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Exchange contribution in Compton scattering on nuclei
Volume 51 B, number 2
PHYSICS LETTERS
22 July 1974
EXCHANGE CONTRIBUTION IN COMPTON SCATTERING ON NUCLEI P. CHRISTILLIN
Istituto di Scienze dell'Informazione, SeuolaNormale Superiore, Pisa,Italy M. ROSA-CLOT
Istituto Nazionale FisicaNucleare, Scuola Normale Superiore, Pisa,Italy Received 10 April 1974 Low energy theorems are used to determine the amplitude for Compton scattering on nuclei. Seagull terms coming from exchange forces are shown to give a sizeable contribution in the region below nucleon resonances. The elastic Compton scattering amplitude on nuclei has been widely discussed to order e 2 in the framework of perturbation theory [1,2]. This approach is based on the knowledge of the interaction Hamiltonian obtained from the nuclear one through the minimal coupling of the electromagnetic (e.m.) field. This gauge condition gives rise, in the absence of velocity dependent potentials, to the well known interaction Hamiltonian: A (l+r/3)1- e
Hint = - ~ ~
e
e2
]
L~mPi/t(ri) + ~m A(r i) Pi , -~m A(~)A(~) j
i=1
(1)
and as a consequence to the perturbative amplitude:
M=e2 ~ m2 n
A, 3 , i ~/=1 (Ol~,i=l~(l+~)(ePi)exp(-ik"~)ln)(n _ ~(1 +7.3) (p/e) exp (ik.5)lO) (En-Eo-ko)
_ +r] ) ( p / . e )' exp ( - i k ' ~ ) 1 0 ) } (OiZiA=l(Pie)exp(ik.rt)~(l+r3)ln)(nlZ/=1½(13 +
( 2)
_2 - m (01
(1
exp
J0>
where we use the standard notations given in ref. [1]. Of course the introduction of velocity dependent terms (for example a space exchange potential) gives rise to a more complicated interaction Hamfltonian [e.g. 3]. Gerasimov [4] has used this approach in order to calculate the full Compton amplitude in the presence of exchange forces. However, this method is rather cumbersome and the result obtained has been recently criticized [5] so that at present a clear discussion of the problem is lacking. In order to overcome these difficulties we evaluate in this work the total amplitude through the use of low energy theorems. To exploit more easily the gauge condition we start from a covafiant formulation even if the result will be non-relativistic. This method has the advantage of being more straightforward and allowing a better understanding of the approximations introduced. The S matrix for the process is, according to standard techniques: Sfi = 6fi - i(27r) 4 64(pi+k-pf-k')e've u Tuv
(3)
where 125
Volume 51B, number 2
PHYSICS LETTERS
- i(27r)4~4(pi+k-pf-k')Tu = fd4xd4y = - i (2zr) ~5(Ei+k o - E f - k'o) ~
fd3x
exp (ikx) exp
fd3y
22 July 1974
(-ik'y)(OIT(/u(x), ]~Cv))10>
( <0 I],(0, y)
exp ( - i k '
"y)ln>~nI].(o, x) exp (ik" k)[o)/(Eo-En+ko)
n
+ (0lju(0, x) exp Here ju(x)
•
t
~.
F
(ik'x)ln)(n ljv(O,y) exp (-lk "y)lO)/,Eo-En-ko)}.
(4)
=-(Jo(X),j(x)) is the full e.m. current operator satisfying the gauge condition
o/.(x) = 0,
(5)
and the sum 1;n runs over a complete set of states. It is, however, easy to prove that, despite eq. (5) gauge invariant. As a matter of fact:
k'vku f d4xd4y exp (ikx) exp ( - i k ' y )
Tuv is not
~OITUu(x),/~0,))10> (6)
= fd4x fd4y
exp (ikx) exp (-ik'y)~5 (Xo-Yo)(0 [[jo(y), [H, ]o(X)] I I0>
whence a counterterm Suv must be added to T.v such that
k.k'v[r.
* S. I = 0 .
