design and calibration of a source and a polarimeter for polarized 14.4 kev mössbauer gamma rays
TRANSCRIPT
Design and calibration of a source and a polarimeter for polarized 14.4 keV Mossbauer gamma rays
J . M. DANIELS, H. Y. LAM, A N D P. L. LI' D C P ( I I ~ / ? I C ? I ~ of Pt~ysics, Urriuersity of Toronto, Toror~to, O ~ I I . , Cancrh M5S / A 7
Received December 18. 19792
A source of polarized 14.4 keV Mijssbauer y rays. using a magnetized foil, and a polarizer using a single crystal of siderite, are described. The procedures for determining the polarization of they rays, and for calibrating and using the polarimeter, are described in detail.
On dicrit une source d e rayons gamma Miissbauerde 14.4 keV utilisant une feuille aimantee et un polariseur utilisant un monocristal de siderite. On decrit aussi en detail les procedures suivies pour la determination d e la polarisation des rayons y ainsi que pour la calibration et I'utilisation du polarimktre.
Can. J . Phys.. 58. IOlO(1980) [Traduit par le journal]
Introduction It has been known for a long time that the inten-
sity of Mossbauer absorption lines depends on the polarization of the y rays, and on their direction relative to the nuclear spin axes of the absorber (1). However, very little use has been made of these facts for the purpose of elucidating properties of the absorber-such as finding out the directions of these spin axes. Recently, Daniels (2) has extended the work of Frauenfelder et al. and has reformu- lated it in a manner more appropriate for the sys- tematic investigation of the absorber. An example of such an investigation is the determination of the spin arrangements in the ordered state of K,Fe(CN), below its Nee1 temperature (3); other similar investigations are being carried out and it appears that this technique may be exploited exten- sively in the future.
Since the information which can be obtained from such experiments comes from the intensities of the absorption lines, it is important to be able to make these measurements accurately and with confidence. It is also important, when using polarized y rays, to know exactly what is the polarization state of the radiation used. These are not trivial matters, and the purpose of this paper is to describe the construction and calibration of a source of polarized y rays for S7Fe Mossbauer spectroscopy, and to describe the construction and calibration of a polarimeter for 14.4keV S7Fe Mossbauer y rays.
The Choice of a Polarized Source Two ways of making a source of polarized
'Present address: Martin Marietta Corporation, P.O. Box 5837. Orlando, FL 32855, U.S.A.
2Revision received March 20, 1980.
Mossbauer y rays have been proposed and put into effect. One way is to take a single line unpolarized source, and to filter out the unwanted component in a manner analogous to the way visible light is polarized with a Polaroid. The other way is to use a source with a split line-either a Zeeman or a quad- rupole splitting-in which the different compo- nents of the line have different polarizations.
Although a single line polarized source appears neater, we would assert that such a source is not suitable for these kinds of experiment for the fol- lowing three reasons:
First, the intensity of an absorption line depends not only on the polarization of the incident y rays, but also on other factors, such as the effective thickness of the absorber, which may change when the position of the absorber is changed. If all these other factors are to be eliminated, it is very desira- ble to have at least two lines of different polariza- tion in the source, and to measure the ratios of intensities of different lines in the same absorption spectrum.
Second, practical filters absorb quite a bit of the component to which they are supposed to be trans- parent. This is because they are rarely ideal, and this is particularly true if the filter is a magnetized iron foil because it is not possible to magnetize such foils to saturation. Filters based on a quadrupole splitting in a single crystal are much better in this respect. There is also the nonresonant absorption which cuts down the intensity of the y rays of the desired polarization, and this can be quite appreci- able unless the filter is depleted in S7Fe. As a result, the intensity of the filtered radiation is cut down, and yet the degree of polarization is disappointingly small. As is well known, if t is the thickness parameter, such that the intensity of the absorbed
ooO8-4204180107 1010-09$0 1 .OO/O @I980 National Research Council of CanadaIConseil national de recherches du Canada
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DANIELS ET AL. 101 1
component is reduced by afactor e-' at the centre of the absorption line, the integrated intensity of this component is e-L12Jo(it/2), which approaches ll(nt)'I2 for large t (4). Some numerical values il- lustrate this point-if t = 4.6, e-' = 0.01 and e-L12Jo(it12) = 0.28; if t = 12, e-( = lop5 and e-t'2Jo(it/2) = 0.17-and it is easy to see that, even using enriched iron, it is not easy to design an efficient polarizing filter.
