Table Of ContentLower bound on the value of the fine-structure constant
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, Israel
and
The Hadassah Institute, Jerusalem 91010, Israel
(Dated: January 10, 2011)
Recently we have proposed the existence of a universal relation between the maximal electric
charge and total mass of any weakly self-gravitating object: Z ≤ Z∗ = α−1/3A2/3, where Z is
the number of protons, A is the total baryon (mass) number, and α = e2/¯hc is the fine-structure
constant. Motivatedbythisnovelbound,weexplorethe(Z,A)-relationofatomicnucleiasdeduced
from the Weizs¨acker semi-empirical mass formula. It is shown that all nuclei, including the meta-
stable maximally charged ones, conform to the upper bound. Moreover, we suggest that the new
charge-mass bound places an interesting constraint on the value of the fine-structure constant:
1 α>∼1/323.
1
0
2 The weak cosmic censorship conjecture (WCCC), in- satisfies the relation Z/A2/3 ≃ 2.67 < α−1/3. Thus, one
n troduced by Penrose forty years ago [1, 2], is one of the finds thatthis nucleus alsorespects the upper bound (1)
a
corner stones of general relativity. This principle asserts [3].
J
that spacetime singularities that arise in gravitational Evenheaviermeta-stablenucleiareexpectedtobepro-
7
collapse are always hidden inside of black holes. The ducedintheforthcomingyearsusingacceleratorproduc-
] elimination of a black-hole horizon is ruled out by this tion techniques. In fact, some calculations suggest that
h
hypothesis because that would expose naked singulari- nucleiofA∼300to476mayhaveverylonglifetimes[7].
t
- ties to distant observers. Could these highly charged nuclei be able to threaten
p
Arguingfromthecosmiccensorshipprinciple,wehave the validity of the cosmic censorship conjecture by vi-
e
h proposed [3] the existence of a universal bound on the olating the (Z,A)-bound (1)? In order to address this
[ charge q of any weakly self-gravitating object of total question, we shall investigate the charge-mass relation
2 energyµ: q ≤µ2/3Ec−1/3,whereEc isthecriticalelectric of atomic nuclei as deduced from the well-known semi-
v field for pair-production [4]. For charged objects with empirical mass formula [8–12].
8 nuclear matter density the upper bound corresponds to Consider an atomic nucleus composed of Z protons
8 and N neutrons. The total baryon number is given by
7 Z ≤Z∗ =α−1/3A2/3 , (1) A=Z+N. The charge and mass of a nucleus are given
4
by
. where Z and A are the number of protons and the to-
9
0 tal baryon number, respectively, and α ≡ e2/¯hc is the q =Z|e| ; µ=Zm +Nm −E ≃Am , (2)
p n B p
0 fine-structureconstant. [We shallhenceforthuse natural
1 units in which c=1.]
where E is the binding energy of the nucleus, which is
: B
v Thisboundwasinferredfromtherequirementthatthe typicallymuchsmallerthanitsmass. Thebindingenergy
i WCCC be respected in a gedanken experiment in which
X EB of a nucleus (that is, the difference between its mass
a charged object falls into a charged black hole. The
andthe sum ofthe masses of its individual constituents)
r
a integrityoftheblack-holehorizonisrespectedprovidedZ iswellapproximatedbythesemi-empiricalmassformula,
isboundedasinEq. (1). Thisrelationlimitsthecharges
also known as Weizs¨acker’s formula [8–12]:
ofobjectssuchasatomicnucleiandquarknuggets[3,5].
