NpNn Scheme Based on New Empirical Formula
for Excitation Energy
Jin-Hee Yoon, Eunja Ha, and Dongwoo Cha∗
Department of Physics, Inha University, Incheon 402-751, Korea
(Dated: July 25, 2007)
Abstract
We examine the NpNn scheme based on a recently proposed simple empirical formula which
is highly valid for the excitation energy of the first excited natural parity even multipole states
in even-even nuclei. We demonstrate explicitly that the NpNn scheme for the excitation energy
emerges from the separate exponential dependence of the excitation energy on the valence nucleon
numbers Np and Nn together with the fact that only a limited set of numbers is allowed for the
Np and Nn of the existing nuclei.
PACS numbers: 21.10.Re, 23.20.Lv
∗Electronic address: dcha@inha.ac.kr; Fax: +82-32-866-2452
http://arxiv.org/abs/0704.1693v2
mailto:dcha@inha.ac.kr
The valence nucleon numbers Np and Nn have been frequently adopted in parameter-
izing various nuclear properties phenomenologically over more than the past four decades.
Hamamoto was the first to point out that the square roots of the ratios of the measured
and the single particle B(E2) values were proportional to the product NpNn [1]. It was
subsequently shown that a very simple pattern emerged whenever the nuclear data concern-
ing the lowest collective states was plotted against NpNn [2]. This phenomenon has been
called the NpNn scheme in the literature [3]. For example, when the measured excitation
energies Ex(2
1 ) of the first excited 2
+ states in even-even nuclei were plotted against the
mass number A (A-plot), we got data points scattered irregularly over the Ex-A plane as
seen in Fig. 1(a). However, we suddenly had a very neat rearrangement of the data points by
just plotting them against the product NpNn (NpNn-plot) as shown in Fig. 1(b). A similar
simplification was observed not only from Ex(2
1 ) but also from the ratio Ex(4
1 )/Ex(2
[5, 6, 7], the transition probability B(E2; 2+1 → 0
+) [8], and the quadrupole deformation
parameter e2 [9].
The chief attraction of the NPNn scheme is twofold. One is the fact that the simplification
in the graph occurs marvelously every time the NpNn plot is drawn. The other attraction
0 100 200 300 400 5000 100 200
4 (b) N
-Plot
(a) A-Plot
Mass Number A
FIG. 1: A typical example demonstrating the NpNn scheme. The excitation energies of the first 2
states in even-even nuclei are plotted (a) against the mass number A and (b) against the product
NpNn. The dashed curve in part (a) represents the bottom contour line which is drawn by the first
term αA−γ of Eq. (1). The excitation energies are quoted from Ref. 4.
0 100 200
0 100 200
Mass Number A
(b) Empirical Formula(a) Data
FIG. 2: Excitation energies of the first excited natural parity even multipole states. Part (a) shows
the measured excitation energies while part (b) shows those calculated by the empirical formula
given by Eq. (1). The measured excitation energies are quoted from the compilation in Raman et
al. for 2+
states [4] and extracted from the Table of Isotopes, 8th-edition by Firestone et al. for
other multipole states [20].
is the universality of the pattern, namely the exactly same sort of graphs appears even at
different mass regions [2]. Since the performance of the NpNn scheme has been so impressive,
many expected that the residual valence proton-neutron (p-n) interaction must have been
the dominant controlling factor in the development of collectivity in nuclei and that the
product NpNn may represent an empirical measure of the integrated valence p-n interaction
strength [3]. Also, the importance of the p-n interaction in determining the structure of
nuclei has long been pointed out by many authors [10, 11, 12, 13, 14, 15, 16].
In the meantime, we have recently proposed a simple empirical formula which describes
the essential trends of the excitation energies Ex(2
1 ) in even-even nuclei throughout the
periodic table [17]. This formula, which depends on the valence nucleon numbers, Np and
Nn, and the mass number A, can be expressed as
Ex = αA
−γ + β [exp(−λNp) + exp(−λNn)] (1)
where the parameters α, β, γ, and λ are fitted from the data. We have also shown that the
source, which governs the 2+1 excitation energy dependence given by Eq. (1) on the valence
nucleon numbers, is the effective particle number participating in the residual interaction
from the Fermi level [18]. Furthermore, the same empirical formula can be applied quite
successfully to the excitation energies of the lowest natural parity even multipole states such
as 4+1 , 6
1 , 8
1 , and 10
1 [19]. It can be confirmed by Fig. 2 where the measured excitation
energies in part (a) are compared with those in part (b) which are calculated by Eq. (1).
