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Calculation of Doppler-Broadened cross sections directly from Doppler- (and resolution-) broadened cross sections corresponding to a lower temperature
NUCLEAR INSTRUMENTS AND METHODS
49 (t967) 89-92; (0 NORTH-HOLLAND PUBLISHING CO .
CALCULATION OF DOPPLER-BROADENED CROSS SECTIONS DIRECTLY FROM DOPPLER- (AND RESOLUTION-) BROADENED CROSS SECTIONS CORRESPONDING TO A LOWER TEWERATURE* F. H. FR45HNER General Atomic DiWsian, General Dynamics Corp 7ration, John Jay Hopkins Laboratory for Pure and Applied Science,
San Diego, California, U.S.A .
Received 24 November 1%6 cross section for a finite temperature r < T, and use essentially the same convolution. The proof is given for the exact free-gas kernel and also for the commonly used approximation to a broadened Breit-Wigner resonance. Some implications are discussed briefly.
Conventionally, a Doppitr-broadened cross section for a given temperature T is calculated from the zero-temperature cross section by nmas of a convolution. A simple proof is presented for the fact that instead of starting from the zero-temperature cross section one can as well start from the Doppler-broadened
1. Introdwfion Conventionally, the, Doppler-broadenedcross section for a temperature Tis defincd and computcd as a convolution of the zero-temperature cross section and the velocity distribution for T of the target particles. Harris') proved that in the case of a Maxwell-Boltzmann distribution one can just as well start from the broadened cross section for a finite temperature T' < T and perform cisentially the same convolution, but with the differcrice T- T' as the temperature paraincter charactenzing the velocity distribution . Howcvcr, Harris' proof ;s indirect and fairly involved, using the concept of adjoint differential equations. This, plus the fact that his proof is rather well-hidden in a technical report may be the reason that it is not as widely known as it should be. A very simple and straightforward proof will be given in this paper. First the case of an arbitrary cross section will be treated. exactly. Then it wilt be shown that the proof can also be given for the widely used narrow-resonance approximation (with the Doppler shape functions and 9). Some implications will be discussed briefly.
mass, and k is Boltzmann's constant). For a gas target, T isjust the sample temperature; for a Debye crystal, T is an effective temperature slightly higher than the actual sample temperature') ; uo = u,,(T) is the most probable value of u. The Doppler-bi oadened cross section for temperature T is defined generally as a(v, T) = ( 1 lv) 1 d'tif(v, T)wnr(w,O),
where v I v I is the speed of the bombarding particle, and )%- = v - u I is the relative speed, a(w, 0) is the unbroadened cross section ; a may be thought of as any cross section, total, partial, differential, etc. Inserting eq . (1) in eq . (2) one finds, using spherical coordinates, a(v,T) = v
g' e x p ( - ij `
it,, =_ (2k"!','
fo dw(u0.~,/n)`[exp eXp
{
or, in terrns of energies, a(E, T) = E -1
2. Doppler brovidening with a Maxwell-Boltzmann distrt1ution In a wide variety of cases one may treat Doppler broadening of atomic or nuclear cross sections by assuming a Maxwell-Boltzmann distribution, 1 (P, T) d - t i r, (2J d 3 11~ (la) with
(2)
fo
X [exp
{_(V _ W)21U12
_(V+ W)2/U }]W2a(W , 0) 0
(3)
dE'I {E'/ (nEO)}'k x
( -(.,IE - IE t)210} _
- exp { -( /E + ,/E )210 }]a(E',0), (4)
where E mv', E' = Imit, 2,0 =,fnk TIM, and cr(E.T) 7-) ; in Is the mass of the bombarding particle .
.,
Dcrivat-ons ofeq . (4) were given by scvcral authors' - '). With the abbreviatiom,
(I b)
for the velocities, u of the im-gvi particles (A! is their Thk 7escarch wa% ,1.1prt)t~tcj 4,toriic 17.ncigA,
cand
commissiort.
G(v,uJ =_ (ti ,/n)-'exp(-v'liio),
(5)
D(v,w ;T)=-[G(v-w,uo)-G(v+iv,uo)](iv'lv'), (6)
89
F. H. FRÖHNER
bc writteii a,-,; C7( V,
T)
diiD(v,w ; T)a(",,O).