It is not a serious disadvantage if they radiation is not completely polarized, but what is much more serious is the distortion of the lineshape produced by the filter. This is the third reason why we main- tain that filtered sources are not suitable for these experiments. Suppose that the intensity distribu- tion, as a function of energy, of the emission line has a Lorentzian profile, ll(1 + x2), where x is the energy difference from the line centre in units of the half width at half maximum. Then the distribution after filtering is given by:
[I] [I/( 1 + x2)l exp [- tl( l + x2)l
Figure 1 shows this distribution for t = 4.6 and it is seen that it has a double peak. Figure 1 also shows how the distribution of the transmitted component is distorted for t = 0.3, a value which is not unrea- sonable for a filter made from a magnetized iron foil. These distortions can pose serious problems in fitting lines to the absorption spectra and in deter- mining intensities.
There are two well-known environments in which lines of different but known polarization are emitted; these are an axial quadrupole splitting and an axial Zeeman splitting. An axial quadrupole splitting gives two lines of which one is fully plane polarized and the other is partially plane polarized.
FIG. 1. ( (1) The intensityof radiation 1/(1 + x 2 ) , as afunction of energy x of a Lorentzian emission line. The units o fx are the half width at half maximum. ( b ) and ( c ) The energy distribution, exp ( - r / { + x 2 } ) / ( 1 + x 2 ) of radiation of the same line after passing through a filter with a Lorentzian absorption profile of the same width and same centre: in ( b ) t = 0.3 , and in ( c ) t = 4.6.
The disadvantages of this source are that there is no possibility of obtaining a circularly polarized com- ponent, and that there is no possibility of perform- ing the internal consistency tests (described later in this paper) which are possible with a Zeeman split source. Also, a single crystal is needed, and suita- ble crystals are rare and, being non-metallic, the manufacturing techniques are not well known (LiNbO, has been used (5)). However, the absorp- tion spectrum obtained with such a source is almost as simple as it can be, consisting of only two superimposed absorption spectra, one for each source line.
A Zeeman split source, on the other hand, has six lines and thus gives six superimposed spectra, which is a complication in the analysis. It also needs a magnet which can be rather heavy. How- ever, it does not have the other disadvantages ofthe quadrupole split source; the technique of making such a source is well known, and the various com- ponents can be bought commercially. For these reasons we chose to make a Zeeman split source.
Description of the Source and Its Calibration Our source was about 20 mCi of S7Co diffused
into a spot 6 mm diameter in the centre of afoil3 cm x 1 cm x 25 pm of iron enriched to 98% in 56Fe. It was mounted across the faces of an Alnico V per- manent horseshoe magnet, and it was supported sandwiched between two pieces of Lucite to pre- vent it vibrating relative to this magnet. These Lu- cite sheets were glued to the magnet with epoxy, and the outer one had a hole in it so as not to obstruct the y rays. The dimensions of the source were as shown in Fig. 2, its weight was 30g, and the field measured half way between the pole faces in the place where the 57Co was to be situated was 270 G. A small mirror was glued on to the side of the magnet to aid in the azimuthal orientation of the source.