The values of the parameters adopted for Fig. 2(b) are listed in Table I.
0 100 200 300 400 500
0 100 200 300 400 500
(I) Z=2~8
 Data
 Empirical Formula
(II) Z=10~20
(III) Z=22~28
(IV) Z=30~50
(V) Z=52~82
(VI) Z=84~126
FIG. 3: The NpNn-plot for the excitation energies of the first 2
+ states using both the data (open
triangles) and the empirical formula (solid circles). The plot is divided into six panels each of which
contains plotted points that come from each one of the proton major shells.
TABLE I: Values adopted for the four parameters in Eq. (1) for the excitation energies of the
following multipole states: 2+
, and 10+
Multipole α(MeV) β(MeV) γ λ
34.9 1.00 1.19 0.36
94.9 1.49 1.15 0.30
441.4 1.51 1.31 0.25
1511.5 1.41 1.46 0.19
2489.0 1.50 1.49 0.17
In this study, we want to further elucidate about our examination of the NpNn scheme
based on the empirical formula, Eq. (1), for Ex(2
1 ). Our goal is to clarify why Ex(2
complies with the NpNn scheme although the empirical formula, which reproduces the data
quite well, does not depend explicitly on the product NpNn.
First, we check how well the empirical formula does meet the requirements of the NpNn
scheme. In Fig. 3, we display the NpNn-plot for the excitation energies of the first 2
states using both the data (empty triangles) and the empirical formula (solid circles). We
show them with six panels. Each panel contains plotted points from nuclei which make up
the following six different proton major shells: (I) 2 ≤ Z ≤ 8, (II) 10 ≤ Z ≤ 20, (III)
TABLE II: The maximum value of NpNn and the minimum value of Ex for each major shell in
Fig. 3 are indicated here. The numbers in the parenthesis represent Ex calculated by the empirical
formula given by Eq. (1).
Major Shell Z Max. NpNn Min. Ex (MeV)
I 2 ∼ 8 8 1.59 (1.85)
II 10 ∼ 20 36 0.67 (0.82)
III 22 ∼ 28 16 0.75 (0.77)
IV 30 ∼ 50 140 0.13 (0.18)
V 52 ∼ 82 308 0.07 (0.08)
VI 84 ∼ 126 540 0.04 (0.05)
0 50 100 150 200 250
0 50 100 150 200 250
Empirical Formula
FIG. 4: Extract from Fig. 3 for some typical nuclei which belong to the rare earth elements.
Different symbols are used to denote excitation energies of individual nuclei.
22 ≤ Z ≤ 28, (IV) 30 ≤ Z ≤ 50, (V) 52 ≤ Z ≤ 82, and (VI) 84 ≤ Z ≤ 126. From this
figure, we can see an intrinsic feature of the NpNn-plot, namely, the plotted points have their
own typical location in the Ex-NpNn plane according to which major shell they belong. For
example, the plotted points of the first three major shells I, II, and III occupy the far left side
part of the Ex-NpNn plane in Fig. 3 since their value of the product NpNn does not exceed
several tens. On the contrary, the plotted points of the last major shell VI extend to the far
right part of the Ex-NpNn plane along the lowest portion in Fig. 3. This is true since their
value of the excitation energy Ex is very small and also their value of NpNn reaches more
than five hundreds. We present specific information such as the maximum value ofNpNn and
the minimum value of Ex in Table II for the plotted points which belong to each major shell
in Fig. 3. There are two numbers for each major shell in the last column of Table II where
one number is determined from the data and the other number in parenthesis is calculated
by the empirical formula. We can find that those two numbers agree reasonably well. We
also find in Fig. 3 that the results, calculated by the empirical formula (solid circles), meet
the requirement of the NpNn scheme very well and agree with the data (empty triangles)
satisfactorily for each and every panel.
In order to make more detailed comparison between the measured and calculated excita-
tion energies, we expand the largest two major shells V and VI of Fig. 3 and redraw them
in Fig. 4 for some typical nuclei which belong to the rare earth elements. The upper part of
Fig. 4 shows the data and the lower part of the same figure exhibits the corresponding cal-
culated excitation energies. We can confirm that the agreement between them is reasonable
even though the calculated excitation energies somewhat overestimate the data and also the
empirical formula can not separate enough to distinguish the excitation energies of the two
isotopes with the same value of the product NpNn for some nuclei.
