Journal of Modern Physics, 2013, 4, 66-77
doi:10.4236/jmp.2013.45B012 Published Online May 2013 (http://www.scirp.org/journal/jmp)
Ground State of 4-7H Considering Internal Collective
Rotation
S. Paschalis1, G. S. Anagnostatos2*
1Lawrence Berkeley National Laboratory, Berkeley, California, 94720 USA
2Institute of Nuclear Physics, National Center for Scientific Research “Demokritos”, Aghia. Paraskevi, 15310 Greece
Received 2013
ABSTRACT
The g.s. of heavy and superheavy hydrogen isotopes, namely 4-7H, are successfully examined by applying the Isomor-
phic Shell Model. Properties examined are binding energies and effective radii. The novelty of the present work is that,
due to the small number of nucleons involved and the subsequently large deformation, an internal collective rotation
appears which is inseparable from the usual internal motion even in the ground states of these nuclei, i.e., for such nu-
clei the adiabatic approximation is not valid. This extra degree of freedom leads to a reduction of binding energies, an
increase of effective radii, and an increase of level widths.
Keywords: Hydrogen Isotopes 4H, 5H, 6H, and 7H; Cluster Models; Isomorphic Shell Model; Drip Lines; Internal
Collective Rotation
1. Introduction
Usage of secondary beams of short-lived radioactive nu-
clei has enabled studies of nuclei near or beyond the lim-
its of nuclear stability. During these studies very inter-
esting new phenomena have been observed. In particular,
the experimental study of heavy and superheavy hydro-
gen isotopes is very interesting for several reasons:
 They are the closest to pure neutron nuclei and
thus their study can provide information on neutron mat-
ter.
 They are the simplest nuclear systems and their
treatment could be relatively simple.
 They lie at the limit of nuclear stability (drip
lines) and thus they may differ from ordinary nuclei.
 Even though they were the subject of many
studies for over 40 years, their study is still incomplete.
 They exhibit an extreme fraction of neutron to
proton ratio.
 They are the only nuclei with the 1s proton shell
incomplete.
 They may provide information for testing and
developing nuclear models.
 They may possess unusual structures, e.g., being
very extended in space.
 They may provide knowledge for the physical
meaning of a possible nuclear halo.
 They may provide a more precise definition of
the limits of nuclear stability.
Among the interesting references concerning hydrogen
isotopes are [1-16]. Nevertheless, one has to admit that
the experimental information and its interpretation
available today are still contradictory and extremely lim-
ited.
Up to now, for the study of heavy and superheavy hy-
drogen isotopes, several theoretical efforts were em-
ployed, using different approaches. The present theoreti-
cal treatment employs the same model, the Isomorphic
Shell Model briefly summarized below in section 2, for
all isotopes examined. This model employs the most
probable forms and the average sizes of all nuclear shells
which are attributed to two further utilized properties of
nucleons, namely, their fermionic nature and their aver-
age sizes. The first property alone is responsible for the
most probable forms of nuclear shells and the second for
their average sizes. Indeed:
First, the anti-symmetric requirement of the wave
function for nucleons results in a distribution for the
maxima of this wave function which is equivalent to that
obtained for the equilibrium of repulsive particles [17].
The repulsive property of nucleons is derived not only by
this antisymmetrization, but also by the repulsive char-
acter of the nuclear force itself being considered as
originating due to the quark structure of the nucleons.
The above equivalence replaces the nuclear many-body
problem with that of finding the distribution of locations
with maximum probability of repulsive particles on spheres
*Corresponding author.

S. PASCHALIS, G. S. ANAGNOSTATOS 67
like the spherical shapes of nuclear shells.
In 1957 J. Leech [18] concluded that this problem has
a solution only for certain numbers of particles. That is,
for different numbers of particles there is no equilibrium
of repulsive particles on a sphere. Such equilibria in three
dimensions are at the vertices or middle of faces or mid-
dle of edges or combinations of these points of regular
polyhedra or their derivative polyhedra [18]. For large
numbers of particles such equilibria are arranged on
concentric spheres or equivalently on concentric poly-
hedra standing for the most probable forms of nuclear
shells [18] which all are in equilibrium by themselves.
The cumulative number of vertices of equilibrium
polyhedra – taken in specific sequence, as we will see
below – precisely reproduce the nuclear magic numbers,
with no use of the strong spin-orbit coupling. Thus, such
polyhedra can be taken as the average forms of nuclear
shells. It is essential to emphasize here that the structures
of these polyhedra accurately possess the quantization of
orbital angular momentum [19-21]. Specifically, charac-
teristic points of these polyhedra, e.g., vertices or center
of faces, precisely form the angles cos-1m/ (1)

 for
all and m with respect to a common quantization axis
for all equilibrium polyhedra employed. This property of
the above equilibrium polyhedra, in the framework of the
Isomorphic Shell Model, permits the assignment of
quantum states to their vertices standing as average posi-
tions of nucleons [19-21].

Secondly, the fact that nucleons have average finite
sizes (bags), i.e., they are not point particles, leads to the
average sizes of the polyhedra employed to present nu-
clear shells, if a) the bags of nucleons are considered at
the vertices of the aforementioned concentric polyhedral
shells and b) these polyhedra acquire their minimum
sizes, i.e., the nucleon bags of an average polyhedral
shell are in contact with the bags of a previous average
polyhedral shell.
The present approach, where the average structure is
derived without reference to the inter-particle forces, is
applicable not only in nuclear physics but also in any
other branch of physics where fermions of definite aver-
age size are the constituent particles, e.g., in cluster
physics where the constituent particles are atoms with
half-integer spins and thus could be considered as atomic
fermions [22]. Thus, the present paper can guide re-
search in other fields, as well as in nuclear physics.
