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Electron capture by proton from muonium
Physica 106C (1981) 135-141 North-Holland Publishing Company
ELECTRON CAPTURE BY PROTON FROM MUONIUM S. BHATTACHARYYA Gokhale College, 1/1 H. Mukheriee Road, Calcutta 700020, India
and L CHATTERJEE and T. ROY Jadavpur University, Calcutta 700032, lndia
Received 22 August 1980
The differential cross-section (DCS) for charge transfer in proton-muonium (M) collisions with the subsequent formation of H(ls) and a muon (~+) is computed. The angular distribution of U+ at energies 40, 50, 100 and 500 eV of the incident proton, is obtained. The DCS is found to decrease with the increase of the scattering angle of the ~+ at all energies of the incident proton.
1. Introduction Muonium is the bound state of the positive muon and an electron. From the viewpoint of muon electrodynamics muonium is of great interest because it provides the simplest bound system in which the interaction of the muon and the electron can be studied. Muonium interaction in gases and condensed matter is of importance to atomic and molecular collision theory and to solid state physics and astrophysics. Muonium can be considered as an isotope of hydrogen in which the proton is replaced by the positive muon. Muonium behaves very much like hydrogen in its atomic and molecular interactions and in its chemical reactions, and it is an active paramagnetic atom. Formation of muonium by collision of the muon with inert gases and measurement of energy levels of muonium are extensively reviewed by Hughes and Kinoshita [1 ]. Different decay modes of muonium have been studied by the authors [2, 3]. Here we have limited our study to the detachment of the electron from muonium by proton impact and the subsequent production of the Is-hydrogen atom. The computation is done considering only second order Coulomb interaction. Total as well as differential cross-sections for reactions like H(1 s) 4- H+ ~ ~ H(n,/) 4- H+, n,l
He + + I-1+ ~ H(ls) + He ++,
Ar + H + -+ ~ Hfn,/)+ Ar+, n,l
have been studied very recently by Kleber and Nagarajan [4], Mukherjee, Sil and Basu [5] ; Kubach and Sidis [6] and Benoit, Kubach, Sidis, Pommier and Barat [7]. We have to determine how the charge transfer reaction cross-section 0378--4~63/81/0000-0000/$2.50 © North-Holland Publishing Company
S. Bhattacharyya et at/Electron capture by proton from muonlum
136
between proton and muonium compares with the above reaction cross-sections. Since the lifetime of this shortlived bound system is of the order of 10 -6 s, there is every possibility for a proton to capture an electron in a low energy encounter before the weak decay of the muon into a positron and two neutrinos. From the reaction crosssection we find that the muonium behaves like an ordinary atom and shows the usual feature of a decrease of the DCS with the increase of the angle of ejection of the muon and also with the increase in the energy of the incident proton.
2. Mathematical formalism The process under consideration is M + H+ -~ ja+ + H(I s).
(1)
Hydrogen and muonium are assumed to be in the ground state. We further assume that the polarisation vector of the muon remains unaltered after interaction. The dominant term in the S-matrix of the process is the second order Coulomb interaction given by
Hc =fOp(r)pe,(r')d3rd3r, + f J
O"
P
P
p_(r)p~,!r) d3r d3,,. ° Ir-rl
Ir-rl
(2)
Op, Pc, Pu are the charge densities of proton, electron and muon, respectively. Let 0(x), if(x) and ~b'(x) be the respective operators of the fields of proton, electron and muon. In the non-relativistic limit, the Pauli-SchriSdinger solutions for them are
O(r) : ~, f bs(P) •s
exp (ipr) d3p,
$
~b(r) = ~ far(P) Xr exp
(ipr) d3p,
y
~b'(r) =
~ fct(P) X't exp (ipr) d3p.
(3)
t
bs(p), ar(P ) and Cr(P) are the annihilation operators for the corresponding fields and ~,, ×, ×' are the respective Pauli spin symbols. The amplitude for the process is given by e2 Mfi-=
< 'ftUcl 'i> - M p,
(4)
where M~p is the contribution from the muon-proton interaction and Mep is the contribution from the electronproton interaction in (2). I~bi) and I~l,f), the initial and final state vectors for the process, are given, respectively, by IqJi>= E f ffa~(/11213) abe
at,(/1) b~ (!2) c~ (/3) 10>,
(5)
S~Bhattacharyyaet aL/Electroncaptureby proton from muonium
137
and
l'f)
y~Se:,(ktk2k3)atr(kl)bts(kgc~(k3)lO).
