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ON THE DIFFRACTION OF THE NUCLEAR RESONANT GAMMA-RAYS BY MOSAIC
CRYSTALS
F. Chukhovskii, I. Perstnev
To cite this version:
F. Chukhovskii, I. Perstnev. ON THE DIFFRACTION OF THE NUCLEAR RESONANT GAMMA- RAYS BY MOSAIC CRYSTALS. Journal de Physique Colloques, 1974, 35 (C6), pp.C6-185-C6-192.
�10.1051/jphyscol:1974619�. �jpa-00215776�
NO VEL AND EX0 TIC APPL ICA TIONS.
ON THE DIFFRACTION OF THE NUCLEAR RESONANT GAMU-RAYS BY MOSAIC CRYSTALS
F. N. CHUKHOVSKII and I. P. PERSTNEV
Institute of Crystallography, Academy of Sciences of the USSR, Moscow, USSR
Rhsumb. - On developpe la theorie de diffraction du rayonnement nucleaire resonnant dans des cristaux idkalement imparfaits en presence d'un ecartement hyperfin des niveaux nuclkaires.
On obtient analytiquement un tenseur de polarisation de l'onde diffractke. On a examine d'une manikre detaillee les phknomknes resultant de la birefringence et de la non-orthogonalit6 des polari- sations des ondes propres dans les cristaux, contenant des noyaux rksonnants.
On a dCduit les formules analytiques pour la definition de la phase de I'amplitude Blectronique de la diffusion suivant la solution asymptotique de I'Cquation intkgrale qui lie 17intensitC expk- rimentale de la reflexion Rl(u) au coefficient spectral de la diffusion Il(w).
Abstract. - A diffraction theory of the resonant nuclear radiation by mosaic crystals is developed in the presence of hyperfine splitting of nuclear levels. The polarization tensor of the diffracted wave is obtained analytically. Particular attention is paid to the interference and polarization pheno- mena, which occur as a result of the birefringence and the non-orthogonality of the eigenwave polarizations in a crystal containing resonant nuclei.
Analytical expressions are obtained for determining the phase of the electronic scattering ampli- tude by the asymptotic solution of the integral equation, which connects the experimental reflecting power Rl(u) with the spectral scattering coefficient Il(w).
1. Introduction. - Along with the phenomenon of the suppression of nuclear reaction in Mossbauer diffraction experiments predicted recently by Kagan and Afanas'ev [I, 21 (see also [3]), the interference of the nuclear resonant and electronic Rayleigh scattering is of special interest. The interference phenomena were experimentally observed by a number of authors [4-91.
The theory of the resonant gamma-ray diffraction is given in ref. [lo].
The investigation of the interference scattering of gamma-rays allows, in principle, to determine the hyperfine splitting parameters and leads to the practical possibility of the direct determination of the electronic structure factor phase [l 1-1 71.
In the general case, under conditions of the hyperfine splitting of the nuclear resonant line the change of the electromagnetic field polarization (the Faraday nuclear magneto-optical effect [18]) essentially complicates the interference picture analysis. Nevertheless, in a number of the important physical cases, the problem of the gamma-ray propagation through a nuclear resonant medium can completely be analysed. In particular, a broad class of phenomena in the presence of hyperfine splitting can be revealed when the suppression effect occurs 119-201. So, a rather peculiar interference picture under the condition of the suppression effect
for two electromagnetic field polarizations is obtain- ed [20], the picture being caused by the difference in the phase velocities and by the non-orthogonality of the polarization vectors of the eigenwaves in a perfect crystal. In practice, the most of crystals have a pro- nounced mosaic structure and the experimental data [4- 81 is in a good agreement with the theory in which the radiation extinction is neglected (the ideal mosaic crystal model). It seems resonable to develop a theory of the resonant gamma-ray diffraction by mosaic crystals for an arbitrary hyperfine splitting.