The model employed here has a minimum number of
parameters and has been successfully applied to many
properties of other nuclei [23-25] spread over the peri-
odic table of the elements. No additional ad hoc assump-
tions are here employed. For the nuclei examined here,
namely 4-7H, however, a new degree of freedom appears
due to their small number of nucleons and the resulting
very large deformation [26,27] which lead to an adiabatic
approximation non-validity.
2. Isomorphic Shell Model
The Isomorphic Shell Model is a microscopic nuclear
structure model that incorporates into a hybrid model the
prominent features of single-particle and collective ap-
proaches in conjunction with the nucleon finite size [28
and references therein]. The model consists of two parts,
namely, the complete quantum mechanical part and the
semiclassical part.
2.1. Semiclassical Part of the Model
Here, we present the semiclassical part of the model,
which has been used many times [23-25] in place of the
quantum mechanical part of the model [28], in the spirit
of the Ehrenfest theorem [29,30] (that for the average
values the laws of Classical Mechanics are valid). Of
course, a semiclassical approach is more easily accepted
for heavier nuclei, but here it is used even for hydrogen
isotopes, where anti-symmetrisation effects play a crucial
role. Indeed, as briefly explained in the introduction [28],
the vertices of the polyhedra of Figure 1 (which is the
space employed by the Isomorphic Shell Model for this
region of nuclei) stand for the distribution of the maxima
of the wave function for nucleons due to the anti-sym-
metric requirement of this function with no limit to the
nuclear size, thus including the hydrogen isotopes.
The Ehrenfest’s theorem for the observables of posi-
tion ( R ) and momentum ( P ) takes the form (see all
details in [30] p.240).
d<R>/dt = (1/m)<P> (1)
and
d<P>/dt = - <V( R )> . (2) 
For simplicity here, the case of a spinless particle in a
scalar stationary potential V(r) is considered.
The quantity <R> represents a set of three time-de-
pendent numbers {<X>, <Y>, <Z>} and the point <R>(t)
is the center of the wave function at the instant t. The set
of those points which correspond to the various values of
t constitutes the trajectory followed by the center of the
wave packet.
From Eqs. (1) and (2) we get
md2<R> /dt2 = - <V(R)> . (3) 
Furthermore, it is known [30] that for special cases of
force, e.g., for the harmonic oscillator potential assumed
by the model, the following relationship is valid:
<

V(R)> = [V(r)]r = <R> , (4) 
where
-[

V(r)]r = <R> = F . (5)
That is, for this potential the average of the force over

S. PASCHALIS, G. S. ANAGNOSTATOS
68
the whole wave function is rigorously equal to the clas-
sical force F at the point where the center of the wave
function is considered. Thus, for the special case of po-
tential considered here, the motion of the center of the
wave function precisely obeys the laws of classical me-
chanics [30]. Any difference between the quantum and
the classical description of the nucleon motion exclu-
sively depends on the degree the wave function may be
approximated by its center. Any such difference would
contribute to deviations between the experimental data
and the predictions of the semiclassical part of the model
employed.
Thus, in the present semiclassical treatment the nu-
clear problem is reduced to that of studying the centers of
the wave functions of the constituent nucleons or, in
other words, of studying the average positions of these
nucleons [26]. This is true without any restriction con-
cerning the number of the constituent particles, i.e., if the
nucleus is very light as here (hydrogen isotopes) or very
heavy.
We further proceed with the help of Figure 1 which is
identical to that figure employed in [23-25], where the
most probable forms and average sizes of the first three
proton and the first three neutron shells are presented. It
is essential to mention that these average sizes so lely
depend on the average size of a proton, rp = 0.860 fm,
and that of a neutron, rn = 0.974 fm. Each occupied ver-
tex configuration of this figure corresponds to a quantum
state configuration with definite angular momentum and
energy. More details of the figure are given in its caption.
The expressions of the two-body (two Yukawa) poten-
tial V employed [31] for the present semi-classical
treatment, of the kinetic energy T [32], of the spin-orbit
energy VLS [33], of the intrinsic energy Eintr , of the rota-
tional energy ER , and of the total binding energy EB are
given in Eqs. (6) - (8) and (10) - (12), respectively. Iso-
spin term in Eq.(10) is not needed since the isospin is
here taken care of by the different shell structure (forms
and sizes) between proton and neutron shells, as apparent
from Figure 1.
Vij= 0.993*1017*e-31.2334rij/rij-241.193* e-1.4546rij/rij (6)
<T>nℓm = (ћ2/2M)[1/R 2max + ℓ (ℓ + 1)/ρ2nℓm] (7)
ΣiVLiSi=λ Σi*( ћωi )2 /(h2/m)*ℓi si
(λ = 0.03, the third parameter of the model (8)
Figure 1. The space of the Isomorphic Shell Model for nuclei up to N = 20 and Z = 20. The equilibrium polyhedra in row 1 (2)
stand for the most probable forms and average sizes of the first three neutron (proton). shells. The vertices of polyhedra
(numbered as shown) stand for average positions of nucleons in definite quantum states (τ , n, ℓ, m, s). Central axes standing
for the quantization of directions of the orbital angular momentum are labelled as nθℓ
m and pass through the points marked
by small solid circles. At the bottom-left of each block the numbering of this polyhedron proceeded by the letter Z (N) for
protons (neutrons) is given. Over this the number of polyhedral vertices and the number of possible unoccupied vertices
(holes, h) are also given. At the bottom-right of each block the radius of polyhedron is listed. Over this the cumulative num-
ber of vertices of all previous polyhedra and of this polyhedron is also given. Finally, at the bottom-center of each block the
distance ρnℓm of the nucleon average position nℓm from the relevant axis nθℓ
m
is given.

S. PASCHALIS, G. S. ANAGNOSTATOS 69
ћωi = (ћ2/M)(n+3/2)/<ri2> (9)
Eintr.= ΣijVij - Σ<T>n m - ΣiVLiSi (10)

ER = (ћ2/2M)I(I + 1) /2Θ (11)
EB = Eintr - ER , (12)
where
• Vij is the potential energy between a pair of nucleons
i, j at a distance rij,
• n, ℓ, m are the quantum numbers characterizing a
polyhedral vertex standing for the average position of a
nucleon at the quantum state n, ℓ, m.
