- Email: [email protected]
Borrmann effect in parametric X-ray radiation
20 September 1999
Physics Letters A 260 Ž1999. 391–394 www.elsevier.nlrlocaterphysleta
Borrmann effect in parametric X-ray radiation Nikolai N. Nasonov
1
Laboratory of Radiation Physics, Belgorod State UniÕersity, 12 Studencheskaya, 308007 Belgorod, Russia Received 13 July 1999; accepted 28 July 1999 Communicated by V.M. Agranovich
Abstract The suppression of the photoabsorption in parametric X-ray radiation by relativistic particles in a crystal is predicted. This dynamical effect caused by the contribution of emitting particle transition radiation to parametric X-ray radiation yield is possible in a crystal with a finite thickness only. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 78.70.-g; 78.70.Ck; 32.30.Rj; 32.80.Cy
1. Introduction Anomalon suppression of the photoabsorption for X-rays propagating in a crystal is one of the most important dynamical diffraction effects known as a Borrmann effect w1x. This effect is realized in the process of a free photon scattering only. The performed analysis of a parametric X-ray radiation ŽPXR. by a relativistic particle moving in a crystal Žsee for example w2x. has shown the negligible role of Borrmann effect in the process of particle Coulomb pseudophoton scattering. The mentioned circumstance is the consequence of the small influence of dynamical diffraction effects upon PXR caused by momentum and energy conservation laws in the emission process. It is important to note that the conclusion w2x was obtained on the basis of the asymptotic expression for the PXR spectral-angular distribution describing PXR yield proportional to a crystal thickness. Dy1
E-mail: [email protected]
namical diffraction effects do not influence practically upon this yield characteristics. A more detailed analysis of a relativistic particle PXR in Laue geometry developed in the presented work has shown the existence of an additional PXR peak appearing due to Bragg scattering of a particle transition radiation Žthis peak is adequate to the well known transition diffracted radiation peak w3x.. A new peak is formed in the process of real photon scattering. Therefore, the dynamical diffraction effects ŽBorrmann effect in particular. can take place in the considered process. The possibility to observe the Borrmann effect in PXR of high enough energy particles crossing a crystal has been shown in the presented work.
2. General expressions Let us consider the structure of an electromagnetic field emitted by the flux of relativistic particles moving in a cristal with a thickness L along the axis
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 4 5 - 9
N.N. NasonoÕr Physics Letters A 260 (1999) 391–394
392
e 1 as is shown in Fig. 1. Here g is the reciprocal lattice vector defining the reflecting crystallographic plane of the crystal, w is the fixed emission angle, the orientational angle Q X counted from the Bragg resonance position describes the possible crystal turning by the use of goniometer. To find the Fourier-transform of an emission electrical field inside a crystal E kcrv s
1
Ž 2p .
yi k rqi v t
3
4
Hdtd rE Ž r ,t . e
l denotes the wave polarization, el k is the polarization vector E kcrv s Ý l el k Elcr0 , E kcrg v s Ý l el k gElcrg . Equations for the fields outside the crystal Elvac 0 Ž . and Elvac g follow from 1 in the limit x 0 , x g ™ 0. Determining the uniform and non-uniform solutions of presented equations separately inside and outside the crystal we find unknown coefficients in uniform solutions by means of ordinary boundary conditions. The final expression for the field Elvac g propagating along the Bragg scattering direction e 2 behind the crystal is given by
we make use of the well-known equations of the two-wave approximation of dynamical diffraction theory w4x,
Elvac g s al d Ž k x y p . ,
Ž k 2 y v 2 y v 2x 0 . Elcro y v 2xyg al Elcrg
al s
iv e s
Ž
2p 2
i v e v 2x g al 2
4p Õ x k g y kl
el k zd Ž v y kz . ,
k g2 y v 2 y v 2x 0
1 y
.
Elcrg y v 2x g al Elcr0 s 0.
