Contribution of $α^2$-terms to the total interaction cross sections of relativistic elementary atoms with atoms of matter

5 Pages·0.15 MB·English

by  L. Afanasyev

Contribution of α2-terms to the total interaction cross sections of relativistic
elementary atoms with atoms of matter
L.Afanasyev,∗ A.Tarasov,† and O.Voskresenskaya‡
Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia
(Dated: February 1, 2008)
Itisshownthatthecorrectionsofα2 ordertothetotalcrosssectionsforinteractionofelementary
hydrogen-likeatomswithtargetatoms,reportedinthepreviouslypublishedpaper[S.Mr´owczyn´ski,
Phys.Rev. D36, 1520 (1987)], do not include some terms of the same order of magnitude. That
results in a significant contribution of these corrections in particular cases. The full α2-corrections
havebeenderivedanditisshownthattheyarereallysmallandcouldbeomittedformostpractical
applications.
2
PACSnumbers: 11.80.Fv,34.50.-s,36.10.-k
0
0
2
I. INTRODUCTION performed conveniently in the rest frame of the projec-

n tile EA (anti-lab frame). As the characteristic transfer
a momentumisoftheorderoftheEABohrmomentum,in
J The experiment DIRAC [1], now under way at PS
CERN’s, aims to measure the lifetime of hydrogen-like this frame after the interactionEAhas a non-relativistic
0
3 elementary atoms (EA) consisting of π+ and π− mesons velocityandthustheinitialandfinalstatesofEAcanbe
(A2π)withanaccuracyof10%. Theinteractionofπ+π−- tInretahtiesdminantneermrtshoefwneolln-k-rneolawtnivdisitffiiccuqlutiaenstoufmthmeercehlaatniviciss-.
2 atoms with matter is of greatimportance for the experi-
v mentasA dissociation(ionization)insuchinteractions tic treatment of bound states can be get round.
2π
8 As in the EA restframe a targetatom moveswith the
isexploitedtoobserveA andtomeasureitslifetime. In
0 the experiment the ratio2πbetween the number of π+π−- relativistic velocity, its electromagnetic field is no longer
2 pure Coulomb. It is described by the 4-vector poten-
pairs fromA dissociationinside a targetandthe num-
9 2π
tial A = (A ,A) with components related to its rest
0 ber of produced atoms will be measured. The lifetime µ 0
Coulomb potential U(r):
1 measurement is based on the comparison of this experi-
0 mental value with its calculated dependence on the life-
A =γU A=γβU . (1)
/ time. The accuracy of the cross sections for interaction 0
h
p ofrelativisticEAwith ordinaryatoms,whicharebehind Here β = v/c, v is the target atom velocity in the EA
- allthese calculations[2], is essentialfor the extractionof restframeandγ isitsLorentz-factor. Thetime-likecom-
p
the lifetime.
e ponentA0 ofthe4-potentialinteractswiththechargesof
h Study of interactions of fast hydrogen-like atoms with the particles forming EA and the space component with
: atoms has a long history starting from Bethe. One of their currents.
v
i theresentcalculationsforhydrogenandone-electronions InthispaperweconsideronlyEAconsistingofspinless
X was published in [3]. Interactions of various relativistic particles (π, K-mesons etc.) which are of interest for
r EA consisting of e±, π±, µ±, K± were considered in the DIRAC experiment. In the Born approximation the
a
differentapproaches[4,5,6,7,8,9,10,11,12,13,14,15, amplitudes of transition from the initial state i to the
16]. In this paper we reconsider corrections of α2 order finalonef duetotheinteractionwithA canbewritten
µ
tothe EAtotalinteractioncrosssectionsobtainedin[7]. as:
(Through this paper α is the fine-structure constant.)