In the present paper a detailed analysis of the reso- nant gamma-ray propagation through an ideal mosaic crystal is carried out. The problem of the gamma-ray diffraction in this case can be solved exactly. The propagation of the transmitted and the diffracted radiation through a crystal is described by analogy with the well-known magneto-optical birefringence effect [21]. However, in the problem under considera- tion, unlike the light optics, the dielectric polarizability tensor is non-Hermitian. This is connected with the resonant absorption of gamma-rays in a Mijssbauer medium. In Section 2 the waves and polarization eigen- vectors of the electromagnetic field inside a mosaic crystal are found. An analytical expression for the polarization tensor of the diffracted wave 1Ysi(o) is
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974619
C6-186 F. N. CHUKHOVSKII AND I. P. PERSTNEV obtained. The interference and polarization pheno-
mena (see Section 3) arising as a result of the bire- fringence and the non-orthogonality of polarization eigenvectors are analysed. Then, the results of this investigation (Section 4) are applied to Mossbauer Bragg diffraction in a-Fe20,, reflection (888) [9], to estimate the influence of the spatial direction of the internal magnetic field at FeS7 nuclei upon the velocity dependence of the Bragg reflecting power R1(v). The problem of phase determination from the Mossbauer diffraction experiments is discussed (Section 5). The phase of the electronic structure factor is derived from the wings and the area of the experimental curve R,(v).
In a mosaic crystal case the situation is simplified.
The propagation of gamma-rays through a crystal is described by the Fresnel equation
In passing from (2.3) to (2.4) we have set Xj$,, = 0 for h f h' and have taken into account that the electro- magnetic field inside a crystal remains practically transversal. Besides, we have used the continuity condition of the tangential component of gamma-ray wave vector at the crystal surface in the form
2. The derivation of the fundamental equations. The kh = K~ + IC Eh - n , yh = cos (K,, n n) Faraday nuclear effect. - Let a monochromatic plane Yh
wave of gamma-rays enters a crystal. To describe the electromagnetic field inside the crystal, the Maxwell and the material equations are used as follows
Here eil" is the unit antisymmetric pseudotensor, K, the wave vector of the incident gamma-quanta, the magnetic susceptibility is taken equal to unity, Xi\, is
the spatial Fourier transform of the crystal polarizabi- lity the summation over repeated indices is carried out as usual.
The complex polarizability tensor X;h, is a simple sum of two components corresponding to the electronic and the resonant nuclear subsystems of a crystal
the electronic tensor XF-h, being proportional to the unit tensor
il a i l
Xh-h* = Xh-h, (2.2b)
and the nuclear tensor gthr depending on the transition type, on the hyperfine splitting and generally not reducing to the unit tensor 6". Note that in the case under conside~ation the polarization properties of the medium, as will be shown below, are determined only by the nuclear polarizability g:;,.
Substituting the first eq. (2.1) in the second one we find the following set of equations
(n is the unit normal to the crystal surface and K~ is the real wave vector of the gamma-quanta in the vacuum).
At the same time the scattering with the transition of a y-quantum from the initial state I k,, eo > into the final one I k,, el > can be described in the first order of the perturbation theory. Namely, the transition amplitude is given by
It is easy to see that now the finding out of the eigen- values E,, and the polarization eigenvectors e,, reduces to the diagonalization of the second-order matrix and therefore two possible polarization eigenvectors e,, (I = 1,2) correspond to each propagation direction K,.
We represent the solution of (2.4) in the form
where ef) and ep) are the mutually orthogonal real unit vectors which are orthogonal to the wave vector
K,. To be definite we have chosen the polarization vectors e r ) perpendicular to the scattering plane (kO> kl).
Making use of eq. (2.4) and (2.5), the following expressions for the eigenvalues E,, and the expansion coefficients ah, and b,, of the polarization eigenvectors are obtained
en 2 - 1,2 ( XE
(;:;I
= (I X a 2 + I 2 E h L - xhh1 ) 2 &h2. - Xhh .(2.3) Here we have introduced the notations which is a generalization of the Fresnel equation [21] = e;) xrrh A e p f , s, s' = o, n .
in spatially periodic media in the crystal optics. If the
Bragg condition is satisfied the transmitted and the The characteristic feature of the solution obtained diffracted waves (h = 0, 1, respectively) are dynami- as compared to one of the crystal optics is that he pola- cally coupled via the set of two vector equations. So the rizability tensor X:i is non-Hermitian. As a result, the problem solution is reduced, in general case, to the polarization eigenvectors eh, are complex and mutually diagonalization of the fourth order matrix 13, 19-20]. non-orthogonal. Thus, in the general case, there are two
ON THE DIFFRACTION OF THE NUCLEAR RESONANT ?-RAYS BY MOSAIC CRYSTALS C6-187 elliptically polarized waves propagating through the
crystal. It should be mentioned that according to eq. (2.5)-(2.7), the polarization eigenvectors do not depend on the electronic part of the crystal polari- zability and the electronic term 3 x0 enters the expres- sion for gh, only.
