Foundations of Physics, VoL 12, No. 3, 1982
On Nuclear Energy Levels and Elementary Particles J. A. de Wet ~ Received April 20, 1981 Considering only exchange forces, the binding energies and excited states of nuclei up to 24Mg are predicted to within charge independence, and there is no reason why the model should not be extended to cover all of the elements. A comparison of theory with experiment shows that the energy of one exchange is 2.56 MeV. Moreover, there is an attractive well of depth 30 Mel/, corresponding to the helium nucleus, before exchange forces become operative. A possible explanation of the origin of mesons is also presented.
I. INTRODUCTION This paper is a logical continuation of the ideas presented in Ref. 1 on the topology of nuclear manifolds. Here nuclear states were constructed by finding the rth Kronecker product of self-representations of the complete homogeneous Lorentz group L 0. The levels themselves emerged as the eigenvalues of a mass operator, so the explicit construction of a Hamiltonian was avoided. In fact, the analysis was shown to be equivalent to considering only the Barlett, Majorana, Heisenberg, and Wigner exchange forces between immediate neighbors and very credible agreement was obtained with the measured energy levels of I°B, 1°Be, ~°C, 12C, and 160 using only one parameter for each nuclear manifold. By a nuclear manifold M we mean the result of a mapping (;: 4 t o N of the 4' coordinates of configuration space (arising from the rth Kronecker product of self-representations of L0) into N observed nuclear states [2], where ~0 factors out all those permutations that include every possible exchange of spin, charge, and coordinate. Each element is therefore characterized by N and the isospin operator T, while the i Witmos, South Africa. 285 0015-9018/82/0300-0285503.00/0~) I982 Plenum PuNishing Corporation
286
de Wet
nuclear states [4] are labeled by additional eigenvalues of the rotationreflection group as discussed in detail in Ref. 1. It was also suggested that if a measure could be defined on the nuclear manifolds, then it should be possible to eliminate parameters altogether and achieve a purely geometric picture where the only contact between geometry and the laboratories of men would be in measured units of energy density corresponding to one exchange of spin, charge, or coordinate. Such a measure is presented in Section 2 and allows us, in principle, to predict not only ground state levels, but also the excited states of all nuclei to within the energy difference between analog states of mirror nuclei. No nuclear model would be complete without some indication of the origin of mesons, and Section 4 attempts to show how they could be incorporated.
2. MEASURE ON A N U C L E A R MANIFOLD As pointed out in the introduction, a nuclear manifold M, with coordinates ~a, is a result of the contraction
(o: 4r ~ N
(1)
of the coordinates x i of flat configuration space into the smaller space of nuclear states [;t] once permutations have been factored out. We will label the set of x i belonging to the same state [i] by X i, so that the Cartesian metric of fiat space is Jtj dX~ dXj" Then if the mapping q~ is an isometry (i.e., preserves distances)
6i.i dXi dX'~ = ga. d~"~d~"
(2a)
where ga~, is the new metric tensor on M. Naturally we cannot expect M to be fiat; it will, in fact, become distorted in such a way that ground states are saddlepoints in the new landscape. It is our task to find a measure in the neighborhood of this saddlepoint and to this end we will need a canonical form of g~t~ in accord with the critical point theory of Morse. (2~ According to Morse theory, it is possible to find a transformation ~, of the neighborhood of a critical (or saddle) point p such that the geometry of the manifold in this region is the hyperboloid °> 1 =
+ ... + (x,)
-
(x'+
.....
(3a)
Here p + q = n is the dimension of the manifold and p is called the index or type number of the critical point and we have written x i for ~. Now
On Nuclear Energy Levels and Elementary Particles
287
following Barut and Raczka, ~4) we find a measure on (3a) by using the biharmonic coordinate system k = 1,2 ..... p,
Xk=X'kCOShS, xp+Z = £t sinh 8,
0@ {0, oo)
(4)
I = 1, 2 ..... q
where x', £ are the coordinates on p- and q-spheres given by (t2) below. For the hyperboloid H p'q the equivalent form of (2a) is g , ~ ( H p'q) = gao(M p'q) c3,~xa(l'2) C3~Xb(O),
C2,~-- ~/C~l ~
(2b)
where gab(M s'q ) is the diagonal metric tensor gll = g2z . . . . .
