Discussion about the effects of Mössbauer source line broadening

Nuclear Instruments and Methods in Physics Research B 269 (2011) 145–152

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Discussion about the effects of Mössbauer source line broadening V. Rusanov a,⇑, V. Gushterov a, L. Tsankov b, L.H. Böttger c, A.X. Trautwein c a

University of Sofia, Faculty of Physics, Department of Atomic Physics, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria University of Sofia, Faculty of Physics, Department of Nuclear Technique and Nuclear Engineering, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria c Universität zu Lübeck, Institut für Physik, Ratzeburger Allee 160, D-23538 Lübeck, Germany b

a r t i c l e

i n f o

Article history: Received 14 July 2010 Received in revised form 14 October 2010 Available online 21 October 2010 Keywords: Mössbauer source Line broadening Age broadening Self-absorption

a b s t r a c t A detailed analysis of the effects of Mössbauer source line broadening is provided. Computer calculations of emission line broadening of Mössbauer sources with various effective thicknesses using the quantum mechanical theory of the emission spectrum proposed by Odeurs and Hoy, Nucl. Instrum. Methods Phys. Res. B, 254 (2007) 143–148 [1] are made. Corresponding calculations are performed using two simpler classical models giving an account of the non-resonant absorption of the radiation. Differences between the results obtained with the theory of Odeurs and Hoy, other model calculations and experimental data, and also inaccuracies, which could give rise to unusual large broadenings, are discussed. Two experimental tests are described, which unambiguously show that the broadenings calculated in [1] as a result of source line self-absorption broadening (age broadening) are huge and unrealistic. These quantitative evaluations of the age broadening yields an increase of line width of about 0.0004 mm/s for the decrease of 57Co/57Fe source activity by 1 mCi. Our conclusion is that the predicted observation of ‘‘hole burning’’ line profile could be possible, but only at huge source activities and thicknesses, which are practically not used. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The journal Nuclear Instruments and Methods in Physics Research has published a few years ago the work of Odeurs and Hoy entitled ‘‘Quantum mechanical theory of the emission spectrum of Mössbauer sources submitted to self-absorption: An exact result for source line broadening’’ [1]. This work deals with the quantum mechanical approach and calculation of the resonance line broadening by self-absorption in the Mössbauer source. Using the coherent-path quantum-mechanical model the authors calculate the emission line shape of source containing different quantities of resonant ground-state nuclei. Three major conclusions are discussed: (1) As the concentration of implanted ground-state resonant nuclei increases, the emission line-shape from such sources becomes increasingly broadened. This is a well known result from many other studies [2–6]. (2) The broadening, calculated at large concentration of 57Fe nuclei in the ground state (1.3  1020 nuclei/ cm3), reaches 1.2 of the natural line width C, which is large, but still remains in the realistic range. (3) For this concentration an emission line with ‘‘hole burning’’ line shape is predicted. The resonance line of the isotope 57Fe has a natural line width C ¼ 0:097 mm/s or 0.48108 eV. There is no spectrometer with ⇑ Corresponding author. Tel.: +359 28161380; fax: +359 29625276. E-mail address: [email protected]fia.bg (V. Rusanov). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.10.010

such high resolution, which could directly measure this line width. The line width broadenings calculated by Odeurs and Hoy [1] are in conflict with experimentally measured line widths and therefore need revision and tests, which is the main goal of this work. 2. Experimental details and calculations 2.1. Model calculations We performed calculations using four different models. The first one is very simple and unrealistic but this is the only case where we can observe ‘‘hole burning’’ line shape. Here all the radioactive nuclei form an infinitely thin layer at the back surface of the source; the gamma-rays have then to be transmitted through this ‘‘absorber’’ which is formed by the already decayed resonant nuclei. In this case there is no isomer shift between the lines of emission and absorption. In the second model all nuclei, in both ground and excited states, are homogenously distributed inside the source matrix. Non-resonant absorption in the matrix is also taken into account. The real nuclear distribution in the matrix can be investigated experimentally. The third type of our calculations is similar to that of Odeurs and Hoy, given by their Eq. (12) in [1]. However, non-resonant absorption is not taken into account in this case. In the fourth

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model we include also the non-resonant absorption within the source. Mathematical details are provided in the Appendix.