The polarization tensor I;L"~(w) of the scattered wave emerging from a mosaic crystal of thickness t along the direction k, is written as
where O(z) is the step function : O(z) = 0 for z < 0 and O(z) = 1 for z > 0 and y, = cos ( ~ , , n ) A is direction cosine of the diffracted wave with respect to the inci-
dent normal. The wave amplitudes Eo(z) and El(z, z') are given by
el(z, z') = z cla(zf) el, exp r
The coefficients clr(z) are proportional to the ampli- tude eo(z) in the first order of the perturbation theory and the constant coefficients c0a are found from the boundary conditions (cf. [17] for details). Now substi- tuting (2.9) in (2.8) and introducing the polarization tensor of the incident radiation I:', after the direct calculations one finds
1;'"(0) = R:~?'(W) I:' (2.10)
ss'
80, - &o,, SIP - &?,,))
1
1 - exp jirt j-- Yo - Y 1 Y 1 < 0x , (2.11)
where we have introduced the reciprocal vectors ch, so
that EhA ehd' = r)lE.p.
The polarization tensor (2.11) may be used to describe the optical properties of the scattered resonant radiation from an absorbing mosaic crystal in the presence of an arbitrary hyperfine splitting.
3. Interference and polarization effects. - It is clear from the preceding considerations that the polarization tensor Is1"(o) depends essentially on the nuclear polari- zability tensor g:h, and in general case, it has a rather complicated form. It is of interest to analyse several particular cases which are general enough. Further we confine ourselves to the case of the nuclear M1- transition when spins of the ground and the excited levels are equal to I, = 3, I = 3 or I, = 9, I = 1 2 ,
respectively (for example, M1-transition of the nuclei Fe57, Sn119, Xe131). Besides, we assume that the hyperfine splitting of the nuclear line is induced by either the axial electric field gradient (EFG) or by the internal magnetic field at the nuclei.
3.1 For crystals, in which the hyperfine splitting is induced by several axial EFG at the nuclei in the unit cell, the tensor h,g: has the form
k n ikm elnp
g;;. = -go - e
8 r2
Here, the summation over the positions of the resonant nuclei in the unit cell is carried out, q j is the unit vector along the direction of EFG axis at jth nucleus, 2 A j is the electric quadrupole splitting ; o0 and r are the mean energy and the total width of the Mossbauer line.
Utilizing (3.1) as well as (2.5)-(2.7) it is easy to prove that in the case, when EFG at the resonant nuclei in the unit cell do not coincide over the quantities and the axis directions, two elliptically polarized eigenwaves propagate through the crystal, which correspond to a certain direction of the wave vector. In opposite cases eigenwaves are linearly polarized and mutually orthogonal. The orthorhombic phase of the ferroelectric boracities [22] offers the example of the
C6-188 F. N. CHUKHOVSKII AND I. P. PERSTNEV first kind. In this phase there are two non-equivalent
positions of the resonant nuclei FeS7 within the unit cell. Sodium nitroprusside crystal, in which the axial EFG have two different directions and the quadrupole splittings Aj are equal [23], is the example of the second kind. Note that if EFG directions q j do not depend on index j, the polarization eigenvectors are found directly from the symmetry
3 . 2 In magnetic crystals the nuclear polarization tensor can be represented in the form (the Ml-tran- sition)
where hj is the unit vector of the magnetic field as the jth nucleus, the other notations are generally known.
If the contributions to the polarizability gL;, of all but one components of the hyperfine splitting are negligible for a given value of the energy of the gamma- quantum, then the polarization eigenvectors are complex and mutually orthogonal, i. e. e:, eh,,=6,,,.