(7)
ightforward proof of the more general
I
a(v,T) ==
0
dwD(v,w ; T-T)cr(w,T),
(8)
ich is the relationship proven indirectly by Harris . proof will involve nothing more complicated thar, the symmetry property of the Gaussian, G(P,u 0 )
= G(- v,u0),
(9)
and its group property 1
~
ce
dv'G(v',u')G(v'-v,uo) 0
J-
= G{v(ti""+ 0
U 2),* } . 0
(10)
Eq. (10) states the familiar fact that the convolution of two GaussianslYields another Gaussian, the squares of the two widths adding up to the squared width of the resulting Gaussiian. We have to show that the right-hand side of eq . (8) actually equals a(v, T) . First we insert eq. (7) into eq . (8)
f
= (V'/V)'
"
1
-T')
t'
0
di~'D(i
. e iht:n cha,nge the or&r of integrations . The w-integration involves only the two D-functions. With u'2 rids
=-
2kTIM,
= 2k(T-T')IM,
)D(vv,v' ; T)
d w [G(ii - r, t4) -- G(ii, + v, uij x
dx[G(x,ti)G(x - v'+ v,u) -
-- G(xti)C,(x + u' + 1,',U')] .
In the last step we used the symmetry property, eq. (9), of the Gaussian and, the fact that x and y are just dummy variables in combining the first andflast, and the second and third, of the four integrals. In so doing we get rid of the finite lower limits of the integrals. This allows us to use the group property of the Gaussian, eq . (10), which immediately yields '100 dwD(v,w ;T-T')D(w,v',T")=D(v',v ;T). (12) J0 Thus the Doppler kernels D have the group property, too : convolution of two D-functions yields another Dfunction . The width parameters, u, u'combine in a way which is equivalent to the addition of the corresponding temperatures. Inserting eq . (12) in eq. (11) and recalling the definition (7) of a(v, T) we get, finally, f "0
divD(v, w -, T - T)a(w,T') =
dwD(v.w ; T - T')o-(w, T')
dwL)('t,,i-v .
J ", *x;
J-0",
dv'D(t,,,v'-,T)a(t,',O)=r ;(t,,T), (13)
which completes tile proof. 3. Resonance cross sections Many atomic and nuclear cross sections show sharp resonance peaks. From very general arguments it can be shown that near a single isolated resonance with resonance energy E. and width F the cross section can often be described by a Breit-Wigner formula of the type with
a(E,O) = ~ A / (I + x2)1 + {Bx / (I + X 2) J + C,
(14)
EJ 1(1, F), L
(V
N - 11, 14) -
dxG(.X,ti)G(V' 1- X + L',
N1, here A, R, C and V are slowly varyiiig Rinoti ons of the energy E. Doppler broadening of such resonances can usually be treated with the foilowing simplificatiOlls : 1 . The second exponential in the Doppler kernel, eq . (4), being small as compared to the first one, can be neglected . 2. In the first exponential, otic uscs the approximation H 4- (E- E)/ E} 4
1
CALCULATION Of DOPPLER-BROADENED CROSS SECTIONS
3. The lower limit ofthe integration is shifted to - oc. . 4. A, B, C and F are treated as constants, as far a, the E'-Integ;- ation is concerned . Once the F-Integration is carried out they are used as varying again, to ensure the correct asymptotic behavior in the wings of the resonance . The result is' , "),
o(E,T) -- A~(x,fl) + B
(15)
The Doppler shape functions are given by6, 7) O(X,P)
=
dx'G(x - x',P)O(x,0)
(1 6a)
(P(X,fl)
=f
dx'G(x - x`,fl)(p(x,0),
(16b)
with
*( X,O) = I /(I + X2),
(I 7a)
9(X,O) = X/ (I +X'),
(I 7b)
The D(-,ppler parameter# is defined as P =- {A(T)}1(jr) =- (41r)(E D kTm1M)1,
(18)
where J(T) Is the Doppler width and I- the resonance width at E =~ L,, . littiman et at ., find that this resonance approximation lead~, to noticeable errors oniv at very low energies (E< I eV for neutron resonances, \vitli 0.0-153 eV). In complete analogy to our proof of eq, (1 31) 1[ can he ~homal fll~lt
11, (x f,~') =
dx"G(x - x%y)~(x
-
dx'G(x-x',[~)~(x,O),
(19)
1r~
4P(V,
Where
Iti fac : . the proof Is iiiiicli simpler, 4cc~~ ,we Ille hroiid,Itliply kerriel iii the ~IPI)r( ,Xitll~it loti is
Me Giussian (;,,tild ils group pr,,-~pcrty, cq. (10), cati be used inimediately . From eqs . (l 5), (19) and (20) one gets, using the fotir a,,st.iiiiption~, again
a(E,T) ;z~
91
dE"G{E" - EJ(T- T')}u(E",T') _- j
dE"G{E-E,A(T)~a(E',O) .