Isomer shift between Fe in Cr

272

and Rh matrix is 0.256 mm/s

57

2.2. Experimental test with 1.4 mCi- and 0.3 mCi- Co[Rh] sources

2.3. Mössbauer spectrum of a 57Co[Rh] source foil used as an absorber A rhodium foil of a Mössbauer source produced in 1994 was prepared as an absorber. This source was chosen not only for its very low activity, but for its polymethyl methacrylate holder as well. During the source production the Rh-foil has been encapsulated in about 0.5 mm two-component resin glue layer on the holder surface. The source was cut and polished from both sites, so that the foil remained encapsulated in a thin polymeric material. This foil was then used as absorber and measured in transmission geometry. For these measurements a commercial WISSEL spectrometer, working in constant acceleration mode, and a proportional counter were used. The Mössbauer source in this case was 57 Co[Cr] with an activity of about 5 mCi. Again the distance between the internal third and fourth lines of the a-Fe sextet spectrum (1.6794 mm/s) was used for calibration. To avoid additional unwanted broadening, caused by the folding of the spectrum, every one of the two velocity ranges corresponding to the forwards and backwards movement was processed separately. If a spectrum is not appropriately folded, maximum of about one channel, i.e. 0.017 mm/s additional line broadening could be expected. We were cautious to avoid disadvantages which may be inherent to this procedure by taking into account the geometry factor, which originates from the different solid angles at v and zero velocity. In our case, because of the large source-sample-detector distance (each of about 10 cm) and the small sample diameter (about 6 mm), this geometry factor is small, about ± 0.2% for the two groups respectively, and was fitted and corrected (see Fig. 1). Additionally, the drift factor related to the fact, that the left hand channels are measured always before the right hand channels in the first group and vice versa in the second, were also fitted. This deviation is very small, in the range of the statistical uncertainty, and was therefore not taken into account in the further calculations. The same fit procedure was applied in the experimental test with the old sources, as described in Section 2.2.

4

Intensity, 10 , counts

270

268

266

a

272

270

268

b

266 Residual deviation in standard diviation units

Two 57Co[Rh] sources with original activities of 50 mCi were used; at the moment of measurements their residual activities were about 1.4 mCi and 0.3 mCi, respectively. The tests were made with a 25 lm thin natural a-Fe foil as absorber. The sextet spectrum of a-Fe shows line intensity 3:2:1:1:2:3. It was accumulated in a narrow velocity range, so that only the internal (the third and the fourth) lines were detected and their widths were measured. Each one of these lines had 1/12 of the whole effective thickness and in an approximation of a thin absorber it should have a line width close to the natural one, CA ¼ 0:097 mm/s. Actually, because of environmental and parasite vibrations broadening and because of the nonzero thickness, a line width CA ffi 0:110 mm/s is expected. The source line width is of the same order. In addition, aging broadening of 0.030.04 mm/s is expected, so experimental line width of about 0.260 mm/s is expected to be finally measured. Additionally, we expected some parasite broadening because of the very long measurement time (more than one month with the weaker source). The authors of [1] predict that at these conditions only the broadening because of ageing should be a few natural line widths. For the Mössbauer measurements a standard spectrometer, working in transmission geometry and constant acceleration mode, and a proportional counter were used. The calibration was made using the distance between the internal third and fourth lines of the a-Fe sextet spectrum, which is known with very high precision, i.e. 1.6794 mm/s.

4 2 0 -2

c 0

50

100

150

200

250

Channel number Fig. 1. Geometry factor correction. (a) The geometry factor deviation could be visualized only if the ordinate scale is strongly stretched. (b) Its form has been fitted and the results correspondingly corrected. (c) The residual deviation in standard deviation units r: There are some systematic deviations in the line profile between channels 90 and 120, but only in the range of 2r .

2.4. Gamma-ray, X-ray spectroscopy and Fe, Co distributions in the Rh matrix The residual activity of the Mössbauer source produced in 1994 was measured with a semiconductor HpGe detector ORTEC with 33.1% relative efficiency and an energy resolution of the 60Co 1332.5 keV line of about 1.7 keV. The iron concentration, iron distribution and element composition have been precisely confirmed by Energy Dispersive X-ray Fluorescence (EDXRF) analysis using an electron probe microanalyzer LYRA I XMU with a detector Quantity 455, X-Flash 5010300 of BRUKER. The energy resolution of the detector at standard line Mn, Ka was 127 eV.