Ž 1.
where k g s k q g, z is the particle’s velocity and x 0 and x g are the structure amplitudes in the expression for the crystal dielectric permeability
´ Ž v ,r . s 1 q x 0 Ž v . q
Ý
xg Ž v . e i g r ,
y
k )2 y k 12
ž
/
el k z
ž
1 k )2 y p 2 q 2 k g y D
Ž 1 y eyi Ž k ) yk 1 . L . 1
1
k )2 y p 2 q 2 k g y D
y
k )2 y k 22
/
= Ž 1 y eyi Ž k ) yk 2 . L . e iŽ k ) yp. L ,
g/0
a 1 s 1, a 2 s cos w , Elcr0 and Elcrg are the ‘primary’ and the ‘diffracted’ waves respectively, the subscript
where
(
p s v 2 y k g2 I ,
Dsg
ž
g 2kg y
y1
k g Is e z k z q e y k g y ,
/
is the so-called resonance defect, Õ x k ) s v q gÕ y y k g I z,
bl2 s Fig. 1. The angular variables used, e 1 is the beam axis, e 2 is the X detector axis, Q is the orientational angle.
v4 k g2 y
(
kl s D 2 q bl2 ,
x g xyg al2 ,
2 k 1,2 s p 2 q v 2x 0 y k g y Ž D " kl . .
Ž 2.
N.N. NasonoÕr Physics Letters A 260 (1999) 391–394
393
To calculate the spectral-angular distribution of emitted quanta the integration ElRad s Hd 3 k g = Ž . Elvac g exp ik g nr should be carried out by means of the stationary phase method. Here n is the unit vector along the direction of the emitted photon propagation. Defining the angular variables by the formulae
and v b s 9rŽ2sin2 Ž wr2.. is the well known Bragg frequency in the vicinity of which the emission spectrum is concentrated.
n s e 2 Ž 1 y 12 Q 2 . q Q ,
Let us make use of the general formula Ž4. for the analysis of emission properties. The first item in the brackets Ž4. proportional to factor Ž 1d y d y1 x 0 . corresponds to the contribution of transition radiation emitted on the in-surface of the crystal and diffracted in the Bragg-scattering direction. The last items describe the Bragg-scattering of an emitting particle Coulomb field Žordinary PXR.. Two branches of the solution of the wave Eq. Ž1. make the contribution to the total emission amplitude but only one of them corresponding to the quantity sq is essential for the ordinary PXR yield formation in a thick crystal when the asymptotic expression may be obtained from the last item in brackets Ž4. by the use of the condition Rew sq x s 0 Žthe equality Rew sy x s 0 is not valid.. This expression coincides with the one obtained earlier w5x taking into account the dynamical diffraction effects. It is well known that the ordinary PXR angular distribution has a deep gap in the range of Q - gy2 y x 0 f
1
ž
z s e1 1 y
2
e 2 Q s 0,
gy2 y 12 C 2 ,
/
e 1C s 0.
Ž 3.
we obtain the final expression for the spectral-angular distribution in the convenient form
v
dNl d v d 2Q
=
ž
s
1
d
e 2al2
X 16p 2 sin4 Ž wr2 . < kl <
1 y
ž
d y x0
ž
y 1y
/½ ž
exp y
ivL
yexp y
< x g2 <
2cos Ž wr2 .
sy
ivL 2cos Ž wr2 .
/5 ž
1 y exp y
sy d y x0
/
Vl2
sq
ivL
/
sy
2cos Ž wr2 . sy
/
ž
q 1y
sq
d y x0
/
ivL w sq 2cos 2
(
(g
y2
2
1 y exp y
3. Borrmann effect
0 , Ž 4.
sq
q v 02rv 2
, where v 0 is the plasma frequency. This circumstance allows us to neglect the contribution of ordinary PXR to the emission yield within the angular range Q F gy1 for the case of a high enough emitting electron’s energy
g 2 < x0 < f where V 1 s Q H yC H , V 2 s 2Q X q Q I qC I , Q I , C I ŽQ H , C H . are the components of the angles Q and C parallel Žperpendicular. to the plane defined by vectors e 1 and e 2 ,
klX s
v Xb
al2
2
žv /
y1 q
4sin4 Ž wr2 .
x g xyg ,
d s gy2 q V 12 q V 22 , s "s d y x 0 y 2sin2 Ž wr2 .
v Xb
žv
y 1 " klX ,
v Xb s v b Ž 1 q Ž Q X q Q I . cot Ž wr2 . . ,
(
/
g 2v 02 v 2b
4 1.