• ℓi and s
i stand for the orbital angular momentum
quantum number ℓ and the intrinsic spin quantum num-
ber s of any nucleon i.
• M is the mass of a proton Mp or of a neutron Mn,
• R
max is the outermost proton or neutron polyhedral
radius (R) plus the relevant average nucleon radius rp for
a proton and rn for a neutron, (i.e., Rmax is the radius of
the nuclear volume in which protons or neutrons are con-
fined),
• ρnℓm is the distance of a nucleon average position at a
quantum state (n, ℓ, m from its orbital angular momen-
tum at the direction nθ
m
,
)

• I is the angular momentum of rotation (Here, of in-
ternal collective rotation), and
• Θ is the moment of inertia of collectively rotating
nucleons.
When only binding energies (and not scattering prop-
erties) are required as here, just the second term of the
above two-body potential of Eq.(6) would be sufficient.
Thus, for non-scattering properties, the parameters of the
model are the following five: the two-size parameters rp
and rn, the two parameters from the second term of Eq.(6),
and the one parameter, λ, from Eq.(8). With the help of
these parameters all quantities Rmax, ρnℓm , ћωi, and Θ in
Eqs.(6) - (12) are obtainable by employing the coordi-
nates of the nucleon average positions given in the cap-
tion of Figure 2. All these will become apparent in sec-
tion 3 dealing with the applications on the different hy-
drogen isotopes.
Interesting applications of this version of the model on
nuclear structure and reactions are included in Refs. [23-
25] and [34-35], respectively.
In general, in order for the present results to have
credibility concerning hydrogen isotopes and to show
that these results do not depend on an ad hoc potential or
on a set of adjustable parameters, we provide, beyond the
applications on other nuclei, a rather detailed discussion
concerning the potential and the parameters used in the
model.
2.2. Two-Body Potential of the Model
The potential of Eq. (6) is slightly stronger than that de-
rived in [31]. This potential consists of one repulsive and
another attractive Yukawa-type component, where each
component involves two parameters describing its
strength and range. The parameters of this potential were
determined by examining its scattering and binding en-
ergy properties. First, the potential was applied to the
classical determination (using Newton’s equation for the
two-body problem) of the first two moments [σ(1) (E) and
σ(2)(E)] of the scattering of free nucleons at high energies.
Second, the potential was applied to the semiclassical
estimation of total binding energies of a set of even-even
nuclei with N = Z in order for its saturation property to
be checked for the correct nuclear size and, hence, nor-
mal density. If just binding energy properties are required,
as here, only the second term of Eq. (6) is sufficient.
It is very interesting to comment on the fact that the
potential employed here derived strictly from nuclear
physics is very similar to that of N. Isgur [presented as an
invited talk at the International Nuclear Physics confer-
ence at Harrogate UK, V2 (1986) p.345] strictly derived
from particle physics. Specifically, the central effective
nucleon-nucleon potentials in the 3S1 and 1S0 channels
arising from residual color forces (comprising the latest
efforts to derive nuclear physics from the Quark Model
with chromodynamics) are very similar to each other and
to the potential of Eq. (6).
Figure 2. Most probable forms and average sizes for the ground states of 4-7H. Numbering of spheres follows that of Figure 1.
Neutrons (protons) are presented by light (dark) spheres. The central axes of internal collective rotations are shown as Ri.c.r.
together with their coordinates. All distances dij between nucleons and ρij from the axes Ri.c.r. are determined from the coor-
dinates x, y, z, i.e., (1): 0.974 cos450 , 0.974 cos450 , 0.000, (2): - 0.974 cos450 ,- 0.974 cos450 , 0.000, (5): 0.000, 2.511, 0.000, (6):
0.000, - 2.511, 0.000, (7): 2.511, 0.000, 0.000, (8): - 2.511, 0.000, 0.000., (3΄): 0.000, 0.000, 1.554. The quantum numbers τ, n, ℓ,
m, s employed for the nucleon average positions in this figure are determined from Fig.1 via the angles nθℓ
m
and are (1): ½, 1,
0, 0, ½, (2): ½, 1, 0, 0, -1/2, (3) or (3’): -1/2, 1, 0, 0, ½, (5): ½, 1, 1, 1, ½, (6): ½, 1, 1, -1, -1/2, (7): ½, 1, 1, 1, -1/2, (8): ½, 1, 1, -1, ½,
(9): ½, 1, 1, 0, ½, (10): ½, 1, 1, 0, -1/2.
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S. PASCHALIS, G. S. ANAGNOSTATOS
70
The potential of Eq. (6) has been derived by employ-
ing the charge independent assumption of nuclear forces,
where nn ~ pp ~ np (all three forces are approximately
the same). Nevertheless, there is certain experimental
evidence supporting the notion that instead of the as-
sumption of charge independent, the charge symmetry of
nuclear forces is a better assumption, where np > nn ~pp.
At this point it is interesting for one to observe from
Figure 1 that the average structures of a neutron and of
the corresponding proton shell in the model are presented
by reciprocal polyhedra, i.e., the average positions of
protons are at the directions through the centers of faces
of the corresponding neutron polyhedron possessing the
same rotational symmetry. This fact makes the np dis-
tances systematically smaller than the nn (or the pp) dis-
tances of this pair of polyhedra. This situation, even us-
ing the same r-dependent potential as in Eq.(6), leads to a
much stronger average np interaction.
Finally, for the potential of Eq. (6), it should be made
clear that this potential is a good approximation for
binding energy properties as in the present application.
