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V ol . 1 0 2 ( 2002)
A CT A P HY SIC A P O LO N IC A A
No . 3
Si ze of M u on iu m H y d rid e
M . Suff czy ¥sk i a , T. K o t o w sk i b an d L. Wo l ni ewic z c
a
In st it ut e of P hysi cs, Po l i sh Ac ademy of Sciences
al . Lo tni k§w 32/ 46, 02- 668 Warszawa , Po l and
b
Insti tute of E xp eri m ental Physi cs, Wa rsaw Uni versi ty
Ho âa 69, 00-681 W arszawa, Pol and
c
Insti tute of Physi cs, Ni col as Cop erni cus Uni versi ty
G rudzi ¨ dzk a 5, 87-100 T oru ¥, Po land
( Recei ved A pr i l 11 , 2002; i n Ùnal for m June 18, 2002)
Bi ndin g energ y and ex pectati on v al ues of t he interparticle distances of
muonium hydride are calculated variationall y w ith a wave function dependent exp onential ly on three interparticle distances .
PAC S num b ers: 36.10.Gv , 61. 72.V v
1. I n t r o d u ct io n
The muo nium hydri de, MuH, a system consisti n g of the muo ni um ato m Mu
and a hydro gen ato m H i s, l i ke the p ositro ni um hydri de and the H 2 m ol ecule,
a four- parti cle neutra l system . The pro to n to positro n m ass ra ti o is m p = m e =
1836: 152 6675 , the pro to n to muo n m ass ra ti o is m p = m ñ = 8 :8 8 0 2 4 4 0 8 [1].
In m easurem ents of the m uon spin precession i n m agneti c Ùeld [2{ 5], muo ni um is observed by means of i ts tri pl et state precessi on [2{ 4], and by the p ositi ve
m uo n spin rel axa ti on [3]. Muo ni um centers are i nvesti gated i n i nsul ato rs [3, 6, 7]
and i n elemental [3, 8] and com pound [3, 7, 9{ 14] semi conducto rs. Muo ni um form ati on i n l i qui ds [2], i n gases [2, 3, 15] and i n condensed rare gases [16], is of i nterest
for a com pari son of Mu form ati on ra te consta nts wi th tho se of the hydro gen ato m.
Muo ni um hydri de app ears i n the Mu reacti on of hydro gen ato m abstra cti on f rom
l i near mol ecules [2]. Muo ni um m ay b ecome i ncorp ora ted i n a vari ety of m ol ecules,
and chemi cal consequences of repl acem ent of hydro gen by muo ni um i n a m ol ecule
can b e observed [2, 3]. Theref ore, considera ti on of m uonium bound to a neutra l
ato m wi th an i m mobi le nucl eus i s al so of i nterest.
(3 51)
352
M . Su˜c zy¥ ski , T . Ko t owski , L. Wol ni ewi cz
2 . T h e w a ve fu n ct io n
W e cal cul ated the gro und sta te of muo ni um hydri de wi th a vari ati ona l wa ve
functi on, symm etri zed i n the coordi nates of the two el ectro ns of the m ol ecule,
as describ ed previ ousl y [17{ 21]. The wave functi on wa s a l i near com bi nati o n of
m ono mi al s i n four i nterpa rti cle di stances, wi th an overa l l exp onenti al facto r dep endent on the three lepto n di stances f rom the most massive parti cle. The most
m assive parti cle was a pro to n i n m uoni um hydri de, MuH, a deutero n wi th mass
m d = 3 6 7 0 : 4 8 2 9 5 5 0 m e i n Mu D [1], a tri to n wi th m ass m t = 5 4 9 6 :9 1 8 m e i n MuT
[22, 23], and an i mmo bi le pro to n in MuH . The correspondi ng reduced masses ñ r
of the muo n are gi ven i n T abl e I. The vari ati onal wav e functi on was [18]
1
` (rij ) = ˆ
!
(1 )
(k rij );
X
ˆ
!
(rij ) = ˆ
!
