one-hole (1 p h ) excitation will directly relate to two correspondingsingle-particle levels to a large extent. Once we can arbitrarily regard the excitation energy E p h as the function oftwo single-particle levels e and e with the standard deviations σ e and σ e , respectively, namely, E ∗ p h = f ( e , e ) . (11) TABLE I: Various parameter sets for Woods-Saxon potentialParameter V (MeV) r c (fm) a c (fm) λ r so (fm) a so (fm)Wahlborn [57] 51.0 1.27 0.67 32.0 1.27 0.67Rost [58] n 49.6 1.347 0.7 31.5 1.280 0.7p 1.275 17.8 0.932Chepurnov [59] 53.3 1.24 0.63 23 . · (1 + 2 I ) 1.24 0.63New [60] n 49.6 1.347 0.7 ∗ ∗ ∗ ∗ Universal [6] n 49.6 1.347 0.7 35.0 1.31 0.7p 1.275 36.0 1.32Optimized [61] n 49.6 1.347 0.7 36.0 1.30 0.7p 1.275Cranking [62, 63] 53.754 1.19 0.637 29.494 1.19 0.637
Regardless of whether or not e and e are independent, the standard uncertainty of such an excited state can bewritten as [51] σ E ∗ p h = (cid:115)(cid:18) ∂f∂e (cid:19) σ e + (cid:18) ∂f∂e (cid:19) σ e + 2 ∂f∂e ∂f∂e ρ ( e , e ) σ e σ e , (12)where the quantity ρ ( e , e ) is the Pearson correlation coefficient, which is given by [52, 53] ρ ( e , e ) = cov ( e , e ) σ e σ e . (13)Such a cross-correlation coefficient measures the strength and direction of a linear relationship between two variables,e.g., e and e . The greater the absolute value of the correlation coefficient, the stronger the relationship. The extremevalues of -1 and 1 indicate a perfectly linear relationship where a change in one variable is accompanied by a perfectlyconsistent change in the other. At these two cases, all of the data points fall on a line. A zero coefficient representsno linear relationship. That is, as one variable increases, there is no tendency in the other variable to either increaseor decrease. When the cross-correlation coefficient is in-between 0 and +1/-1, there will be a relationship, but all thepoints don’t fall on a line. The sign of the correlation coefficient represents the direction of the linear relationship.Positive coefficients indicate that when the value of one variable increases, the value of the other variable also tendsto increase. Positive relationships produce an upward slope on a scatterplot. Negative coefficients represent the cases:when the value of one variable increases, the value of the other variable tends to decrease. Correspondingly, negativerelationships produce a downward slope. It should be noted that Pearson’s correlation coefficient, which measuresonly linear relationship between two variables, will not detect a curvilinear relationship. For instance, when thescatterplot of two variables shows a symmetric distribution, the relationship may exist but the correlation coefficientis zero. III. RESULTS AND DISCUSSIONA. Evaluation of Woods-Saxon potential parameters
In the phenomenological nuclear mean field, the realistic Woods-Saxon potential has shown certain advantagesand is still used widely. For instance, it provides a good description of not only the ground-state properties butalso the excited-state properties of nuclei. Nowadays, many authors are still working in different issues with theWoods-Saxon potential. Such a simple nuclear mean field has been successfully applied to explain and predict thenuclear equilibrium deformations, the hight- K isomer, the nucleon binding energies, the fission barriers, numerioussingle-particle effects for superdeformed and fast rotating nuclei, and so on. As shown in Table I, there exist variousparametrizations of the Woods-Saxon potential (cf. Ref. [6] and references therein), which are usually obtained byfitting the available single-particle data (or part of them, namely, one of the subdatabases) or other observables.Indeed, based on the same mathematical modeling and different sample databases/subdatabases, different parametersets can be obtained. It can be seen that these parameter sets are somewhat different, even rather different for somequantities among them. Correspondingly, the different parameter sets are just suitable for a certain nuclear massregion. Sometimes, the difference of the corresponding quantity (e.g., single-particle energies) calculated theoreticallyusing different parameter sets is referred to as model discrepancy, which can be evaluated by using different modelsand/or different parameter sets. The universal parameter set of the Woods-Saxon potential is one of the most commomparameter sets. In principle, it can be used for the ‘global’ calculation for nuclei . In present paper, we will performour investigation based on the universal parameter set. - 6- 4- 20246- 1 5 %- 1 0 %- 5 %0 %5 %1 0 % ( a ) D E / MeV
N Z ( b )
PD(p1)
C a O C a
N i
Z r
S n
G d
P b
N u c l e u s
FIG. 1: (Color online) (a) Discrepancies between the available experimental data and the calculated single-particle energiesusing the Woods-Saxon Universal parameter set for even-even nuclei O, Ca, Ca, Ni, Zr,