gpp = --1,
gp+l,p+l . . . . .
gp+q,p+q -~- 1
of the flat Minkowski space (3a) in which n p'q is embedded and O are the internal coordinates {r/' ..... r/p+q-I or ~ok, 0 k, 0} on the space H p'q. Using (4) and (2a), it is easily seen that g ~ is diagonal and the Riemannian measure is d/a = I g,~l 1/z dO = cosh p- 10 sinh q- 1 0 dO
(5)
In order to evaluate (5), we shall have to obtain p, q from the metric gn~ of the nuclear manifold and determine the boundaries of the cap of the quadratic hypersurface (3a) in order to fix the range of O. We will approach the second problem first, and to this end will follow Wallace's excursion into the neighborhood of a critical point. (3) Essentially he considers the hypersurface ~ Ci(xi) 2-.~- 1 (C i = ~ 1 ) to define a noncritical level M a just below the critical point p and the hypersurface ~ ei(xi) 2 = - I to define a noncritical level M~ just above P. Here =
2 +
2 +
...
-
(xp+W
....
(x"y
= c
Ob)
and if we choose c = 1, X p + I = X p + 2 . . . . . X'~ = 0, we get a ( p - 1)-sphere similarly, if e = - l , xl=x 2 ..... x P = O , we get an S n - p - 1 on M b. Then members of the family F of orthogonal trajectories starting on M a - S p-1 all end on M b - - S " - p - 1 and vice versa, while members of F through points of S p - I or S n-p-~ all end on P. The latter constitute the cells E p and E "-p, with boundaries S p - 1 and S "-p-I, respectively, which are mutually orthogonat. Moreover, the projection of E "-p on the hypersurface (3a) will be seen to define the cap of interest because its boundaries are the intersections with M a of the limiting trajectories still contained in a maximum neighborhood S n - p - 1 × E p of M b . S p-1 on Ma; (n-p-1)-sphere
,,~.¢
Fig. 1.
Neighborhood of a critical point.
E)"
J
l
r~
On Nuclear Energy Levels and Elementary Particles
289
Figure 1 is an illustration of the xPx p÷I plane, where the limiting trajectory is XPX;+I = V/2
(6)
In hyperbolic coordinates the point A corresponds to sinh 0 = 1 - sinh 00
(7a)
which determines the maximum value of 0 in (5). However, since it is possible to consider trajectories through the boundaries of smaller neighborhoods of M b, we shall find in Section 3 that [(V/5 - 1)/2] 1/~ < sinh 00 ~ 1
(7b)
Note that the choice x p, x p+~ in (6) is not unique and any pair from the sets {x 1..... x p }, {xp+ 1..... x" } could have been chosen. Turning now to the determination of the index p, we shall need the metric ga, of the nuclear manifold M. Our task is much simplified by the fact that M equipped with ga, is a symmetric space. (4) These spaces have some very special properties and are defined as Riemannian manifolds for which the curvature tensor is invariant under all parallel translations. (5) They were completely classified by E. Cartan, who reduced the problem to the classification of simple Lie algebras, of which there are four that are the infinitesimal rings of the classical matrix groups S U ( n + 1), S O ( 2 n + 1), Sp(n), and SO(2n), and a further five exceptional algebras. It can be argued that these nine are the only algebras available for modeling a meaningful physical situation. Another important property which we shall need is the fact that the invariance of the curvature tensor under parallel transformations is equivalent to the condition that the geodesic symmetry with respect to each point be an isometry (see Section 4). Equation (2) is an isometry and the corresponding symmetry a separates M into just the sets of rotational and vibrational energy levels in a way that is transparently illustrated by examining the pattern of gnu for a°B given in Fig. 2a. This is simply a canonical form of Fig. 3 of Ref. 1, where we shall not need numerical values to see that the sets labeled by R, V transform into one another. In other words, the 12 X 12 matrix decomposes into two 6 X 6 matrices as in Fig. 2b, where numerical values
have been inserted to show that the matrices are nonsingular. Here [2] - [21222324] is an observed nuclear state such that 22 + 23 is the number of nucleons with a given spin, 23 + 24 the number with a given
290
de Wet
R
,
R
5 ....