According to the empirical formula given by Eq. (1), the excitation energy Ex is deter-
mined by two components: one is the first term αA−γ which depends only on the mass
number A and the other is the second term β[exp(−λNp)+exp(−λNn)] which depends only
on the valence nucleon numbers, Np and Nn. Let us first draw the NpNn-plot of Ex(2
1 ) by
using only the first term αA−γ. The results are shown in Fig. 5(a) where we can find that
the plotted points fill the lower left corner of the Ex-NpNn plane leaving almost no empty
spots. These results simply reflect the fact that a large number of nuclei with different mass
numbers, values of A, can have the same value of NpNn. Now we draw the same NpNn-plot
by using both of the two terms in Eq. (1). We display the plot of the calculated excitation
energies in Fig. 5(b) which is just the same sort of graph of the measured excitation energies
shown in Fig. 1(b) except that the type of scale for Ex is changed from linear to log. By
comparing Fig. 5 (a) and (b), we find that the second term of Eq. (1), which depends on the
valence nucleon numbers, Np and Nn, pushes the plotted points up in the direction of higher
excitation energies and arranges them to comply with the NpNn scheme.
It is worthwhile to note the difference between the A-plot and the NpNn-plot. The graph
drawn by using only the first term of Eq. (1) becomes a single curve in the A-plot as shown
in Fig. 1(a) with the dashed curve. It becomes scattered plotted points in the NpNn-plot as
can be seen from Fig. 5(a). Now, by adding the second term of Eq. (1) in the A-plot, the
plotted points are dispersed as shown in the top graph of Fig. 2(b) which corresponds to the
measured data points in Fig. 1(a); while by adding the same second term in NpNn-plot, we
find a very neat rearrangement of the plotted points as shown in Fig. 5(b). Thus, the same
second term plays the role of spreading plotted points in the A-plot while it plays the role
of collecting them in the NpNn-plot.
However, this mechanism of the second term alone is not sufficient to explain why the
empirical formula given by Eq. (1) which obviously does not depend on NpNn at all, can
0 100 200 300 400 500
0 100 200 300 400 500
(a) Only the first term (b) Both terms
FIG. 5: The NpNn-plot of the calculated first excitation energy Ex of 2
+ states. The excitation
energies Ex are calculated by (a) using only the first term and (b) using both terms of Eq. (1).
show the characteristic feature of the NpNn scheme. In order to shed light on this question,
we calculate the excitation energy Ex(2
1 ) by the following three different conditions on the
exponents, Np and Nn, of the second term in Eq. (1). First, let Np and Nn have any even
numbers as long as they satisfy Np +Nn ≤ A. The resulting excitation energy Ex is plotted
against NpNn in Fig. 6(a). Next, let Np and Nn have any numbers that are allowed for the
valence nucleon numbers. For example, suppose the three numbers of a plotted point are
A = 90, Np = 40, and Nn = 50 in the previous case. For the fourth major shell IV in Table II,
the valence proton number for the nucleus with the atomic number Z = 40 is 10 and the
valence neutron number for the nucleus with the neutron number N = 50 is 0. Therefore,
we assign Np = 10 and Nn = 0 instead of 40 and 50, respectively. The excitation energy
Ex, calculated under such a condition, is plotted against NpNn in Fig. 6(b). Last, we take
only those excitation energies which are actually measured among the excitation energies
shown in Fig. 6(b). The results are shown in Fig. 6(c), which is, of course, exactly the same
as shown in Fig. 5(b). From Fig. 6(d) where all the three previous plots (a), (b), and (c) are
placed together, we can observe how the NpNn scheme emerges from the empirical formula
given by Eq. (1) even though this equation does not depend on the product NpNn at all.
On one hand, the two exponential terms which depend on Np and Nn separately push the
excitation energy Ex upward as discussed with respect to Fig. 5. On the other hand, the
restriction on the values of the valence nucleon numbers Np and Nn of the actually existing
nuclei determines the upper bound of the excitation energy Ex as discussed regarding Fig. 6.