Ž 5.
On the other hand the width of an additional emission peak described by the first item in brackets Ž4. has a value of the order of gy1 in the emitting particle energy range Ž5., as follows directly from Ž4.. Thus the condition Ž5. allows to select the contribution of a particle transition radiation to PXR yield by the use of a photon-collimator with angular size of the order of Qc ; gy1 . Taking into account the principal aspects of the considered problem we have used the simplest crystal model with elementary cell containing one atom:
N.N. NasonoÕr Physics Letters A 260 (1999) 391–394
394
x 0 s yv 02rv 2 q i x 0XX , x g s yv g2rv 2 q i x gXX ' xyg , v g2 s v 02 F Ž g .rZ Ž F Ž g . is the atom formfactor, Z is the number of electrons in an atom.. A final expression for the collimated PXR yield has the very simple form v
dNl dv d Q =
vb L
e 2 Vl2
s
2
4p
2
d
For this reason one should determine the dependence of the total number of emitted quanta upon crystal thickness L. Assuming the emitting flux angular spread to be small enough ŽC 2 F gy2 . we obtain from Ž6. the following expression:
žv /
½ ž ž ž
qexp y
ž
(ž
=
cos Ž wr2 .
cos Ž wr2 .
sqXX
syXX
vL
y2exp y
=cos
vL vL
cos Ž wr2 .
/
Nl ,
/ /
v
XX s" s x 0XX .
2
/
(ž
v
)
4v g4 al2 g4
2
/
y1 q
al cos Ž wr2 . 2pv b L x gXX
vb L cos Ž wr2 .
½
Ž x 0XX y x gXX al .
= ln Ž 1 q g 2Qc2 . y
g 2Qc2 1 q g 2Qc2
5
,
Ž 7.
predicting the dependence NlŽ L. ; Ly1 r2 in the crystal thickness region
/5
,
2 v g2 al2rg 2
v Xb
2g
2
x0
cos Ž wr2 . y1 q
e 2v g2
=exp
2 v Lsin2 Ž wr2 .
v X0
x 0XX Ž v b . y x gXX Ž v b . al F 1.
cos Ž wr2 .
2
4v g4 al2rg 2 2 v Xb y 1 q 4v g4 al2rg 4
= exp y
character on the condition of the Borrmann effect manifestation vb L x XX Ž v b . 4 1, cos Ž wr2 . 0
4v g4 al2 4
LF
x gXX .
Ž 6.
cos Ž wr2 .
vb
Ž x 0XX y x gXX a L .
y1
corresponding to the anomalous photoabsorption suppression.
g
The derived formula demonstrates the very narrow angular distribution of considered emission peak Ž DQ ; gy1 .. Due to this the maximum spectral width Dv Žin the case Qc G gy1 . does not exced the value of the order of gy1v b . The first and second items in the brackets Ž6. correspond to the contribution of different branches to the emission yield. The last one describes the interferences between these branches. The main result of this work consists in the prediction of the Borrmann effect existence in the considered emission process. It is easy to see that the Borrmann effect is described by the factor expŽyŽ v Lrcos w2 . sqXX . in Ž6. and shows up in the case of x gXX f x 0XX . According to the expression for sqXX in Ž6. the dependence of the absorption length upon an emitted photon energy has a resonance
Acknowledgements The author is very thankful to Professor B.M. Bolotovsky for helpful discussion on the obtained result. It is a great pleasure to acknowledge the support from Russian Foundation for Basic Researches under grant 99–02–16920.
References w1x G. Borrmann, Z. Phys. 42 Ž1941. 157. w2x V. Bazylev, N. Zhevago, Fast Particle Emission in Matter, Nauka, Moscow, 1987. w3x A. Caticha, Phys. Rev. B 45 Ž1992. 9541. w4x Z. Pinsker, Dynamical Scattering of X-Rays in Crystals, Springer, Berlin, 1984. w5x N. Nasonov, A. Safronov, Proc. RREPS–93, Tomsk, 1993, p. 134.