Sn,
Gd and
Pb. Thedata are taken from Refs. [54, 55]. (b) Percentage differences between the ‘best’ and the ‘optimal’ p (namely, V ) parameters.See text for more details. In order to evaluate the universal potential parameters, Figure. 1(a) shows the discrepancies ∆ E ( ≡ e theo .i − e exp .i ) of the calclated single-particle energies from the available data (e.g., eight spherical nuclei [54, 55]: O, Ca, Ca, Ni, Zr,
Sn,
Gd and
Pb). The discrepancies show us that the single-particle levels generated bythe universal parameters, in fact, cannot agree with the data very well (similar to the mass calculation [56], the questfor some possibly missing interactions and ‘better’ mathematic modeling will never stop). Moreover, most of thevalues are smaller or larger than the data. For instance, as can be seen, there exist the systematic overestimationand underestimation for protons and neutrons, respectively, especially in the lighter nuclei. To see the statisticalproperties of the parameters, the percentage difference,
P D ( p j ), of the model parameter p j (as an example, j = 1here) extracted from the experimental data is presented in Fig. 1(b), which is defined by P D ( p j ) = p b j − p o jp b j + p o j × , (14)where the p o j means the j th ‘optimal’ (universal) value of { p } parameters; the p b j parameter denotes the so-called‘best’ value which can be obtained based on the following method. For a certain model parameter, e.g. the potentialdepth p (namely, V ) of the Woods-Saxon parameters, we calculate the corresponding single-particle energies of agiven nucleus by varying the value of this parameter p around its optimal value p o1 and keeping other parameterswith universal values unchanged. If the discrepancy of the calculated single-particle energy for a certain nucleus fromthe corresponding experimental data equals zero, the “best” value p b1 of this parameter p for this nucleus is thereforeobtained. In principle, for a large sample, we can extract the standard deviation σ p with 68 .
3% confidence level forthe parameter p . From Fig. 1(b), it is found that the percentage differences distribute between ± { σ p } ≡ { σ p , σ p , σ p , σ p , σ p , σ p } for the parameters { p } withinreasonable domains, taking the universal parameters { p o1 , p o2 , p o3 , p o4 , p o5 , p o6 } as the corresponding mean values. B. Producing pseudo-data of potential parameters
FIG. 2: (Color online) Two-dimensional scatter plots, together with their corresponding correlation coefficients, between 6independent WS model parameters.
Based on the given mean values { p o } and the corresponding standard deviations { σ p } , the Gaussian-distributedrandom sets { p } can, in principle, be numerically generated in the spirit of the Monte Carlo approach. Consideringthe uncertainty estimations of the Woods-Saxon parameters and the sensitivity coefficients of single-particle levels,we, in practice, use a set of percentage coefficients { c } ≡ { c , c , c , c , c , c } = { . , . , , , , } tocalibrate the standard deviations { σ p } during the calculations. That is, the standard deviations are given by σ p σ p σ p σ p σ p σ p = (cid:16) c c c c c c (cid:17) p o1 p o2 p o3 p o4 p o5 p o6 . (15)Such a set { σ p } may deviate from the ‘true’ values to an extent but does not affect the conclusion of our investigationsince they lie in the reasonable domains. Moreover, the strong overlaps of the ‘peaks’ of single-particle levels can beavoided (as seen below). We perform the Woods-Saxon single-particle-level calculations with 10 000 samples for { p } ,which is large engough to suppress the error coming from stochastic choices. To show the quality of the normallydistrubited random quantities { p } , Figure 2 presents the two-dimensional scatter plots related to the 6 Woods-Saxonparameter sampling of neutrons, together with the corresponding correlation coefficients. Note that it is similar forprotons. For comparison, the normal distribution N ( p o i , σ p i ) is transformed into the standard normal distribution N (0 ,
1) by defining the dimensionless parameter x i = ( p i − p o i ) /σ p i in Fig. 2. One can see the Gaussian-distributedand independent properties of these parameters. In addition, the calculated skewness and kurtosis values are zero, asexpected, indicating the Gaussian-type distributions as well. C. Uncertainties of single-particle energies
With the sampling { p } , the uncertainties of the single-particle energies will be able to be exactly evaluated. Indeed,this is the advantage of the Monte Carlo method. For convenience, using the similar γ − γ coincidence techniquewhich is widely used for experimentally deducing the nuclear level scheme, we construct a level-level coincidencematrix (namely, a two dimensional histogram). Each axis of the matrix corresponds to the energy of the calculatedsingle-particle levels. The matrix has a dimension of 4096 × − .