For the transitions with M - Mo = 0 the situation is absolutely analogous to the case of the quadrupole splitting of the second kind whereas for the transi- tions with M - Mo = + 1 the eigenwaves are ellipti- cally polarized. It is easy to see that for the single component of the hyperfine splitting the tensor g;LT
can be represented in the bi-vector form [3]
and from the symmetry consideration one immediately obtained the polarization eigenvectors (see (3.2))
In accordance with (3.4) one of the eigenpolariza- tions of the transmitted and diffracted waves (1 = ,u = 1) does not interact with nuclei. Therefore, in the expres- sion (2.11) only one term remains and the polarization tensor ~:'"(o) has no oscillations depending on the crystal thickness provided one can neglect the gamma- quantum interaction with electrons. Thus, the polari- zation of the diffracted radiation is strictly fixed and does not depend on the polarization of the incident wave. In particular, if the direction of the magnetic field h is perpendicular to the wave vector k 1 the diffracted wave is linearly polarized parallel to h for the components with M - M , = + 1 and perpendicular to h for the ones with M - Mo = 0. If h Il k1 the diffracted wave is circularly polarized. It is of interest
to note that if all components of the hyperfine splitting are taken into account, the case, when the vector h is parallel to the wave vector k,, is analogous to the Faraday effect in crystal optics. Indeed, two diffracted eigenwaves, which propagate in the medium are cir- cularly polarized with the opposite directions of the rotation. However, the diffracted wave emerging from a crystal is now elliptically polarized as a result of diffe- rent absorption of the eigenwaves in the crystal.
3.3 Now, we consider a case when the polarization eigenvectors are mutually nonorthogonal and the intensity of the diffracted wave Z,(w) oscillates with the crystal thickness. To observe this interference effect it is necessary that the crystal thickness should be less or of the order of the absorption length
and simultaneously the inequality
should hold.
Let us analyse the symmetrical diffraction of the resonant gamma-rays (yo = y,) by the crystal film of natural iron. The diffraction geometry is shown in figure 1. The sample is magnetized along the easy magnetization axis [OOl].
FIG. 1. - The scattering plane and the direction of the magneti- zation with respect to the crystallographic directions in Fe. The
direction 11 101 is the normal to the crystal surface.
For a given direction of the internal magnetic field at the nuclei the eigenvalues (2.6) in accordance with (3.3) are independent of the index h. The conditions (3.5) being fulfilled one obtains for the intensity of the diffracted wave
ON THE DIFFRACTION OF THE NUCLEAR RESONANT y-RAYS BY MOSAIC CRYSTALS C6-189
(In deriving (3.6) we have neglected the terms of the order of I rct Re (sh, - shJ I 4 1.)
In the case under consideration the nuclear scattering amplitude is much smaller than the electronic one. The off-diagonal matrix element of the Fourier compo- nent of the crystal polarizability is
Here ro is the classical radius of an electron, F(K,) is the atomic scattering factor ; K, = k , - ko is the momen- tum transferred ; Vo is the unit cell volume.
Suppose now that the gamma quanta energy o is close to the energy of the second line of the hyperfine splitting with M - Mo = 0 (but, nevertheless, the inequality I o - 0, 1 % r assumes to be fulfilled).
Then, in the zero-order approximation the eigen- polarizations are orthogonal and are determined by the relations (3.4) with A, = A, = [k, h], while the non-orthogonality correction for the polarization vec- tors can be calculated in the first order of the perturba- tion theory. For this purpose it is most suitable to use just the initial set of eq. (2.4). The direct calculations give
sin 8 (3.7) (eg2 eel) = - (eT2 el,) = i --
c0s2 0 l . Substituting (3.7) in (3.6) gives finally
2 K t
+ exp ( - - Y o Im so,)
where ( is the dimensionless non-orthogonality para- meter :
It follows from (3.8) that the amplitude of the inten- sity oscillations is proportional to t2 and the inter- ference effect has the order of for the case of the natural iron. The value of is apparently the lower estimate of the effect as the iron has a relatively large hyperfine splitting. If one utilizes the non-magnetic crystals (e. g. white tin) in a proper magnetic field the splitting being about 5 r (instead of 50 r for Fe) the interference effect will be about one percent of the total intensity. Note that the smallness of the interference
effect in the case of a mosaic crystal as compared to a perfect crystal case when it is of the order of unity [20]
is compensated by the lack of the high requirements to the crystal perfection and t o the angular collimation of the incident beam.
4. Energy dependence of the reflecting power. - In ref. [9] the energy curves of the diffracted wave intensity for the different Bragg reflections from the hematite crystal a-Fe,O, have been investigated in detail. I t is of interest to make reasonable estimate of the spatial directions of the internal magnetic field at the positions of nuclei using the Mossbauer diffraction data from ref. [9]. We consider a case of the Bragg reflection (888).