= ~
(6)
clef The g function is the Fourier transform in momentum space of the solution of the Schr~Sdinger equation for the systems of particles contained in the initial and the final states [8]. In the initial state, there are one free proton and one electron bound to a muon. In the final state there are one free muon and a hydrogen atom in the ground state• Substituting (2), (5) and (6) into (4) we get, after computing for vacuum expectation values, a product of five Dirac delta functions and making proper integrations to eliminate them, we obtain e2 Mvp =~'.t (2n)3 ~
•
abc,de/
X exp (i(k 2 Irl
fgraffc(lll21~grje:(llk2k~63(k2-12 + k 3 - / 3 )
12)r)
Xs Xt' ~* Xt'*, d 3 r d 3 i 1 d3/2 d313d3k2 d3k3 ,
(7)
e2
MeP =~" (2n)3abc, ~ de:fgzs~(II1219~Je~klk213)63(kl -II + k2-12) X
exp {i(k 2 - l ~ r } Irl
.
Xr As X~ ~s' d3r d311 d312 d313 d3kl d3k2 •
(8)
Breaking into centre-of-mass (C.M.) and relative coordinates we get for the initial system of the muonium-proton wave function 1
g~aS~(il121~-(2rr)3/253(ll +12 +13 - Pc) F m ( 1 3 - ~ i l )
X Gp ( / 2 - M / I )
8arSbs~ct,
(9)
where F m and G. are the wave functions of the muonium and the proton in momentum space, while M,/a and m are the masses ofVproton, muon and electron, respectively• Similarly for the final system of the muon and the hydrogen atom
(lO) t
F~I and Gv are the ground state wave functions of the hydrogen atom and the free muon in momentum space, respectively. In the following Pc and Qc will denote the C.M. momenta of the initial and final systems, respectively. Substituting (9) and (10) in (7) and changing into coordinate space we get after a lengthy calculation:
S. BhattacharyyaetaL /Electron capture by proton from muontum
138
e2 (r la Mup ='~.T(2~r)3~$3(Pc - Qc) ~CPls 1 - m
l exp X R'*(, 2 + O) 7"7, IPl
~_rip:M+"c1.m
(r 1 + r2)
,,
R(rl)Ckls(r2)
+m~--t°}]
X(a~bc,de.t.Saa~Sbe,cf 1 d3rld3r2d3p.
(11)
Since no spin flip is assumed to occur and the interacting beams are unpolarised
E abe,
5ad5be6cf = 8.
def
~ls(r) and ¢{s (r) are the wave functions in the ground states of the muonium and the hydrogen atom, respectively.
R(r) and R'(r) are the respective plane waves for proton and muon. Similarly we get for Mep
e2 1 I ~ b l s ( r l ) R { ~ _ . ~m Mep=_~.(2n)363(Pc_Qc)~p
# - - - r l );aM + m
, ,( M+la+m rn p) X (~lr(r2)R - M + m r 1 + r 2 + M + m
Xexp
rl
p +m M + m r 1 + r 2 + ~M
(12)
X(a~,def6adrSbe6cf) d3rld3r2d3# • Taking
~pls(r)= t--~) exp(-aur), au= (~---~)-~, a31/2 (Mm)e 2 ~Is(r)=I--~-H l exp(--aHr), all=~ "~' ,
a u and a H being the inverse Bohr radius of the muonium and the hydrogen atom, respectively, we get after integrating (1 l) and 02):
:
m.p = 7, (2~)3#fec - Qc)
{"7:
(aHa~)S/2(XYZ) -2
t
,
(13)
S. Bhattacharyya et al /Electron capture by proton from muonium e2 ( 32(2rr)3 "" 2 MeP ='~.(27r)363(Pc-Qc)~ n (aHa~)~/Z(a~ + S2)-2(a2H + T2)-2} '
139 (14)
where
X
=
R-
S-
~-~L
M+# +m
M+m
M+l~+m
U M+m
(
I+
p - Q
+P-O
M
)
M+t~+m
P
,
,
T=
M+la+m
M+#+m M+m
Q,
Q.
(15)
P is the incident momentum of the proton relative to muonium and Q and K are the momenta of the outgoing muon and the hydrogen, respectively. We have used the relation that
]kip Pc = Qc -
M+~+rn
IzQ + (M + m)K Iz+M+m
Finally the matrix element for the whole process is Mfi = (e2/2[)(2n)363(P C - Qc)(2n)3(32) (8/lr) (aHau)5/2(A - B),
(l 6)
where
A : (XYZ) -2, and
s --R-2(a.2+ s2)-2(.~ + r2) -2. Assuming the muonium to be at rest relative to the proton, the phase space factor for the reaction is 1
(2~r)6 M/~I Q dE
~ d ~ d3K.