The intriguingfeatureofourderivation[3]isthatituses Z2
E (A,Z)=a A−a A2/3−a
a principle whose very meaning stems from gravitation B V S CA1/3
(the cosmic censorship principle) to derive a universal 2
(A−2Z)
boundwhichhas nothing to do withgravitation[written −aA . (3)
A
out fully, the bound (1) would involve ¯h and c, but not
G]. This provides a striking illustration of the unity of This well-known formula is based on the liquid drop
physics. model which treats the nucleus as a drop of incompress-
Itisofconsiderableinteresttocheckthevalidityofthe iblenuclearfluidcomposedofprotonsandneutrons. The
208
new charge-mass bound (1). For instance, Lead Pb, four terms on the r.h.s of Eq. (3) correspond to the co-
82
the largest known completely stable nucleus satisfies the hesive binding of all the nucleons by the strong nuclear
relation Z/A2/3 ≃ 2.33 < α−1/3. Thus, this nucleus force, a surface energy term (which represents the fact
conforms to the upper bound (1). The largest known thatsurfacenucleons arelesstightly bound ascompared
artificially made nucleus contains Z = 118 protons and to bulk nucleons), the electrostatic mutual repulsion of
a total number of A = 294 nucleons [6]. This nucleus the protons, and an asymmetry term (which represents
2
the fact that protons and neutrons occupy independent Substituting the experimentally measured values of the
quantum momentum states) [8–12]. The coefficients in coefficients aV and aC [12], one finds Zmax(A) <
thesemi-empiricalmassformulaarecalculatedbyfitting 4.71A2/3. We therefore obtain the series of inequali-
to experimentally measured masses of nuclei [12]: ties Z(A)/Z∗(A) ≤ Zmax(A)/Z∗(A) < 4.71/α−1/3 <
0.91 < 1. This implies that all atomic nuclei, includ-
aV =15.36(3) ; aS =16.42(8) ; ing the meta-stable maximally charged ones (with the
a =0.691(2) ; a =22.53(7) . (4) maximally allowedelectric chargeaccordingto the semi-
C A
empirical mass formula. Most of these nuclei are yet to
To a rough approximation, the nucleus can be con- beproducedartificially)conformtothenewZ−Abound
sidered a sphere of uniform charge density. The po- (1).
tential energy of such a charge distribution is given by In figure 1 we depict the actual ratio Zmax(A)/Z∗(A)
Ep = 3Q2/5R. It is well-known that the radii of atomic as a function of the mass number A, where Zmax(A)
nuclei are well approximated by the empirical relation is calculated from the full expression of the Weizs¨acker
R(A) = 1.219× A1/3 fm [12]. This is a direct conse- semi-empirical mass formula (3). One indeed finds
quence of the fact that the size of an individual nucleon Zmax/Z∗ < 1 for all nuclei, with a maximal value of
is roughly given by its Compton length. Thus, one can ≃0.85.
write R(A) ≃ A1/3 × ξh¯ fm, where m is the proton’s ∗
mp p In figure 1 we also depict the ratio Zstable(A)/Z (A)
mass and ξ is a constant of order unity (empirically one as a function of the mass number A, where Zstable(A) is
finds ξ ≃ 5.788). Substituting R(A) and Q = Z|e| into thenumberofprotonsofthemoststablenucleusofmass
the expression of the potential energy Ep, one finds number A. Here Zstable(A) is obtained by maximizing
the binding energy E (A,Z) with respect to Z. This
B
3m α
a ≃ p ≡cα , (5) yields [3]
C
5ξ
A 1
where c ≡ 3mp/5ξ ≃ 97.38. For α ≃ 1/137.036 [14] Zstable(A)= 2 1+ aC A2/3 . (7)
one finds aC ≃ 0.711. This estimated value of aC is 4aA
astonishingly close to the empirically measured one (less
(For light nuclei this expression reduces to the canonical
than 3% difference), see Eq. (4). Below we shall come
∗
back to this observation. relation Z = A/2.) One finds that the ratio Zstable/Z
has a maximal value of ≃0.56 [3].