(7)
Eq. (7) determines Suu uniquely up to terms of zeroth order in k, k'. The total amplitude in the radiation gauge is therefore: I
M = etem{Ttm + Sire}.
(8)
We remark that the value of the double commutator in eq. (6) depends on the explicit form ofju(x ) and H. In particular it is different from zero when a non-relativistic limit is used for both. Since with this choice ju(x) has matrix elements only between positive energy states, this corresponds to a truncation in the sum of eq. (4) and as a consequence to a loss of its gauge invafiance. Let us now explicitly evaluate eq. (8) in two cases. a) The nuclear Hamiltonian is given by a kinetic term IgA=lp2/2m plus a velocity independent potential. In this case we know that the current is given to O(k/m) [7] by
Jo(X)=e ~ ~ ( x - ~ ) , i=1
j(x)=e~] - - ' 7 i=1
~(x-5)+8(x-~) ~
(ga,b)
Eqs. (9) are approximated in two respects: first they are reliable only below pion production because no real particles other than nucleons are allowed for their matrix elements and second, magnetic contributions are neglected. Using the above form of the charge density we easily find e2
A
Sire = - - ~ 8tin ( 0 1 ~ ( 1 i=1
+r/3) exp {i(k-k')'ri}l
O) + O(k/m, k' /m).
(10)
b) The nuclear Hamiltonian contains also an exchange potential given by Vex = vA Dex v" i] where p~x is the z"i
126
Volume 51B, number 2 e2
PHYSICS LETTERS
22 July 1974
A
Slrn "~ - --m 81m (0t/~1"= ½(1 +7"3) exp {i(k-k').r/}10)
(11)
A _ e2
53)2(
m
exp
-
exp
exp
- 5 exp 0k-ge,
x %J0>.
We reproduce with eqs. (4, 8, 10) the result of perturbation theory eq. (2), i.e., the Thomson amplitude at low energy and the incoherent scattering, coming from the seagull term above nuclear resonances. The use of the current ] + AJ and of eq. (4, 8, 11) gives the generalized amplitude in the presence of exchange potentials. Our result appears as a power expansion in the parameter k/m. As a consequence it holds only where a power expansion in the energy is meaningful and the first term gives the main contribution, i.e., far below the energy of the nucleon resonances. We further remark that it is valid to any order in (kr) since we assume to know the nuclear Hamiltonian and therefore the contribution of the different multipoles. In conclusion our analysis combines a standard treatment of the nuclear structure [8] with low energy theorems below nucleon resonances. Let us now analyze the behaviour of the amplitude in case b). At low energy, below the first nuclear levels, it is easy to show that the T ordered product can be written as:
2 ~, (Eo-En)l(OleDIn)(n le'DI 0) = (01 [[H, eD], e'D] 10>,
(12)
n
where D = Z A i =1i1(1 +r~3) [ r . - E sA=1 rs/A]" This term combines with the seagull to give the energy independent part of the total amplitude: A e2(01 [[H,
i=1
-
A
cO], e'O] - e2(01[ [H, ~ (erl)(1 +@)/2] ~ (e'~.)(l +r?)/2110} "
•
(13)
/=1
e2
(e'e')m { - ZN/A +Z} = - (e.e')(Ze)2/A m
which reproduces the Thomson limit. This is indeed a gratifying result since it shows that for long wavelengths the gamma explores only the macroscopic features of the target irrelevant of the details of the nucleonic interactions. Above nuclear resonances the T ordered product is hindered by the energy denominators and the only important contribution comes from the seagull term. So we have for the forward scattering amplitude: A =(e-e')e 2
{-Z/m-
~
i
(~3- r3)2 [4 *r~-2(ri'r/) cosk(~-r,)]Piexvi]lO)} *
*
(14)
= (e.e') e2{- Z/m -/'(k)}. Eq. (I 4) holds in the region between nuclear and nucleon resonances which, as can be seen from experimental data on photonuclear reactions [9], is well defined and ranges roughly from 50 to 150 MeV. It is apparent, beside incoherent scattering the presence of the term depending on the exchange potential. The function f(k) increases with the energy starting from f(O)=(OI
A ~ i<]--1
(r3-r3)2(ri-9)2VoXVi/[O).