(22)
The interpretation is, of course, the same as that of eq.(13) : The Doppler-broadened cross section for a temperature T can be obtained by: 1 . broadening the zero-temperature cross section using T as the parameter of the broadening kernel, or 2. further broadening a Doppler-broadened cross section for a lower temperature Tusing the temperature difference T- T' to calculate the kernel . It is not quite trivial that this property of the exact eq. (3), or eq. (4), should apply to the approximation (15) also. 4. Ds*seussion In transmission, scatterin2g . capture, etc., measurements with thin samples resonance cross sections are obtained directly, broadened by instrumental resolution and Doppler effect . Now the formalism of section 3 can be extended without any changes, to the treatment of instrumental resolution also, whenever the res-lution function can be approximated by a ~3aussian") . In these cases the measured (resolution-broadened) cross section is equal to the pure Doppler-broadened cross section correzpnnding11 to .,i higher temperature T,., gi,.cn by T, = [ ~J(T )2 ~_ ;l , 2 k~ ,1 ),'A(T) JT, (23) where T P, the effectivc ternperature of the sample. J(F) i.,, the Doppler will'th, and WIN the l,'e-width of the resolution fanction . From such an observed cross section one can directly calculate Doppler broadened cross sections for temperatures T> T,,, without even taking out the reso-lution effect first . The importance of these results, is quite evident for neutron transport calculations and calculations of Doppler coefficients, where tow energy cross sections are often supplied as simple lists of values measured at some non/cro temperature . Furthermore . these restilts c,tn be applied to tile cross sections of fissile muctides in tLe resonance region, where the single-level Brett\\iigne i forintila ( 14) i .~ I ii~ideqiiale and I he correct pariCohen' ') 11611/~oJ 011's pos"IhOlly rcck,111 ,1~ It, c~dci.ilatc Doppler broadened cross sLcnons fo, 23 5 t - ,ci , 1 500 ai .d '1 500 K directiv froni N,,l icha Udo 1, 12 cross .."'ection data . I arn indebted to C . A. Stevens for bringing Hari is' report to niv atiention .
F. H. 1FR8HNER
1) DI. R. Harris, Be, tis-Techn . Rev. WAPD-BT-30 (1964). 2) W. F- Lamb, Jr., Phys. Rev. 55 (1939) 190. 1) G, GmrtreI,P-,cvc . Firm Intern. Conf, Peaceft;l Uses Atomic 'Enerv 4 (1955) 471. 4) Q L Dal,loos-Alaum Report LA-2322 (1959) . 5) A. W. Solbtig, Jr., Nuct. Sci . Enjr 101(1961) 167. 6) 13orn, ,CW& (Wnger-Vtdag, Berlin, 63a). ,
I
7) H. Bethe and G. Placzek, Phys. Rev . 51 (1937) 450 . 8) G. W. Hinrnan, G. F. Kuncir, J. B-. Sampson and G. B. West, Nucl. Sci . Eng. 16 (1963) 202, 9) J. E. Lynn, Phys. Rev . Letters 13 (1964) 412. 19),FAR. friBhner and !E. Haddad~ Nuet.'Physics 71, (1965) 129. 1 1) .S. C. Cohen, General Atornic Report ' GA-7391 (1-966). 12) X Michaudón, Saelay Report CEA-Ik 2552 (1964) and A. Mkhatjdön et al~,l.'Phys . Rädiutü 21 (1960) 429.