Physics Letters A 260 Ž1999. 391–394 www.elsevier.nlrlocaterphysleta
Borrmann effect in parametric X-ray radiation Nikolai N. Nasonov
1
Laboratory of Radiation Physics, Belgorod State UniÕersity, 12 Studencheskaya, 308007 Belgorod, Russia Received 13 July 1999; accepted 28 July 1999 Communicated by V.M. Agranovich
Abstract The suppression of the photoabsorption in parametric X-ray radiation by relativistic particles in a crystal is predicted. This dynamical effect caused by the contribution of emitting particle transition radiation to parametric X-ray radiation yield is possible in a crystal with a finite thickness only. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 78.70.-g; 78.70.Ck; 32.30.Rj; 32.80.Cy
1. Introduction Anomalon suppression of the photoabsorption for X-rays propagating in a crystal is one of the most important dynamical diffraction effects known as a Borrmann effect w1x. This effect is realized in the process of a free photon scattering only. The performed analysis of a parametric X-ray radiation ŽPXR. by a relativistic particle moving in a crystal Žsee for example w2x. has shown the negligible role of Borrmann effect in the process of particle Coulomb pseudophoton scattering. The mentioned circumstance is the consequence of the small influence of dynamical diffraction effects upon PXR caused by momentum and energy conservation laws in the emission process. It is important to note that the conclusion w2x was obtained on the basis of the asymptotic expression for the PXR spectral-angular distribution describing PXR yield proportional to a crystal thickness. Dy1
E-mail: [email protected]
namical diffraction effects do not influence practically upon this yield characteristics. A more detailed analysis of a relativistic particle PXR in Laue geometry developed in the presented work has shown the existence of an additional PXR peak appearing due to Bragg scattering of a particle transition radiation Žthis peak is adequate to the well known transition diffracted radiation peak w3x.. A new peak is formed in the process of real photon scattering. Therefore, the dynamical diffraction effects ŽBorrmann effect in particular. can take place in the considered process. The possibility to observe the Borrmann effect in PXR of high enough energy particles crossing a crystal has been shown in the presented work.
2. General expressions Let us consider the structure of an electromagnetic field emitted by the flux of relativistic particles moving in a cristal with a thickness L along the axis
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 4 5 - 9
N.N. NasonoÕr Physics Letters A 260 (1999) 391–394
392
e 1 as is shown in Fig. 1. Here g is the reciprocal lattice vector defining the reflecting crystallographic plane of the crystal, w is the fixed emission angle, the orientational angle Q X counted from the Bragg resonance position describes the possible crystal turning by the use of goniometer. To find the Fourier-transform of an emission electrical field inside a crystal E kcrv s
1
Ž 2p .
yi k rqi v t
3
4
Hdtd rE Ž r ,t . e
l denotes the wave polarization, el k is the polarization vector E kcrv s Ý l el k Elcr0 , E kcrg v s Ý l el k gElcrg . Equations for the fields outside the crystal Elvac 0 Ž . and Elvac g follow from 1 in the limit x 0 , x g ™ 0. Determining the uniform and non-uniform solutions of presented equations separately inside and outside the crystal we find unknown coefficients in uniform solutions by means of ordinary boundary conditions. The final expression for the field Elvac g propagating along the Bragg scattering direction e 2 behind the crystal is given by
we make use of the well-known equations of the two-wave approximation of dynamical diffraction theory w4x,
Elvac g s al d Ž k x y p . ,
Ž k 2 y v 2 y v 2x 0 . Elcro y v 2xyg al Elcrg
al s
iv e s
Ž
2p 2
i v e v 2x g al 2
4p Õ x k g y kl
el k zd Ž v y kz . ,
k g2 y v 2 y v 2x 0
1 y
.
Elcrg y v 2x g al Elcr0 s 0.