A =U(Q)a (q), (2)
fi fi
∞
sinQr
II. GENERAL FORMULAS U(Q)=2 U(r) rdr , (3)
Z Q
0
As shown in [7] analysis of the relativistic EA inter- a (q)=ρ (q) βj (q). (4)
fi fi fi
−
action with the Coulomb field of target atoms can be
The transitiondensities ρ (q) andtransitioncurrents
fi
j (q) are expressed via the the EA wave functions ψ
fi i
and ψ for the the initial and final states:
f
∗Also at: CERN EP Division, CH-1211 Geneva 23, Switzerland;
Electronicaddress: [email protected] ρ (q)= ρ (r) eiq1r e−iq2r d3r, (5)
†Current address: Institute for Theoretical Phisics, University of fi Z fi −
Heidelberg, Philosophenweg 19, D-69120, Heidelberg, Germany; (cid:0) (cid:1)
µ µ
Electronicaddress: [email protected] j (q)= j (r) eiq1r+ e−iq2r d3r , (6)
‡Electronicaddress: [email protected] fi Z fi (cid:18)m m (cid:19)
1 2
2
ρ (r)=ψ∗(r)ψ (r) (7) energy ε , see bellow). Then, using the completeness
fi f i f
relation
i
j (r)= ψ (r)∇ψ∗(r) ψ∗(r)∇ψ (r) . (8)
fi 2µ i f − f i f f =1, (14)
(cid:2) (cid:3) | ih |
Xf
The EA wave functions ψ and the binding energies
i,f
ε obey the Schr¨odinger equation we can writte sum (12) in the form:
i,f
1
Hψi,f =εi,fψi,f (9) σitot = β2 Z hi|A∗(q)A(q)|iid2qT . (15)
∆
H = +V(r) b b
−2µ
III. SIMPLIFIED APPROACH
with the Hamiltonian H. It is worth noting that the
explicit form of the potential V(r) of the interaction be-
One should take some caution when passing from the
tween the EA components have no influence on the final
exact expressions (2–10) for the transition amplitudes,
result of this paper.
withexplicitdependenceontheε (throughthetime-like
f
In the above equations m are masses of EA com-
1,2 q and longitudinal q components of 4-vector q), to the
ponents, q = (q ,q) is the transfer 4-momentum. All 0 L
0 approximate one without such dependence. Otherwise,
other kinematic variables are related by the following
it is possible to obtain a physically improper result as it
equations:
has happened to the authors of the paper [7] at deriving
µ µ m m of the sum rules for the total cross section of interaction
q1 = q, q2 = q, µ= 1 2 , M =m1+m2, of ultrarelativistic EA (β =1) with targetatoms. Below
m m M
1 2
we discuss this problem in detail.
q =(q ,q), q =(q ,q ),
0 L T The most essential simplification, that arises in the
Q2 case of β = 1 is that Q2 = q2. Thus U(Q) = U(q )
q =ω + =βq=βq , ω =ε ε , (10) T T
0 fi 2M L fi f − i [see (10)] and only Afi in (2) depends on εf throughthe
Q= Q2 , Q2 =q2 q2 =q2 +q2(1 β2). exponentialfactorsexp(iq1r)andexp( iq2r)in(5)and
− 0 T L − (6) −
p
The differential and integral cross sections of the EA µ µ
q r = qr = (q z+q r ), (16)
transition from the initial state i to the final state f due 1,2 m m L T T
1,2 1,2
tointeractionwiththeelectromagneticfieldofthetarget
atom are related to amplitudes (2): where qL =ωfi+qT2/2M if β =1.
Now let us take into account the fact that the typical
dσ 1 valueofz inthese expressionsis ofthe orderofthe Bohr
fi = A (q)2
dqT β2| fi | radius rB = 1/µα and the typical qL ∼ ωfi ∼ µα2, thus
the product q z is of the order of α. Then it seems
1 L
σfi = β2 Z |Afi(q)|2d2qT . (11) natural to neglect the qL-dependence of afi:
a (q) a (q ). (17)
fi fi T
Formulae(2–11)allowtocalculatethetransition(par- ≈
tial) cross sections in the Born approximation. But for and consider this case as the zero order approximation
applications (for example see [2]) the total crosssections to the problem [7]. It corresponds to the choice of the
oftheEAinteractionwithtargetatomsarealsorequired. operator A in the form:
BecausetheBornamplitudesoftheEAelasticscattering
are pure real values, the optical theorem cannot be used A(q)b=U(qT) eiq1TrT e−iq2TrT
− − (18)
tocalculatethetotalcrosssections. Thustheyshouldbe b (eiq1Tr(cid:2)T/m1+e−iq2TrT/m2)βp .
calculated as the sum of all partial cross sections:
(cid:3)
Here p= i∇ is the momentum operator. b
−
σtot = σ . (12) Substituting (18) in (15) results in the following sum
i fi rules b[7], where the total cross section is expressed as
Xf
the sum of the “electric” σel and “magnetic” σmag cross
Togetaclosedexpressionforthesumofthisinfinitese- sections:
ries (the so-called “sum rule”) the transition amplitudes σtot =σel+σmag , (19)
(2) are usually rewriten as:
σel = U2(q )M(q )d2q , (20)
T T T
A (q)= f A(q)i , (13) Z
fi
h | | i
M(q )=2(1 S(q ))
T T
b −
wheretheoperatorA(q)doesnotcontainexplicitdepen-
S(q )= ψ(r)2eiqTrd3r ;
dence on the EA final state variables (for example, its T Z | |
b
3
σmag = U2(q )K(q )d2q , (21) thecorrectvaluesofσmagdonotdependonEAmassesas
T T T
Z it follows from the simplified approximation result (25).