3. Results and discussion 3.1. Some preliminary results and discussion There is only one possibility to evaluate (not to measure) experimentally the self-absorption broadening of the resonance line. In Mössbauer studies the measured quantity is the experimental line width Cexp. It includes: (1) The intrinsic source line width, which is only theoretically equal to the natural line width. (2) The so-called source environmental broadening, because atoms in different locations have slightly different environments. (3) The source selfabsorption broadening, because there are 57Fe nuclei in the ground state. (4) The intrinsic absorber line width, which is only theoretically equal to the natural line width. (5) The so-called absorber environmental broadening, because atoms in different locations have slightly different environments. (6) The absorber self-absorption

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line shape. Most probably, even if existing, the ‘‘hole burning’’ line shape is not so easy observable because the experimental line in the Mössbauer spectra would be a convolution from the ‘‘hole burning’’ line shape of the source and the Lorentzian line shape with nearly natural line width of the absorber. The result would be a much broadened line. 3.2. Model calculations At first we present a simplified model where all radioactive nuclei form an infinitely thin layer. The emitted gamma-rays have then to be transmitted in the forward direction through the ‘‘absorber’’ which is formed by the already decayed resonant nuclei in the ground state. Calculations are performed according to Eq. (1).

Ib ðEÞ ¼ N0 f

C=2p 2

C2 =4

b

2

ðE  E0 Þ þ C =4

ele d e

ðEE0 Þ2 þC2 =4

ð1Þ

here b is the effective thickness of the layer, containing groundstate nuclei, f is the resonance emission probability (so called Lamb–Mössbauer factor), C is the natural line width and N0 represents the number of gamma-quanta emitted. As expected, in this case ‘‘hole burning’’ is seen at effective thickness of 2 (Fig. 1, left), which corresponds to a source with an original activity of nearly 200 mCi, after all the nuclei have reached their stable ground state. This first model is relatively simple. In a real situation the radioactive nuclei and also these in the ground state are uniformly distributed in the source matrix. Then calculations have to be performed with Eq. (2). For both models the non-resonant absorption is taken into account. In the latter case even with the enormous effective thickness of 20 no ‘‘hole burning’’ effect is observed. At effective thickness of 2 the line is only slightly broadened (Fig. 2, right), while in the first model the ‘‘hole burning’’ lineshape effect was observed (Fig. 2, left).

Ib ðEÞ ¼

N0 f C=2p d ðE  E0 Þ2 þ C2 =4

Z

d

ele x e

bdx

C2 =4 ðEE0 Þ2 þC2 =4

dx

ð2Þ

0

The third model follows strictly the results of the quantummechanical calculations described in [1]. The non-resonant absorption in the source is not taken into account in [1] (see Eq. (12) in [1]):

0 1 2

1.0

Intensity

broadening, because there are 57Fe nuclei in the ground state. (7) Parasitic vibrational broadening of both source and absorber lines. Detailed studies on the broadening of the experimental line width are scarce. In [7] the sum of the experimental line widths of the two lines from a quadrupole doublet extrapolated to the zero thickness (hence the self-absorption effect in the absorber is amended) is Cexp ¼ 0:427 mm/s or for each single line 0.213 mm/s. The latter width is larger than the theoretical minimum 2C ¼ 0:194 mm/s. Thus environmentally-caused broadening and broadening due to source ageing, the so-called resonance selfabsorption broadening, cause this deviation of 2C. In [7] the authors refer to the source certificate based on a test with a 25 lm thin natural a-Fe foil absorber [8]. The source line width (at activity about 50 mCi) was evaluated to be CS ¼ 0:103 mm/s. A fresh source has an environmental broadening of about 0.006 mm/s. The line width of the fresh sources from CYCLOTRON, Co., Ltd, Obninsk, has been estimated for the standard sources 50 and 100 mCi to be CS ffi 0:105ð3Þ mm/s and 0.108(3) mm/s, respectively. For higher specific activity the line width can reach 0.150 mm/s [9]. Even real single-crystal absorbers show environmental broadening and a line width of the same order CA ffi 0:103 mm/s can be expected. For the minimum of the experimental line width at zero source and absorber thicknesses a value of about Cexp ð0Þ ffi 0:206 mm/s could be expected. This value can be obtained only in case the Mössbauer spectrometer moves perfectly and there are no parasitic vibrations. The latter, though, do exist and are in the range of 0.002 mm/s, so the line width could reach the value of 0.208 mm/s. Following the analysis of Rusanov et al. [7] the minimum experimental line width measured is Cexp ð0Þ ffi 0:213 mm/s. The difference between 0.208 and 0.213 mm/s, i.e. 0.005 mm/s, must be ascribed to the self-absorption broadening when the source ages from 50 to about 40 mCi. The conclusion then is that the self-absorption broadening when the source ages could be 0.0005 mm/(s mCi) or even less 0.0004 mm/(s mCi), since parasitic vibrations are usually bigger. With a source of 100 mCi (3.7  109 Bq) 57Co, half life t1=2 ¼ 271 d, decay constant k ¼ 2:96  108 s1, there are N r ¼ 1:25  1017 active nuclei in the source. The 57Co active part is prepared by electro-deposition of high-purity carrier-free 57Co into a thin (about 7 lm thick, an active area about 20 mm2 as the real example, see Appendix) metal matrix of Rh (or Pd, Pt, Cr) followed with controlled annealing (diffusion) process. Then the nuclei density is about n ¼ 9  1020 nuclei/cm3. At the end of the recommended working life most of the 57Co will be transformed into 57Fe. At this concentration of 57Fe atoms, according to Odeurs and Hoy [1], the source should exhibit a huge line width because still in iron concentration of 1.3  1020 nuclei/cm3, the broadening only reaches 1.2 natural line widths C, the full source line width is 2.2C and a ‘‘hole burning’’ line shape have to be expected. According to CYCLOTRON Ltd. catalogue [9] and the findings described by Rusanov et al. [7] the source has a starting line width of 0.108 mm/s, and after the decay of 95 mCi activity it reaches additionally a self-absorption broadening of 0.038 mm/s and, thus, a source line width of about 0.146 mm/s (1.5C). This estimate is in complete contradiction to the result described by Odeurs and Hoy [1] and should not be passed without comments. We have never used a source stronger than 70 mCi in our experiment. Some colleagues [10] have used sources with 100 and more mCi activity but nobody has reported to observe the ‘‘hole burning’’ line shape. For the Mars missions even stronger source activities, over 0.5 Ci, are mounted on the Spirits and Opportunity Rovers. On June 10, 2003 the first Mars Exploration Rover spacecraft was launched on a Delta II rocket from Cape Canaveral, Florida. It had functioned for more than six years, already has a broad line, but it has not come to our attention that it has shown a ‘‘hole burning’’