(15)
A rough estimate of eq. (15) can be found in the literature [10]: 127
Volume 51B, number 2 f(O) ~ (0.5 - 1.0)
PHYSICS LETTERS
ZN/Am.
22 July 1974 (16)
We then conclude that the exchange contribution can be o f the same order o f magnitude o f the incoherent scattering. In a naive way it can be thought of as the measurement o f the quantity " ~iel/~i 2 where the sum is extended to every charged particle other than protons in the nucleus. In this scheme, assuming that pions are responsible for the main part of the exchange potential, eq. (16) suggests that the number o f pions seen b y the photon in this energy region is about 10% o f the nucleon number.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
128
R. Silbar, C. Werntz and H.13beraU, Nud. Phys. A 107 (1968) 655. T.E.O. Ericson and J. Hiifner, Nucl. Phys. B 57 (1973) 604. R.G.Sachs and N. Austern, Phys. Rev. 81 (1951) 710. S.B. Gerasimov, Phys. Lett. 13 (1964) 240. T. Matsuura and K. Yazaki, Phys. Lett. 46B (1973) 17. For a more detailed discussion of this well known point see for example (in the case of Compton scattering on electrons) J.J. Sakurai, Advanced quantum mechanics (Addison Wesley Reading 1967) pp. 135-137. R.H. Dalitz, Phys. Rev. 95 (1954) 799. R. Silbar and H. UberaU, Nucl. Phys. A 109 (1968) 146. B. Ziegler et al., Proc. Intern. Conf. on Nuclear structure studies using electron scattering and photoreaction, Senday (1972) p, 213. H.A. Bethe and J.S. Levinger, Phys. Rev. 78 (1950) 115; W.T. Weng, T.T.S. Kuo and G.E. Brown, Phys. Lett. 46B (1973) 329.
PHYSICS LETTERS
22 July 1974
EXCHANGE CONTRIBUTION IN COMPTON SCATTERING ON NUCLEI P. CHRISTILLIN
Istituto di Scienze dell'Informazione, SeuolaNormale Superiore, Pisa,Italy M. ROSA-CLOT
Istituto Nazionale FisicaNucleare, Scuola Normale Superiore, Pisa,Italy Received 10 April 1974 Low energy theorems are used to determine the amplitude for Compton scattering on nuclei. Seagull terms coming from exchange forces are shown to give a sizeable contribution in the region below nucleon resonances. The elastic Compton scattering amplitude on nuclei has been widely discussed to order e 2 in the framework of perturbation theory [1,2]. This approach is based on the knowledge of the interaction Hamiltonian obtained from the nuclear one through the minimal coupling of the electromagnetic (e.m.) field. This gauge condition gives rise, in the absence of velocity dependent potentials, to the well known interaction Hamiltonian: A (l+r/3)1- e
Hint = - ~ ~
e
e2
]
L~mPi/t(ri) + ~m A(r i) Pi , -~m A(~)A(~) j
i=1
(1)
and as a consequence to the perturbative amplitude:
M=e2 ~ m2 n
A, 3 , i ~/=1 (Ol~,i=l~(l+~)(ePi)exp(-ik"~)ln)(n _ ~(1 +7.3) (p/e) exp (ik.5)lO) (En-Eo-ko)
_ +r] ) ( p / . e )' exp ( - i k ' ~ ) 1 0 ) } (OiZiA=l(Pie)exp(ik.rt)~(l+r3)ln)(nlZ/=1½(13 +
( 2)
_2 - m (01
(1
exp
J0>
where we use the standard notations given in ref. [1]. Of course the introduction of velocity dependent terms (for example a space exchange potential) gives rise to a more complicated interaction Hamfltonian [e.g. 3]. Gerasimov [4] has used this approach in order to calculate the full Compton amplitude in the presence of exchange forces. However, this method is rather cumbersome and the result obtained has been recently criticized [5] so that at present a clear discussion of the problem is lacking. In order to overcome these difficulties we evaluate in this work the total amplitude through the use of low energy theorems. To exploit more easily the gauge condition we start from a covafiant formulation even if the result will be non-relativistic. This method has the advantage of being more straightforward and allowing a better understanding of the approximations introduced. The S matrix for the process is, according to standard techniques: Sfi = 6fi - i(27r) 4 64(pi+k-pf-k')e've u Tuv
(3)
where 125
Volume 51B, number 2
PHYSICS LETTERS
- i(27r)4~4(pi+k-pf-k')Tu = fd4xd4y = - i (2zr) ~5(Ei+k o - E f - k'o) ~
fd3x
exp (ikx) exp
fd3y
22 July 1974
(-ik'y)(OIT(/u(x), ]~Cv))10>
( <0 I],(0, y)
exp ( - i k '
"y)ln>~nI].(o, x) exp (ik" k)[o)/(Eo-En+ko)
n
+ (0lju(0, x) exp Here ju(x)
•
t
~.