Finally, we show the NpNn-plots of the first excitation energies for (a) 4
1 , (b) 6
1 , (c)
8+1 , and (d) 10
1 states in Fig. 7. The measured excitation energies are represented by the
empty triangles and the calculated ones from the empirical formula, Eq. (1), are denoted by
solid circles. These graphs are just the NpNn-plot versions of the A-plot shown in Fig. 2
with exactly the same set of plotted points. We can learn from Fig. 7 that the same kind
of NpNn scheme observed in the excitation energy of 2
1 states is also functioning in the
excitation energies of other natural parity even multipole states. We can also find from
Fig. 7 that the calculated results, using the empirical formula, agree with the measured data
quite well. Moreover, it is interesting to find from Fig. 7 that the width in the central part
of the NpNn-plot is enlarged as the multipole of the state is increased. The origin of this
enlargement in the empirical formula can be traced to the parameter α of the first term in
Eq. (1). The value of α is monotonously increased from 34.9MeV for Ex(2
1 ) to 2489.0MeV
for Ex(10
1 ) as can be seen in Table I.
0 100 200 300 400 5000 100 200 300 400 500
(a) All Possible Values
(b) All Possible N
's and N
(d) All Three Cases
(c) Only Existing N
's and N
FIG. 6: The NpNn-plot of the first excitation energy of the 2
+ states calculated by the empirical
formula given by Eq. (1) using the following three different conditions on the exponent Np and Nn:
(a) Np and Nn can have any even numbers as long as they satisfy Np +Nn ≤ A. (b) Np and Nn
can have any number that is allowed for the valence nucleon numbers. (c) Np and Nn can have
numbers which are allowed for the actually existing nuclei. (d) All of the previous three cases are
shown together.
0 100 200 300 400 500
0 100 200 300 400 500
(b) E
(c) E
(d) E
) (a) E
 Data 
 Empirical Formula
FIG. 7: The NpNn-plot for the first excitation energies of the natural parity even multipole states
(a) 4+
, (b) 6+
, (c) 8+
, and (d) 10+
using both the measured data (open triangles) and the empirical
formula (solid circles). These graphs are just the NpNn-plot versions of the A-plot shown in Fig. 2
with exactly the same set of data points.
In summary, we have examined how the recently proposed empirical formula, Eq. (1),
for the excitation energy Ex(2
1 ) of the first 2
1 state meets the requirement of the NpNn
scheme even though it does not depend on the product NpNn at all. We have demonstrated
explicitly that the structure of the empirical formula itself together with the restriction on
the values of the valence nucleon numbers Np and Nn of the actually existing nuclei make
the characteristic feature of the NpNn scheme appear. Furthermore, our result shows that
the composition of the empirical formula, Eq. (1), is in fact ideal for revealing the NpNn
scheme. Therefore it is better to regard the NpNn scheme as a strong signature suggesting
that this empirical formula is indeed the right one. As a matter of fact, this study about
the NpNn scheme has incidentally exposed the significance of the empirical formula given by
Eq. (1) as a universal expression for the lowest collective excitation energy. A more detailed
account of the empirical formula for the first excitation energy of the natural parity even
multipole states in even-even nuclei will be published elsewhere [19]. However, it has been
well established that the NpNn scheme holds not only for the lowest excitation energies
1 ) but also for the transition strength B(E2) [8]. Unfortunately, our empirical study
intended to express only the excitation energies in terms of the valence nucleon numbers.
The extension of our study to include the B(E2) values in our parametrization is in progress.
Acknowledgments
This work was supported by an Inha University research grant.
[1] I. Hamamoto, Nucl. Phys. 73, 225 (1965).
[2] R. F. Casten, Nucl. Phys. A443, 1 (1985).
[3] For a review of the NpNn scheme, see R. F. Casten and N. V. Zamfir, J. Phys. G22, 1521
(1996).
[4] S. Raman, C. W. Nestor, Jr., and P. Tikkanen, At. Data Nucl. Data Tables 78, 1 (2001).
[5] R. F. Casten, Phys. Rev. Lett. 54, 1991 (1985).
[6] R. F. Casten, D. S. Brenner, and P. E. Haustein, Phys. Rev. Lett. 58, 658 (1987).
[7] R. B. Cakirli and R. F. Casten, Phys. Rev. Lett. 96, 132501 (2006).
[8] R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 70, 402 (1993).
[9] Y. M. Zhao, R. F. Casten, and A. Arima, Phys. Rev. Lett. 85, 720 (2000).
[10] A. de-Shalit and M. Goldhaber, Phys. Rev. 92, 1211 (1953).
[11] I. Talmi, Rev. Mod. Phys. 34, 704 (1962).
[12] K. Heyde, P. Vanisacker, R. F. Casten, and J. L. Wood, Phys. Lett. 155B, 303 (1985).
[13] R. F. Casten, K. Heyde, and A. Wolf, Phys. Lett. 208B, 33 (1988).