The geometry of the problem us shown in figure 2. In
FIG. 2. - The orientation of the internal magnetic field at the nuclei in hematite with respect to the scattering plane.
the experiment the direction of the magnetic field at the Fe-nuclei is perpendicular to the crystal axis [l 1 I]. The hematite has a collinear antiferromagnetic structure and the hyperfine splitting does not depend on the index j of the resonant nucleus in the unit cell. Then, in the expression (3.3) with h = h' the term which depends linearly on the direction of the magnetic field, vanishes. Assuming the energy of the incident quanta is close to the extreme left (right) resonance with M - Mo = + 1 and the contribution of other lines t o the polarizability can be neglected, the eigenvalues and the polarization vectors become
e,, = (sin2 (p sin2 0 + cos2 (p)-'I2
sin (p sin 0
C6-190 F. N. CHUKHOVSKII AND I. P. PERSTNEV
So, these are functions of the angle cp between the magnetic field direction h and the normal p to the scattering plane (k,, k,).
Simple calculations lead to the following expression for the reflecting power (cf. (2.10)-(2.1 I), y, = - yo, t + m )
I el, ;lo eon 1
zlio) =
2 Im (eon + el,). (4.3) Assuming cp to be small we estimate now the influence of the magnetic field direction on the energy depen- dence of I,(o). Neglecting, for simplicity, a small deviation of the Bragg angle 8 from 7114 one finds from (4.3)
within the accuracy of terms - cp4 < 1.
As follows from (4.4), when cp = 0 , near to the exact resonance o = o, the curve has a dip which is equivalent to the additional absorption in this energy region.
When the angle cp being increased, this dip vanishes.
The condition of the lack of the dip in the energy curve has obviously the form
I l ( 4 2 Z l ( ~ 0 )
which gives the following estimate for cp
Substituting the numerical values X , = - 16.4 + i 3.3, Im xo = 5.0, I g, ,,, I = 430, Im g,,, = 1 150 corres- ponding to the experiment [9] in ( 4 . 5 ) one obtains
cp 2 250 .
The numerical calculation using eq. (2. lo), (2.11) and (3.3) with account of the exact value of the Bragg angle and all the lines of the hyperfine splitting do not change the estimate (4.5). The results of these calcula- tions of the energy dependence of Z,(W) for three values
rp = 00,200 and 450 are presented in figure 3. Only the left parts of the curves are shown, since the whole picture is almost symmetrical with respect to the middle point.
It is easy to see that the essential variations of the energy curve as a function of the angle cp occurs for the extreme left and the extreme right lines with M - M o = + 1. This is connected with the physical circumstance that, the scattering of the a-polarized gamma quanta on the Bragg angle 6 LX z/4 is negligibly small whereas z-polarized quanta do not interact with the nuclei at all when cp = 0. While with the increase
FIG. 3. - The theoretical curves of the symmetric Bragg reflect- ing power. Hematite crystal, reflection (888), 14.4 keV-gamma rays ; 9 being the angle between the direction of the internal magnetic field and the normal to the scattering plane : a ) dot- and-dash line is the curve for 9 = O0 ; b) dash line - yl = 20° ;
c) solid line - 9 = 450.
of the angle cp the scattering of the gamma quanta with eigenpolarizations neighbouring to the vector ef) increases rapidly.
5. Determination of the phase of the electronic scat- tering amplitude. - Up to now we considered the monochromatic incident beam. This is obviously correct when the width of the energy spectrum Is(v, o ) of the source can be neglected as compared with the distance between the hyperfine splitting lines of the crystal-scatterer. If this condition does not hold the experimental reflecting power R,(u) is related to the spectral scattering power Il(w) by the integral equa- tion [15, 161
wheref, is the Mossbauer factor of the source, v being the source velocity with respect to the scatterer.
Expression (5.1) is basic for determining the phase of the electronic scattering amplitude cp, which enters
W - W o
I,(o) as a parameter only. When --- r goes to infi- nity, the nuclear resonant scattering is practically negligible and Z l ( o ) is determined by the pure electronic scattering. This means that I , ( m ) contains no infor- mation about the phase cp.
Suppose now that the source spectrum Z,(v, o) has the Lorentz form
ON THE DIFFRACTION OF THE NUCLEAR RESONANT 7-RAYS BY MOSAIC CRYSTALS C6-191 (eq. (5.2) assumes that the source line is not split and
that the self-absorption in the source can be neglected).