0 2 10 20

1.0

0.8

0.8

0.6

0.6

0.4

0.4

20 10 2

0.2

2

0.2 0 1 0.0

0 0.0

-4

-2

0

2

4

-4

-2

0

2

4

Energy in Γ Fig. 2. (left) Form of the source resonance line calculated by formula (1); (right) form of the source resonance line calculated by formula (2) for different effective thicknesses b.

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C=2ph 2

þ

N 1 X

57

C=2ph

4

If non-resonant absorption is included and the normalized energy 0 e ¼ EE C=2 is used instead of the frequency x, then Eq. (3) takes the form:

( ( (  N1 m  X X m  h 1 1 1d le 2N le 2mþ1 d 2N SN ðeÞ ¼ e þ e þ Np 1 þ e2 m¼1 1 þ e2 n¼1 n  n h  i 1 Re ð1 þ ieÞn1 ð4Þ   1þa The explanation of the different physical quantities is provided in the Appendix. Results from both model calculations are shown in Fig. 3. The term N ¼ 2fbCCr (effective thickness equivalent) is introduced for the purpose of quantum mechanical calculations; it represents the number of the resonant nuclei in the ground state that radiation meets in its way. Line broadenings in both cases are very large. One can see ‘‘hole burning’’ line shape in the first case at N ffi 20, but when the non-resonant absorption is considered, at N ffi 25. A point of a great interest is the comparison of results from different calculations, i.e. for sources with original activity of 50 and 100 mCi, with all the 57Co nuclei having decayed into the 57Fe ground state. Such sources would have effective thickness b of 0.6 and 1.2, respectively. Some comparative results are shown in Fig. 4. Calculated broadenings by Odeurs and Hoy [1] are very large. One of the reasons about these large broadenings might be the presumption of unreal source matrix thickness, close to 100 lm. Note that typical thicknesses are in the range 6–8 lm. By decreasing the matrix thickness, the observed line widths according to [1] remain larger than those calculated according to Eq. (2) (Fig. 4). In case of a 13 times thinner matrix the results from the two calculations become close. Calculations made by Eq. (2) for the example of 100 mCi source provide a width of 1.31C (0.133 mm/s). Taking the above-mentioned value of 0.0004 mm/ (s mCi) for the increase of line width which accompanies the decrease of activity per 1 mCi, 95% of nuclei would be in a ground state at the end; then the line width will reach the value 0.135 mm/s, which is in good agreement with the calculated result by Eq. (2). Actually, the line width is bigger as the width of a source

