How to Use the QND Software (Windows and Mac OSX versions)
Norman D. Cook
[email protected]
1. 40Ca: A small nucleus to illustrate diverse nuclear models
Start the QND program by double-clicking the icon. Basic
information concerning the display mode is shown on the screen,
but no nucleus is visible. Once a nucleus is specified, many
display options can be chosen under the Viewing Mode and
Display items in the menu bar or by typing keyboard commands
(see the “Keyboard” item under Help in the menu bar or type “K”
to toggle an on-screen summary) (Fig. 1).
To display the 40Ca nucleus in a less billiard-ball-like
style, type “C” to select the “probability cloud” depiction of
nucleons. Type "c" several times to change the density of the
clouds and “F7” and “F8” to alter the nucleon size (Fig. 3).
Figure 3: The 40Ca nucleus with nucleons represented as
“probability clouds”, wireframe off (“w”), nucleons enlarged,
and the background color set to black (under “Display”).
Nuclei can be built using the Function Keys. The default
coloring scheme is isospin (“i”). Protons (yellow) are added
one-by-one by typing “F2”; neutrons (blue) by typing “F4”; and
nucleon shells by typing “F6”. These can be removed with “F1”,
“F3” and “F5”, respectively. Let us begin by constructing one of
the most interesting of the small nuclei, Calcium-40: Type “F6”
3 times (or, equivalently, “F2” and “F4” 20 times each), which
gives the default ground-state structure for 40Ca (Fig. 2).
Return to the billiard-ball lattice by typing “F”. Next, type
“n” to color the nucleons according to their principal
quantum number, n. It is known (from both experiment and
theory) that a maximum of 4 nucleons (2 protons, 2 neutrons)
are in the lowest energy shell (red), 12 nucleons are in the
next shell (yellow) and 24 nucleons in the next (purple).
These are the first 3 closed shells in the lattice. They coincide
with the closure of the first 3 doubly-magic nuclei, 4He, 16O
and 40Ca of the shell model. To emphasize the structure of
each these 3 shells, type “1”, “2” and “3” in turn (Figs. 4-6).
Type “8” to return to solid-sphere depictions of all nucleons.
Figure 2: The 40Ca nucleus in the fcc lattice viewing mode.
Figure 4: 40Ca with emphasis shown on n=0 nucleons.
Figure 1: Blank screen with keyboard Help (“K”) displayed
1 The local nucleon interactions can be emphasized by
drawing the bonds between nearest-neighbors. Typing “F12”
will toggle among several types of bond display (Fig. 8).
Figure 5: 40Ca with emphasis shown on n=1 nucleons.
Figure 8: 40Ca with embedded shells and all bonds shown.
These 3 closed shells in the lattice correspond to 4, 16 and
40 nucleons - i.e., 2, 8 and 20 protons and neutrons. In other
words, the first 3 “magic” numbers of the shell model have a
remarkably simple geometrical representation within the
antiferromagnetic fcc lattice with alternating proton and
neutron layers. Because the lattice is close-packed, it of course
has the texture of a (frozen) liquid-drop. To display a less rigid
structure dominated by nearest-neighbor bonding (a “liquid
drop”), type “L” (Fig. 9). Typing “8” repeatedly will toggle
the translucent spheres, and typing “F7” a few times will
reduce the size of the nucleons, making the internal texture
more evident. By alternating between “F” and “L”, the liquiddrop and shell structure of the 40Ca lattice becomes evident.
Figure 6: 40Ca with emphasis on n=2 nucleons.
The triaxial (roughly spherical) symmetry of these shells is
best viewed by displaying the coordinate axes (type “x”, “y”
and “z”). By typing “8” once again, the first 3 closed shells are
displayed within translucent spheres (Fig. 7). Use the mouse to
confirm that the same symmetries are found along all 3 axes.
Figure 9: 40Ca depicted as a liquid-drop with isospin coloring
and nucleons drawn at realistic sizes (0.86 fm radii).
To see the simultaneous alpha-particle substructure of this
nucleus, type “A” once (to turn on the Alpha-cluster mode),
and then “a” 5 times to view various alpha-particle
configurations. Remove all bonds by typing “F12” twice. Note
Figure 7: 40Ca with embedded shells displayed.