1 2 The ratio values confirmthe above estimation about the
K(q )= + (eiqr 1) β ∇ ψ (r)2d3r .
T Z (cid:20)µ2 m m − (cid:21)| i | “magnetic” term contribution. Thus, inaccuracy in the
1 2
calculations did not allow the authors of [7] to observe
These results reproduce the ones obtained in [7] and such a significant contribution of σmag in their results.
differ from the sum rules used in [2] by the additional
termσmag. Forbeginningletusconsideritscontribution
TABLE I: The “electric” (el) and “magnetic” (mag) total
qualitatively. For this purpose the target atom poten-
crosssectionsinunitsofcm2andtheirratio(mag/el)in%for
tial U(r) can be approximatedby the screenedCoulomb
EAconsistingofπandKmesons(A2π,AπK,A2K)andtarget
potential: materials with the atomic number Z. The values published
in [7] are given in parentheses.
Zα
U(r)= e−λr, λ m αZ1/3, (22)
r ∼ e Z A2π AπK A2K
6 el 3.03·10−22 1.37·10−22 3.08·10−23
where m is the electron mass and Z is the atomic num-
e
ber of the target. The pure Coulomb wave function can (3.1·10−22) (1.4·10−22) (3.0·10−23)
beusedforψ (i.e. thecontributionofthestronginterac- 6 mag 6.73·10−24 6.73·10−24 6.73·10−24
i
tion between the EA components is neglected, see [17]). (2.5·10−24) (1.3·10−24) (0.3·10−24)
For the ground state it is written as: 6 mag/el 2.22% 4.90% 21.9%
13 el 1.33·10−21 6.08·10−22 1.37·10−22
µα3/2
ψ (r)= e−µαr . (23) (1.3·10−21) (6.2·10−22) (1.4·10−22)
i
√π
13 mag 1.89·10−23 1.89·10−23 1.89·10−23
Under such assumptions for the ground state the follow- (0.96·10−23) (0.55·10−23) (0.15·10−23)
ing results can be easily obtained: 13 mag/el 1.41% 3.10% 13.7%
29 el 6.17·10−21 2.84·10−21 6.48·10−22
σel = 8πZ2 ln 2µ 3 , (24) (6.1·10−21) (2.9·10−21) (6.7·10−22)
µ2 (cid:20) (cid:18)Z1/3me(cid:19)−4(cid:21) 29 mag 5.50·10−23 5.50·10−23 5.50·10−23
4π Zα 2 (3.6·10−23) (2.3·10−23) (0.68·10−23)
σmag = +O(α2σel)=
3 (cid:18) λ (cid:19) 29 mag/el 0.891% 1.94% 8.49%
4πZ4/3α2 47 el 1.55·10−20 7.15·10−21 1.64·10−21
= 3m2 +O(α2σel). (25) (1.5·10−20) (7.3·10−21) (1.7·10−21)
e 47 mag 1.05·10−22 1.05·10−22 1.05·10−22
It is seen that in spite of α2 in the numerator of σmag (0.79·10−22) (0.52·10−22) (0.17·10−22)
the electron mass square in the denominator makes the 47 mag/el 0.676% 1.46% 6.37%
contributionofthe“magnetic”termin(19)notnegligible 82 el 4.46·10−20 2.07·10−20 4.81·10−21
with respect to the “electric” one, especially for the case
(4.4·10−20) (2.1·10−20) (5.1·10−21)
of EA consisting of heavy hadrons and low Z values.