1 10 20 25

Intensity

0.8

0.8

0.6

0.4

0.4

0.2 20 1 10 -4

-2

0

2

1 10 20 25

1.0

0.6

0.0

0

2

2 C C ðx  x0 Þ2 þ 4 m¼1 ðx  x0 Þ þ 4h2 h2 ( !)#     m  m 1 X Cr n C n1 Re iðx0  x  i Þ  þ ð3Þ p n¼1 n 2h 2h

1.0

0.2

25

4

-4

3

-2

0

2

4

6

8

10

12

14

16

18

1.9 1.8

e

Hole burning effect 10

1.7 1.6

100 mCi

s

tim

13

50 mCi

1.5

10 μ m

thiner source

Odeurs and Hoy [1] 100 μ m matrix

7.5 μm

es t im

1.4

This work Eq. (2)

1.3 1.2 1.1 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Effective thickness , β Fig. 4. Results according to the model of Odeurs and Hoy [1] with a 100 lm Rh foil, assuming a matrix thickness 10 and 13 time smaller, and results from our calculation based on Eq. (2) with thickness 7 lm.

with an initial activity of 100 mCi, according to the catalogue from CYCLOTRON, Co., Ltd, Obninsk is CS ffi 0:108ð3Þ mm/s [9], which deviates somewhat from the natural line width C ¼ 0:097 mm/s. 3.3. Results from the experimental test with 1.4 mCi- and 0.3 mCi-57Co[Rh] sources We have made two additional tests to show experimentally that the predicted broadenings in [1] are unrealistically high. The first one was performed with Mössbauer source with low residual activity (ca. 1.4 mCi; original activity ca. 50 mCi). The Mössbauer spectrum with the second source (residual activity ca. 0.3 mCi; original activity ca. 50 mCi) has been accumulated for about six weeks. The result obtained with an a-Fe foil (25 lm thick) is shown in Fig. 5. To avoid additional broadening effects, caused by the folding of the spectrum, every one of the two velocity ranges was processed separately. The measured experimental line widths are in the range 0.260.27 mm/s. However, at this concentration of 57 Fe source nuclei in the ground state should have, according to Odeurs and Hoy [1], a huge line width (Fig. 4), and ‘‘hole burning’’ line shape should be observable. 57

Co[Rh] source foil used as

We have performed an additional Mössbauer transmission experiment using a 57Co[Rh] source foil as absorber (original activity 50 mCi from 1994 and very low residual activity). This absorber foil contains about 0.62  1017 57Fe nuclei in their ground state. The value corresponds to an effective thickness of about 0.6. This estimate yields an iron concentration of less than 1 at.% in the cubic closed packed crystal structure of Rh. The fit of the measured spectrum (Fig. 6) provides, using single Lorentzian line profile, the values Cexp = 0.267(5) mm/s and Cexp = 0.260(5) mm/s.

20 10

Energy in Γ

2

2.0

3.4. Transmission Mössbauer study of a absorber

25 1

0.0

20

Concentration of ground-state Fe nuclei x10 / cm

Source emission line width in Γ

SðxÞ ¼

2

4

Fig. 3. Results after calculations according to the model of Odeurs and Hoy [1]: (left) non-resonant absorption is not taken into account as in [1]; (right) nonresonant absorption of radiation in the rhodium matrix is included.

3.5. Results from the gamma-ray, X-ray spectroscopy and study of the Fe, Co distributions in the Rh matrix The rhodium source foil gamma-spectroscopy measurements showed a minimal residual activity of about 18(1) nCi for 57Co

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Intensity, 10 4 , counts

V. Rusanov et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 145–152

118

118

116

116

114

114

112

112

110

110

1.6794 mm/s

1.6794 mm/s

108

108

0

50

100

150

200

0

250

132

132

130

130

128

128

126

126

124

124

1.6794 mm/s

122

Γ3 = Γ4 = 0.262(5) mm/s

106

Γ3 = Γ4 = 0.267(5) mm/s

106

50

100

150

200

250

1.6794 mm/s

122 120

120

Γ3 = Γ4 = 0.267(5) mm/s

118 0

50

100

150

200

Γ3 = Γ4 = 0.268(5) mm/s

118 0

250

50

100

150

200

250

Channel number

3

Intensity, 10 counts

Fig. 5. Mössbauer spectra of 25 lm thick a-Fe foil collected with old sources. In a narrow velocity range only the internal third and forth line are detected and their line widths are measured. The two velocity groups are shown separately. (top) Source with residual activity 1.4 mCi, (bottom) source with residual activity only 0.3 mCi.