For these properties for finite size nucleons only the tail
of the potential is used. That is, for the size rp = 0.860 fm
and rn = 0.974 fm employed here only the tail of the po-
tential after 2rp = 1.720 fm (minimum possible distance
of two proton bags in contact) is used in determining
binding energies.
2.3. Parameters of the Model
All five parameters employed by the model are numeri-
cal (universal), i.e., they are not adjustable and thus they
maintain the same values for all properties in all nuclei.
Specifically:
The two size parameters of the proton and of the neu-
tron average sizes, i.e., rp = 0.860 fm and rn = 0.974 fm
(both consistent with our knowledge from QCD that the
average nucleon size is about 1 fm and from particle
physics where also their relative size is supported) are
sufficient, due to symmetry employed by the model, for a
complete determination of all linear distances (rij) needed
in Eq.(6) and all quantities R, ρ, and ћω involved in Eqs.
(7-9). These parameters have possessed the same physi-
cal meaning and the same numerical values since 1982
[31], dealing with properties of many nuclei throughout
the periodic table. For example, see the extreme cases in
[43] for neutron nuclei and in [28] for super heavy nu-
clei.
The two potential parameters VA= 241.193 MeV and
RA= 1.4546 fm {see Eq.(6)} are almost the same since
1982 [compare 31, 42] when we first estimated the
two-body (two-Yukawa) potential.
Only the spin-orbit parameter λ = 0.03 {see Eq. (8)} is
employed later, but it has still kept a constant numerical
value since 2003 [42].
The coordinates of nucleon average positions given in
the caption on Fig.2 have been determined and published
in 1982 [31] and are identically employed in all publica-
tions thereafter {e.g., [23-26, 31-32, 34-35, 39, 41-43].
2.4. The Model for Very Light Nuclei
In dealing with very light nuclei, one must recall the sec-
tion of the quantum mechanical treatment of the model
referring to these nuclei [26]. The relevant Hamiltonian
includes rotation [36-38] as described in Eq.(13)
H = H0(r΄) + Hrot + H΄. (13)
The three terms on the right-hand side of this equation
describe the motion of the internal degrees of freedom
(of shell model type), the rotation of the nucleus, and the
coupling between the rotation and the internal motion,
respectively.
If the total angular momentum I is written as the sum
in Eq.(14)
I = R + J, (14)
where R is the angular momentum of the rotation and J
is the angular momentum associated with the internal
degrees of freedom, the rotational term in Eq.(13) takes
the form of Eq.(15)
Hrot = (ћ2/2Θ) R2 = (ћ2/2Θ) J2
+ (ћ2/2Θ) I2 - (ћ2/2Θ)2IJ, (15)
and the Hamiltonian of Εq.(13) reaches the form of
Eq.(16)
H = H0(r΄) + (ћ2/2Θ) J2 + (ћ2/2Θ) I2
– (ћ2/ Θ) IJ2 + H΄. (16)
Considering that H΄ = 0, the last three terms in Eq.(16)
become zero, e.g., for the ground state of even-even nu-
clei where I = 0. Then, the Hamiltonian of Eq.(16) is
simplified [26] to the Hamiltonian of Eq.(17).
H = H0(r΄) + (ћ2/2Θ) J2, (17)
where both terms on the right-hand side of this equation
refer to the internal motion of the nucleons.
Now, the internal wave function (up to a normalization
factor) is given by Eq.(18)
Ψ∞ (r΄)
, (18)
xK

0
and can be assumed to be an eigenfunction of the Hamil-
tonian of Eq.(17). In Eq.(18) K is the projection of the
total angular momentum on the axis (z΄) and τ stands for
the rest of the quantum numbers.
At this point one may argue that the large deformation
of light nuclei (due to their small number of nucleons)
does not favour perfect pairing, a fact which could lead
to an internal angular momentum J ≠ 0. In addition, the
internal collective rotation R cannot be separated from
the usual internal motion since the adiabatic approxima-
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S. PASCHALIS, G. S. ANAGNOSTATOS 71
tion is not valid for very light nuclei, where ωintr. ≈ ωrot
[39]. That is, the existence of J ≠ 0 in Eq.(17) implies
that a sort of non-adiabaticity is included in this Hamil-
tonian. This is in contrast to what happens to nuclei of
the well-deformed region where ωintr. ≈ 100 ωrot and thus
the total wave function can be written as a product of the
internal wave function and the rotational wave function.
In different wording, even in the ground state of an
even-even nucleus where I = 0, there is an additional
term in the Hamiltonian [see Eq.(17)]. That is, if the
spins (s) or the individual total angular momenta (j) of
certain or all nucleons do not pair perfectly but lead to an
internal total angular momentum J, then an internal rota-
tion R is needed to compensate for J, which leads to a
total angular momentum I = 0, e.g., for an even-even
nucleus. This situation reduces the nuclear binding en-
ergy [by an amount equal to the internal collective rota-
tional energy, as shown in Eq.(12)] and increases [26] the
radius according to Eq.(19) and (20), respectively:
Erot = (ћ2/2m) 2(2 + 1)/2Θrot , (19)
<r2>eff. = <r2>intr. + <r2>rot , (20)
where:
Θrot =. Σ ρi2 + 0.165Arot., (21)
Arot
i1
Arot. is the number of rotating nucleons,
. ρi is the distance of a rotating nucleon i from the
relevant axis of rotation, and
. 0.165 fm2 is the contribution of the nucleon finite
size to the moment of inertia [23-25].
Thus, our effective radii are derived through Eq.(20)
which has two terms. The first term called intrinsic
comes, as usual, from the square integrable of the wave
function which here is originated from the first term of
Eq.(17), i.e., from the term H0(r΄). The second term of
Eq.(20) comes from the second term of Eq.(17), i.e.,
from the term (ћ2/2Θ) J2. Thus, our radii via Eq.(20)
come directly from the Hamiltonian of Eq.(17) and not
from the square integrable of the wave function of
Eq.(18). That is, for radii we follow the same reasoning
as in [26].