82 mag 2.20·10−22 2.20·10−22 2.20·10−22
To obtain exact numerical values we have precisely
(1.9·10−22) (1.3·10−22) (0.48·10−22)
repeated the calculations made in [7]. More accurate
presentation of the target atom potential, namely, the 82 mag/el 0.493% 1.06% 4.58%
Moli´ere parametrization of the Thomas-Fermi potential
[18] was used as in [7]:
It is clear that such strong enhancement of the mag-
netic termin(19)is the consequenceofits inversepower
3 c e−λir
i dependence(25)onthesmallscreeningparameterλ. Itis
U(r)=Zα ; (26)
r also easy to see that the originof such unnatural depen-
Xi=1
denceisinthebehaviourofthefactorK(q )atsmallq
c =0.35, c =0.55, c =0.1; T T
1 2 3 in(21). Thisfactor,contraryto M(q )in(20),doesnot
T
λ =0.3λ , λ =1.2λ , λ =6λ , λ =m αZ1/3/0.885. approach zero at q 0. But at β = 1 such behaviour
1 0 2 0 3 0 0 e T
→
of K(q ) is in conflict with some general properties of
T
The values of the “electric” (el) and “magnetic” (mag) transitionamplitudes(4),whichfollowfromthecontinu-
total cross sections (in units of cm2) and their ratio ity equation:
(mag/el) are presented in Table I for various EA and
targetmaterials. Thevalues publishedin[7]aregivenin ω ρ (q) qj (q)=0. (27)
fi fi fi
−
parentheses. It is seen that the “electric” cross sections
coincide within the given accuracy, but the “magnetic” (ThelattercanbederivedfromtheSchr¨odingerequation
ones are underestimated in [7]. It is worth noting that (9)). Indeed, rewriting the continuity equation in the
4
form: while the second one
ω ρ (q) q βj (q) q j (q)=
ωfi[ρfi β−j L(q)f]i q2−βjT(fqi)/2M q j (q)=0, ρ(f1i) =iqLZ ψfE(qT,rT)zψid3r (35)
fi fi− fi − T fi − T fi
(28) wasomittedaccordingtothereasoningofapproximation
(17). In equations (34), (35) E(q ,r ) denotes:
it is easy obtain T T
µ µ
a (q)=ρ (q) βj (q) E(q ,r )= eiq1TrT + e−iq2TrT . (36)
fi fi − fi T T m1 m2
1 (29)
= ω qT2βjfi(q)/2M +qTjfi(q) . Let us consider this neglected part in detail. As it is
fi(cid:2) (cid:3) proportional to q =ω +q2/2M and therefore explic-
L fi T
Thus all transition amplitudes become zero at qT = 0. itly depends onεf, onecannotuse completenessrelation
Therefore, any transition cross section (11) can depend (14) to calculate its contribution to the total cross sec-
on the screening parameter λ at least only logarithmi- tion directly. First we need to transform it to the form
cally,butneverlikeinversepowerofthisparameter. The free of such dependence. It can be done with the help of
same is valid for the sum (12) of this quantities, i.e the Schr¨odinger equation (9).
total cross section.
ε ψ∗(r)E(q ,r )zψ (r)d3r=
fiZ f T T i
IV. ACCURATE FORMULAS
ψ∗(r) ε E(q ,r )z ε E(q ,r )z ψ (r)d3r =
Z f { f T T − i T T } i
Since the λ-dependence of the magnetic term in (25)
is contradictory to the general result, we must conclude ψ∗(r)[H,E(q ,r )z]ψ (r)d3r (37)
Z f T T i
that there is a fallacy in the deriving of sum rules (19)
somewhere. To understand the origin of the error, made The commutator in this relation is easily calculated and
by the authors of [7], let us go back to quantities (5),(6) after simple algebra we get the following result:
and expand them in powers of the longitudinal momen-
i ∂ψ (r)
tum transfer qL: ρ(1)(q)= ψ∗(r)E(q ,r ) i d3r+∆ρ(1)(q)
fi −µZ f T T ∂z fi
ρ = ∞ ρ(n) , ρ(n) = qLn dn ρ , (30) (38)
fi nX=0 fi fi n!(cid:18)dqLn fi(cid:19)(cid:12)(cid:12)qL=0 ∆ρ(1)(q)=
(cid:12) fi
jfi =nX∞=0jf(ni) , jf(ni) = qnLn!(cid:18)ddqnLnjfi(cid:19)(cid:12)(cid:12)(cid:12)qL=0. (31) iZ ψf∗(r)(cid:20)mµ1eiq1TrTO1+ mµ2e−iq2TrTO2(cid:21)zψi(r)d3r
(cid:12) b b (39)
(cid:12)
It is easily shown that terms of these expansions obey
the following estimation: O = qT2 ±2qTp , p= i∇. (40)
1,2
2m −
1,2
ρ(fni) ∝αn , jf(ni) ∝αn+1 . (32) It is seen tbhat “large” (nonbvanisbhing at q =0) parts
T
The additionalpowerofα inthe currentexpansioncoef- (1)
oftwoterms(34)and(38),contributingtoa ,areequal
ficients, in comparison with the density one, reflects the fi
and opposite in sign, so that in the resulting expression
ordinary relationbetween the values of currentand den-
they cancel each other, leaving only the term with the
sity in the hydrogen-like atoms.