2 that each of the 40 nucleons is now bound to one and only one
of the 10 alpha particles (Fig. 10). Use the mouse (left button)
to view this structure from various angles and type “v” (10
times) to display the alphas as tetrahedra connecting the
nucleons, or type “V” to enlarge the alphas. Remarkably, a
tetrahedron of alphas (green) inside of an octahedron of alphas
(light blue) is the traditional alpha-cluster model for 40Ca.
m - the azimuthal subshell within a given j-subshell
(m = +1/2, +3/2, +5/2, +7/2, +9/2, …)
s - the spin-up or spin-down state of the nucleons (s = +1/2)
i - the isospin (proton or neutron) character of the nucleons
(i = +1)
p - the parity of the nucleons (even = 0, odd = 1)
In the software, individual quantum values for all
nucleons in a given nucleus can be displayed by typing the
quantum number labels (n, j, m, s, i, p). These are shown for
the nucleons of the fictitious 140Yt nucleus containing 70
protons and 70 neutrons in Figs. 11-16. The geometry is best
examined with the size of the nucleons reduced (type “F7”
several times) and the stripes on the nucleons removed (type
“F9” twice). It is seen that the n-value has spherical
symmetry (Fig. 11), the j-value has cylindrical symmetry (j is
dependent on the distance of the nucleon from the nuclear
spin axis) (Fig. 12), and the m-value shows a conic symmetry
relative to the y-z plane (Fig. 13). Spin and isospin show
alternating layered structure (Figs. 14 and 15), and parity
shows alternating even and odd parity, in accordance with the
even- and odd-numbered n-shells (Fig. 16).
Figure 10: The 10 alphas inherent to the
alpha contains 2 protons and 2 neutrons.
40
Ca structure. Each
In summary, the lattice representation of 40Ca is of special
interest because it demonstrates how the 3 main, historicallyimportant, models of nuclear structure theory (the shell,
liquid-drop and alpha-cluster models) can be reproduced in a
unified approach. There is consequently no need to go backand-forth between mutually-contradictory models. In the
lattice, nuclei have the properties of a liquid-drop because
nearest-neighbor nucleon interactions predominate, but adding
nucleons to lattice sites around a centrally-lying tetrahedron
produces doubly-magic shells, as well as alpha cluster
configurations that are inherent to a close-packed lattice.
The unification of the diverse nuclear models is of possible
relevance in the context of nuclear “modeling”, but a more
important question concerns the relationship of the lattice to
the quantum mechanical description of nuclei. The central idea
behind QND is that the lattice symmetries quite simply are the
symmetries of quantum mechanics. Let us examine that idea
by returning to the default lattice structure. First, type “A” to
exit the Alpha Mode, and then “D” to return to the default
graphics settings and the default nucleon build-up.
Figure 11: The principal energy (n) shells of 140Yt. 2. Quantum Numbers: The connection between the lattice
and quantum mechanics
As is known from nuclear applications of quantum mechanics,
the quantal states specified by nucleon quantum numbers
indicate the basic energy levels of the nucleons. As a
consequence, every nucleon can be described uniquely by a
set of 5 distinct eigenvalues (and parity), as follows:
n - the principal energy level (n = 0, 1, 2, 3, …)
j - the total angular momentum subshell within a given n-shell
(j = 1/2, 3/2, 5/2, 7/2, 9/2, …)
Figure 12: The total angular momentum (j) subshells of 140Yt.
3 uniquely in the fcc lattice. In other words, the lattice provides
a unique geometrical representation of the known symmetries
of nuclear “quantum space”.
Figure 13: The magnetic moment (m) subshells of 140Yt. Figure 16: The parity symmetries in 140Yt.
3. Quantum Nucleodynamics (QND): Every nuclear state
has a corresponding lattice configuration
The quantum symmetries in the lattice model imply that each
nucleon with a unique set of quantum numbers has a unique
position in nuclear Cartesian space. Experimentally, what is
normally known for a given nucleus are: the number of
protons and neutrons, the total angular momentum, the
magnetic moment, and the total parity. Those experimental
facts can often be used to deduce the quantum numbers of the
last proton/neutron and therefore its lattice position. To build
a specific nucleus, use the “Picking” mode (“P”). Protons and
neutrons can be added, deleted and moved to other locations
by clicking on the small translucent lattice sites (Fig. 17). The
properties of the lattice structure can then be compared to the
experimental data displayed on the screen. Good luck!
Figure 14: The spin (s) layers in 140Yt. Figure 15: The isospin (i) layers in 140Yt. The occupancy of each of these shells and subshells is
theoretically known and has been experimentally confirmed
many times. This is the undeniable bedrock of nuclear
theory! It is of interest, therefore, that the exact same set of
eigenvalues with the known occupancies is reproduced
Figure 17: The properties of the last proton (orange arrow) in
15
N are shown by clicking on specific sites and comparing the
experimental and theoretical values. Many lattice structures
are possible, but only a few match the experimental data.
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