According to Eq.(20) the effective radius depending
on the measured, experimental cross section increases
due to internal collective rotation, while the real nuclear
radius (average geometrical radius) remains the same.
The mean square radii of charge and mass are given in
Eqs.(22) - (25) below:
ch<r2>intr. = [Σι=1Ζ <ri2> + Z(0.842)2
– N(0.34)2] / Z , (22)
ch<r2>rot. = Σι=1Ζrot. <ρi2>/Z, (23)
m<r2>intr. = [Σι=1Ζ <ri2> + Z(0.8)2
+ Σι=1N <ri2>+ N(0.91)2 ] / A , (24)
m<r2>rot. = [Σι=1Ζrot. <ρi2> + Σι=1Nrot. <ρi2>] / A, (25)
where 0.842 fm and 0.34 fm in Eq.(22) are the absolute
values of the average charge radius of a proton and of a
neutron, and 0.8 fm and 0.91 fm in Eq.(24) are the aver-
age mass radius of a proton and of a neutron, respectively
[40].
All ri in Eqs.(22, 24) are radii from the nuclear center
and are equal to the radii R of the polyhedra given in
Figure 1. They can be derived from the coordinates [31,
32] of the nucleon average positions presented in this
figure or from the coordinates given in the caption of
Figure 2.
All ρi in Eqs.(23, 25) are distances of the rotating nu-
cleon average positions from the relevant axis of internal
collective rotation. They are derived by employing the
same coordinates [31,32] as above and the coordinates of
the relevant axis of internal collective rotation shown in
Figure 2. The numerical values of ρi for each hydrogen
isotope are also specified in the next section.
An interesting application of this part of the model on
4He is included in Ref.[26].
3. Results and Discussion
In this section, we make extensive use of the material in
sections 2.1 and 2.2. We realize, of course, that the heavy
hydrogen isotopes are of unbound nature. They only exist
as broad resonances with a very short lifetime (in general,
their widths are up to several MeV). However, in the
framework of the present model, resonances and excited
states are treated in the same space of Figure 1, as the
ground states of any nucleus. We argue that no matter
what their life time is (extremely short or infinite), it is
enough to know that their quantum description includes s
and p states and thus their average structures are pre-
sented as vertex configurations of Figure 1. Their rele-
vant properties come from their different vertex configu-
rations of Figure 1 when the same equations (see section
2.1) are used. Thus, no matter if they are resonances,
excited or ground states. Their unbound nature and their
large width make no difference in the present treatment.
The knowledge of their quantum states is enough. This is
the advantage of the present model in comparison with
other models. Our above reasoning is further supported
by the very good agreements between model predictions
and relevant experimental data, as will become apparent
from our results shortly.
Here, the observed broadness of resonances is under-
stood as due to the two different, independent compo-
nents of binding energy for each isotope, that of Eq.(10)
and that of Eq.(19). The first component comes (as usual)
from the contribution of all nucleons, while the second
from the contribution only of the rotating nucleons. The
existence of the aforementioned two components of en-
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S. PASCHALIS, G. S. ANAGNOSTATOS
72
ergy is due to invalidity of the adiabatic approximation as
earlier explained. Each of these two components has its
own width around its own center and the observed broad
width is their superimposition.
As stated in the introduction, the space of the model,
i.e., here that of Figure 1, is a necessary consequence of
the anti-symmetrisation requirement of the total wave
function for fermions (nucleons) and, as it should be, this
space accurately possesses the quantization of orbital
angular momentum [19-21]. Hence, Quantum Mechanics
is inherent in the model space. Only the equations used
(see section 2.1.1) are semiclassical. The same space of
the model is employed for its purely quantum mechanical
treatment [28]. This is a basic difference between the
present semiclassical model and any other semiclassical
treatment.
In Figure 2 we provide the average g.s. structures ac-
cording to the Isomorphic Shell Model for 4-7H, one iso-
tope in each block of the figure. As apparent, these aver-
age structures have cluster forms derived from Fig.1 by
requiring maximum binding energy. All average neutron
positions employed in this figure are identical to those of
Figure 1 and are characterized with the same numbers.
Two of the neutrons fill the two average positions in the
neutron 1s shell (N1) numbered 1 and 2, while the va-
lence neutrons 1 for 4H, 2 for 5H, 3 for 6H, and 4 for7H
occupy average positions among the four equivalent av-
erage positions in the neutron 1p shell (N2), numbered
from 5 to 8. The 1s proton average position in Figure 1
numbered 3 is in the direction h – h of Z2 and
pre-assumes that the proton shell Z2 is complete. How-
ever, for the hydrogen isotopes examined here the proton
average position numbered 3΄ is on the z axis as shown in
all parts of Figure 2. This choice maximizes the dis-
tances between the proton average position and the neu-
tron average positions in 1p shell N2 as required by the
antisymmetrization condition of the total wave function,
which leads (as mentioned in the introduction [17]) to
equivalent results with those of repulsive particles. In
both Figures 1 and 2 the x axis passes through the points
7 and 8, while the y axis through the points 5 and 6. The
quantization axis labelled q is defined by the average
neutron positions 1 and 2, bisects the right angle of x, y
axes [19-21], and passes through the middles of the
edges 5, 7 and 11, 12, as shown in Figure 1 (see blocks
N2 and Z2, respectively).
Due to the very small number of particles, as seen
from all parts of Figrue 2, the deformation is very large
for the g.s. of all four isotopes examined. Thus, in these
isotopes there are conditions for the internal angular
momentum J which favour J ≠ 0 (section 2.2 ). Specifi-
cally for each isotope:
4H. Here, we assume that the individual spins of each
of the four nucleons of 4H couple to J = 2+, as internal
angular momentum. This J ≠ 0 is responsible for the
creation of an internal collective rotation R = 2+ to com-
pensate for J, i.e., R = -J. The internal collective rotation
R involves all four nucleons and the relevant axis of ro-
tation is defined as follows.