2720

2720

2700

2700

2680

2680

2660

Γ exp = 0.267(5) mm/s

2660

2640

2640

2620

2620

2600

2600

2580

2580

2560

2560 0

50

100

150

200

250

Γ exp = 0.260(5) mm/s

0

50

100

150

200

250

Channel number Fig. 6. Mössbauer spectra in a narrow velocity range of a rhodium source foil in which, since 1994, 57Co nuclei decayed into separately. The inset shows the rhodium foil (diameter about 6 mm) encapsulates in thin polymeric material.

and an additional activity of about 16(1) nCi for 60Co as impurities (Fig. 7 (a). All the other lines are very weak background lines arising from the natural radioactive families 232Th, 238U and 40K. In Fig. 7 b the spectrum of a characteristic X-ray radiation is shown. At electron beam energy of 20 keV, R-shell ionisation of the rhodium is not possible. This is the reason to observe, with high inten-

57

Fe. The two velocity groups are shown

sity, the L-series of rhodium. Some weak lines of impurities are seen: Cr and Cu are present as impurities, the carbon is from the cover C layer, deposited to avoid any electrostatic charge on the sample surface. The silicon signal is due to the semiconductor Si detector. Small quantities of Al and O are also detected, probably as impurities from the Al2O3 powder used as a polishing material.

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Fig. 7. (a) Results from the gamma-spectroscopy study of a 57Co[Rh] Mössbauer source foil produced in 1994. The main lines of 57Co, 60Co and some very weak background lines from patterns of the radioactivity families 232Th, 238U and 40K are detected. (b) Results from EDXRF analysis of the rhodium foil. The Rh L-series are detected with very high intensity. Small quantities of impurities C, O, Si, Al, Cr and Cu are detected, too. The measured iron concentration is very low (< 1 at.%). (c) In the Rh foil the black square indicates the area from which the EDXRF spectrum was collected. (d) Scanning electron-microscope photo at small magnification, shows that the Rh foil has an approximate thickness of about 8 lm. (e) Part of the Rh foil at larger magnification. The white line represents one of the directions where the iron distribution in the Rh foil is examined. (f) Example for the distribution of the iron concentration in the Rh foil.

Three scanning electron-microscope photos (Fig. 7 c, d, e) show that the rhodium foil thickness is close to 8 lm. The measured iron concentration is very low and is under 1 at.%. It is worth to pay attention on two more facts: (1). No signal is detected from Co, which means, that practically all of the originally available 57Co nuclei have been entirely decayed into 57Fe. (2). Regardless of the high efficiency of the semiconductor detector in the scanning electron microscope for the 14.4 keV energy, even trace intensity at the Mössbauer line energy was not detected (Fig. 7 b). One of the goals of the present study was to investigate the iron distribution in the rhodium foil. The latter is important for the mathematical integration procedure in our model calculations. We have assumed that the distribution is homogeneous, i.e. rectangular. Fig. 7 f represents one of the many studies made on iron distribution in the rhodium foil. Its form is very close to rectangular, corresponding to a homogeneous distribution. There is a small negative slope, which however remains after a sample rotation on 180° angle. This finding indicates a geometry effect, connected with the small detector take-off angle of 20°. As a whole the assumption of a homogeneous distribution of the iron concentration in the source matrix seems rightful. 4. Conclusions The main results from this study are summarized as follow: 1. With the use of the quantum mechanical emission theory, described by Odeurs and Hoy [1], computer calculations of the emission line width of Mössbauer sources with different effective thicknesses were made. Also, the same calculations were performed taking into account the non-resonant radiation absorption, which is not taken into account in the original publication [1].

2. Computer calculations of the emission line width of Mössbauer sources with different effective thicknesses using two simple classical models, with non-resonant absorption included were carried out. 3. The line widths computed were compared with experimental and catalogue data of Mössbauer source producers. 4. The differences between the results of Odeurs and Hoy, our model calculations and experimental data are described. The unrealistically large line widths according to the model of Odeurs and Hoy [1] are ascribed to: – The non-resonant absorption is not taken into account, which by our evaluations (see Eqs. (3) and (4) and Fig. 3) leads to only insignificant decrease in the line widths. – Inaccuracy between N and b in the relation N ¼ 2fbCCr ; the so called ‘‘precise relationship’’ (1) in [1], which was earlier discussed in detail in Hoy’s work [11]. A real picture is provided for the value N ¼ 1. Then the effective thickness is b ¼ 0:16; which corresponds to a concentration of 12  1019 resonant nuclei/cm3. Even at a lower concentration of 8.51019, the calculations of Odeurs and Hoy yield a spectrum that shows signs of ‘‘hole burning’’ (Fig. 2 from page 147 [1]). Instead, the line has natural line width at N ¼ 1 according to Eq. (12) [1]. – From the correlation (p. 146 in [1]) N ¼ 5 $ concentration 4.2  1019 57Fe nuclei/cm3, a foil thickness of d ¼ 100 lm is derived. In [1] it is said ‘‘the source samples are at most about 103 m thick’’ (p. 145 [1]). Note that real foils used for source matrixes are close to 8 lm thickness. 5. The two tests with 1.4 mCi- and 0.3 mCi-57Co[Rh] sources having an initial activity of about 50 mCi, used earlier as sources in Mössbauer transmission experiments, do not confirm the large broadenings predicted in [1]. On the contrary, experimentally measured widths (source line width + absorber line width + any