R
3
R
R
V
5005
x
2
1
5023
x
4114
x
x
x x
5032
x
4123
x
x x
O
50/~1
x
4132
x
X
X
X
x
X X
x x
x x
5005
x
o
F
O
.4
F 1o
I
5F
5041
4132
x
x
3232
5023 4ii4
x x
5050 4141
gs
R
V
x
3223
V
R
W
V
V
V
ioF 55/~
5/~ I
h
1oF
-~'z3--t . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! _i 5o~4 K~" ~ 5F 5032 lo ~ ioj~ i v 41~3 i o F 5J~ 50 io"5/~ 5050 JsI 414~ I ~J3- 55/~: B Fig. 2,
Metric tensor for I°B.
IY 5 F
50
5J~ lo~3-o
On Nuclear Energy Levels and Elementary Particles
291
charge (in this case, five), and 24 + 22 that number with a positive or negative parity. There are N[zl = r!/21 ! 22 ! 23 ! /]'4 !
(9a)
equivalent states (which are the same up to a permutation) for each of the N partitions 21 +22 +)],3 + 24 = r of the number of nucleons r (ten for I°B). Here ~ j for this example = N
is Ntaj normalized and
(9b)
JNE,oo,j
The index p is found for the ground state [32321, according to the prescription of Ref. 1, by deleting the rows and columns of ga, corresponding to that state and then diagonalizing the remaining matrix MY. From the pattern (2b) it is evident that p=q=5
(10a)
since that is the rank of MY. It is also the number of rotational states R and vibrational states V, where these sets have been chosen to agree with the allocations generally given to even-even nuclei such as 160, 24Mg.~6'v) Actually, in the latter case the number of rotational states is always less than the number of vibrational states, so that the rank is determined by the former. Only in the case of nuclei with equal odd numbers of protons and neutrons do we find the symmetry of (Fig. 2b) where in fact the rotational and vibrational states are coupled because the pairs A IA zA 3A 4, A 1A 2A 4A 3 have the same energies (but different spins). If the numbers Z of protons and N of neutrons are both equal and even, the pattern of ga. taken from Ref. 1 shows that the number of rotational states is
r 2lr(+ 4
1+2+3+...+--~-=
+1
)
(10b)
where r = A = number of nucleons; and this is also the type number p = q since the ground state falls among the vibrational levels (for reasons to be discussed presently) and therefore may be eliminated without reducing the rank of M y. This is not the case with I°B, or indeed with any of the elements with Z = N = odd, when theoretically p=q=
1 + 2 + 3 + .-. + ¼(r+ 2)-- 1 = @ ( r +
2 ) ( r + 6)-- 1
(10c)
292
de Wet
but in practice, in the case of 14N, it is necessary to eliminate another unobserved state intertwined with the ground state. This will be discussed in more detail in the next section, when we will also consider isotopes. 3. C O M P A R I S O N W I T H E X P E R I M E N T
We may readily integrate (5) over the cap of the hyperboloid (3a) bounded by (7b) with the type numbers (10) by means of tables, for example those of Ryshik and Gradstein. ~8) The results are set out in Table I, where measured nuclear ground states, or binding energies, are also compared to theoretical energy densities. According to (9), the normalized number of exchanges is Jlrgs = Ngs/NtAI (9c) if A is the state with the least symmetry, so that NtA 1 is the smallest Ntaj. 2 If only the exchange forces between immediate neighbors are considered, then the energy of the state [/t.] is theoretically proportional to V/~a~ 't) and the energy density would be proportional to
~/~.,q/#(M)
(11)
Table I ...............................