[14] J.-Y. Zhang, R. F. Casten, and D. S. Brenner, Phys. Lett. 227B, 1 (1989).
[15] P. Federmann and S. Pittel, Phys. Lett. 69B, 385 (1977).
[16] J. Dobaczewski, W. Nazarewicz, J. Skalski, and T. Werner, Phys. Rev. Lett. 60, 2254 (1988).
[17] E. Ha and D. Cha, J. Korean Phys. Soc. 50, 1172 (2007).
[18] E. Ha and D. Cha, Phys. Rev. C 75, 057304 (2007).
[19] D. Kim, E. Ha, and D. Cha, arXiv:0705.4620[nucl-th].
[20] R. B. Firestone, V. S. Shirley, C. M. Baglin, S. Y. Frank Chu, and J. Zipkin, Table of Isotopes
(Wiley, New York, 1999).
http://arxiv.org/abs/0705.4620
	Acknowledgments
	References

  We examine the $N_p N_n$ scheme based on a recently proposed simple empirical
formula which is highly valid for the excitation energy of the first excited
natural parity even multipole states in even-even nuclei. We demonstrate
explicitly that the $N_p N_n$ scheme for the excitation energy emerges from the
separate exponential dependence of the excitation energy on the valence nucleon
numbers $N_p$ and $N_n$ together with the fact that only a limited set of
numbers is allowed for the $N_p$ and $N_n$ of the existing nuclei.

NpNn Scheme Based on New Empirical Formula
for Excitation Energy
Jin-Hee Yoon, Eunja Ha, and Dongwoo Cha∗
Department of Physics, Inha University, Incheon 402-751, Korea
(Dated: July 25, 2007)
Abstract
We examine the NpNn scheme based on a recently proposed simple empirical formula which
is highly valid for the excitation energy of the first excited natural parity even multipole states
in even-even nuclei. We demonstrate explicitly that the NpNn scheme for the excitation energy
emerges from the separate exponential dependence of the excitation energy on the valence nucleon
numbers Np and Nn together with the fact that only a limited set of numbers is allowed for the
Np and Nn of the existing nuclei.
PACS numbers: 21.10.Re, 23.20.Lv
∗Electronic address: dcha@inha.ac.kr; Fax: +82-32-866-2452
http://arxiv.org/abs/0704.1693v2
mailto:dcha@inha.ac.kr
The valence nucleon numbers Np and Nn have been frequently adopted in parameter-
izing various nuclear properties phenomenologically over more than the past four decades.
Hamamoto was the first to point out that the square roots of the ratios of the measured
and the single particle B(E2) values were proportional to the product NpNn [1]. It was
subsequently shown that a very simple pattern emerged whenever the nuclear data concern-
ing the lowest collective states was plotted against NpNn [2]. This phenomenon has been
called the NpNn scheme in the literature [3]. For example, when the measured excitation
energies Ex(2
1 ) of the first excited 2
+ states in even-even nuclei were plotted against the
mass number A (A-plot), we got data points scattered irregularly over the Ex-A plane as
seen in Fig. 1(a). However, we suddenly had a very neat rearrangement of the data points by
just plotting them against the product NpNn (NpNn-plot) as shown in Fig. 1(b). A similar
simplification was observed not only from Ex(2
1 ) but also from the ratio Ex(4
1 )/Ex(2
[5, 6, 7], the transition probability B(E2; 2+1 → 0
+) [8], and the quadrupole deformation
parameter e2 [9].
The chief attraction of the NPNn scheme is twofold. One is the fact that the simplification
in the graph occurs marvelously every time the NpNn plot is drawn. The other attraction
0 100 200 300 400 5000 100 200
4 (b) N
-Plot
(a) A-Plot
Mass Number A
FIG. 1: A typical example demonstrating the NpNn scheme. The excitation energies of the first 2
states in even-even nuclei are plotted (a) against the mass number A and (b) against the product
NpNn. The dashed curve in part (a) represents the bottom contour line which is drawn by the first
term αA−γ of Eq. (1). The excitation energies are quoted from Ref. 4.
