Fractionally-charged atomic systems and the Stanford quark-search experiments

Volume 112B, number 3 PHYSICS LETTERS 13 May 1982

FRACTIONALLY-CHARGED ATOMIC SYSTEMS AND THE STANFORD QUARK-SEARCH EXPERIMENTS

Virendra GUPTA and Vtrendra SINGH Tata Institute of Fundamental Research, Bombay 400005, lndta

Recewed 22 December 1981

The binding energies for the one- and tv, o-electron systems (feD, (fe-e-), (fte-) and (ffe-e-) containing fractmnaUy-charged e 1 2 4 objects f are calculated R suits are gwen for the fractional-charge values 7, 5 and g A possible explanatmn of the expcrtrnen-

2 tal results of the Stanford group is gwen m terms ot these Our results ~eem to prefer the value g suggesting that one of the posslbflmes for f ts that Is might be a hberated u-quark

Experimental evidence for the existence of fractional charges has been recently reported by Larue et al [1] They made forty measurements of the residual charge on 13 niobium balls (mass < 9 × 10- 5 gin) and found charges consistent with 0 or -+1/3 (in units of the proton charge e) Of these only five balls yielded + 1/3 (9 times), - 1 / 3 (5 tunes) and zero (8 tunes) In a total of 22 measurements On the remaining balls no fractional charge was found

If one takes their results at face value then one has to assume that they are due to fractionaUy-charged massive objects f being present on some of the moblum balls The possible values o f the electric charge, Z, wluch f may carry, consistent with the observations at f'lrst sight appear to be -+1/3, -+2/3, -+4/3 In view of the ease with Much the mobxum balls change their residual charge it would seem that f is not strongly bound to a moblum nucleus This would suggest that only positively charged f's or atormc systems con- raining them are being seen Consequently, It IS inter- esting to consider bound fractionally-charged atoms and molecules In this note we study the simplest one- and two-electron systems containing f namely F - ( f e - ) , F - -= ( f e - e ), F~ = ( f f e - ) and F 2 -- ( f fe - e - ) Our notat ion is based on that used for the hydrogen atom, molecule etc where f has replaced the proton

We first present the new results for the ground-state binding energies for these four systems for Z = 1/3, 2/3 and 4/3 We then use these to speculate on a pos-

sible explanation of the Stanford results [1 ] and later on the nature of the object f

[:sttrnates o f the ground-state energy O) k-atom The hydrogen-ltke atom F -- ( f e ) re-

quires no work, the exact ground-state energy

E = - Z 2 e o , (1)

where e 0 = 1 Rydberg = 13 6 eV Here and throughout we assume that f is massive compared to the electron and neglect its motion

(n) F - - sys t em The hehum-llke atom F - = ( f e e -) is more complicated Hylleraas [2] has given a semi- empirical formula for the ground-state energy, E- , for such atoms, as a function of the nuclear charge and which works very well for integral values However, the formula revolves inverse powers of the nuclear charge and as such is not expected to work for small fractional values Instead one can use the one-parameter varm- tional calculation of Bhadurl and Nogami [2] wtuch gwes a fairly close upper bound to the actual ~ - for integral nuclear charge The results using the Hylleraas formula for Z = 4/3, (labelled Hy) and those using the variational calculation for all Z = 1/3, 2/3 and 4/3 (labelled BN) are summarlsed m table 1

(lU) 'I~2-system The system F~ - ( f i e - ) is Sunllar to the ionized hydrogen molecule H~- In this case one can see directly from the hamdtonlan that ItS energy E ~ ( Z , = R) less the nuclear repulsion energy is Z 2 tunes

0 031-9163/82 /0000-0000/$02 75 © 1982 North-Holland 251

Volume 112B, number 3 PHYSICS LE I'IT RS 13 May 1982

Table 1 The results ot the bmdmg energy (m Rydbergs) ~alculatlons for one- and two-electron st stems comammg the fraclmnally- charged obJeCt t with charge Z ForZ = l, f ls just the proton resulting m the experimentally measured h~drogemc systems For further explanation ~ee text The negative of the nuln- bers m the table are upper hmlts to the ground-state energy l-or further explanatmn see text