Element
Egs MeV
4He 8Be
28.295 56.498
x°C
60.361
l°B 1°Be
64.750 64.977
nC
V/~s
p
V~ 6
1 3
V~ 10 ~
t2(M) 0.8814 0.4202
~/~,Jfd(M)
sinh 0 o
2.78 14.29
1 1
0.5167
21.20
1
0.4516 0.5167
22.15 21.20
1 1
1 2 2 2 2 5 13
12C 14N t60
6 6 6 8 10
0.5167 0.5167 0.5167 0.7464 1.1675
27.37 27.37 38.78 46.89 59.96
1 1 t 1 1
2°Ne 24Mg
160.646 198.258
252 924
15 21
3.103 8.760
81.21 105.5
0.9855 0.9675
i
N4
5 6
2V/~0 2V/200 20 35 70
i iiiiiiii
i ii,,ll
6
73.443 76.206 92.163 104.657 127.620
lib
,,
i
144 2584 i
2 Actually the state corresponding only to charge exchange, so charge exchanges are factored out in (9c) as might be expected since Nta J is a constant for each element and therefore should not affect the excited states. This is another example of a geodesic symmetry.
On Nuclear Energy Levels and Elementary Particles
293
/
/
/
/
/
200
1.6
15o
~÷/N/~
100
t2
p --~- C
,OBej ~
E (l~IeV) 8
/~¢m
5o B e
O 50
i0o
/~(M) Fig. 3. Nuclear binding energies and excited states.
zSo
294
de Wet
where /l(M) is the measure on the nuclear manifold (corresponding to a particular element) given by integrating (5). Figure 3 compares the measured binding energies to (11), and we find a straight line intersecting the ordinate axis at 30 MeV with a slope of 1.6 in units corresponding to one exchange. The energy of one exchange is therefore 1.62 = 2.56 MeV, which compares very favorably with the extra binding that can be attributed to the pairing of two particles. ~9) Moreover, there is an attractive well of depth 30MeV beyond which exchange forces becomes operative in accord with accepted theory (Ref. 9, Ch. 11, § 10). Furthermore, the excited states fall on the same straight line and lie between the ground state of the element concerned and its neighbor (with one proton less) if this point corresponds to ~ = N~a1/N~A ~ = 1 in the spectra of excited states calculated in Ref. 1. Since the latter also corresponds to the changeover from T a = ½(Z - N) = 0 to T = 1, it may easily be identified, and a survey of the tables of Ref. 10 shows the above condition to be satisfied closely. It will be noticed from Table I that the upper unit of integration sinh 00 falls below unity for A > 16. This may be associated with the fact that the volume of the sphere S p- ~ also decreases for p > 8, as can be seen from Fig. 4. Here we have used a biharmonic coordinate system xi=x'isinO
k,
i = 1,2 ..... 2 k - 2 ,
k<.p/2
x 2k-1 = cos (0k cos 0 k,
~0k ~ [0, 2~z),
k = 2, 3 ..... p / 2
x 2k = sin ~0k cos 0 k,
v~k E [0, z~/2),
k = 2, 3 ..... p / 2
(12a)
where x'~ = cos ~o1,
x'2 = sin tp1,
rpIE [0, 2zr]
on S p-1 (p even) in order to construct the measure ~4) p/2
dlg(S p - ' ) = t g{ 1/2 daJ = ~ I cos ok sin~2k-3~Ok ae~ k=2
]-[ dq'
(t2b)
i=1
which may be integrated over the sphere to obtain
u(s
1) = (2
y/2/(p _ 2)!t
(13a)
i f p = 2n, otherwise /l(S ~-1) = [(2z)P/2/(p -- 2)!!] t "/2 sin~2"-l)0 "+1 dO ~+~ :0
( 3b)
On Nuclear Energy Levels and Elementary Particles
295
4O
3O
p
eve~
i
u'b 20
\
10
p
odd
S°
I
I0
P
20
3O
Fig. 4. Measure on a p-sphere.