Ž 1.
where k g s k q g, z is the particle’s velocity and x 0 and x g are the structure amplitudes in the expression for the crystal dielectric permeability
´ Ž v ,r . s 1 q x 0 Ž v . q
Ý
xg Ž v . e i g r ,
y
k )2 y k 12
ž
/
el k z
ž
1 k )2 y p 2 q 2 k g y D
Ž 1 y eyi Ž k ) yk 1 . L . 1
1
k )2 y p 2 q 2 k g y D
y
k )2 y k 22
/
= Ž 1 y eyi Ž k ) yk 2 . L . e iŽ k ) yp. L ,
g/0
a 1 s 1, a 2 s cos w , Elcr0 and Elcrg are the ‘primary’ and the ‘diffracted’ waves respectively, the subscript
where
(
p s v 2 y k g2 I ,
Dsg
ž
g 2kg y
y1
k g Is e z k z q e y k g y ,
/
is the so-called resonance defect, Õ x k ) s v q gÕ y y k g I z,
bl2 s Fig. 1. The angular variables used, e 1 is the beam axis, e 2 is the X detector axis, Q is the orientational angle.
v4 k g2 y
(
kl s D 2 q bl2 ,
x g xyg al2 ,
2 k 1,2 s p 2 q v 2x 0 y k g y Ž D " kl . .
Ž 2.
N.N. NasonoÕr Physics Letters A 260 (1999) 391–394
393
To calculate the spectral-angular distribution of emitted quanta the integration ElRad s Hd 3 k g = Ž . Elvac g exp ik g nr should be carried out by means of the stationary phase method. Here n is the unit vector along the direction of the emitted photon propagation. Defining the angular variables by the formulae
and v b s 9rŽ2sin2 Ž wr2.. is the well known Bragg frequency in the vicinity of which the emission spectrum is concentrated.
n s e 2 Ž 1 y 12 Q 2 . q Q ,
Let us make use of the general formula Ž4. for the analysis of emission properties. The first item in the brackets Ž4. proportional to factor Ž 1d y d y1 x 0 . corresponds to the contribution of transition radiation emitted on the in-surface of the crystal and diffracted in the Bragg-scattering direction. The last items describe the Bragg-scattering of an emitting particle Coulomb field Žordinary PXR.. Two branches of the solution of the wave Eq. Ž1. make the contribution to the total emission amplitude but only one of them corresponding to the quantity sq is essential for the ordinary PXR yield formation in a thick crystal when the asymptotic expression may be obtained from the last item in brackets Ž4. by the use of the condition Rew sq x s 0 Žthe equality Rew sy x s 0 is not valid.. This expression coincides with the one obtained earlier w5x taking into account the dynamical diffraction effects. It is well known that the ordinary PXR angular distribution has a deep gap in the range of Q - gy2 y x 0 f
1
ž
z s e1 1 y
2
e 2 Q s 0,
gy2 y 12 C 2 ,
/
e 1C s 0.
Ž 3.
we obtain the final expression for the spectral-angular distribution in the convenient form
v
dNl d v d 2Q
=
ž
s
1
d
e 2al2
X 16p 2 sin4 Ž wr2 . < kl <
1 y
ž
d y x0
ž
y 1y
/½ ž
exp y
ivL
yexp y
< x g2 <
2cos Ž wr2 .
sy
ivL 2cos Ž wr2 .
/5 ž
1 y exp y
sy d y x0
/
Vl2
sq
ivL
/
sy
2cos Ž wr2 . sy
/
ž
q 1y
sq
d y x0
/
ivL w sq 2cos 2
(
(g
y2
2
1 y exp y
3. Borrmann effect
0 , Ž 4.
sq
q v 02rv 2
, where v 0 is the plasma frequency. This circumstance allows us to neglect the contribution of ordinary PXR to the emission yield within the angular range Q F gy1 for the case of a high enough emitting electron’s energy
g 2 < x0 < f where V 1 s Q H yC H , V 2 s 2Q X q Q I qC I , Q I , C I ŽQ H , C H . are the components of the angles Q and C parallel Žperpendicular. to the plane defined by vectors e 1 and e 2 ,
klX s
v Xb
al2
2
žv /
y1 q
4sin4 Ž wr2 .
x g xyg ,
d s gy2 q V 12 q V 22 , s "s d y x 0 y 2sin2 Ž wr2 .
v Xb
žv
y 1 " klX ,
v Xb s v b Ž 1 q Ž Q X q Q I . cot Ž wr2 . . ,
(
/
g 2v 02 v 2b
4 1.