96 to 0 .
00 MeV, covering the range of the single-particle energies (e.g.,all the bounded ones for neutrons) that we care for. By using the gated spectra on different level-level matrice, one canconveniently analyze the peak distributions of single-particle levels and even their correlation properties at differentconditions. s =(0,0,0) s =(1,0,0) (b)(d) s =(1,1,1) (e)(c) c oun t s ( ) channel s =(0,1,0) s =(0,0,1) FIG. 3: (Color online) Calculated single-neutron levels labeled as { nl } in Pb (gated at the 1 s level). The dotted lines areprovided to guide the eye. See the text for further calculated details. As is well known, there are three typical types of single-particle levels during the evolution of nuclear models (forinstance, from the harmonic oscillator model, adding strong spin-orbit coupling to obtain the shell model, and axialdeformation to give the collective model). In this work, we use a more realistic Woods-Saxon potential (lies between theharmonic oscillator potential and the finite square well) to produce these three kinds of single-particle levels and studytheir energy uncertainties originated from model parameters. Similar to the parameter space ( p , p , p , p , p , p ), letus define a correspondingly 6-dimensional ‘switch’ space ( s , s , s , s , s , s ), where s i = 0 or 1 (for i = 1 , , · · · , s i = 0, the universal parameter p o i is always adopted (that is, the standard deviation σ p i is not used). For s i = 1, it indicates that the sampling p i value is adopted (in other words, the parameter σ p i is opened). Obviously, / / / / s / f7 / f5 / / / / / / / / / f7 / i / f5 / / i / j / s =(1,1,1,1,1,1)(a) s =(0,0,0,0,0,0) c oun t s ( ) channel FIG. 4: (Color online) Similar to Fig. 3, but the spin-orbit coupling is considered (gated at the 1 s / level). we can evaluate the effects of different parameter uncertainties and their combinations on single-particle levels bycalculating at different points s = ( s , s , s , s , s , s ).In Fig.3, we show the spherical single-particle levels (labeled as { nl } quantum numbers) calculated using the Woods-Saxon potential without the inclusion of the spin-orbit coupling. Note that in spectroscopic notation the bound statesfor angular momentum states with l = 0 , , , , , , · · · are indicated with the letter s, p, d, f, g, h, · · · , respectively.The projection spectra at different s points are obtained by gating at the 1 s level. Figure 4 shows the second kindof single-particle levels (labeled as { nlj } ) calculated at two s points using the spherically Woods-Saxon potentialwith the spin-orbit part. In this case, the l orbital will be splitted into two j = l ± substates. Similar to Fig.4,in Fig. 5, we show the deformed Woods-Saxon single-particle levels (labeled as { Ω[ N n z Λ] } , the so-called Nilssonquantum numbers) calculated at β = 0 .
1, an arbitrarily selected axial deformation value. In Fig. 5, one can seethat the peak heights of the deformed single-particle levels are all same, with the sampling value, 10 000, since thetwo-fold degenerate levels { Ω[ N n z Λ] } are not degenerate any more. However, the levels labeled as { nl } and { nlj } (a) s =(0,0,0,0,0,0)(b) s =(1,1,1,1,1,1) channel c oun t s ( ) FIG. 5: (Color online) Similar to Fig. 4, but the axail deformation is added (gated at the 1 / { Ω[ Nn z Λ] } quantum numbers are given due to space limitations. n + 1)- and ( j + )-fold degeneracies, respectively, due to the spherical symmetry of the Woods-Saxonpotential. As seen in Figs. 3(a) and 4(a), the counts dividing by 10 000 indicate the degrees of degeneracy of thecorresponding levels. Based on these gated spectra at different s points, we can analyze the distributed properties ofthe single-particle levels without the strong overlap. For instance, it is convenient to fit the distributions in Figs.3and 4 while it is difficult in the right part of Fig.5 since the distributions overlap strongly. - - c oun t s ( ) channel s =(1,0,0,0,0,0) s =(1,1,0,0,0,0) s =(1,1,1,0,0,0) s =(1,1,1,1,0,0) s =(1,1,1,1,1,0) s =(1,1,1,1,1,1) i spherical(b)FWHM1: 80.562 keV FWHM2: 384.230 keV : 34.209 keV : 163.167keV s =(1,0,0,0,0,0) s =(1,1,1,1,1,1) (a) FIG. 6: (Color online) (a) Distributions of the spherical i / level calculated at different s = ( s , s , s , s , s , s ) points. (b)The Gaussian fits to the distributions at s = (1 , , , , ,
0) and (1 , , , , ,
1) points.