In order to describe the source and the experi- ments which were done with it, we need a coordi- nate system which is shown in Fig. 2. (x, y, z) is a Cartesian system where y and z lie in the plane of the foil and where Oz is the direction of the magne- tic field at the S7Fe nuclei. The direction of a y ray emitted from the source at the origin 0 is described by the usual polar angles 8 and 4. In order to de- scribe the polarization of they ray it is necessary to specify the angle of rotation about the direction Oy . This angle is denoted by \Ir and is measured anti- clockwise, as seen by an observer looking back towards 0, from the direction PX which is the direction of increasing 8 in the plane zOy.
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FIG. 2. Diagram of the source and the coordinate system used to describe the experiments. The dimensions were a = 30 mm, b = 15 mm, c = 20 mm, d = 10 mrn, and e = 9 mm. M is a mirror, 4 mm x 4 mm, stuck on to the side of the magnet using epoxy, and L is a Lucite support for the iron foil; the outer Lucite support is not shown. x , y and z are Cartesian axes whose origin is the centre O of the source. Oy is the direction of emission of a y-ray. PX and PM are two directions perpendicular to Oy, and PX lies in the plane zOy.
The method we use to describe the polarization is to specify oi, the density matrix in photon spin space. We choose as basis states the state of posi- tive helicity (i.e., the photon spin is directed along the direction of propagation Oy) denoted by I+) or (A), and the state of negative helicity denoted by I-) or ( y ) , and the phases are chosen so that in each state at time zero the electric vector lies at an angle $ = 90". With this convention, the state of elliptic polarization shown in Fig. 3 can be specified by the state vector
(sin p + cos p) e-'a (sin p - cos p) eiu I
or by the density matrix
(1 + sin 2P) - cos 2P e-2i" 6. = -
-cos 2p e2'" (1 - sin 2P) 1 It is convenient to write this matrix in the form
a + b c - i d c + i d a - b 1
and to specify the values of the four (real) expan- sion coefficients a , b, c, and d. Thus, for the elliptic polarization shown in Fig. 3, a = i, b = 3 sin 2P, c = - 3 cos 2P cos 2a, and d = -3 cos 2P sin 2a. (It is seen that these are the same, to within a small numerical factor, as the Stokes parameters as usu- ally defined for the description of polarized light.)
A Zeeman split source emits 6 lines, and these
FIG. 3. Definition of the angles a and P used to denote a state of elliptic polarization, where the tip of the electric vector fol- lows the ellipse in the direction of the arrow.
are conventionally numbered from 1 to 6, line 1 being the line of highest energy corresponding to the transition +3/2 t, + 112. If the axis of quantiza- tion of the nuclei were along the z-axis, the density matrices of lines 1 and 2 would be:
fo r l ine1 :a l=f r ( l+cosZ9) , b l = c o s 9 c, = -&(I - cos29), d l = 0
for line 2: a, = i (1 - cos2 91, b, = 0 c, = $(I - cos2 9), d, = 0
Not only in this ideal case, but also in all cases, the density matrix of line 5 is the same as that of line 2, the density matrix of line 6 is the same as that of line 1 but with the sign of b reversed, and the density matrices of lines 4 and 3 are the same as those of lines 1 and 6, respectively, but with all the elements divided by 3. However, the axes of quantization do not all lie along the z-axis because the foil cannot be magnetized to saturation with a small permanent magnet. To take account of this, we suppose that the distribution of these axes is represented by a function p(9, 4), where the probability that an axis can be found in a solid angle df2 in the direction (9, 4 ) is p(9, 4 ) df2. This function p(9, 4 ) is then expanded in a series of zonal harmonics. Now, since the 14.4 keV y ray is MI , only harmonics up to the second order need be considered, and also, since the x-z and y-z planes are mirror planes for p(9, 4) . The coefficients of several of these har- monics vanish. The only nonvanishing coefficients which matter are
AoO = Jp(9 ,4) df2 = 1
[21 A,O = Jp(B,+) cos 9 df2
A20 = Jp(9 ,4)($cos2 9 - 3) df2
A,, = Jp(9 ,4)(<3/2) sin2 9 cos 241 df2
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DANIELS ET AL. 1013
Thus, the source is characterized by the three coefficients AlO, A20, and A,', and the polarizations of the six lines are known, in all directions, if these three coefficients are known. Formulae expressing the density matrices in terms of AI0, A20, and have been given by Daniels (20); they are:
for line 1:
+ (<3/6) sin2 8 cos 24
x AZ2(1 + cos2 8) cos 2 4 d l = ( f i 1 3 ) A , ~ cos 8 sin 24
for line 2: a , = # - * 9 A20($ cos2 8 - 8 )
- ( 2 f i / 9 ) sin2 8 cos 24
x AZ2(1 + cos2 8) cos 24
d2 = - (4f i19) cos 8 sin 24
The absorption of y rays can be described by a density matrixo, , and the intensity of the absorbed y rays is given by Tr (oioa) where o is the density matrix of the incident radiation. Ifo, has expansion coefficients A, B, C , and D, then Tr (oioa) = 2(Aa + Bb + Cc + Dd). In general, A, B, C , and D are functions of the direction which the y rays take in passing through the absorber, but for an isotropic (powder) absorber B = C = D = 0 and A depends only on the thickness.