We shall first consider nuclei with the largest possible
The dimensionless fine-structure constant α is one of
electric charge, Zmax(A), for a given value of the baryon
the fundamental parameters of the standard model of
number A. These nuclei pose the greatest challenge to
particle physics. It has puzzled many scientists since its
the charge-mass bound (1). A nucleus may in principle
introduction by Sommerfeld [15] almost a century ago.
be produced (and live even for a short duration of time)
The question of how to derive the numerical value of α
if it has a positive binding energy. The maximal values
from some underlying theory has been one of the most
Zmax(A) are determined from Eq. (3) with the require-
important open questions in modern physics [16–24]. As
ment EB(A,Zmax) ≥ 0 [with EB(A,Zmax +1) < 0]. It
weshallnowargue,onemaydeduceaboundonthevalue
should be emphasized that, although such hypothetical
of the fine-structure constant α from the novel charge-
nuclei are characterizedby positive binding energy (that
is, their masses are smaller than the sum of the masses mass bound (1). To that end, let us assume that α is a
free parameter in Eqs. (1) and (5).
of their individual constituents), they are expected to
∗
be short-lived. This is due to the fact that their bind- The requirement Zstable(A) ≤ Z (A) yields the
ing energies are smaller than the corresponding binding quadratic equation
energies of the stable nuclei, see Eq. (7) below. Thus,
a
wedonotexpecttofindsuchnucleiinnature. Neverthe- C A2/3−α1/3A1/3+2≥0 . (8)
2a
less,suchnucleicouldinprinciplebeproducedartificially A
(andliveforashortdurationoftime), anditistherefore
Theinequality(8)wouldbesatisfiedforallAvaluespro-
ofinteresttostudythesenucleiinthecontextofthenew
vided the discriminant is non-positive [25]:
charge-mass bound (1).
Noting that the second and fourth terms in the mass 4a
α2/3− C ≤0 . (9)
formula (3) are negative, one realizes that the binding
a
A
energyE (A,Z) willbecome negativeonce the Coulomb
B
energyovercomesthevolumeenergy[13]. Thus,onemay Substituting a = cα from Eq. (5), one obtains the
C
obtain a simple upper bound on the value of Zmax(A): inequality:
Zmax(A)<(cid:16)aaVC(cid:17)1/2A2/3 . (6) α≥(cid:16)a4Ac(cid:17)3 . (10)
3
Substituting the experimentally measured values of the [2] R. Penrose, Riv. Nuovo Cimento bf 1, 252 (1969); in
coefficientsa andc [12],one findsa lowerboundonthe GeneralRelativity,anEinsteinCentenarySurvey,edited
A
value of the fine structure constant: by S. W. Hawking and W. Israel (Cambridge University
Press, Cambridge, England, 1979).
α>∼ 51611.7 . (11) [[43]] FS..HSoaudt,ePr,hyZs..PLhetyts..B696,9374,233(919(3210)1;0)W[a.rHXeivis:e1n0b09er.3g6a9n5]d.
H. Euler, Z. Phys. 98, 714 (1936); J. Schwinger, Phys.
We shall now consider atomic nuclei with the largest Rev. 82, 664 (1951).
possible electric charge for a given mass number A. In
[5] J. Madsen, Phys. Rev.Lett. bf 100, 151102 (2008).
orderto avoida violationofthe bound (1) (which would [6] Y.T.Oganessianet.al.,Phys.Rev.C74,044602(2006).
ultimately lead to a violation of the WCCC, see [3]), [7] See J. B. Natowitz, Physics 1, 12 (2008) and references
we must demand that any collection of nucleons which therein.
seems to violate (1) is actually unstable. More explic- [8] J.W.Rohlf,ModernPhysicsfromatoZ0(Wiley,1994).
[9] E. Segr´e, Nuclei and Particles (Second Edition, W. A.
itly, we must demand that any collection of Z protons
and A−Z neutrons with Z > α−1/3A2/3 is character- Benjamin, 1977).
[10] N.Cook,ModelsoftheAtomicNucleus(SpringerVerlag,
∗
ized by a negative binding energy. Substituting Z =
2006).
α−1/3A2/3 from the bound (1) into (3) and demanding [11] Seealsohttp://en.wikipedia.org/wiki/Semi-empirical mass formula.
that E (A,Z∗+1)<0, one obtains the quadratic equa- [12] M. W. Kirson, Nucl. Phys.A 798, 29 (2008).