F
(ik'x)ln)(n ljv(O,y) exp (-lk "y)lO)/,Eo-En-ko)}.
(4)
=-(Jo(X),j(x)) is the full e.m. current operator satisfying the gauge condition
o/.(x) = 0,
(5)
and the sum 1;n runs over a complete set of states. It is, however, easy to prove that, despite eq. (5) gauge invariant. As a matter of fact:
k'vku f d4xd4y exp (ikx) exp ( - i k ' y )
Tuv is not
~OITUu(x),/~0,))10> (6)
= fd4x fd4y
exp (ikx) exp (-ik'y)~5 (Xo-Yo)(0 [[jo(y), [H, ]o(X)] I I0>
whence a counterterm Suv must be added to T.v such that
k.k'v[r.
* S. I = 0 .
(7)
Eq. (7) determines Suu uniquely up to terms of zeroth order in k, k'. The total amplitude in the radiation gauge is therefore: I
M = etem{Ttm + Sire}.
(8)
We remark that the value of the double commutator in eq. (6) depends on the explicit form ofju(x ) and H. In particular it is different from zero when a non-relativistic limit is used for both. Since with this choice ju(x) has matrix elements only between positive energy states, this corresponds to a truncation in the sum of eq. (4) and as a consequence to a loss of its gauge invafiance. Let us now explicitly evaluate eq. (8) in two cases. a) The nuclear Hamiltonian is given by a kinetic term IgA=lp2/2m plus a velocity independent potential. In this case we know that the current is given to O(k/m) [7] by
Jo(X)=e ~ ~ ( x - ~ ) , i=1
j(x)=e~] - - ' 7 i=1
~(x-5)+8(x-~) ~
(ga,b)
Eqs. (9) are approximated in two respects: first they are reliable only below pion production because no real particles other than nucleons are allowed for their matrix elements and second, magnetic contributions are neglected. Using the above form of the charge density we easily find e2
A
Sire = - - ~ 8tin ( 0 1 ~ ( 1 i=1
+r/3) exp {i(k-k')'ri}l
O) + O(k/m, k' /m).
(10)
b) The nuclear Hamiltonian contains also an exchange potential given by Vex = vA Dex v" i] where p~x is the z"i
126
Volume 51B, number 2 e2
PHYSICS LETTERS
22 July 1974
A
Slrn "~ - --m 81m (0t/~1"= ½(1 +7"3) exp {i(k-k').r/}10)
(11)
A _ e2
53)2(
m
exp
-
exp
exp
- 5 exp 0k-ge,
x %J0>.