If we now obtain the spectrum using a single line powder absorber, and with the source oriented so that they rays which pass through the absorber are emitted in the x-z plane (4 = 0 but 8 $ O), we find from [3] that the intensities of the lines are:
for lines 1 and 6:
for lines 2 and 5:
[4b] $ - # A20($cos2 8 - +) - (2<3/9) sin2 8
and that the intensity of lines 3 and 4 is one third that of lines 1 and 6. Thus, two measurements of the ratio of the intensity of line 1 to that of line 2, made at two different values of the angle 8 , are sufficient to determine the quantities A20 and AZ2. The deter- mination ofA ,O presents considerable experimental
difficulty, for it is not easy to find an absorber whose value of B is known with certainty. How- ever, if the source foil is magnetized almost to saturation, so that p(8, 4 ) = 0 except near the z-axis, we can use the Taylor expansion of cos 8. Then A20 is the average of 1 - 302/2 and AI0 is the average of 1 - 02/2, so that 1 - A10 = +(l - A20).
We measured spectra at angles 8 = 0, k30 , k45, and &6O0 using an enriched sodium ferrocyanide absorber which was moved in the constant acceler- ation mode. The total spectrum was spread over 512 channels, 256 in each direction, and was ac- cumulated until there were about 3 x lo6 counts per channel. For each of the 14 spectra so obtained, the intensities of lines 1 and 6 were averaged, as were also the intensities of lines 2 and 5, and the ratio of these averaged intensities was found. Thus, using expression 4, a linear equation for A20 and was found from each spectrum, and this set of 14 simultaneous linear equations was solved by the method of least squares to find the best values of A20 and A,'.
The determination of the intensities presented some problems. There is a test available to show whether the intensities have been determined cor- rectly, that the intensities of lines 1 and 6 must be three times that of lines 3 and 4. Now since we are measuring intensities of different emission lines, saturation in the absorber does not matter since it is the same for each line. However, the ratio of the heights of lines 1 and 6 to those of lines 3 and 4 was about 2.5; and this indicated that the emission lines were not all of the same width. In a case like this, the ratio of the intensities is equal to the ratio of the areas of the absorption lines, and we attempted to determine these areas by fitting various approxi- mations to the lineshape integral using the method of least squares. There was a little improvement, the ratio was typically about 2.8 which is not good, and the fit was quite poor as determined by the chi-squared test. We concluded from this that the shape of the emission lines is not Lorentzian and therefore any attempt to fit theoretical curves based on this lineshape is futile.