B
tion [13] A similar argument was raised by A. Gal (private com-
munication).
a 4a 4a
(a − C −a )A2/3+( A−a )A1/3− A ≤0. (12) [14] D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev.
V α2/3 A α1/3 S α2/3 Lett. 100, 120801 (2008).
[15] A. Sommerfeld, Atombau und Spectrallinien, (Friedr.
This inequality would be respected by all nuclei (by all
Vieweg & Sohn, Brounschweig, 1951).
A values) provided the discriminant is non-positive [26]: [16] P.A.M.Dirac,Proc.R.Soc.(London)A133,60(1931).
[17] A. S. Eddington, Fundamental Theory (Cambridge Uni-
4a 16a a
A 2 A C
( −a ) + (a − −a )≤0 . (13) versity Press, Cambridge 1946).
α1/3 S α2/3 V α2/3 A [18] A. Das, and C. V. Coffman, J. Math. Phys. 8, 1720
Substituting a = cα from Eq. (5), the inequality (13) (1967).
can be writtenCas a quadratic equation for α1/3: [19] T.H. Boyer, Phys.Rev. 174, 1764 (1968).
[20] A. Wyler, C.R. Acad. Sci. A269, 743 (1969).
a2α2/3−8a (2c+a )α1/3+16a a ≤0 . (14) [21] G. Rosen, Phys.Rev.D13, 830 (1976).
S A S A V [22] G. F. Chew, Phys. Rev.Lett. 47, 764 (1981).
This yields the lower bound [23] D. K. Ross, Int.J. Theor. Phys. 25, 1 (1986).
[24] P. C. W.Davies, Int.J. Theor. Phys.47, 1949 (2008).
4a (2c+a )−4 a2(2c+a )2−a2a a [25] Note that if α is identically zero, then condition (8) be-
α1/3 ≥ A S p aA2 S S A V . comes a trivial one: 2≥0. Thus, our arguments cannot
S rule out this trivial solution.
(15)
[26] Note that if α ≡ 0, then condition (12) is satisfied triv-
Substituting the experimentally measured values of the
ially. Thus, our arguments cannot rule out this trivial
coefficients aV, aS, aA, and c [12], one finds the lower solution.
bound [27] [27] TheothersolutionofEq.(13)yieldsthetrivialinequality
α1/3 <∼140.9.
1
α>∼ . (16) [28] We note again that our arguments cannot rule out the
323.6 trivial solution α≡0. In thiscase thebound(1) is valid
trivially.
This bound is necessary for the validity of the charge-
mass relation (1) [28]. Remarkably, the bound (16) on
the value of the fine-structure constant is of the same
orderofmagnitudeastheexperimentallymeasuredvalue
[14].
ACKNOWLEDGMENTS
This research is supported by the Meltzer Science
Foundation. I thank Oded Hod, Yael Oren and Arbel
M. Ongo for helpful discussions.
[1] S.W.HawkingandR.Penrose, Proc. R.Soc.London A
314, 529 (1970).
4
1 ∗ ∗
FIG. 1: The ratios Zmax(A)/Z (A) and Zstable(A)/Z (A) as
a function of the mass number A. Here Zmax(A), Zstable(A),
er andZ∗(A)arethemaximallyallowednumberofprotonsina
b 0.8
m nucleusof mass numberA according totheWeizs¨acker semi-
nu empiricalmassformula(3),thenumberofprotonsofthemost
n stable nucleus of atomic mass A, and the maximally allowed
o 0.6
ot numberofprotonsforagivenmassnumberAaccordingtothe
pr novel upper bound (1), respectively. One realizes that these
malized 0.4 ZZmax / /Z Z** rhpaertoaidvoyuscanerudec)sl,emwiawitllihethratpmhoasanixtii1mvefaoblrivanaldlluinneugcoelfeni≃e(rig0ny.c8l5wu.dhiinchgahryepyoetthettoicbael
stable
r 0.2
o
N
0
500 1000 1500 2000 2500 3000
Mass number A