if p = 2n + 1. Although tJ(S p-l) soon tends to zero [because of the term sin~2k-3)O in (12b)], there is a lower limit to sinh 0 0 which may be obtained by looking at the first term in the expansion for Oo
g(g) = ;o c°shV- ~0 sinh v- ~OdO ~ (sinh 0 o cosh 0o) v-2 sinh 20o 2p-2
(14)
Here p may be very large, so (sinh 0o cosh 0o) will be just greater than unity
296
de Wet
in order to avoid very large numbers, i.e,, a lower bound of sinh 00 is [(x/r5 - I)/2] 1/2 as shown by Fig. 1. The point is that if the functional behavior of 0 o with p were properly understood, it would be possible to predict all nuclear energy levels labeled by T 3 = ½ ( Z - N) to within charge independence of nuclear forces, so that the problem is reduced to one of geometry) The rank p is intimately associated with the number of observed states, so that its determination is a stringent test of the logical consistency of the theory, and here it must be admitted that there is one as yet unobserved spin-1 state in the case of 1°C, 12C, 160, and 24Mg and one state that is unobserved and must also be eliminated to get the correct rank in the case of 1aN. These are all "ghost states" intertwined with the ground state in the sense that when they are eliminated part of all of the ground state also disappears from the matrix M s. Even though the ground state is chosen to lie among the vibrational levels, the ghost state is always rotational and moreover has an energy nearest to the ground state value. When isotopes such as 1°Be, 1°C, HB, ~C are analyzed the correct rank is precisely obtained by eliminating states that do not satisfy the fundamental criterion ~-1 ) (42,43,44).
4. PARTICLE THEORY Returning to Section 2, the real homogeneous linear transformations that leave the quadratic form (3a) invariant belong to the pseudo-orthogonal group SO(p, p), and the Caftan decomposition of its Lie algebra so(p, p) is precisely into the rotational and vibrational states, as already found by using the properties of the symmetric space X = SO(p, p)/SO(p - 1, p). Rotations transform the sets {xl,...,xP}, {xp+I ..... x 2p} separately among themselves and the generators of the rotation group O(p) are the maximal compact subalgebra K in the Cartan decomposition
so(p, p) = K + P
(15)
This is the subgroup of SO(p, p) with determinant equal 1 and is called the group of proper pseudo-orthogonal transformations. All other transformations among the x f are noncompact and P is the vector space spanned by 3 It may be shown, by direct calculation, that if sinh 00 is chosen such that the even-even nuclei fit Fig. 3 then there is a linear relationship between [~t(S~- 1)]1/p and sinh 0o for those nuclei with 4n >~36. In this case p is given by (10b) and the straight line passes through the points 1, 1, and 0, 0.8045, proving that sin 0o is indeed proportional to a length characterizing the sphere on the hyperboloidbut cannot decrease below 0.8045 > [(V~ - t)/211/~.
e~
Fig. 5.
Meson box.
Tr*~
/
i
J
O
B
Z
298
de Wet Table II.
Dynkin Diagram for
S0(4, 4)
III
IIIIII
n
© e~
©
t
--2
0
+1
0
e. I e_~ e _y
0 +1 0
--2 +1 0
+1 -2 +1
0 +1 -2
IIIIII
III
III
IIIII
IIIII
I
I
the noncompact generators, namely the space of the vibrational states. Since the latter are also critical levels, this suggests that we consider representations of the noncompact group SO(p, p). [Note that the rotational levels cannot be critical since if they are eliminated the rank of the matrix M s is reduced by one, which changes the measure p(M) for that element.] We will find that representations of the n- and p-mesons and A-particles readily emerge, so according to this model, the bombardment of a nucleus will raise it to a vibrational level E~ which then decays, releasing new particles with energies determined by the available energy E v. 4 We will employ Joseph's graphical method (11) to find the representations of SO(p, p), if p is even, and the ladder representation of Fig. 5 illustrates the case for p = 4. Here the quantum numbers {~, r/,fl, 7} of the four commuting Hermitian operators hg, h n, h~, h~, are associated with the lowering operators e_~, e_ n, and e r, which generate only a finite number of weights m, while e ~ spans the change of metric of (3) and generates an infinite string of weights. Thus we tentatively identify ~ and q with spin and isospin, respectively, 7 with a classification index, and/? with time, in that it will seem to indicate the direction of reactions involving particles labeled by 7. The Dynkin diagram and the action of the lowering operators (as given by the commutation relations) is given in TabtelI. This summarizes the structure of the Lie algebra so(4, 4) and is all we need to construct Fig. 5. Then ~ = - co corresponds to the set of two protons (spin o = ±½, isospin T 3 = +½) and two neutrons (a = ±~, T 3 = _1). This is the a-particle observed in so many decays. If the process is allowed to proceed, fl = -co - 2 could correspond to a box containing n-mesons (or = 0, T 3 = 1,
4 It has been pointed out that the model should at this stage be only regard as tentative. However the intention is to show how the theory can also account for other nuclear particles.