Ž 5.
On the other hand the width of an additional emission peak described by the first item in brackets Ž4. has a value of the order of gy1 in the emitting particle energy range Ž5., as follows directly from Ž4.. Thus the condition Ž5. allows to select the contribution of a particle transition radiation to PXR yield by the use of a photon-collimator with angular size of the order of Qc ; gy1 . Taking into account the principal aspects of the considered problem we have used the simplest crystal model with elementary cell containing one atom:
N.N. NasonoÕr Physics Letters A 260 (1999) 391–394
394
x 0 s yv 02rv 2 q i x 0XX , x g s yv g2rv 2 q i x gXX ' xyg , v g2 s v 02 F Ž g .rZ Ž F Ž g . is the atom formfactor, Z is the number of electrons in an atom.. A final expression for the collimated PXR yield has the very simple form v
dNl dv d Q =
vb L
e 2 Vl2
s
2
4p
2
d
For this reason one should determine the dependence of the total number of emitted quanta upon crystal thickness L. Assuming the emitting flux angular spread to be small enough ŽC 2 F gy2 . we obtain from Ž6. the following expression:
žv /
½ ž ž ž
qexp y
ž
(ž
=
cos Ž wr2 .
cos Ž wr2 .
sqXX
syXX
vL
y2exp y
=cos
vL vL
cos Ž wr2 .
/
Nl ,
/ /
v
XX s" s x 0XX .
2
/
(ž
v
)
4v g4 al2 g4
2
/
y1 q
al cos Ž wr2 . 2pv b L x gXX
vb L cos Ž wr2 .
½
Ž x 0XX y x gXX al .
= ln Ž 1 q g 2Qc2 . y
g 2Qc2 1 q g 2Qc2
5
,
Ž 7.
predicting the dependence NlŽ L. ; Ly1 r2 in the crystal thickness region
/5
,
2 v g2 al2rg 2
v Xb
2g
2
x0
cos Ž wr2 . y1 q
e 2v g2
=exp
2 v Lsin2 Ž wr2 .
v X0
x 0XX Ž v b . y x gXX Ž v b . al F 1.
cos Ž wr2 .
2
4v g4 al2rg 2 2 v Xb y 1 q 4v g4 al2rg 4
= exp y
character on the condition of the Borrmann effect manifestation vb L x XX Ž v b . 4 1, cos Ž wr2 . 0
4v g4 al2 4
LF
x gXX .
Ž 6.
cos Ž wr2 .
vb
Ž x 0XX y x gXX a L .
y1
corresponding to the anomalous photoabsorption suppression.
g
The derived formula demonstrates the very narrow angular distribution of considered emission peak Ž DQ ; gy1 .. Due to this the maximum spectral width Dv Žin the case Qc G gy1 . does not exced the value of the order of gy1v b . The first and second items in the brackets Ž6. correspond to the contribution of different branches to the emission yield. The last one describes the interferences between these branches. The main result of this work consists in the prediction of the Borrmann effect existence in the considered emission process. It is easy to see that the Borrmann effect is described by the factor expŽyŽ v Lrcos w2 . sqXX . in Ž6. and shows up in the case of x gXX f x 0XX . According to the expression for sqXX in Ž6. the dependence of the absorption length upon an emitted photon energy has a resonance
Acknowledgements The author is very thankful to Professor B.M. Bolotovsky for helpful discussion on the obtained result. It is a great pleasure to acknowledge the support from Russian Foundation for Basic Researches under grant 99–02–16920.
References w1x G. Borrmann, Z. Phys. 42 Ž1941. 157. w2x V. Bazylev, N. Zhevago, Fast Particle Emission in Matter, Nauka, Moscow, 1987. w3x A. Caticha, Phys. Rev. B 45 Ž1992. 9541. w4x Z. Pinsker, Dynamical Scattering of X-Rays in Crystals, Springer, Berlin, 1984. w5x N. Nasonov, A. Safronov, Proc. RREPS–93, Tomsk, 1993, p. 134.