Thus the problem of finding the phase of the elec- tronic scattering amplitude from the Mossbauer diffraction data reduces to the solution of the integral eq. (5.1) with the kernel (5.2).
In the dimensionless variables
(0, and r being the center and the total width of the Mossbauer line in the crystal) the solution has the form [16-171
Il(u) - I 1 ( m ) = f,-I cos
i-
r-2)
[Rl(u) - R,(co)] + + n f ~ sin ) (T: v, plm
d U r R I ( u ' ) - R ~ ( m ) .- 03 u-U'
Here COS sin (T:) denotes symbolically the series of appropriate trigonometric functions in powers of the differential operator ? a/au.
Using the asymptotic solution of (5.4) far from the nuclear resonance we can obtain equations for the direct determination of the phase cp. Indeed, putting
1 u 1 % 1 formally in (5.4) we obtain
The coefficients A and B depend on the phase cp.
Therefore the problem of the electronic phase determination may in practice be reduced to finding two real parameters, A and B, which are in turn determined from the asymptotic behaviour and the area under the experimental interference curve Rl(u) in accordance with eq. (5.5) and (5.6) (cf. 1161 for details). The coefficient A is universal and does not depend on the magnitude and direction of the internal fields at the nuclei. In order to determine cp uniquely (except when cp = 0 and cp = n) it is essential to use the value of the coefficient B which is determined not only by the asymptotic of R,(u), but also by the area under the experimental curve R,(u) - R,(co). The form of B depends to a large extent on the parameters of the Mossbauer line splitting. Note that the area under the experimental reflection curve can be expressed in terms of the integral of the spectral scattering power I,(o)
over the gamma quantum energy. From (5.1) and (5.2) we have
'Then (5.6) transforms to (1 u 1 + 1 )
from which in principle it is possible to determine the phase cp using only the asymptotic expansion of the experimental curve R,(u). However, the integral in (5.7) now also depends on the phase cp. I t can easily be seen that for an unsplitted Mossbauer line it is better to use (5.7) for finding the coefficients A and B.
Now we apply (5.5)-(5.6) to calculate the phase cp for the Bragg diffraction of the resonant gamma quanta from the (020) and (040) planes of potassium ferri- cyanide K ,Fe(CN), [I 51. An analysis of the Mbsbauer- effect data for this crystal has shown that there is an EFG at the Fe nuclei which produces a splitting of the excited state of Fe57 nucleus into two levels.
Assuming that the EFG is axial and using (3.1) after the direct calculation one can obtain (cf. 1171)
where 2 A is the magnitude of the quadrupole splitting, q , and q , are the unit vectors in the directions of the axes of the EFG at the first and the second nucleus in the unit cell respectively.
Substituting the numerical values of the parameters corresponding to the conditions of the experiment [IS]
in ( 5 . 8 ) and (5.9), we find
Aoz0 = 3.08 cos cpO2, , A,,, = 1.02 cos yo,, (5.10) Bozo = - 1.79 - 3.08 sin cp,,, -
- 2.50[3(q1 n)' + 3(q2 n)' - I] cos qo2,
B,,, = - 3.90 - 1.02 sin cp,,, -
- 0.83[3(q1 n)' + t ( q 2 nI2 - 11 cos cp,,, .
C6-192 F. N. CHUKHOVSKII AND I. P. PERSTNEV It can be seen from (5.11) that the theoretical value of
B highly depends on the direction of the axes of the EFG and it is possible, in principle, to judge these directions from the values of B.
The experimental values of A and B obtained from (5.7) by a least squares fit to the R,(v) inter- ference curves [15] are given in Table I. The first row of Table I shows the number of experimental points on the left wing of the R,(v) interference curve. The points on the right wing were chosen symmetrically about the exact resonance v = c(o, - o,)/o,. The fourth and seventh rows give the values of cos q0,, and cos 9040 calculated by A.
One sees that the values of A and B obtained from the R,(v) curves are stable and almost correct (should be cos q = I), while for B the spread of values is
rather large although the sign and the order of magni- tude are correct. It should be noted that in the present case the phase of the electronic scattering amplitude is sufficiently well defined by the coefficient A alone and the uncertainty A 9 + 200 is obtained.
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