In Fig. 6(a), as an example, we show that the uncertainty evolution of the selected spherical i / level as moreand more uncertainty parameters are opened. It can be seen that the energy uncertainty of this level increaseswith increasing ‘1’ in the ‘switch’ space. The results of the Gaussian fits to the peaks at s = (1 , , , , ,
0) and(1 , , , , ,
1) points are presented in Fig. 6(b), including the standard deviations and the full width at half maximum(FWHM). The FWHM is a parameter commonly used to describe the width of a “bump”, e.g., on a function curvewhich is given by the distance between points on the curve at which the function reaches half its maximum value.The FWHM can be used for describing the width of any distribution. For a normal distribution N ( µ, σ ), its FWHMis 2 √ σ ≈ . σ ). In principle, we can extract the standard deviation σ e i for each single-particle level e i andfurther find the possible evoluation law. It is found that, in practice, the correct fitting will be rather difficult to beperformed once the peak is not ‘pure’ though we try to limit the amplitudes of the given standard deviations { σ p } .Fortunately, we find that the single-particle energiy e i depends linearly on the potential parameters within theuncertainty domain near the universal parameters. That is to say, it is accuate enough to use the first-order Talorapproximation in Eq. (8), which means that we can approximate the function e i = e i ( p j ) using its tangent line atthe p o j point. Therefore, we can calculate analytically the energy uncertainty σ e i according to Eq. (8). The partialderivatives (sensitivity coefficients) of the single-particle energies e i with respect to the potential parameters { p } at { p o } can be numerically calculated by the finite-difference formula, ∂e i ∂p j (cid:39) e i ( p + j ) − e i ( p − j ) p + j − p − j , (16)with values of p + j and p − j suitably close to p j . For convenience, we define an adjusted sensitivity coefficient as ∂ j e i ≡ ∂e i ∂p j σ p j . (17)1 -40 -30 -20 -10-0.14-0.070.000.07-0.14-0.070.000.07-0.14-0.070.000.070.14 -40 -30 -20 -10 0 -40 -30 -20 -10 0-0.14-0.070.000.07-0.14-0.070.000.07-0.14-0.070.000.07 {n l j}{n l} { [Nn ]} W L z (a) e ¶
1 i (b) ¶
2 i e ¶
3 i e(c) (d) ¶
1 i e(e) ¶
2 i e(f) ¶
3 i e(g) ¶
4 i e(h) ¶
5 i e(I) ¶
6 i e (j) ¶
1 i e(k) ¶
2 i e(l) ¶
3 i e(m) ¶
4 i e(n) ¶
5 i e(o) ¶
6 i e FIG. 7: (Color online) The adjusted sensitivity coefficients ∂ j e i of three types of typical neutron single-particle levels labeledby { nl } , { nlj } and { Ω[ Nn z Λ] } in Pb. See the text for more details.
By giving a set of suitable { σ p } , the adjusted sensitivity coefficients { ∂e i } ≡ { ∂ e i , ∂ e i , ∂ e i , ∂ e i , ∂ e i , ∂ e i } will havethe similar order of magnitude. Figure 7 shows the adjusted sensitivity coefficients for the three tpyes of the calculatedneutron single-particle levels labeled respectively by { nl } , { nlj } and { Ω[ N n z Λ] } in Pb. From this figure, it canbe seen that the adjusted sensitivity coefficients show us the regular evolution trends. In particular, the spectrumenvelopes, e.g., in Figs. 7(g)-7(i) and Figs. 7(m)-7(o), show different but interesting properties. It will be meaningfulto reveal the physics behind them. Based on these sensitivity coefficients and the standard deviations of these modelparameters or their combinations (namely, the adjusted sensivity coefficients), we can calculate the energy uncertainty e i analytically. Indeed, for the i / level, the analytical result coincides with the fitting value of the peak generatedby the Monte-Carlo method. The typical error between the calculated and fitted values is less than 3%.Based on the above method, we analytically calculate the overall uncertainties of the three kinds of levels mentionedabove both for neutrons and for protons in Pb. In the calculations, all the parameter uncertainties are taken intoaccount, which indicates that calculations are done at s = (1 , ,
1) point for { nl } levels and at s = (1 , , , , ,
1) pointfor { nlj } and { Ω[ N n z Λ] } ones. As seen in Fig 8, one can notice that the changing trends of the standard deviationsare similar for neutrons and protons. For the { nl } single-particle levels, there is no obvious change with changing n, l quantum numbers or single-particle energies. For the { nlj } and { Ω[ N n z Λ] } single-particle levels, it seems that theincreasing trends of the energy uncertainties appear with increasing energies or angular momentum j for a given n ,e.g, n = 1. Note that one spherical j mean-field orbital will split into ( j + ) deformed substates e.g., at β = 0 . N e u t r o n ( a ) { n l } ( b ) { n l j } - 4 0 - 3 0 - 2 0 - 1 0 00 . 00 . 10 . 2 ( e ) { n l j }( c ) { W [ N n z L ] } ( d ) { n l } s ei / MeV E / M e V
P r o t o n - 4 0 - 3 0 - 2 0 - 1 0 0 ( f ) { W [ N n z L ] } FIG. 8: (Color online) The standard deviation σ e i of three kinds of typically single neutron (left) and proton (right) energiesin Pb.FIG. 9: (Color online) Scatter plots (with correlation coefficients ρ ) between three pairs of arbitrarily selected single-neutronenergies, 1 i / ⊕ j / i / ⊕ p / g / ⊕ i / Pb. Fromthe left to the right, the uncertainties of more and more parameters are considered as indicated in the text.