The procedure, which gave the correct value of 3.0 (within the limits of statistical error which, in our experiments, was about k0.02) for this ratio, and which therefore gave us confidence that the unknown ratio of the intensity of lines 1 and 6 to that of lines 2 and 5 was correct, was the following. The spectrum was fitted, by the method of least squares, with a parabolic background and with a function of the form
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1014 CAN. J . PHYS. VOL. 58. 1980
for each line. This function is a Lorentzian profile integrated over a channel of unit width; 12 is the channel number, no is the line centre, 6 is the half width at half maximum expressed in channels, and h is the height of the line. Usually the fit is quite poor at the peaks, as is obvious to the eye, but is good on the wings where all likely lineshapes have the same asymptotic form, a l/(n - no)2. For a given line, a point on each wing was chosen outside of which the fit was considered to be satisfactory, and the area under each wing was calculated from the parameters of the fit. Also, from the parameters of the fit, the background under the peak was found; this included the parabolic background common to the whole spectrum and a contribution from the wings of neighbouring lines. Then the area of the peak was obtained by summing, for each channel in the peak, the difference between this background and the actual channel count.
The error in the intensity of a line was calculated in the following way. The least-squares fitting routine, which fitted the background and the pre- liminary lineshape (eq. [5]), gave the standard de- viations of all the fitted parameters. The intensity is the sum of four terms: the calculated area under the two wings, the calculated area under the background in the line centre, and minus the total channel count in the line centre. The standard de- viations of the first three terms are calculated in the usual way from the standard deviations of the parameters as given by the least-squares fitting routine, and the standard deviation of the fourth term is the square root of the total channel count. The standard deviation of the intensity is then the square root of the sum of the squares of these four standard deviations.
The final result for this calibration of our source was3
An Analyser for Plane Polarized Radiation A single crystal of siderite, FeC03, may be used
to analyse plane polarized Mossbauer radiation in a manner analogous to the way a Polaroid is used for visible light. Siderite is a crystal with trigonal sym- metry and with one Fe atom per unit cell, and therefore, from symmetry considerations, the electric field gradient at the Fe nuclei must have axial symmetry. This has been confirmed by sev-
eral investigations on this substance (6, 7). The Mossbauer spectrum of siderite is a well-resolved quadrupole doublet: one line, which we shall call line no. 1, is at 2.138 mm s-I and is the transition f 2 t,f Q; the other line, which we shall call line no. 2, is at 0.340 mm s-I and is the transition fi t, 23. The density matrices for these two lines are: for line 1:
for line 2:
and it is easily seen that the maximum sensitivity to polarized radiation occurs when 0 = 90". In this ca seA,= l , C , = - 1 , A 2 = 5 , a n d C 2 = 1.
Our analyser was a slice cut from a single crystal of siderite obtained from a mine at St. Hilare, P.Q. The slice was 40 pm thick and it was mounted on a Lucite sheet. It appeared, both from examination in polarized light and from Laue backscatter photo- graphs, that the trigonal axis of the crystal lay in the plane of the slice to within The slice was placed behind a circular mask with a hole 2 cm diameter, and was mounted in a graduated housing which could be rotated about a horizontal axis perpen- dicular to the plane of the slice. The whole mount- ing could also be rotated about a vertical axis, also provided with a scale of degrees, and the complete assembly rested on an optical bench. The trans- ducer was mounted on the same table as the optical bench, and the polarized source was fixed to the transducer with the magnetic field (z-axis) as nearly as possible vertical. The mounting was shimmed so that the plane of the source was perpendicular to the axis of the transducer. The azimuthal orienta- tion of the source was checked by observing the reflection of a scale in the mirror fastened to the side of the source magnet using a telescope. With the source in this orientation,
It is easily seen how the siderite can be used as an
'These figures are the result of a computer calculation and have not been rounded off. We do not claim the significance which is implied in retaining so many figures, but it is convenient to postpone rounding off until the final result is obtained.
4We are indebted to Prof. H. Gorman of the Department of Geology, University of Toronto, for supplying the crystal of siderite, and for arranging to have it cut in the Geology Depart- ment laboratories.