The valence neutron average position in the relevant
block of Figure 2 is assigned to the position 8, however,
it could be equivalently assigned to the average position
7 (both on x axis; see Figure 1), or even better it could
be assigned in both positions with occupation probability
50% each. Then, the axis z is an axis of symmetry of the
whole nucleus. Now, as usual, the axis of internal collec-
tive rotation should be perpendicular to the axis of sym-
metry and is taken in coincidence with the axis y. The
relevant radii ρi for determining the moment of inertia via
Eq.(21), according to Figure 2 and the rotating axis y,
are: ρ3 = 1.554 fm, ρ1 = ρ2 = (0.974)cos450 ≈ 0.689 fm,
and ρ8 = 2.511 fm. As seen from column 10 of Table1
the total moment of inertia is 10.33fm2 and this value is
analysed in 2.58 fm2 for the proton and 7.75 fm2 for the
three neutrons.
5H: Here, the individual spins of the four neutrons
could couple to an internal angular momentum J = 2+,
while the orbital angular momentum of the two 1p neu-
trons (average positions 7 and 8 or equivalently 5 and 6)
could couple to zero. Since J ≠ 0, an internal collective
rotation R = 2+ is needed to compensate for J. This in-
ternal collective rotation involves all five nucleon aver-
age positions of 5H, i.e., the four responsible for the crea-
tion of J = 2+ and one which follows the rotation (due to
the average structure of the whole nucleus shown in
Figure 2) of the previous four nucleon average positions.
The axis of symmetry for the average structure of 5H
shown in Figure 2 is again the axis z and now the axis of
rotation could either be the y axis if the average positions
of the two 1p neutrons are the 7 and 8, or equivalently,
could be the x axis if the average positions of the two 1p
neutrons are the 5 and 6. This double possibility results
in an in-between axis as axis of rotation, that passing
through the points 1 and 2, which is symbolised by xy
and coincides with the quantization axis q. This axis, as
should be, is perpendicular to the axis of symmetry z.
The relevant radii ρi, for Eq.(21), according to Figure 2
and the rotating axis xy, are: ρ3 = 1.554 fm, ρ1 = ρ2 = 0,
and ρ7 = ρ8 = (2.511)cos450 = 1.776 fm and as seen from
column 10 of Table1 the total moment of inertia is 9.55
fm2 analysed in 2.58 fm2 for the proton and 6.97 fm2 for
the four neutrons.
6H: Here, the two diametrically opposite neutrons in
3/2- state (average positions 7 and 8) could couple to J =
2+ {Figure 1k of Ref.[41]}. The internal collective rota-
tion R = 2+ to compensate for J comes from only the
average nucleon positions 7 and 8, which are responsible
for this rotation. This axis of rotation is specified in
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S. PASCHALIS, G. S. ANAGNOSTATOS
73
Ref.[41] (see Figure 1k) and passes through the origin
and the point with coordinates 1, -1, 1. The relevant radii
ρi for use in Eq.(21), according to Figure 2, are: ρ6 = ρ7 =
2.050 fm and the moment of inertia is 8.74 fm2 for the
five neutrons. The proton does not participate in the in-
ternal collective rotation and thus the rms charge radius
does not increase due to this rotation.
7H. In this nucleus again two diametrically opposite
neutrons (e.g., average positions 7 and 8), as discussed
above, could couple to J = 2+, while the other two 3/2-
neutrons (average positions 5 and 6) could couple to 0+.
The internal collective rotation R and the relevant radii ρi
are identical to those for 6H above. Again the proton does
not participate in the internal collective rotation and thus
does not increase the rms charge radius.
In Table 1 the results obtained here are listed. Spe-
cifically, in the col.1 of the table the four hydrogen iso-
topes are listed, while in col.2 the numbers of the nu-
cleon average positions of Figrue 1 occupied for each
isotope are given by following Figure 2. In col.3 the
total potential energy for each isotope is given by apply-
ing Eq.(6) among all pairs of nucleons. In col.4 the cor-
responding total energy due to spin orbit force is listed
[Eqs.(8) and (9)], while in col.5 the total kinetic energy
for each isotope is listed [Eq.(7)]. In col.6 the intrinsic
energy is listed [Eq.(10)]. In col.12 the energy due to
internal collective rotation is given [Eqs.(19, 21)] for the
rotating nucleons listed in col.7 around the axis specified
in col.8 and thus leading to the moment of inertia listed
in col.9 of the table. In cols.13 and 14 the model binding
energies in MeV [Eq.(12)] (4H/5.65, 5H/5.56, 6H/6.00,
and 7H/7.68) and those of experiments [3, 7, 1, 15]
(4H/~5.66, 5H/~5.48, 6H/5.38-6.18, and 7H/7.49-8.12) are
shown, respectively. The very good agreements between
these two groups of binding energies are apparent.
In Table 1 we have employed binding energies instead
of resonance energies which, of course, can be derived
from the listed binding energies by subtracting the bind-
ing energy of 3H equal to 8.48 MeV. For example for 4H
the g.s. model binding energy from Table1 is 5.65 MeV,
thus the resonance energy is 8.48 – 5.65 = 2.83 MeV
which is in very good agreement with the predictio n
3.05 ± 0.19 of [13].
In cols.10 and 11 of Table 1 the effective, average ra-
dii for charge and mass [Eqs.(22-23) and (24-25)], re-
spectively, are listed. There are no experimental values
for comparison. However, the large contribution to the
relevant radii of the rotational component is apparent,
particularly for the effective charge radii. Notice the
great difference of the charge radii for 4H and 5H (i.e.,
2.26 fm and 2.24 fm, respectively) in comparison to
those of 6H and 7H (i.e., 1.57 fm and 1.54 fm, respec-
tively), where for the first two nuclei the proton partici-
pates in the internal collective rotation, while in the sec-
ond two it does not.