49 (t967) 89-92; (0 NORTH-HOLLAND PUBLISHING CO .
CALCULATION OF DOPPLER-BROADENED CROSS SECTIONS DIRECTLY FROM DOPPLER- (AND RESOLUTION-) BROADENED CROSS SECTIONS CORRESPONDING TO A LOWER TEWERATURE* F. H. FR45HNER General Atomic DiWsian, General Dynamics Corp 7ration, John Jay Hopkins Laboratory for Pure and Applied Science,
San Diego, California, U.S.A .
Received 24 November 1%6 cross section for a finite temperature r < T, and use essentially the same convolution. The proof is given for the exact free-gas kernel and also for the commonly used approximation to a broadened Breit-Wigner resonance. Some implications are discussed briefly.
Conventionally, a Doppitr-broadened cross section for a given temperature T is calculated from the zero-temperature cross section by nmas of a convolution. A simple proof is presented for the fact that instead of starting from the zero-temperature cross section one can as well start from the Doppler-broadened
1. Introdwfion Conventionally, the, Doppler-broadenedcross section for a temperature Tis defincd and computcd as a convolution of the zero-temperature cross section and the velocity distribution for T of the target particles. Harris') proved that in the case of a Maxwell-Boltzmann distribution one can just as well start from the broadened cross section for a finite temperature T' < T and perform cisentially the same convolution, but with the differcrice T- T' as the temperature paraincter charactenzing the velocity distribution . Howcvcr, Harris' proof ;s indirect and fairly involved, using the concept of adjoint differential equations. This, plus the fact that his proof is rather well-hidden in a technical report may be the reason that it is not as widely known as it should be. A very simple and straightforward proof will be given in this paper. First the case of an arbitrary cross section will be treated. exactly. Then it wilt be shown that the proof can also be given for the widely used narrow-resonance approximation (with the Doppler shape functions and 9). Some implications will be discussed briefly.
mass, and k is Boltzmann's constant). For a gas target, T isjust the sample temperature; for a Debye crystal, T is an effective temperature slightly higher than the actual sample temperature') ; uo = u,,(T) is the most probable value of u. The Doppler-bi oadened cross section for temperature T is defined generally as a(v, T) = ( 1 lv) 1 d'tif(v, T)wnr(w,O),
where v I v I is the speed of the bombarding particle, and )%- = v - u I is the relative speed, a(w, 0) is the unbroadened cross section ; a may be thought of as any cross section, total, partial, differential, etc. Inserting eq . (1) in eq . (2) one finds, using spherical coordinates, a(v,T) = v
g' e x p ( - ij `
it,, =_ (2k"!','
fo dw(u0.~,/n)`[exp eXp
{
or, in terrns of energies, a(E, T) = E -1
2. Doppler brovidening with a Maxwell-Boltzmann distrt1ution In a wide variety of cases one may treat Doppler broadening of atomic or nuclear cross sections by assuming a Maxwell-Boltzmann distribution, 1 (P, T) d - t i r, (2J d 3 11~ (la) with
(2)
fo
X [exp
{_(V _ W)21U12
_(V+ W)2/U }]W2a(W , 0) 0
(3)
dE'I {E'/ (nEO)}'k x
( -(.,IE - IE t)210} _
- exp { -( /E + ,/E )210 }]a(E',0), (4)
where E mv', E' = Imit, 2,0 =,fnk TIM, and cr(E.T) 7-) ; in Is the mass of the bombarding particle .
.,
Dcrivat-ons ofeq . (4) were given by scvcral authors' - '). With the abbreviatiom,
(I b)
for the velocities, u of the im-gvi particles (A! is their Thk 7escarch wa% ,1.1prt)t~tcj 4,toriic 17.ncigA,
cand
commissiort.
G(v,uJ =_ (ti ,/n)-'exp(-v'liio),
(5)
D(v,w ;T)=-[G(v-w,uo)-G(v+iv,uo)](iv'lv'), (6)
89
F. H. FRÖHNER
bc writteii a,-,; C7( V,
T)
diiD(v,w ; T)a(",,O).