On Nuclear Energy Levels and Elementary Particles
299
0 , - 1 ) , p-mesons ( a = ±1, 7'3= 1,0,--1), co-mesons ( a = t, 0,--1, T 3 = 0 ), the t/-meson (a = 0, T 3 = 0), and other similar particles shown on the sheet labeled by 7 = - - 1 . A typical reaction indicated by e_~ would be p + n - ~ n + zc°
(16a)
when the fl-eigenvalues on both sides are (-~o + 1 ) + (-~o + 1) --* (-~o + 2) + -co or
-2co + 2 ~ -20) + 2
(16b)
which balance, since we have a crossed interaction, However, if we consider /? = - - c o - 4 to represent a box of zf-particles, typical interactions indicated by e ~ would be
p+zr+--*A ++,
n+n+ ~ 3 +,
p + n - ~ A °,
n+~r-~A-
(17a)
where now the//-eigenvalues on both sides of all four reactions are 09 --, 3
(17b)
indicating a sequential interaction over three distinct multiplets. Moreover, it will be noticed that there is almost no mass splitting within these multiplets, The interaction ~+ + zc+ --,p (18) implies that co = 0 within a sheet labeled by y. Thus the representations of the noncompact group SO(4, 4) provide a model for many features of hadron behavior, and presumably to accommodate strangeness we would need to go to S0(5, 5) to model K-mesons and hyperons. Unfortunately, the method of Re/'. 11 cannot be used for this case, but one can follow Patera et al. ~2) to find the number of times so(4, 4) is contained in so(p, p), since the ladder diagrams for p >~ 6 are not easy to interpret. We will use the relationship
Npp=Np_l,p_ 1 + Np_2,p_2+ 1, p e v e n Np_,,p.., + Np_2,p_z,
podd
(19a) (19b)
for the number Npp of subalgebras of so(p, p) to see that so(5, 5) contains so(4, 4) once; so(6, 6) contains so(5, 5) once and so(4, 4) once; i.e., so(6, 6) contains so(4, 4) twice. Continuing in this way, we find the last column of Table I for the number N 4 of times that so(4, 4) occurs in so(p, p).
300
de Wet
REFERENCES
1. J. A. de Wet, Found. Phys. 11, 155 (1981). 2. M. Morse and S. S. Cairns, Critical Point Theory in Global Analysis and Differential Topology (Academic Press, New York, 1969). 3. A. Wallace, Differential Topology (Benjamin, New York, 1968), Chapter 5. 4. A. O. Barut and R. Raczka, Theory of Group Representations and Applications (PWN, Warsaw, 1977), Chapter 15. 5. S. Helgason, Differential Topology, Lie Groups and Symmetric Spaces (Academic Press, New York, 1978). 6. D. Robson, Nucl. Phys. A 30[~, 381 (1978). 7. P. S. Hauge and S. A. Williams, Phys. Rev. C 4, 1044 (1971). 8. L M, Ryshik and I. S. Gradstein, Tables of Series, Products and Integrals (Academic Press, New York, 1965). 9. E. B. Paul, Nuclear and Particle Physics (North-Holland, Amsterdam, 1969). 10. F. Ajzenberg-Selove and T. Lauritsen, Nuel. Phys. A 114, 1 (1968); 116, 1 (1971). 11. D. W. Joseph, J. Math. Phys. 11, 1249 (1970). 12. J. Patera, P. Winternitz, and H. Zassenhaus, J. Math. Phys. 11, t932 (1974).