(17)
Hence the differential cross-section becomes d~(0) e 4 (2n) 6 2 5 df~u - 4 M# - - - ~ (256) (auaH) ( . 4 - B) 2.
(18)
S~ Bhattacharyya et aL/Electron capture by proton from muonium
140
Table I Differential cross-section (in atomic units) for the reaction M + H+ -* tz+ + H at different energies of the incident proton Angle of scattering of u + (deg)
Energy of incident proton (eV) 40
50
100
S00
0 5 10 20 30 40 50 60 70
0.1638 -5* 0.1595 -5 0.1477 -5 0.1105 -5 0.7179 -6 0.4247 -6 0.2435 -6 0.1349 -6 0.7052 -7
0.1763 -6 0.1718 -6 0.1591-6 0.1190 -6 0.7730 -7 0.4607 -7 0.2623 -7 0.1455 -7 0.7634 -8
0.1732 -9 0.1688 -9 0.1563 -9 0.1169 -9 0.7591-1° 0.4281-1° 0.2579 -10 0.1434 - l ° 0.7575 -11
0.1782 -16 0.1736 -16 0.1607 -16 0.1202 -16 0.7809 -17 0.4652 -17 0.2655 -17 0.1479 -17 0.7853 -18
*Superscripts indicate powers of 10.
3. Results We have computed the differential cross-section given by (18) for energies 40, 50, 100 and 500 eV of the incident proton. The m o m e n t u m and energy conservation relations are given by
-f=-d + x-', p2 = Q 2
2M
(19) K2 (20)
i.t2 + 2(M ~- m---~)+ e"
e is the difference o f the binding energy between hydrogen and muonium. If we confine our observation of the ejected muon, only in the plane o f P and K, we get from (19) and (20):
[Q[ =__-7=7-'.--uIPI cos 0 +
M+ta
1 2M+/a
r-x/(2/alpi cos 0) 2 - 8(M + ta)M/ae,
(21)
where 0 is the angle between P and Q. From (21) it is evident that ejection of the muon after charge transfer is limited to 0 ° ~ 0 < 90 °. The angular distribution o f the muons giving rise to the formation of hydrogen atoms in the ground state is shown in table I.
4. Discussion The expression (18) for the DCS contains a factor (,4 - B) 2. A is proportional to the m u o n - p r o t o n interaction amplitude (MuD) and B is proportional to the e l e c t r o n - p r o t o n Coulomb amplitude ( M ~ ) . At the incident p r o t o n energies under'consideration the value o f A comes out to be o f the order of 10 - 3 0 and less, whereas the order o f magnitude o f B is 10 -15 and more. This shows that the m u o n - p r o t o n Coulomb repulsion plays an insignificant role compared to the e l e c t r o n - p r o t o n Coulomb attraction in this charge transfer reaction. However, when the energy o f the incident proton reaches the MeV range, Coulomb repulsion between the muonium and the proton cannot be ignored, because the term A then gives an appreciable contribution compared to B. The cross-section for p r o t o n - h y d r o g e n charge exchange phenomena as given by Kubach and Sidis [6] is much higher compared to the
S. Bhattacharyya et aL/Electron capture by proton from muonium
141
m u o n i u m - p r o t o n charge exchange cross-section computed here. The masses of the interacting particles, which comes in the form of a product in the phase space factor and also in the expression for the amplitude may be responsible for this large difference in the magnitude of the cross-sections. Finally an interesting fact is that the characteristic feature of the gradual decrease of the DCS, with increasing energy of the incident proton and with increasing angle of ejection of the muon is maintained here as in all other cases.
Acknowledgements S. Bhattacharyya would like to thank UGC for the award of research grant, and L. Chatterjee would like to thank C.S.I.R. for the award of a Senior Fellowship.