( r 1; r 2; r
12;
r3 ;r
13;
r
23)
=
c mn
pq
(1 + P 1 2 ) j m n p q i ;
(2 )
m n pq
j
mnpq
m
i
n
p q
13r 3r 12
= r1 r
exp ( À
˜ r
1
À
Ùr
2
À
(3 )
r3 )
wi th nonneg ati ve i ntegers m; n; p; q , and the sum m + n + p + q ç ! . The r 1 and r 2
are the pro to n{ el ectro n di sta nces, r 1 2 i s the di sta nce b etween the two el ectro ns, r 1 3
and r 2 3 are the m uo n{el ectro n di sta nces, and r 3 i s the pro to n{ muon di sta nce. P 1 2
p erm utes the indi ces 1, 2. The va lues of the nonl i near pa ra meters ˜ , Ù and of the
scali ng facto r k were determ i ned by extra p ol ati on fro m v alues of Refs. [18, 19, 21]
and then by successi ve tri al s.
3 . Th e g r o u n d st at e en er g y
The gro und sta te energy o f the m uoni um m ol ecule was com puted wi th a
wa ve functi on com pri sing al l m onom ials wi th the i nterpa rti cle di sta nce p owers
who se sum ! di d not exceed 7, whi ch renders 330 l i near term s.
T A BLE I
T he ground state energy of the muonium
ñ r
À
E
molecules.
D
[eV ]
Mu H
185. 841
1.1389
3. 8527
Mu D
195. 742
1.1395
3. 8646
199. 273
1.1397
3. 8686
206.76827
1.1401
3. 8766
MuT
MuH
1
T abl e I gi ves, for the muo n reduced mass ñ r (i n uni ts of electro n m ass),
the com puted gro und sta te energy E of the muo ni um m olecul e (i n ato m ic uni ts,
1 a: u : = 27: 211 383 4 eV [1]), and the energy D , i n eV, of di ssociati on i nto a
hydro gen ato m and a muoni um ato m . For the tri to n mass we used the l arg er of
Si ze of Muoni um H ydr i de
353
the two quo ted [22, 23] values, but thi s do es no t a˜ect the val ue of the muo n
reduced mass wi thi n the di gi ts reta i ned by us.
4 . Th e i n ter par t ic l e d i st an ces
T abl e II di splays the exp ectati on val ues of the i nterpa rti cle di sta nces i n the
m uo nium hydri de com puted wi th the opti mi zed wave functi on, i n uni ts of the
el ectro n Bo hr ra di us a B = 0: 529 177 2083 È 10 8 cm [1].
À
T ABLE II
Exp ectation v alues of the interparticle
distances.
Inverses of the exp ectation values of the inverse distances
hr
1
À
1
i
À
1
r
r
r
MuH
1.115
1.781
1.157
1.466
MuH
1.114
1.778
1.155
1.462
Exp ectation v alues of the Ùrst pow ers of distances
r
r
r
r
MuH
1.600
2.283
1.670
1.534
MuH
1.597
2.279
1.666
1.528
Exp ecta ti on val ues of the i nverse i nterpa rti cle di sta nces, h r i , were com puted i n the cal cul ati on of the Coul omb p otenti al energy of the system , wi th the
330- term wa ve functi o n wi th opti m i zed nonl inear pa ra meters. Exp ecta ti o n val ues
of the Ùrst p ositi ve power of di sta nces, h r i , were com puted wi th ! = 6 , i .e. wi th
210- term wave functi on.
The computed h r
i
i n MuH i s l arg er tha n i n H by 3% [24]. The e˜ecti ve
m uo n{el ectro n separa ti on esti m ated fro m a p oi nt- di p ol e m odel of ani sotro py of
the hyp erÙne coupl ing consta nt [25] for Mu i n Si is b etween 1.8 and 2.1 ¡A, and
l arg er i n I I{ VI com p ound semi conducto rs [14], suggesti ve of an extended el ectro n
wa ve functi on.
At concl usion of the previ ous com puta ti on [26] was stressed the desira bi l i ty
of a com puta ti on wi th a n i mpro ved precision. The present computa ti ons were p erform ed wi th a REAL * 16 preci sion. The results of the computa ti ons show tha t the
m uo nium hydri de i s i n i ts size sim i lar to the neutra l hydro gen m olecule [18, 21,
24, 26] ra ther tha n to a p ositro ni um hydri de [20]. They qua nti ta ti vely corro bora te the opi ni on [3], supp orted by experi m enta l evidence: \ The i m p orta nce of
i m pl an ted- muon studi es li es i n the i nfo rm atio n tha t can b e obta i ned vi a compa ri son of m uon and pro to n b ehavi our. In aspects of chem i cal physi cs, the equi val ent
compa ri son i s b etween muo ni um and hydro gen" .
À
À
À
354
M. Su˜czy¥ski , T . Kot owski , L. Wol ni ewi cz
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