We reproduce with eqs. (4, 8, 10) the result of perturbation theory eq. (2), i.e., the Thomson amplitude at low energy and the incoherent scattering, coming from the seagull term above nuclear resonances. The use of the current ] + AJ and of eq. (4, 8, 11) gives the generalized amplitude in the presence of exchange potentials. Our result appears as a power expansion in the parameter k/m. As a consequence it holds only where a power expansion in the energy is meaningful and the first term gives the main contribution, i.e., far below the energy of the nucleon resonances. We further remark that it is valid to any order in (kr) since we assume to know the nuclear Hamiltonian and therefore the contribution of the different multipoles. In conclusion our analysis combines a standard treatment of the nuclear structure [8] with low energy theorems below nucleon resonances. Let us now analyze the behaviour of the amplitude in case b). At low energy, below the first nuclear levels, it is easy to show that the T ordered product can be written as:
2 ~, (Eo-En)l(OleDIn)(n le'DI 0) = (01 [[H, eD], e'D] 10>,
(12)
n
where D = Z A i =1i1(1 +r~3) [ r . - E sA=1 rs/A]" This term combines with the seagull to give the energy independent part of the total amplitude: A e2(01 [[H,
i=1
-
A
cO], e'O] - e2(01[ [H, ~ (erl)(1 +@)/2] ~ (e'~.)(l +r?)/2110} "
•
(13)
/=1
e2
(e'e')m { - ZN/A +Z} = - (e.e')(Ze)2/A m
which reproduces the Thomson limit. This is indeed a gratifying result since it shows that for long wavelengths the gamma explores only the macroscopic features of the target irrelevant of the details of the nucleonic interactions. Above nuclear resonances the T ordered product is hindered by the energy denominators and the only important contribution comes from the seagull term. So we have for the forward scattering amplitude: A =(e-e')e 2
{-Z/m-
~
i
(~3- r3)2 [4 *r~-2(ri'r/) cosk(~-r,)]Piexvi]lO)} *
*
(14)
= (e.e') e2{- Z/m -/'(k)}. Eq. (I 4) holds in the region between nuclear and nucleon resonances which, as can be seen from experimental data on photonuclear reactions [9], is well defined and ranges roughly from 50 to 150 MeV. It is apparent, beside incoherent scattering the presence of the term depending on the exchange potential. The function f(k) increases with the energy starting from f(O)=(OI
A ~ i<]--1
(r3-r3)2(ri-9)2VoXVi/[O).
(15)
A rough estimate of eq. (15) can be found in the literature [10]: 127
Volume 51B, number 2 f(O) ~ (0.5 - 1.0)
PHYSICS LETTERS
ZN/Am.
22 July 1974 (16)
We then conclude that the exchange contribution can be o f the same order o f magnitude o f the incoherent scattering. In a naive way it can be thought of as the measurement o f the quantity " ~iel/~i 2 where the sum is extended to every charged particle other than protons in the nucleus. In this scheme, assuming that pions are responsible for the main part of the exchange potential, eq. (16) suggests that the number o f pions seen b y the photon in this energy region is about 10% o f the nucleon number.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
128
R. Silbar, C. Werntz and H.13beraU, Nud. Phys. A 107 (1968) 655. T.E.O. Ericson and J. Hiifner, Nucl. Phys. B 57 (1973) 604. R.G.Sachs and N. Austern, Phys. Rev. 81 (1951) 710. S.B. Gerasimov, Phys. Lett. 13 (1964) 240. T. Matsuura and K. Yazaki, Phys. Lett. 46B (1973) 17. For a more detailed discussion of this well known point see for example (in the case of Compton scattering on electrons) J.J. Sakurai, Advanced quantum mechanics (Addison Wesley Reading 1967) pp. 135-137. R.H. Dalitz, Phys. Rev. 95 (1954) 799. R. Silbar and H. UberaU, Nucl. Phys. A 109 (1968) 146. B. Ziegler et al., Proc. Intern. Conf. on Nuclear structure studies using electron scattering and photoreaction, Senday (1972) p, 213. H.A. Bethe and J.S. Levinger, Phys. Rev. 78 (1950) 115; W.T. Weng, T.T.S. Kuo and G.E. Brown, Phys. Lett. 46B (1973) 329.