Finally, from Table 1 it is interesting for one to com-
pare the intrinsic energies (col.6) with the rotational en-
ergies (col.12). These two components of total binding
energy (col. 13) for each hydrogen isotope are rather far
apart. This remark supports the broad structure of g.s.
resonances for 4-7H to be of the size of their total binding
energies (cols. 13 and 14). The experimental widths are
for 4H 5.14 ± 1.38 MeV [13], for 5H 5.4 ± 0.5 MeV [9],
and for 6H 5.8 ± 2.0 MeV [16]. Up to now for 7H there
are only indications for its production. Thus, there is not
available an experimental value of width. The aforemen-
tioned values of widths compare very well with our pre-
dicted values 5.56, 5.45, and 5.85 MeV, respectively (col.
13 of Table 1).
In addition, it is considered instructive to give below
some more details concerning the calculations of the total
kinetic energies in column five of Table 1 by applying
Eq.(7) for each nucleon.
The kinetic energy for the 1s proton is calculated ex-
plicitly in Eq.(26) below.
T1p in 1s state =( ћ2/2Mp)[(1-0.04)/(1.554+0.860)2
+ 2(2+1)0.04/(1.5542+0.165)=5.349 MeV.(26)
The first comment on this equation is that according to
Ref.[42] for 3H, in order to obtain the experimental point
proton momentum distribution of this nucleus, one must
assume a mixture of d state to the predominant s state
equal to x = 0.04 [42]. This value of x is kept constant for
all hydrogen isotopes. This is the origin of the factor
(1-0.04) in the first term of Eq.(26) and the physical
meaning of the second term in the same equation with
factor 0.04. The quantity 0.165 fm2, which, as mentioned
earlier, is added to the moment of inertia ρ2 = (1.554)2
fm2 for the proton rotating either around the axis y or the
axis q, stands for the contribution to the moment of iner-
tia of the finite proton size [23-25]. The quantity Rmax in
Eq.(26) for the proton, as discussed in section 2.1.1, is
Table 1. Components of energy (MeV) and effective radii (fm) for 4-7H. Moments of inertia, ΘR in fm2.
Is. Occupied ΣijVij ΣiVLiSi Σ<T> Eintr Rot. Rot. ΘR ch<r2>1/2m<r2>1/2 Erot. ΕB,mod EB,exp. Ref.
aver. pos. pos. Axis
4H 1-3΄, 8 34.56 0.10 17.04 17.62 1-3΄, 8 y 10.33 2.26 2.45 12.04 5.58 ~ 5.66 [3]
5H 1-3΄, 7-8 43.62 0.20 25.32 18.50 1-3΄, 7-8 xy 9.55 2.24 2.47 13.03 5.47 ~5.48 [7]
6H 1-3΄, 6-8 53.42 0.29 33.60 20.11 7-8 1,-1,1 8.74 1.59 2.46 14.24 5.87 5.38-6.18 [1]
7H 1-3΄, 5-8 63.25 0.39 41.88 21.76 7-8 1,-1,1 8.74 1.56 2.49 14.24 7.52 7.49-8.12 [15]

S. PASCHALIS, G. S. ANAGNOSTATOS
74
the radius of the outermost proton polyhedron equal to
1.554 fm from Figure 1 plus the average finite size of a
proton equal to 0.860 fm, i.e., Rmax = 1.554 + 0.860 fm.
For each of the two neutrons in the1s state the Rmax in
Eq.(27) below is the radius of the outermost neutron
polyhedron equal to 2.511 fm from Figure 1 plus the
average finite size of a neutron equal to 0.974 fm. That
is,
T1n in 1s state = (ћ2/2mn)[1/(2.511+0.974)2] MeV (27)
Similarly, for each of the neutrons in the 1p state the
Rmax in Eq.(28) below is again 2.511+0.974 fm since the
outermost neutron polyhedron is the same as for the 1s
neutrons. However, for each of these neutrons, according
to Eq.(7), since ℓ ≠ 0 there is a second term in Eq.(28)
where ℓ = 1 and ρ = 2.511 fm are taken again from Fi-
grue 1 (bottom of the relevant block). That is,
T1n in 1p state = (ћ2/2m)[1/(2.511+0,974)2
+ 1(1+1)/(2.511)2]=8.279 MeV. (28)
Thus, for use in column 5 of Table 1, according to
Eqs.(26-28):
T4H to 7H = T1p in 1s state + T1n in 1s state*2
+ T1n in 1p state*(1 to 4). (29)
The quantity ћωi in Eq.(8) is estimated by employing
Eq.(9) for all cases of nucleons where ℓs ≠ 0, i.e., for the
valence neutrons 5-8 the common ri in Eq.(9) from Fig-
ure 1 is ri=RN2=2.511 fm.
There is a supplementary sheet, available on request,
where all calculations related to the present work are
presented in detail.
4. Conclusions
All four hydrogen isotopes examined here, i.e., 4-7H, have
been treated within the same model, namely, the semi-
classical part of the Isomorphic Shell Model. The two-
body potential, the expressions of the kinetic and spin-
orbit energies come from previously published works
(see [31-33], respectively). No ad hoc assumption is
made here and the totally five parameters involved are
universal, i.e., they have numerical values identical to
those employed in all previous publications. That is, the
two size parameters rn = 0.974 fm, rp = 0.860 fm, the two
potential parameters 241.193 and 1.4534 of the second
term in Eq.(6), and the spin-orbit parameter λ = 0.03 of
Eq.(8) are the same for all properties in all nuclei.
The present work together with other works, i.e., on
4He [26] and that on possible neutron nuclei [43], consti-
tute a successful application of the model to very light
nuclei (section 2.2). It is noticeable that the neutron
structures in 5H and 7H are identica l to those for 4n and
6n, respectively, which have been predicted as possible
particle stable neutron nuclei [43].