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DANIELS ET AL.
analyser in the ideal case where it is a perfect crys- tal and a thin absorber. If it is placed with its tri- gonal axis along $ = 0°, the ratio Ro of the intensity of absorption line 1 to that of absorption line 2, resulting from a single emission line whose polari- zation parameters are n , b , c, and d, is given by
cln = (3 - 5Ro)l(3 + 3Ro)
Similarly, if the siderite is placed with its axis along $ = 45", the ratio R,, of these two lines is given by
therefore
dla = (3 - 5R4,)/(3 + 3R4,)
Thus, measurements of the ratio of the intensity of line 1 to that of line 2 , made for $ = 0 and $ = 45", give the plane polarization components of the inci- dent radiation. However, in our case we had to take into account the fact that our siderite slice was not thin, and might not be perfect, nor be correctly aligned in its housing. In this case, the polarization of the incident radiation is given by a similar for- mula. However, we proceeded to calibrate our siderite specimen ab initio.
Consider the first absorption line. Since the ab- sorber is physically the same when it is inverted, B = 0. Also, an axis can be found for which D = 0 , so that this line can be characterized by three parameters, A , , C , , and $, which is the value of $ for this axis when the scale on the analyser housing is set at zero. The second line can be similarly characterized by parameters A,, C,, and $,'. Now, if the two lines were superimposed, the resulting line would be unpolarized, and so C , = - C , and $,' = $,. The absorber is thus characterized by the four parameters A , , A,, C , and $,.
Now consider the analyser set at an angle $, and take the axis of the analyser as the reference axis for the description of polarization. In this system,
the density matrix for line 1 is [;I :]and the
density matrix of the incident radiation is
a + c cos 2($ + $,) ic sin 2($ + $,) - ic sln 2($ + $,) a - c cos 2($ + $,) 1
This indicates that the incident radiation is to be decomposed into two rays of orthogonal plane polarizations; one whose intensity is n + c cos 2($ + $,), is absorbed exponentially with an attenua- tion coefficient proportional to A , + C , and the other, whose intensity is n - c cos 2($ + $,), is absorbed with an attenuation coefficient propor- tional to A , - C . These are the familiar ordinary and extraordinary rays of a uniaxial crystal; by the symmetry of the crystal, the real and imaginary parts of the permeability tensor have the same eigenvectors, so birefringence does not mix these two rays as they pass through the absorber. Each absorption line thus consists of two superimposed absorption lines, one for each ray. Let the strengths of the absorption of each ray in the first line, which is a function only of A , , C and the thickness of the absorber, be F, , and F,- , respectively. Then the total strength of the absorption is
+ { a - c cos 2($ + $,)}F,-
and the strength of the other absorption line is given by a similar formula.
Now if the absorption coefficient has a Lorent- zian form, t6,1(x2 + 6,), where 6 is the half width at half maximum, x is the deviation from the line centre, and t is a thickness parameter such that the intensity transmitted at the line centre is e-' of the incident intensity, then the area S of the absorption line is independent of the energy distribution of the source radiation and is given by (8)
Here, H is the strength of the recoilless incident radiation, so that nH6. is an area with the same units as S . In order to calculate H, the recoilless fraction f of the source was assumed to be 0.91. Putting SInH6. = F ( t ) , the series for F ( t ) can be inverted so that t can be found from measured values of SInH6.:
The next step is to diagonalize
for the absorber; the result is A , - C
the incident radiation is and, in this representation, the density matrix for +- ( - y4+ 2(q5+ ...