D. Correlation Coefficients between single-particle energies
As mentioned above, the single-particle levels are usually the input quantities in further theoretical calculations(cf.,e.g., Refs. [64, 65]), e.g., the calculations of High- K isomers, shell correction, pairing correction, etc. Both the energyuncertainties and the correlation effects are of importance for further uncertainty predictions. Based on Eq. 13 thePearson correlation coefficients will be able to be calculated between any two levels. Further, we can investigatethe correlation effects among them within the ‘small’ energy domains associated with parameter uncertainties, { σ p } .Figure 9 shows the two-dimensional scatter plots between three pairs of arbitrarily selected { nlj } single-neutron3 -40-30-20-10 0-40 -30 -20 -10 0 (a) E / M e V E/MeV -0.4 ≤ρ <-0.2-0.2 ≤ρ <0.0 0.2 ≤ρ <0.40.0 ≤ρ <0.2 -40 -30 -20 -10 0 (b) ≤ρ <0.60.6 ≤ρ <0.8 0.8 ≤ρ FIG. 10: (Color online) Color-coded plot of the calculated correlation coefficients between single-particle energy levels forneutrons (a) and protons (b) in
Pb. energies,1 i / ⊕ j / , 1 i / ⊕ p / and 1 g / ⊕ i /
2, near the Fermi surface. From the left to the right sidein this figure, the calculations are performed at s = (1 , , , , , , , , , , , , , , , , , , , , , , , , ,
0) and (1 , , , , ,
1) points, respectively. It should be noted that, similar to the operation in the Fig-ure 2 plot, before plotting, the normal distributions of the selected single-particle levels are transfermed into thestandard normal distributions by defining a dimensionless parameter, x i = [ e i (p) − e i (p o )] /σ e i . In Fig. 9, the dimen-sionless parameters x µ , for µ = 1 , , i / , j / , p / and 1 g / . One can clearly see the evolutions of the correlation coefficients and the scatterplotdistributions with opening more and more uncertainty parameters. In particular, it is found that the positive, zero,and negative values appear in the correlation coefficients.In order to give an overall investigation, we show that the color-coded plot of the calculated correlation coefficientsbetween single-particle levels with energy e i < IV. SUMMARY
Taking the
Pb nucleus as the carrier, we have investigated the single-particle energy uncertainties and thestatistical correlations of different levels due to the uncertainty propagation of independent model parameters, whichare important for further theoretical predictions, e.g., the K isomer calculation. The adjusted sensitivity coefficientsare introduced and discussed for three types of single-particle levels. The overall standard deviations of the single-particle levels in the Woods-Saxon nuclear mean field are shown and the evolution properties are briefly discussed.It is also found that the correlation coefficients involve a rather wide domain, which are of importance for furthertheoretical uncertainty predictions relying on the single-particle levels. Noted that the practical energy uncertaintieswill depend on the practical standard deviations of model parameters during the further calculations whereas theevolution laws of parameter uncertainty propagations and the correlation properties of single-particle levels are stillsimilar and valid. Next, we will further investigate the uncertainty propagation of model parameters with partialcorrelation effects. It would be also interesting to extend this study to other phenomenological or self-consistentmodels in nuclear physics, even other fields.4 Acknowledgments
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