0 100 200
0 100 200
Mass Number A
(b) Empirical Formula(a) Data
FIG. 2: Excitation energies of the first excited natural parity even multipole states. Part (a) shows
the measured excitation energies while part (b) shows those calculated by the empirical formula
given by Eq. (1). The measured excitation energies are quoted from the compilation in Raman et
al. for 2+
states [4] and extracted from the Table of Isotopes, 8th-edition by Firestone et al. for
other multipole states [20].
is the universality of the pattern, namely the exactly same sort of graphs appears even at
different mass regions [2]. Since the performance of the NpNn scheme has been so impressive,
many expected that the residual valence proton-neutron (p-n) interaction must have been
the dominant controlling factor in the development of collectivity in nuclei and that the
product NpNn may represent an empirical measure of the integrated valence p-n interaction
strength [3]. Also, the importance of the p-n interaction in determining the structure of
nuclei has long been pointed out by many authors [10, 11, 12, 13, 14, 15, 16].
In the meantime, we have recently proposed a simple empirical formula which describes
the essential trends of the excitation energies Ex(2
1 ) in even-even nuclei throughout the
periodic table [17]. This formula, which depends on the valence nucleon numbers, Np and
Nn, and the mass number A, can be expressed as
Ex = αA
−γ + β [exp(−λNp) + exp(−λNn)] (1)
where the parameters α, β, γ, and λ are fitted from the data. We have also shown that the
source, which governs the 2+1 excitation energy dependence given by Eq. (1) on the valence
nucleon numbers, is the effective particle number participating in the residual interaction
from the Fermi level [18]. Furthermore, the same empirical formula can be applied quite
successfully to the excitation energies of the lowest natural parity even multipole states such
as 4+1 , 6
1 , 8
1 , and 10
1 [19]. It can be confirmed by Fig. 2 where the measured excitation
energies in part (a) are compared with those in part (b) which are calculated by Eq. (1).
The values of the parameters adopted for Fig. 2(b) are listed in Table I.
0 100 200 300 400 500
0 100 200 300 400 500
(I) Z=2~8
 Data
 Empirical Formula
(II) Z=10~20
(III) Z=22~28
(IV) Z=30~50
(V) Z=52~82
(VI) Z=84~126
FIG. 3: The NpNn-plot for the excitation energies of the first 2
+ states using both the data (open
triangles) and the empirical formula (solid circles). The plot is divided into six panels each of which
contains plotted points that come from each one of the proton major shells.
TABLE I: Values adopted for the four parameters in Eq. (1) for the excitation energies of the
following multipole states: 2+
, and 10+
Multipole α(MeV) β(MeV) γ λ
34.9 1.00 1.19 0.36
94.9 1.49 1.15 0.30
441.4 1.51 1.31 0.25
1511.5 1.41 1.46 0.19
2489.0 1.50 1.49 0.17
In this study, we want to further elucidate about our examination of the NpNn scheme
based on the empirical formula, Eq. (1), for Ex(2
1 ). Our goal is to clarify why Ex(2
complies with the NpNn scheme although the empirical formula, which reproduces the data
quite well, does not depend explicitly on the product NpNn.
First, we check how well the empirical formula does meet the requirements of the NpNn
scheme. In Fig. 3, we display the NpNn-plot for the excitation energies of the first 2
states using both the data (empty triangles) and the empirical formula (solid circles). We
show them with six panels. Each panel contains plotted points from nuclei which make up
the following six different proton major shells: (I) 2 ≤ Z ≤ 8, (II) 10 ≤ Z ≤ 20, (III)
TABLE II: The maximum value of NpNn and the minimum value of Ex for each major shell in
Fig. 3 are indicated here. The numbers in the parenthesis represent Ex calculated by the empirical
formula given by Eq. (1).
Major Shell Z Max. NpNn Min. Ex (MeV)
I 2 ∼ 8 8 1.59 (1.85)
II 10 ∼ 20 36 0.67 (0.82)
III 22 ∼ 28 16 0.75 (0.77)
IV 30 ∼ 50 140 0.13 (0.18)
V 52 ∼ 82 308 0.07 (0.08)
VI 84 ∼ 126 540 0.04 (0.05)
0 50 100 150 200 250
0 50 100 150 200 250
Empirical Formula
FIG. 4: Extract from Fig. 3 for some typical nuclei which belong to the rare earth elements.
Different symbols are used to denote excitation energies of individual nuclei.