“correct” behaviour at small q :
T
Expanding (4) and taking into account (32) it seems
reasonabletogrouptermswiththesameorderofαrather a(1) =∆ρ(1)(q) (41)
fi fi
than q as was done in [7]. Then the successive terms of
L
the a expansion in powers of α are (n)
fi The same is valid for any a . Applying Schr¨odinger
fi
a = a(n) equation (9) to reduce by one the power of qL in the
fi fi expression:
Xn (33)
a(n) =ρ(n) βj(n−1) . (iq )n µ n
fi fi − fi ρ(n)(q)= L ψ∗(r) eiq1TrT +
fi n! Z f (cid:20)(cid:18)m (cid:19)
From above it is clear that in the “natural” approxi- 1
mation (17) includes a(f0i) and only one part of the term + ( 1)n+1 µ ne−iq2TrT znψi(r)d3r , (42)
a(1) of expansion (33), namely: − (cid:18)m2(cid:19) (cid:21)
fi
one can represent it in the form:
i ∂ψ
βj(0) = ψ∗E(q ,r ) id3r , (34)
fi −µZ f T T ∂z ρ(n)(q)=βj(n−1)(q)+∆ρ(n)(q) (43)
fi fi fi
5
i(iq )n−1 µ n
∆ρ(n)(q)= L ψ∗(r) eiq1TrTO +
fi n! Z f (cid:20)(cid:18)m (cid:19) 1
1
+ µ ne−iq2TrTO znψ (r)d3r .b σ(2) =−Z U2(qT)W(qT)d2qT +O(α4), (47)
2 i
(cid:18)m (cid:19) (cid:21)
2 1
b (44) W(qT)= z2 q4 ψi(r)2
4m m Z | | −
1 2 (cid:2)
Finally one has 2q pψ (r)2 eiqrd3r .
T i
− | | i
a(fni) =∆ρ(fni)(q). (45) The “correct”qT-dependence of tbhe last integrand ex-
cludes a possibility of some extra λ-dependence arising,
That confirms the qualitative result (29), derived with whichcoulddramaticallyenhancethecontributionofthis
help of continuity equation (27). term (as happened to the σmag term in [7]). This can
The remaining εf-dependence of right-hand side of be illustrated by the explicit expression for the case of
(44) can be removed by repeatedly applying the Schr¨o- the screenedCoulombpotential (22)andthe EAground
dinger equation (9), which allows the transition ampli- state (23):
tudes to be represented in the form (13).
From z-dependence of the integrand in (44) it is easy 8π(Zα)2 2µ 4
toderivethata(f2ik) =0forodd∆(lm)fi,anda(f2ik+1) =0 σ(2) =− 5Mµ (cid:20)ln(cid:18)Z1/3me(cid:19)− 5(cid:21) . (48)
for even ∆(lm) , where ∆(lm) = (l l ) (m
fi fi f i f
m ), and l , l , m , m are the orbital−and m−agnet−ic Because of numerical smallness of α2 this term can be
i i f i f
quantum numbers of the initial i and final f states (the successfully neglected compared to (24).
quantization axis is supposed to be z-axis). Thus “odd” Thus in most practical applications, in which the re-
and “even” terms of expansion (33) do not interfere and quired relative accuracy is less than 10−4, only the zero
therefore in the expansion of the σtot in the powers of α order term, that considers the pure Coulomb interaction
and only the transverse transfer momentum, should be
∞ takenintoaccountforcalculationoftherelativisticatom-
σtot = σ(n) , σ(n) αn (46)
atomcrosssections. Thisresultwarrantstheusageofthe
∝
nX=0 simple expression:
only even powers are present.
The structure of the zero order term of this expansion σtot =2 U2(q )[1 S(q )]d2q (49)
T T T
Z −
is well established [see (20)]. In view of the above dis-
cussion one can be assured that the higher order terms
for the total cross section calculation for the Born ap-
are numerically negligible and should not be discussed
proximationin[2]andfortheGlauberextensionsin[15].
in detail. Nevertheless, for completeness of the consid-
The authors express their gratitude to professors
eration we present the expression for the contribution of
S.Mr´owczyn´ski, L.Nemenov and D.Trautmann for help-
the α2-term to the total cross section which includes the
ful discussions. This work is partially supported by
2
term a(1) and the interference term a(0)a(2). RFBR grant 01–02–17756.
fi fi fi
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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