Z System

(fe-) i

4/3 16/9

I - 1 ~ 1 2 (re-e-) (fie-) (fie-e-)

2 40(lly) 1 66(J) 3 80 2 36(BN) 1 55(S)

0 31(BN) 0 720) 0 95 0 68(s)

0 02(BN) 0 25(J) 0 12 0 21(S)

1 053(Iiy) 1 205(J) 2 26 1 016(BN) 1 2052(1"xp ) 2 35(kxp ) 1 056(Lxp )

2/3 4/9

1/3 1/9

1 1

a function 4~ o f x = ZR, where R is the internuclear distance m umts of the Bohr radms for the hydrogen atom We therefore have the scahng law

E~(Z, R) /e 0 - 2Z2 /R = Z2~b(x) (2)

Alternatwely, an umts of e 0,

Z - 21'~ ~(Z, R) = E~'(J, x) + (2[xXZ - 1 ) (3)

The first term is just the energy E(H~) for H~" for x = R0, the actual separation of the two protons The actual minimum of E~(Z, R) would be at some value x = x 1 say Thus, ff we choose to evaluate the right-hand side of eq (3) at x = R 0 then we would get an upper bound, namely

E~(Z, R) < Z 2 [£(I-I~') + (2/Ro)(Z - I)] (4)

Stmdar scahng upper ltrnlts can be obtained for the electronic excited states which are not considered here For the ground state the experxmental values F(H~-) = - 1 2052 e 0 and R 0 = 1 06 X 10 -8 cm have been used to evaluate the upper limit A better upper limit for the ground-state energy can be obtained by using the two-parameter varxatlonal wavefunctlon used by James [3] for the H~ case which gives/z(H~-) correct to three decimal places v The upper bounds from the scaling result eq (4) (labelled by S) and the James

wavefunctlon (labelled by J) are tabulated m table l For the latter we expect a very good upper hmxt though one notices that the simple scaling bound does not do too badly

(w) F2-system The system F 2 - ( f i e - e - ) clearly corresponds to the hydrogen molecule H 2 Owing to the electromc repulsion no simple scahng argument can be used Thls is a comphcated system and many elaborate treatments are avatlable for H 2 Our object here Is not to get the best result but a reasonable first emmate For this purpose the molecular orbital method for H 2 given by Slater [4] is adequate and the upper hmlt for the ground-state binding energy Is gwen m table 1

Apart from the atom, the entries m table I for the other systems are upper limits Consequently the actual experimental numbers as well as the upper hmJt for H , H~2 and H 2 (Z = 1) are tabulated to give the reader an xdea of how close the upper hmlt is expected to be to the actual value Moreover, the numbers have been rounded off to two decunal places except for Z = 1 Some important features of the results of table 1 are as

follows (a) In view of the accuracy of the James' varlatlonal

function for H~, one expects that for Z = 4/3 also the upper llnut quoted ~s nearly exact Consequently, one expects that for Z = 4]3, F~ would be unstable and break up into F + f

(b) For H- and He the Bhadun-Nogarm [2] upper hnut is only a few percent above the actual values Consequently, one would expect that for Z = 1/3 and 2/3 F - would be unstable and tend to break up into

F + e - mstead (c) The stabdlty of the molecule F 2 is assured if the

upper hmlt on its energy is less than that for F~ + e - , F + f or F + F It is clear from the table that the molecule F 2 is stable for Z = 4[3 and 2[3 but unstable for

Z = 1/3 (d) A further point to be noticed is that F + F and

F 2 have comparable energy for Z --- 4/3 and 2/3 In contrast, to the fractional-charge case, all the four

H, tt , H~ and H 2 are stable

Posszble imphcanons for the Stanford experiment Keeping the above features m mind, we venture a possible explanation of the experimental evadence of Larue et al [ 1 ]

Their experiments maply that mobmm contains frac-

252

Volume 112B, number 3 PHYSICS LETTFRS 13 May 1982

tlonally-charged particles, f, m the propor t ion of about 1 to 1019 -1020 nucleons The non-observance of fractional charges m Iron by Morpurgo et al [5] Is con- slstent with mobmm being about 15 time richer m f's compared to iron One cannot explain this difference at present as the enrichment processes of various ores are not yet understood, but such a factor as not un- plausible m geochemistry