References [ 1] [2] [3] [4] [5] [6] [7] [8]
V.W. Hughes and T. Kinoshita, Muon Physics, VoL III (1977). L. Chatterjee, S. Bhattachaxyya and T. Roy, Acta Phys. Polon. B11 (1980) 635. L. Chatterjee, S. Bhattachaxyya and T. Roy, Communicated for publication (1980). M. Kleber and M. A. Nagarajan, J. Phys. B 8 (1975) 643. S. Mukherjee, N. C. Sil and D. Basu, J. Phys. B 12 (1979) 12.59. C. Kubaeh and V. Sidis, J. Phys. B 8 (1975) 638. C. Benoit, C. Kubach, V. Sidis, J. Pommier and M. Barat, J. Phys. B 10 (1977) 1661. Sujata Bhattaeharyya and T. Roy, Bull. Calcutta Math. Soc. 70 (1978) 89.
ELECTRON CAPTURE BY PROTON FROM MUONIUM S. BHATTACHARYYA Gokhale College, 1/1 H. Mukheriee Road, Calcutta 700020, India
and L CHATTERJEE and T. ROY Jadavpur University, Calcutta 700032, lndia
Received 22 August 1980
The differential cross-section (DCS) for charge transfer in proton-muonium (M) collisions with the subsequent formation of H(ls) and a muon (~+) is computed. The angular distribution of U+ at energies 40, 50, 100 and 500 eV of the incident proton, is obtained. The DCS is found to decrease with the increase of the scattering angle of the ~+ at all energies of the incident proton.
1. Introduction Muonium is the bound state of the positive muon and an electron. From the viewpoint of muon electrodynamics muonium is of great interest because it provides the simplest bound system in which the interaction of the muon and the electron can be studied. Muonium interaction in gases and condensed matter is of importance to atomic and molecular collision theory and to solid state physics and astrophysics. Muonium can be considered as an isotope of hydrogen in which the proton is replaced by the positive muon. Muonium behaves very much like hydrogen in its atomic and molecular interactions and in its chemical reactions, and it is an active paramagnetic atom. Formation of muonium by collision of the muon with inert gases and measurement of energy levels of muonium are extensively reviewed by Hughes and Kinoshita [1 ]. Different decay modes of muonium have been studied by the authors [2, 3]. Here we have limited our study to the detachment of the electron from muonium by proton impact and the subsequent production of the Is-hydrogen atom. The computation is done considering only second order Coulomb interaction. Total as well as differential cross-sections for reactions like H(1 s) 4- H+ ~ ~ H(n,/) 4- H+, n,l
He + + I-1+ ~ H(ls) + He ++,
Ar + H + -+ ~ Hfn,/)+ Ar+, n,l
have been studied very recently by Kleber and Nagarajan [4], Mukherjee, Sil and Basu [5] ; Kubach and Sidis [6] and Benoit, Kubach, Sidis, Pommier and Barat [7]. We have to determine how the charge transfer reaction cross-section 0378--4~63/81/0000-0000/$2.50 © North-Holland Publishing Company
S. Bhattacharyya et at/Electron capture by proton from muonlum
136
between proton and muonium compares with the above reaction cross-sections. Since the lifetime of this shortlived bound system is of the order of 10 -6 s, there is every possibility for a proton to capture an electron in a low energy encounter before the weak decay of the muon into a positron and two neutrinos. From the reaction crosssection we find that the muonium behaves like an ordinary atom and shows the usual feature of a decrease of the DCS with the increase of the angle of ejection of the muon and also with the increase in the energy of the incident proton.
2. Mathematical formalism The process under consideration is M + H+ -~ ja+ + H(I s).
(1)
Hydrogen and muonium are assumed to be in the ground state. We further assume that the polarisation vector of the muon remains unaltered after interaction. The dominant term in the S-matrix of the process is the second order Coulomb interaction given by
Hc =fOp(r)pe,(r')d3rd3r, + f J
O"
P
P
p_(r)p~,!r) d3r d3,,. ° Ir-rl
Ir-rl
(2)
Op, Pc, Pu are the charge densities of proton, electron and muon, respectively. Let 0(x), if(x) and ~b'(x) be the respective operators of the fields of proton, electron and muon. In the non-relativistic limit, the Pauli-SchriSdinger solutions for them are
O(r) : ~, f bs(P) •s
exp (ipr) d3p,
$
~b(r) = ~ far(P) Xr exp
(ipr) d3p,
y
~b'(r) =
~ fct(P) X't exp (ipr) d3p.
(3)
t
bs(p), ar(P ) and Cr(P) are the annihilation operators for the corresponding fields and ~,, ×, ×' are the respective Pauli spin symbols. The amplitude for the process is given by e2 Mfi-=
< 'ftUcl 'i> - M p,
(4)
where M~p is the contribution from the muon-proton interaction and Mep is the contribution from the electronproton interaction in (2). I~bi) and I~l,f), the initial and final state vectors for the process, are given, respectively, by IqJi>= E f ffa~(/11213) abe
at,(/1) b~ (!2) c~ (/3) 10>,
(5)
S~Bhattacharyyaet aL/Electroncaptureby proton from muonium
137
and
l'f)
y~Se:,(ktk2k3)atr(kl)bts(kgc~(k3)lO).