384 nH6 3072 nH6
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If we now insert this notation into the expression [ 6 ] , we find the following formulae for the areas S , and S, of lines 1 and 2:
S,InHG = { a + c cos 2($ + $o) }F( t l+)
191 + { a - c cos 2($ + $,) }F(t ,-)
S21nHG = { a + c cos 2($ + $,)}F(t ,+)
+ { a - c cos 2($ + $,)}F(t , - )
where the thickness parameters are given by
t l + = t , ( l + CIA, ) , t , - = t , ( l - CIA, )
[ l o ] t2+ = t2(l - CIA,), t2- = t2( l + CIA,)
t l = f 1 N o I , t2 = f ' N o 2
Here, f ' is the recoilless fraction for the ab- sorber, N is the number of absorbing nuclei per unit area of the absorber, and o , and o, are the cross sections for unpolarized radiation for lines 1 and 2, respectively. The value off ' for radiation incident perpendicular to the trigonal axis is 0.44 + 0.05 (7 ) . The value of 6 to be used in [9] is half the natural emission line width, or 0.04857 mm s-I.
We assumed that our apparatus was well enough constructed that intensities of lines in different spectra could be compared without introducing significant systematic errors. Spectra were ob- tained with the axis of the siderite at 0 , 30, 45, 60, and 90" to the vertical as defined by the scale on the housing. The areas under each of the 12 lines in the spectrum were obtained by the method described in the last section. In this process, complications arose because 10 of the 12 lines overlap partially in pairs (see Fig. 4) . The method used to disentangle these pairs was the following. The end lines, ( 1 - 1 ) and (6-2) (where (n-m) denotes the line produced by line n of the source and line m of the absorber) are isolated lines, and so their areas are easily found. Next the area of the pair (6-1) and (5-2) is found and, since the areas of lines ( 1 - 1 ) and (6-1) are supposed to be the same, the area of line ( 1 - 1 ) is subtracted from that of the pair to give the area of line (5-2). In a similar way, the area of line (2- 1 ) is found by subtracting the area of line (6-2) from the area of the pair (2-1) and (1-2). This process is continued until the areas of the lines (6-1) and (1-2) are given independently, which serves as a check on the accuracy of the procedure.
The values of the averages ( 1 - 1 ) and ( 6 - l ) , (2-1) and (5- l ) , (1-2) and (6-2), and (2-2) and (5-2) were put into [ 9 ] , and, by the method ofleast squares, the best values were found of the unknown constants F ( t l + ) , F ( t , - ) , F(t ,+), F( t2 - ) , and $,. Figure 5 is a plot of the fit of [ 9 ] . Then, from the values of the
VOL. 58, 1980
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FIG. 4. A typical spectrum taken with the siderite absorber showing the overlap of the lines in pairs. The stick spectra are the absorptions corresponding to lines 1 and 2 of the siderite, respectively.
FIG. 5. The measured areas of the absorption lines plotted as a function of$, the orientation of the siderite crystal. The curves are the result of a simultaneous least-squares fit of [9]. ( a ) is a fit of line (2-1) (averaged with (2-6), (2-31, and (2-4) to obtain higheraccuracy). ( b ) , ( c ) and (d) are fits oflines (1-2). (2-2), and (1-1) similarly averaged, respectively.
F's, the values of thicknesses t , and t , and the ratios CIA, and A,IA, were found.
The whole experiment was then repeated with the siderite rotated through 180" about its vertical axis. The final values oft , , t , , CIA, , and A,IA, were taken as the average of these two results, and, from the two values of $, obtained, we could find the misalignment of the source magnet and of the siderite. The results are shown in Table 1.
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TABLE I . Summary of measurements made to calibrate the siderite polarimeter
Siderite facing Siderite facing source away from source Average
Although no errors are quoted in Table 1 , the statistical error on all derived quantities is of the order of 0.5%, except for t , where it is somewhat greater. All the angles of misalignment turned out to be less than lo, which is quite gratifying.