22 ≤ Z ≤ 28, (IV) 30 ≤ Z ≤ 50, (V) 52 ≤ Z ≤ 82, and (VI) 84 ≤ Z ≤ 126. From this
figure, we can see an intrinsic feature of the NpNn-plot, namely, the plotted points have their
own typical location in the Ex-NpNn plane according to which major shell they belong. For
example, the plotted points of the first three major shells I, II, and III occupy the far left side
part of the Ex-NpNn plane in Fig. 3 since their value of the product NpNn does not exceed
several tens. On the contrary, the plotted points of the last major shell VI extend to the far
right part of the Ex-NpNn plane along the lowest portion in Fig. 3. This is true since their
value of the excitation energy Ex is very small and also their value of NpNn reaches more
than five hundreds. We present specific information such as the maximum value ofNpNn and
the minimum value of Ex in Table II for the plotted points which belong to each major shell
in Fig. 3. There are two numbers for each major shell in the last column of Table II where
one number is determined from the data and the other number in parenthesis is calculated
by the empirical formula. We can find that those two numbers agree reasonably well. We
also find in Fig. 3 that the results, calculated by the empirical formula (solid circles), meet
the requirement of the NpNn scheme very well and agree with the data (empty triangles)
satisfactorily for each and every panel.
In order to make more detailed comparison between the measured and calculated excita-
tion energies, we expand the largest two major shells V and VI of Fig. 3 and redraw them
in Fig. 4 for some typical nuclei which belong to the rare earth elements. The upper part of
Fig. 4 shows the data and the lower part of the same figure exhibits the corresponding cal-
culated excitation energies. We can confirm that the agreement between them is reasonable
even though the calculated excitation energies somewhat overestimate the data and also the
empirical formula can not separate enough to distinguish the excitation energies of the two
isotopes with the same value of the product NpNn for some nuclei.
According to the empirical formula given by Eq. (1), the excitation energy Ex is deter-
mined by two components: one is the first term αA−γ which depends only on the mass
number A and the other is the second term β[exp(−λNp)+exp(−λNn)] which depends only
on the valence nucleon numbers, Np and Nn. Let us first draw the NpNn-plot of Ex(2
1 ) by
using only the first term αA−γ. The results are shown in Fig. 5(a) where we can find that
the plotted points fill the lower left corner of the Ex-NpNn plane leaving almost no empty
spots. These results simply reflect the fact that a large number of nuclei with different mass
numbers, values of A, can have the same value of NpNn. Now we draw the same NpNn-plot
by using both of the two terms in Eq. (1). We display the plot of the calculated excitation
energies in Fig. 5(b) which is just the same sort of graph of the measured excitation energies
shown in Fig. 1(b) except that the type of scale for Ex is changed from linear to log. By
comparing Fig. 5 (a) and (b), we find that the second term of Eq. (1), which depends on the
valence nucleon numbers, Np and Nn, pushes the plotted points up in the direction of higher
excitation energies and arranges them to comply with the NpNn scheme.
It is worthwhile to note the difference between the A-plot and the NpNn-plot. The graph
drawn by using only the first term of Eq. (1) becomes a single curve in the A-plot as shown
in Fig. 1(a) with the dashed curve. It becomes scattered plotted points in the NpNn-plot as
can be seen from Fig. 5(a). Now, by adding the second term of Eq. (1) in the A-plot, the
plotted points are dispersed as shown in the top graph of Fig. 2(b) which corresponds to the
measured data points in Fig. 1(a); while by adding the same second term in NpNn-plot, we
find a very neat rearrangement of the plotted points as shown in Fig. 5(b). Thus, the same
second term plays the role of spreading plotted points in the A-plot while it plays the role
of collecting them in the NpNn-plot.
However, this mechanism of the second term alone is not sufficient to explain why the
empirical formula given by Eq. (1) which obviously does not depend on NpNn at all, can
0 100 200 300 400 500
0 100 200 300 400 500
(a) Only the first term (b) Both terms
FIG. 5: The NpNn-plot of the calculated first excitation energy Ex of 2
+ states. The excitation
energies Ex are calculated by (a) using only the first term and (b) using both terms of Eq. (1).
show the characteristic feature of the NpNn scheme. In order to shed light on this question,
we calculate the excitation energy Ex(2
1 ) by the following three different conditions on the
exponents, Np and Nn, of the second term in Eq. (1). First, let Np and Nn have any even
numbers as long as they satisfy Np +Nn ≤ A. The resulting excitation energy Ex is plotted
against NpNn in Fig. 6(a). Next, let Np and Nn have any numbers that are allowed for the
valence nucleon numbers. For example, suppose the three numbers of a plotted point are
A = 90, Np = 40, and Nn = 50 in the previous case. For the fourth major shell IV in Table II,
the valence proton number for the nucleus with the atomic number Z = 40 is 10 and the
valence neutron number for the nucleus with the neutron number N = 50 is 0. Therefore,
we assign Np = 10 and Nn = 0 instead of 40 and 50, respectively. The excitation energy
Ex, calculated under such a condition, is plotted against NpNn in Fig. 6(b). Last, we take
only those excitation energies which are actually measured among the excitation energies
shown in Fig. 6(b). The results are shown in Fig. 6(c), which is, of course, exactly the same
as shown in Fig. 5(b). From Fig. 6(d) where all the three previous plots (a), (b), and (c) are
placed together, we can observe how the NpNn scheme emerges from the empirical formula
given by Eq. (1) even though this equation does not depend on the product NpNn at all.