= ~
(6)
clef The g function is the Fourier transform in momentum space of the solution of the Schr~Sdinger equation for the systems of particles contained in the initial and the final states [8]. In the initial state, there are one free proton and one electron bound to a muon. In the final state there are one free muon and a hydrogen atom in the ground state• Substituting (2), (5) and (6) into (4) we get, after computing for vacuum expectation values, a product of five Dirac delta functions and making proper integrations to eliminate them, we obtain e2 Mvp =~'.t (2n)3 ~
•
abc,de/
X exp (i(k 2 Irl
fgraffc(lll21~grje:(llk2k~63(k2-12 + k 3 - / 3 )
12)r)
Xs Xt' ~* Xt'*, d 3 r d 3 i 1 d3/2 d313d3k2 d3k3 ,
(7)
e2
MeP =~" (2n)3abc, ~ de:fgzs~(II1219~Je~klk213)63(kl -II + k2-12) X
exp {i(k 2 - l ~ r } Irl
.
Xr As X~ ~s' d3r d311 d312 d313 d3kl d3k2 •
(8)
Breaking into centre-of-mass (C.M.) and relative coordinates we get for the initial system of the muonium-proton wave function 1
g~aS~(il121~-(2rr)3/253(ll +12 +13 - Pc) F m ( 1 3 - ~ i l )
X Gp ( / 2 - M / I )
8arSbs~ct,
(9)
where F m and G. are the wave functions of the muonium and the proton in momentum space, while M,/a and m are the masses ofVproton, muon and electron, respectively• Similarly for the final system of the muon and the hydrogen atom
(lO) t
F~I and Gv are the ground state wave functions of the hydrogen atom and the free muon in momentum space, respectively. In the following Pc and Qc will denote the C.M. momenta of the initial and final systems, respectively. Substituting (9) and (10) in (7) and changing into coordinate space we get after a lengthy calculation:
S. BhattacharyyaetaL /Electron capture by proton from muontum
138
e2 (r la Mup ='~.T(2~r)3~$3(Pc - Qc) ~CPls 1 - m
l exp X R'*(, 2 + O) 7"7, IPl
~_rip:M+"c1.m
(r 1 + r2)
,,
R(rl)Ckls(r2)
+m~--t°}]
X(a~bc,de.t.Saa~Sbe,cf 1 d3rld3r2d3p.
(11)
Since no spin flip is assumed to occur and the interacting beams are unpolarised
E abe,
5ad5be6cf = 8.
def
~ls(r) and ¢{s (r) are the wave functions in the ground states of the muonium and the hydrogen atom, respectively.
R(r) and R'(r) are the respective plane waves for proton and muon. Similarly we get for Mep
e2 1 I ~ b l s ( r l ) R { ~ _ . ~m Mep=_~.(2n)363(Pc_Qc)~p
# - - - r l );aM + m
, ,( M+la+m rn p) X (~lr(r2)R - M + m r 1 + r 2 + M + m
Xexp
rl
p +m M + m r 1 + r 2 + ~M
(12)
X(a~,def6adrSbe6cf) d3rld3r2d3# • Taking
~pls(r)= t--~) exp(-aur), au= (~---~)-~, a31/2 (Mm)e 2 ~Is(r)=I--~-H l exp(--aHr), all=~ "~' ,
a u and a H being the inverse Bohr radius of the muonium and the hydrogen atom, respectively, we get after integrating (1 l) and 02):
:
m.p = 7, (2~)3#fec - Qc)
{"7:
(aHa~)S/2(XYZ) -2
t
,
(13)
S. Bhattacharyya et al /Electron capture by proton from muonium e2 ( 32(2rr)3 "" 2 MeP ='~.(27r)363(Pc-Qc)~ n (aHa~)~/Z(a~ + S2)-2(a2H + T2)-2} '
139 (14)
where
X
=
R-
S-
~-~L
M+# +m
M+m
M+l~+m
U M+m
(
I+
p - Q
+P-O
M
)
M+t~+m
P
,
,
T=
M+la+m
M+#+m M+m
Q,
Q.