These measurements do, however, show some discrepancies when compared with the ideal model. First of all, the total thickness parameter, 1.373, is rather less than the calculated value of 1.820, but this can be understood if the siderite crystal is not pure FeCO,. The values of CIA, and A21A, also show considerable deviation from the ideal values of - I and 513. We believe that these measured values are the correct values for our
crystal, and are not the result offaulty analysis. To support this contention, we note that the value of A J A , , 1.71 13 k 0.018 obtained from these experi- ments, is in agreement with the value 1.736 f 0.01 obtained from another experiment using an un- polarized source. These discrepancies cannot be explained on the hypothesis that the siderite crystal was badly cut. Apart from this fact that such a hypothesis appears untenable when the results of examining the crystal by X rays and by polarized light are considered (the value of CIA, actually found would require the trigonal axis of the siderite to be 13" out of the plane of the slice), misalignment would cause the value A , / A , to be less than 513, not more. A possible explanation of a value of A,A, greater than 513 is that line 2 has another un- polarized line hidden underneath it, and we had suspected this from the observation that the profile ofline 2 was slightly asymmetric. We therefore see no reason for any other point of view than to accept as a fact that our siderite crystal is not perfect, but that its properties have been measured accurately enough for it to be used as an analyser of plane polarized Mossbauer radiation.
The following are the formulae which relate the polarization parameters, cla and d l a , to the ratios R, and R,, of the area of line 1 to that of line 2 , when the axis of our siderite is set at \jr = 0" and \jr = 45", respectively
and dla is given by the same formula with R,, in place of R,.
Conclusion The principles underlying the production and use
of polarized Mossbauer radiation have been known for quite a long time, and many experiments have been reported which demonstrate the correctness of these principles. Other experiments have also been reported which exploit the properties of polarized Mossbauer radiation. However, such ex- periments as have been published have, for the greater part, been those which demonstrate that a measured effect is, more or less, what theory pre- dicts, or which point out some gross feature of a Mossbauer system, such as the sign of a magnetic moment or an electric field gradient. The exploita- tion of polarized Mossbauer radiation will, in the future, require accurate measurements of polariza- tion parameters and of line strengths, and will re-
quire procedures for deducing the quantities sought directly from the measurements, rather than veri- fying that the measured values are in reasonable agreement with some model. What we have done in this article is to describe how a source of polarized Mossbauer y rays was constructed, how the polari- zation of the y rays was measured, and how an analyser was constructed and calibrated, all from first principles. Although we have described how to analyse only for the plane polarization compo- nents, it is our intention to report later on how to analyse directly for the circularly polarized com- ponent.
Acknowledgments We are indebted to the Natural Sciences and
Engineering Research Council of Canada for finan- cial support, and to the Department of Physics, University of Toronto for a substantial contribu- tion to the cost of our on-line computer for control-
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1018 CAN. J. PHYS. VOL. 58, 1980
ling our exuerimental eauiument. Most of the work 4. S. RUBY and J. M. HICKS. Rev. Sci. Instrum. 33,27(1962). - described here was carried out during the tenure by 5. U. GONSER. H. SAKAI. and W. KEUNE. J. Electrochem.
Soc. 123. 1915 (1976). of us (J'M'D') of a Guggenheim 6, U, GONSER, R. M , HOUSLEY. and R, W, G R A N . ~ , Phys,
I. H. F R A U ~ N F ~ L D E R . D. E. NAGLE.R. D. TAYLOR. D. R. F. Lett. A. 29. 36(1969). 7. V. I. GOL'DANSKII, E. F. MAKAROV. I. P. S U Z D A L ~ V , and
CoCHRANt' and W' M' VISSCHtR' Phys' Rev' I. A . VINOGRADOV. Zh. Eksp. Teor. Fiz. 58. 760 (1970), (1962). Sov. Phys. JETP, 31,407 (1970). 2. ( ( 1 ) J . M. DANICLS. NucI. Instrum. Methods. 128. 483 8, G, A , B~~~~ and pHAM ZUY H I E N , Zh, EkSp.Teor, Fiz. (1975); ( b ) Can. J. Phys. 57,263 (1979).
3. M. T . HIRVONCN. A . P. JAUHO. T . E. KATILA, K. J. RISKI, 43.909 (1962); Sov. Phys. JETP. 16,646 (1962).
and J. M. DANIELS. Phys. Rev. B, 15. 1445 (1977).
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