On one hand, the two exponential terms which depend on Np and Nn separately push the
excitation energy Ex upward as discussed with respect to Fig. 5. On the other hand, the
restriction on the values of the valence nucleon numbers Np and Nn of the actually existing
nuclei determines the upper bound of the excitation energy Ex as discussed regarding Fig. 6.
Finally, we show the NpNn-plots of the first excitation energies for (a) 4
1 , (b) 6
1 , (c)
8+1 , and (d) 10
1 states in Fig. 7. The measured excitation energies are represented by the
empty triangles and the calculated ones from the empirical formula, Eq. (1), are denoted by
solid circles. These graphs are just the NpNn-plot versions of the A-plot shown in Fig. 2
with exactly the same set of plotted points. We can learn from Fig. 7 that the same kind
of NpNn scheme observed in the excitation energy of 2
1 states is also functioning in the
excitation energies of other natural parity even multipole states. We can also find from
Fig. 7 that the calculated results, using the empirical formula, agree with the measured data
quite well. Moreover, it is interesting to find from Fig. 7 that the width in the central part
of the NpNn-plot is enlarged as the multipole of the state is increased. The origin of this
enlargement in the empirical formula can be traced to the parameter α of the first term in
Eq. (1). The value of α is monotonously increased from 34.9MeV for Ex(2
1 ) to 2489.0MeV
for Ex(10
1 ) as can be seen in Table I.
0 100 200 300 400 5000 100 200 300 400 500
(a) All Possible Values
(b) All Possible N
's and N
(d) All Three Cases
(c) Only Existing N
's and N
FIG. 6: The NpNn-plot of the first excitation energy of the 2
+ states calculated by the empirical
formula given by Eq. (1) using the following three different conditions on the exponent Np and Nn:
(a) Np and Nn can have any even numbers as long as they satisfy Np +Nn ≤ A. (b) Np and Nn
can have any number that is allowed for the valence nucleon numbers. (c) Np and Nn can have
numbers which are allowed for the actually existing nuclei. (d) All of the previous three cases are
shown together.
0 100 200 300 400 500
0 100 200 300 400 500
(b) E
(c) E
(d) E
) (a) E
 Data 
 Empirical Formula
FIG. 7: The NpNn-plot for the first excitation energies of the natural parity even multipole states
(a) 4+
, (b) 6+
, (c) 8+
, and (d) 10+
using both the measured data (open triangles) and the empirical
formula (solid circles). These graphs are just the NpNn-plot versions of the A-plot shown in Fig. 2
with exactly the same set of data points.
In summary, we have examined how the recently proposed empirical formula, Eq. (1),
for the excitation energy Ex(2
1 ) of the first 2
1 state meets the requirement of the NpNn
scheme even though it does not depend on the product NpNn at all. We have demonstrated
explicitly that the structure of the empirical formula itself together with the restriction on
the values of the valence nucleon numbers Np and Nn of the actually existing nuclei make
the characteristic feature of the NpNn scheme appear. Furthermore, our result shows that
the composition of the empirical formula, Eq. (1), is in fact ideal for revealing the NpNn
scheme. Therefore it is better to regard the NpNn scheme as a strong signature suggesting
that this empirical formula is indeed the right one. As a matter of fact, this study about
the NpNn scheme has incidentally exposed the significance of the empirical formula given by
Eq. (1) as a universal expression for the lowest collective excitation energy. A more detailed
account of the empirical formula for the first excitation energy of the natural parity even
multipole states in even-even nuclei will be published elsewhere [19]. However, it has been
well established that the NpNn scheme holds not only for the lowest excitation energies
1 ) but also for the transition strength B(E2) [8]. Unfortunately, our empirical study
intended to express only the excitation energies in terms of the valence nucleon numbers.
The extension of our study to include the B(E2) values in our parametrization is in progress.
Acknowledgments
This work was supported by an Inha University research grant.
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http://arxiv.org/abs/0705.4620
	Acknowledgments
	References