(15)
P is the incident momentum of the proton relative to muonium and Q and K are the momenta of the outgoing muon and the hydrogen, respectively. We have used the relation that
]kip Pc = Qc -
M+~+rn
IzQ + (M + m)K Iz+M+m
Finally the matrix element for the whole process is Mfi = (e2/2[)(2n)363(P C - Qc)(2n)3(32) (8/lr) (aHau)5/2(A - B),
(l 6)
where
A : (XYZ) -2, and
s --R-2(a.2+ s2)-2(.~ + r2) -2. Assuming the muonium to be at rest relative to the proton, the phase space factor for the reaction is 1
(2~r)6 M/~I Q dE
~ d ~ d3K.
(17)
Hence the differential cross-section becomes d~(0) e 4 (2n) 6 2 5 df~u - 4 M# - - - ~ (256) (auaH) ( . 4 - B) 2.
(18)
S~ Bhattacharyya et aL/Electron capture by proton from muonium
140
Table I Differential cross-section (in atomic units) for the reaction M + H+ -* tz+ + H at different energies of the incident proton Angle of scattering of u + (deg)
Energy of incident proton (eV) 40
50
100
S00
0 5 10 20 30 40 50 60 70
0.1638 -5* 0.1595 -5 0.1477 -5 0.1105 -5 0.7179 -6 0.4247 -6 0.2435 -6 0.1349 -6 0.7052 -7
0.1763 -6 0.1718 -6 0.1591-6 0.1190 -6 0.7730 -7 0.4607 -7 0.2623 -7 0.1455 -7 0.7634 -8
0.1732 -9 0.1688 -9 0.1563 -9 0.1169 -9 0.7591-1° 0.4281-1° 0.2579 -10 0.1434 - l ° 0.7575 -11
0.1782 -16 0.1736 -16 0.1607 -16 0.1202 -16 0.7809 -17 0.4652 -17 0.2655 -17 0.1479 -17 0.7853 -18
*Superscripts indicate powers of 10.
3. Results We have computed the differential cross-section given by (18) for energies 40, 50, 100 and 500 eV of the incident proton. The m o m e n t u m and energy conservation relations are given by
-f=-d + x-', p2 = Q 2
2M
(19) K2 (20)
i.t2 + 2(M ~- m---~)+ e"
e is the difference o f the binding energy between hydrogen and muonium. If we confine our observation of the ejected muon, only in the plane o f P and K, we get from (19) and (20):
[Q[ =__-7=7-'.--uIPI cos 0 +
M+ta
1 2M+/a
r-x/(2/alpi cos 0) 2 - 8(M + ta)M/ae,
(21)
where 0 is the angle between P and Q. From (21) it is evident that ejection of the muon after charge transfer is limited to 0 ° ~ 0 < 90 °. The angular distribution o f the muons giving rise to the formation of hydrogen atoms in the ground state is shown in table I.
4. Discussion The expression (18) for the DCS contains a factor (,4 - B) 2. A is proportional to the m u o n - p r o t o n interaction amplitude (MuD) and B is proportional to the e l e c t r o n - p r o t o n Coulomb amplitude ( M ~ ) . At the incident p r o t o n energies under'consideration the value o f A comes out to be o f the order of 10 - 3 0 and less, whereas the order o f magnitude o f B is 10 -15 and more. This shows that the m u o n - p r o t o n Coulomb repulsion plays an insignificant role compared to the e l e c t r o n - p r o t o n Coulomb attraction in this charge transfer reaction. However, when the energy o f the incident proton reaches the MeV range, Coulomb repulsion between the muonium and the proton cannot be ignored, because the term A then gives an appreciable contribution compared to B. The cross-section for p r o t o n - h y d r o g e n charge exchange phenomena as given by Kubach and Sidis [6] is much higher compared to the
S. Bhattacharyya et aL/Electron capture by proton from muonium
141
m u o n i u m - p r o t o n charge exchange cross-section computed here. The masses of the interacting particles, which comes in the form of a product in the phase space factor and also in the expression for the amplitude may be responsible for this large difference in the magnitude of the cross-sections. Finally an interesting fact is that the characteristic feature of the gradual decrease of the DCS, with increasing energy of the incident proton and with increasing angle of ejection of the muon is maintained here as in all other cases.
Acknowledgements S. Bhattacharyya would like to thank UGC for the award of research grant, and L. Chatterjee would like to thank C.S.I.R. for the award of a Senior Fellowship.
References [ 1] [2] [3] [4] [5] [6] [7] [8]
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