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effects of broadening) do not exceed 0.27 mm/s. Single Lorentzian lines with a line width of 0.26 mm/s have been measured with a rhodium foil from an old Mössbauer source produced in 1994, which was examined as an absorber in a Mössbauer transmission experiment. 6. Because of environmental broadening the initial line width of modern Mössbauer sources with an activity between 25 and 100 mCi is in the range 103–108 lm/s [9]. The decrease of source activity by 1 mCi is accompanied by an increase of the line width by about 0.0004 mm/(s mCi) due to source selfabsorption broadening (age broadening). For example, a source with initial activity of 50 mCi will have, after complete decay, a line width close to 1.28 C. For this case, an extrapolation of the results of Odeurs and Hoy [1] from Fig. 4 yields a value of 5.5 C, which is in drastic contradiction. 7. The predicted observation of a ‘‘hole burning’’ line profile in [1] could, in principle, be due to huge source activity and thickness of the source; however, this is because of practical reasons an unlikely situation.

Acknowledgements We (V. R. and A. X. T.) thank the Alexander von Humboldt Foundation for the kindly given fellowship research grant. Special thanks are given to Dr. G. Tsutsumanova for the help with EDXRF analyses and the contract No. DO 02-56/2008, National research found, Bulgaria, the financial support of which has made the scanning electron microprobe studies available.

Integration is taken over the whole source thickness d. The integral has an analytical solution and takes the form:

 1   N0 f 1 b 1 l dþ b  l  e e 1þe2  1 e 2 2 d 1þe pd 1 þ e

Ib ðeÞ ¼

ðA5Þ

2) Odeurs and Hoy have treated the problem quantum– mechanically and have obtained the following equation for the source self-absorption spectrum

C=2ph

SðxÞ ¼

C2

2

þ

N1 X

2

C=2ph

4

2

C ðx  x0 Þ þ 4h2 m¼1 ðx  x0 Þ2 þ 4 h2 ( !)#        n m X m 1 Cr C n1 þ Re i x0  x  i  p n¼1 n 2 h 2h

where N is the number of the resonant nuclei in the ground state, which take part in resonant absorption; Cr is the partial radiation width of the line (for gamma-ray emission). In order to compare the spectra at different N, a normalizing factor 1/N has to be introduced. If we separate the whole expresP sion into the form SðxÞ ¼ A þ N1 m¼1 ðA þ Bm Þ; then we receive:

A¼

¼

1 ph

1 2 h

h

2 C2 =4 i¼ 1 ph h2 C ðE  E0 Þ2 þ C2 =4 ðhx  hx0 Þ2 þ C2 =4

C=2

C2 =4

2h

pC ðE  E0 Þ2 þ C2 =4

 n1   m  m 1X Cr n C Re  þ iðx0  xÞ p n¼1 n 2h 2h  n1 !  n  n1 m  X m 1 Cr 1 C ¼ Re  þ iðE0  EÞ h p n¼1 n 2h 2 n1  n  1  m  X m 1 Cr 1 C Re þ iðE0  EÞ  ¼ h p n¼1 n 2 2 n   n1 ! m  X m h Cr C  ¼ Re þ iðE0  EÞ p n¼1 n 2 2

Bm ¼ Appendix 1) For a thin source without self absorption, with a fixed absorber in front, the resonant radiation emitted, which passes through the absorber, would has the following energy distribution:

Ib ðEÞ ¼ N0 f

C=2p

b

ðE  E0 Þ2 þ C2 =4

ele d e

C2 =4 ðEE0 Þ2 þC2 =4

ðA1Þ

;

where N0 is the number of the gamma-quanta emitted, f is the Lamb–Mössbauer resonance emission factor, C is the source full line width at a half maximum, E0 is the energy at the maximum, le is the non-resonant linear absorption coefficient of the absorber material, d is the absorber thickness, b ¼ nr0 f 0 d is the effective thickness of the absorber. f 0 is the Lamb–Mössbauer resonant absorption factor, r0 is the maximum absorption cross-section, and n is the number of resonant nuclei in a volume unit. 0 Introducing the variable e ¼ EE C=2 the energy distribution Ib reads as