(7)
ightforward proof of the more general
I
a(v,T) ==
0
dwD(v,w ; T-T)cr(w,T),
(8)
ich is the relationship proven indirectly by Harris . proof will involve nothing more complicated thar, the symmetry property of the Gaussian, G(P,u 0 )
= G(- v,u0),
(9)
and its group property 1
~
ce
dv'G(v',u')G(v'-v,uo) 0
J-
= G{v(ti""+ 0
U 2),* } . 0
(10)
Eq. (10) states the familiar fact that the convolution of two GaussianslYields another Gaussian, the squares of the two widths adding up to the squared width of the resulting Gaussiian. We have to show that the right-hand side of eq . (8) actually equals a(v, T) . First we insert eq. (7) into eq . (8)
f
= (V'/V)'
"
1
-T')
t'
0
di~'D(i
. e iht:n cha,nge the or&r of integrations . The w-integration involves only the two D-functions. With u'2 rids
=-
2kTIM,
= 2k(T-T')IM,
)D(vv,v' ; T)
d w [G(ii - r, t4) -- G(ii, + v, uij x
dx[G(x,ti)G(x - v'+ v,u) -
-- G(xti)C,(x + u' + 1,',U')] .
In the last step we used the symmetry property, eq. (9), of the Gaussian and, the fact that x and y are just dummy variables in combining the first andflast, and the second and third, of the four integrals. In so doing we get rid of the finite lower limits of the integrals. This allows us to use the group property of the Gaussian, eq . (10), which immediately yields '100 dwD(v,w ;T-T')D(w,v',T")=D(v',v ;T). (12) J0 Thus the Doppler kernels D have the group property, too : convolution of two D-functions yields another Dfunction . The width parameters, u, u'combine in a way which is equivalent to the addition of the corresponding temperatures. Inserting eq . (12) in eq. (11) and recalling the definition (7) of a(v, T) we get, finally, f "0
divD(v, w -, T - T)a(w,T') =
dwD(v.w ; T - T')o-(w, T')
dwL)('t,,i-v .
J ", *x;
J-0",
dv'D(t,,,v'-,T)a(t,',O)=r ;(t,,T), (13)
which completes tile proof. 3. Resonance cross sections Many atomic and nuclear cross sections show sharp resonance peaks. From very general arguments it can be shown that near a single isolated resonance with resonance energy E. and width F the cross section can often be described by a Breit-Wigner formula of the type with
a(E,O) = ~ A / (I + x2)1 + {Bx / (I + X 2) J + C,
(14)
EJ 1(1, F), L
(V
N - 11, 14) -
dxG(.X,ti)G(V' 1- X + L',
N1, here A, R, C and V are slowly varyiiig Rinoti ons of the energy E. Doppler broadening of such resonances can usually be treated with the foilowing simplificatiOlls : 1 . The second exponential in the Doppler kernel, eq . (4), being small as compared to the first one, can be neglected . 2. In the first exponential, otic uscs the approximation H 4- (E- E)/ E} 4
1
CALCULATION Of DOPPLER-BROADENED CROSS SECTIONS
3. The lower limit ofthe integration is shifted to - oc. . 4. A, B, C and F are treated as constants, as far a, the E'-Integ;- ation is concerned . Once the F-Integration is carried out they are used as varying again, to ensure the correct asymptotic behavior in the wings of the resonance . The result is' , "),
o(E,T) -- A~(x,fl) + B
(15)
The Doppler shape functions are given by6, 7) O(X,P)
=
dx'G(x - x',P)O(x,0)
(1 6a)
(P(X,fl)
=f
dx'G(x - x`,fl)(p(x,0),
(16b)
with
*( X,O) = I /(I + X2),
(I 7a)
9(X,O) = X/ (I +X'),
(I 7b)
The D(-,ppler parameter# is defined as P =- {A(T)}1(jr) =- (41r)(E D kTm1M)1,
(18)
where J(T) Is the Doppler width and I- the resonance width at E =~ L,, . littiman et at ., find that this resonance approximation lead~, to noticeable errors oniv at very low energies (E< I eV for neutron resonances, \vitli 0.0-153 eV). In complete analogy to our proof of eq, (1 31) 1[ can he ~homal fll~lt
11, (x f,~') =
dx"G(x - x%y)~(x
-
dx'G(x-x',[~)~(x,O),
(19)
1r~
4P(V,
Where
Iti fac : . the proof Is iiiiicli simpler, 4cc~~ ,we Ille hroiid,Itliply kerriel iii the ~IPI)r( ,Xitll~it loti is
Me Giussian (;,,tild ils group pr,,-~pcrty, cq. (10), cati be used inimediately . From eqs . (l 5), (19) and (20) one gets, using the fotir a,,st.iiiiption~, again
a(E,T) ;z~
91
dE"G{E" - EJ(T- T')}u(E",T') _- j
dE"G{E-E,A(T)~a(E',O) .