As noted by Lame et al [ 1 ], the fractional charges reside near the surface of the mobmm balls since their residual charge can be changed with ease, either by contact with a plastic holder or by washing the balls with acetone or alcohol It is then most hkely, as noted earher, that it is the atomic or molecular systems con- taming f which reside m the interstitial sites at the surface The removal of a f bound mslde the electromc K-shell of mobmm would be rather hard to accomphsh in this way

In view of the rarity of the fractionally charged objects, probably remnants from the p re - lep ton-hadron era of the very early umverse, we assume that at most one such atomac or molecular system containing f, is present on a gwen mobmm ball Moreover, these systems must be stable m the free state so as not to have decayed by now

It is also reasonable to assume that only one kmd of f ~s revolved smce of all the fractionally-charged objects, wtuch may exist, only the least massive one would be absolutely stable by charge conservation T 1 In this sltuataon, to explain the residual charge o f - 1 / 3 as well as +1/3, it is clear that some balls must have one f wtule the others must have two f's Further, with only 1 f per 1019 nucleons, it is extremely unhkely that two atomic systems (hke F or F - ) will be packed

independently onto the same ball A stable molecule containing two f's, however, is a more reasonable pos- slbdlty A further auxdlary argument favouring molecules may be their preferentml surface adsorption such as occurs for molecular hydrogen on metals

Dascountmg the clean plastic containers as a hkely

source, the observed fractional charge on the bails must be due to primordial f's at tached to the nlobmm itself The balls contalmng the f's may, on contact with

the plastic containers or the capacitor plates, loose the f's and later pick them up again For further discussion, we use F(Z), F (Z), F~'(Z) and F2(Z ) to exphcttty m- d~cate the charge Z of the f fomamg the system In thas notation, smce Larue et al record the residual charge modulo 1, the stable systems wtuch will appear with effective charge +1/3 are F(4/3 F - ( 4 / 3 ) , F~'(2/3), F2(2/3 ) and F(1/3), whale those with effectwe charge - 1 / 3 will be F2(4/3 ), F (2 /3 ) and F ~ ( I / 3 ) For the balls whtch exhibit a charge +1/3 and 0 or - 1 / 3 and 0 clearly a number of posslbdltles are there However, the 13 measurements on ball 6 seems to demand a scenario with Z = 2/3 The reason is that the first two measurements gave a value of +1/3 whale subsequent measurements ytelded zero (five times), - 1 / 3 (two tunes) and +1/3 (two tunes) Thus, ball 6 exhibits a fracttonal change of 2/3 umts m its charge unhke the others It ~s then natural to assume that it started wath a molecule having two f's, with an effective charge +1/3 The most attractwe posslbthty * 2 is that the molecule F2(2/3 ) was mmal ly attached to ball 6 The ball could then lose F2(2/3 )ent t re ly or F2(2/3 ) could dlssocmte, under the prevaflmg ambient and/or surface fields mto two F(2/3) atoms and one of these could be left behind m the plastic contamer In these cases it would exhablt an effective charge of 0 and --1/3, respecttvely m a later measurement Alter- natwely, the fractionally-charged object left behind an the plastic contamer can be picked up again by the ball to exhibit a charge of +1/3

If the above nawe explanation for the pecuhar sequence o f values for the measured charge on ball 6 is sensible, then one can predict the expected values for the residual charge on repeated measurements Assum- mg, only one kmd of f with Z = 2/3, one would expect ball 9 to behave hke ball 6 as the only two measurements on it so far have yielded a value of +1/3 In contrast, the only two measurements made on ball 3 yielded f'trst - 1 / 3 and then zero Thas would mean that, in our picture, at started with F(2/3) and so subsequent measurements on it cannot yield a value of +1/3 These predictions would be altered ff the observations of fractional charge is due to more than

t l In case there IS some other exactly conserved quantum number (unknown to us at present) then It Is possible that, m addition, other heavier but stable fractionally-charged particles may exast

t 2 "i his mechamsm can also work if one starts with t- 2 or l- wlthZ = 2/3 (modulo I) lfone is wdhng to admit two kinds of l's then a combination of say I-~(1/3) and 1- 2 (4/3) can work