Ib ðeÞ ¼

N0 f

p

1  b ele d e 1þe2 1 þ e2

ðA2Þ

The case mentioned above describes the spectrum of a source with active nuclei distributed in a thin layer on the surface; the radiation passes an absorber and is then registered by the detector. In Eqs. (A1) and (A2) coefficients are chosen such that the integral under the emission line becomes N 0 f : Here the transition coefficient C=2ðdE ¼ de C2 Þ is used. If the nuclei have uniform distribution across the material, then the above equations are rewritten:

Ib ðEÞ ¼

Ib ðeÞ ¼

N0 f C=2p d ðE  E0 Þ2 þ C2 =4 N0 f 1 pd 1 þ e2

Z 0

d

Z

ele x e

bdx

C2 =4 ðEE0 Þ2 þC2 =4

dx

ðA3Þ

0

b d 1þe2

dx

ele x e

d

dx

ðA4Þ

So, the whole expression takes the following form:

1 SN ðEÞ ¼ N

(

2h

C2 =4

pC ðE  E0 Þ þ C =4 2

2

þ

N1 X m¼1

(

2h

C2 =4

pC ðE  E0 Þ2 þ C2 =4

 n1 !))   m  m h X Cr n C Re  þ iðE0  EÞ þ p n¼1 n 2 2 SN(e) represents a simpler form for the calculations:

(

N1 X 1 2h 1 þ pC 1 þ e2 m¼1 pC 1 þ e2 ))     m  i m h X Cr n C n1 h n1 Re ð1 þ ieÞ  þ p n¼1 n 2 2 ( (  N1 m  X m 1 2h 1 2h 1 h 2 X ¼ þ þ N pC 1 þ e2 m¼1 pC 1 þ e2 p C n¼1 n  n h i 1 Re ð1 þ ieÞn1   1þa ( (  N1 m  X X m 2h 1 1 þ þ ¼ NpC 1 þ e2 m¼1 1 þ e2 n¼1 n  n h i 1 Re ð1 þ ieÞn1   1þa

1 SN ðeÞ ¼ N

2h

Here the definitions C ¼ Cr þ Ce ¼ Cr þ aCr ¼ ð1 þ aÞCr and 0 e ¼ EE C=2 are used.

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V. Rusanov et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 145–152

To give an account of the non-resonant absorption, it is necessary to consider different ‘‘layers’’ in the formula with weights, relevant to the decreased intensity as a result of this absorption:

(

SN ðeÞ ¼

N1 X

(

(

2h 1 1 2mþ1 1 ele 2Nd þ ele 2N d þ NpC 1 þ e2 m¼1 1 þ e2  n h  i 1 Re ð1 þ ieÞn1   1þa

decay of the source studied, it will have an effective thickness b ¼ nr0 fd ¼ 8:93  1020  2:56  1018  7  104  0:75 ¼ 1:2 and a source with initial activity of 50 mCi will reach b ¼ 0:6, respectively. References

m  X

m

n¼1

n



3) For the computation of particular line parameters we have used a concrete standard source with initial activity of 100 mCi, active area size of 20 mm2 (the same 5 mCi/mm2), and a Rh matrix. lðRhÞ ¼ 43:6 cm2/g for the radiation with an energy E ¼ 14:4 keV. The substance density is qðRhÞ ¼ 12:41 g/cm3. So le ¼ 541:1 cm1. The source thickness is d ¼ 7 lm, f has the value of 0.75. It is known that the maximum value for resonant cross-section is r0 ¼ 2:56 1018 cm2 and a ¼ 8:2 for the internal conversion coefficient. The half life time of 57Co is t1=2 ¼ 271 days. For the radioactive decay constant we have k ¼ 1s ¼ t1=21ln 2 ¼ 0617 ; thus k ¼ 2:96  108 s1. So, the t 1=2 number of resonant 57Fe nuclei in a 100 mCi (or 3.7  109 Bq) source is N r ¼ DN=k ¼ 3:7  109 =2:96 108 ¼ 1:25  1017 . The volume of the source is V ¼ 0:0007 0:2 ¼ 14  105 cm3, i.e. the active nuclei density is n ¼ 8:93  1020 nuclei/cm3. Then after the whole radioactive

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