The novelty of the present work and of previous ap-
plications of the model to very light nuclei [26] and [42]
(dealing with 7Be-12Be isotopes) is that for these nuclei
(due to the very small number of nucleons and the sub-
sequently very large deformation, as seen from Fig.2)
there is an extra degree of freedom (that of internal col-
lective rotation) even for their ground states. In these
nuclei, according to the present model, the adiabatic ap-
proximation is not valid. This extra degree of freedom
reduces the binding energy and increases the effective
radii of very light nuclei, as seen from Eq.(12) and
Eq.(20), respectively, and Table1, and in addition in-
creases the level width of the observed resonances (see
Table 1).
The very good agreements between the experimental
total binding energies and those derived by the present
model for all four hydrogen isotopes are apparent from
Table 1. In addition, an explanation is here given for the
broad width of these resonances as due to a superimposi-
tion of two far apart components of energy, one coming
from the usual intrinsic motion (of shell model type) and
the other from the internal collective rotation. All these
results constitute a justification of the whole procedure
followed in the present work in the frame work of the
Isomorphic Shell Model.
Comparisons of the present predictions for nuclear ra-
dii cannot be made since experimental data do not exist.
However, from Table 1 the significant contribution of the
rotational component (where it exists) to the predicted
values of radii is obvious, as a result of Eq.(20). In all
cases the geometrical, intrinsic radius resulting from the
cluster structure of the g.s. for each hydrogen isotope
(Figure 2) is much smaller than the effective radius
where an internal collective rotation exists.
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S. PASCHALIS, G. S. ANAGNOSTATOS 77
Supplementary sheet
Calculations of Table1 concerning the manuscript:
Ground state of 4-7H considering internal collective rotation
Coordinates of all nucleon average positions appearing in Fig.2 in units Fermi
For their identification the numbering here is the same as in the figure.
(3΄) 0.000, 0.000, 1.554
(1) 0.974 cos450, 0.974 cos450, 0.000 (2) - 0.974 cos450, - 0.974 cos450, 0.000
(5) 0.000, 2.511, 0.000 (6) 0.000, -2.511, 0.000
(7) 2.511, 0.000, 0.000 (8) -2.511, 0.000, 0.000
Distances among all nucleon average positions, dij, corresponding nucleon
potential energies, Vij of Eq.(6), and frequency of appearance in Fig.2 (in Fermi).
d
ij Vij 4H 5H 6H 7H
d1-2 = d1-5 = d1-7 = d2-6 = d2-8 = 1.948 7.2808 2 3 4 5
d1-6 = d1-8 = d2-5 = d2-7 = 3.273 0.6306 1 2 3 4
d1-3΄= d2-3΄ = 1.834 9.1281 2 2 2 2
d5-3΄= d6-3΄= d7-3΄= d8-3΄ = 2.953 1.1133 1 2 3 4
d5-6 = d7-8 = 5.022 0.0323 1 1 2
d5-7 = d5-8 = d6-7 = d6-8 = 3.551 0.3879 2 4 Col.3
4H:ΣijVij = 7.2808*2+0.6306*1+9.1281*2+1.1133*1 = 34.56
5H: 7.2808*3+0.6306*2+9.1281*2+1.1133*2+0.0323*1 = 43.62
6H: 7.2808*4+0.6306*3+9.1281*2+1.1133*3+0.0323*1+0.3879*2 = 53.42
7H: 7.2808*5+0.6306*4+9.1281*2+1.1133*4+0.0323*2+0.3879*4 = 63.25
Application of Eqs.(9) and (8) for an1p neutron state. Energies of Table1 in MeV
ћωn1p = (ћ2/m)(n+3/2)/<ri2> = 41.444(1+3/2)/2.511^2 = 16.4327
VLiSi=λ( ћωi )2 /(h2/m)*ℓi si = 0.03(16.4327)2/(41.444)*1/2 = 0.0977 ≈ 0.10
Col.4: 4
H ΣiVLiSi = 0.10, 5H 0.10*2= 0 20, 6H 0.0977*3=0.29,
7H 0.0977*4=0.39
Total kinetic energy according to Eq.(7) as already applied via Eqs.(25) – (28)
Col.5
T1p in 1s state = 5.349 MeV 4H: Σ<T>nm = 5.349+1.706*2+8.279 = 17.04

T1n in 1s state = 1.706 MeV 5H: 5.349+1.706*2+8.279*2 = 25.32
T1n in 1p state = 8.279 MeV 6H: 5.349+1.706*2+8.279*3 = 33.60
7H: 5.349+1.706*2+8.279*4 = 41.88
Internal collective rotation for each isotope according to Eqs.(20) and (18)
Θrot = Σ ρi2 + 0.165Arot Col.8
Arot
i1
4H: Θrot,z= 2r+r +4*0.165=2(0.974cos45)2+(2.511)2+(1.554)2+4*0.165=10.329
2
1z2 8z
5H: Θrot,z= 2r+2r 2
8 +5*0.165 =2(2.511cos45)2+(1.554)2+5*0.165 = 9.545
2
1z z
6H: Θrot,(1,-1,1) = 7H: Θrot,(1,-1,1)= (ρpoint(0, 2.511, 0), axis(1,-1,1)=2(2.050)2 +2*0.165 = 8.735
Erot = (ћ2/2m) 2(2 + 1)/2Θrot =124.332/ Θrot
Col.11: 4H Erot = 12.04, 5H Erot = 13.03, 6H Erot =14.23, 7H Erot = 14.23
Total binding energy for each hydrogen isotope:EB=Σ ijVij - Σ<T>nm-Σ iVLiSi -ER 
Col.12: 4H EB=34.56-17.04+0.10-12.04=5.58 5H EB=43.62-25.32+0.20-13.03=5.47
: EB=53.42-33.60+0.29-14.24=5.87 7H: EB=63.25-41.88+0.39-14 .24=7.52
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