253

Volume 112B, number 3 PHYSICS LL FTLRS 13 May 1982

one kind of f For example, xf ball 3 started with a

F2(4/3 ) then it could yield a value of +I/3 m a later measurement

Slmdarly, one expects that a ball which yields zero

on the f'trst measurement wtll continue to do so, this is exemphfied by the 2, 3 and 6 and measurements on ball

numbers 1, 7 and 10 respectively However, the observed sequence of values 0, 0, +1/3 for ball 11 defies explanation unless it came from the capacitor plates where it could have been left behind by ball 6 [1] We have lmphcltly assumed that the extremely troy balls are kept m separate containers between experiments

for unambaguous ldenhficatlon One can speculate on what this object f may be and

its other properties There are a number of posslbdmes 0) It could be a hberated quark assuming the com-

plete breakdown of colour symmetry [6] The value Z = 2/3 would lead one to identify it with the u-quark * 3 This IS mce as it as the hghtest quark and would be thus eminently stable to survwe from the early unwerse

(u) It could be a hberated "glow" smglet dvantl- quark dd or (u + gluon) in a model with partially

broken colour symmetry [ 10] 0u) It could be a new kind of object, lepton-hke m

its interactions, with spin 0 or 1/2 - a free q [ 11 ] (w) It could be a fractionally-charged hadron ff

there are new quarks with charges other than +2/3 modulo integer [ 11 ] In the last two possabdlhes colour symmetry would be exact m consonance with the

prevailing ideas of colour confinement It is too early to be able to decxde on the nature of

f without more evidence Further, it IS conceivable that

' 3 An explanation m terms of u and d quarks ot the Stanford results prior to September 1978 was suggested by Orear 17 ] I or the expected chemistry and geochemistry of fractionally-charged objects see the work o! Lackner and Zwelg [8] and the reviews by Jorgensen [9]

the present evadence for fractional charges may not be sustained by future expertments In that case our re- suits for the fractional-charged atomic systems may sttllbe of some theorehcal interest However, it as amus- ing that the Stanford results can be understood m terms of a single object of +2/3 charge

We thank Shrl P K Jha for numerical computations

References

11] GS Larue J D PhdhpsandVv M ralrbank Phys Rev Lett 46 (1981) 967 42 (1979) 142, 38 (1977) 1011

[2] RK BhadurlandY Nogaml, Phys Rev A13(1976) 1986, 1 A llylleraas, 2" Ph~,~ 65 (1930) 209

[3] II M James as quoted m L Pauhng and E B Wilson, Introduction to quantum mechamcs, 1 st Ed (McGraw- 11111 Ne~ York, 1935) section 42b, p 333

[4 ] J C Slater, Quantum theory of matter, 2nd Ed (McGraw-lhU, New York, 1968) section 21-2, p 416

[5] M Marmelh and G Morpurgo, lnvated paper, Furopean Physical Society Conl (Lisbon, July 1981), Phys Lett 98B (1981) 465

[6] A dc Rujula, R (,lies and R Jafft., Phys Rcv D17 (1978) 285

[71 J Orear, Phys Rcv D20 (1979) 1736 [81 K S Lackner and G /welg, fhe chemistry of free

quarks, Cal Tech preprmt CALT-68-781 (1980), lntro- ductlon to the chemistry ot fractionally charged atoms, CalTech preprmt CALT-68-865 (1981)

[9] C K Jorgensen, Structure and bonding (Springer, Berhn) 34 (1978) 19, 43 (1981) 1

[10] R Slan~ky J Goldman and (, L Shaw, Observable lractlonal electric charge m broeken QCD, Los Alamos National Laboratory, preprmt No LA-U R-81-1378 (1981)

[ 11 ] V Gupta and P K Kabtr, Fractional charges and quark confinement, Lmverslty of Virgmxa, preprmt (1981 ), to be pubhshed, Phys Rev D, L-F LI and F Wllczek, Prlce of tractlonally charged paltlcles m a umfied model, Umverslty of Cahtorma, Santa Barbara, preprmt No NSI--ITP-81-55 (1981)

254

Documents

Fractionally-charged atomic systems and the Stanford quark-search experiments