(22)
The interpretation is, of course, the same as that of eq.(13) : The Doppler-broadened cross section for a temperature T can be obtained by: 1 . broadening the zero-temperature cross section using T as the parameter of the broadening kernel, or 2. further broadening a Doppler-broadened cross section for a lower temperature Tusing the temperature difference T- T' to calculate the kernel . It is not quite trivial that this property of the exact eq. (3), or eq. (4), should apply to the approximation (15) also. 4. Ds*seussion In transmission, scatterin2g . capture, etc., measurements with thin samples resonance cross sections are obtained directly, broadened by instrumental resolution and Doppler effect . Now the formalism of section 3 can be extended without any changes, to the treatment of instrumental resolution also, whenever the res-lution function can be approximated by a ~3aussian") . In these cases the measured (resolution-broadened) cross section is equal to the pure Doppler-broadened cross section correzpnnding11 to .,i higher temperature T,., gi,.cn by T, = [ ~J(T )2 ~_ ;l , 2 k~ ,1 ),'A(T) JT, (23) where T P, the effectivc ternperature of the sample. J(F) i.,, the Doppler will'th, and WIN the l,'e-width of the resolution fanction . From such an observed cross section one can directly calculate Doppler broadened cross sections for temperatures T> T,,, without even taking out the reso-lution effect first . The importance of these results, is quite evident for neutron transport calculations and calculations of Doppler coefficients, where tow energy cross sections are often supplied as simple lists of values measured at some non/cro temperature . Furthermore . these restilts c,tn be applied to tile cross sections of fissile muctides in tLe resonance region, where the single-level Brett\\iigne i forintila ( 14) i .~ I ii~ideqiiale and I he correct pariCohen' ') 11611/~oJ 011's pos"IhOlly rcck,111 ,1~ It, c~dci.ilatc Doppler broadened cross sLcnons fo, 23 5 t - ,ci , 1 500 ai .d '1 500 K directiv froni N,,l icha Udo 1, 12 cross .."'ection data . I arn indebted to C . A. Stevens for bringing Hari is' report to niv atiention .
F. H. 1FR8HNER
1) DI. R. Harris, Be, tis-Techn . Rev. WAPD-BT-30 (1964). 2) W. F- Lamb, Jr., Phys. Rev. 55 (1939) 190. 1) G, GmrtreI,P-,cvc . Firm Intern. Conf, Peaceft;l Uses Atomic 'Enerv 4 (1955) 471. 4) Q L Dal,loos-Alaum Report LA-2322 (1959) . 5) A. W. Solbtig, Jr., Nuct. Sci . Enjr 101(1961) 167. 6) 13orn, ,CW& (Wnger-Vtdag, Berlin, 63a). ,
I
7) H. Bethe and G. Placzek, Phys. Rev . 51 (1937) 450 . 8) G. W. Hinrnan, G. F. Kuncir, J. B-. Sampson and G. B. West, Nucl. Sci . Eng. 16 (1963) 202, 9) J. E. Lynn, Phys. Rev . Letters 13 (1964) 412. 19),FAR. friBhner and !E. Haddad~ Nuet.'Physics 71, (1965) 129. 1 1) .S. C. Cohen, General Atornic Report ' GA-7391 (1-966). 12) X Michaudón, Saelay Report CEA-Ik 2552 (1964) and A. Mkhatjdön et al~,l.'Phys . Rädiutü 21 (1960) 429.