ELSEVIER

Nuclear Instruments and Methods in Physics Research B 95 (1995) 131-140

L 7 v 7 ........... B Beam Interactions

with Materials & Atoms

Harmonic analysis of M6ssbauer spectra J. Cieglak, S.M. Dubiel *

Facul~. of Physics and Nuclear Techniques, The University of Mining and Metallurgy; al. Mickiewicza 30, 30-059 Krakdw, Poland

Received 16 May 1994; revised form received 4 October 1994

Abstract A new iterative procedure for a numerical analysis of M6ssbauer spectra in terms of higher-order harmonics is outlined

and tested for X19Sn simulated and measured M6ssbauer spectra of chromium. Its usefulness to study spin- and charge-density waves of chromium is demonstrated.

1. Introduction

Antiferromagnetism of chromium is known to be consti- tuted by the so-called spin-density waves (SDW) [ 1 ] which are incommensurate with the underlying cubic lattice. Ac- cording to Young and Sokoloff [2], SDWs can be expended in a series of odd-harmonics:

oo

SDW = Z S2i-1 s in[(2i - 1)ce], i=1

(1)

where a = Qr (Q being the wave vector and r the position vector of the SDW) and S2i-~ is the amplitude of the ith harmonic.

If Szi-t is positive, the corresponding higher-order har- monic is in phase with the fundamental (i=1) SDW; if S2i-~ is negative, it is in antiphase with &.

The higher-order harmonics are inherently associated with the incommensurate nature of the SDW and they play a key role in the understanding of various properties of SDWs [3,41.

Experimentally, the evidence for the existence of $3 was for the first time found from neutron-diffraction experiments [5]. The 3Q-harmonic was revealed to be in phase with Sl and its relative amplitude R = S3/& = 1.45 x l0 -2 at 144 K and 1.65 x 10 -2 at 200 K. Theoretical calculations by Teraoka and Kanamori gave IRI = 2.1 x 10 -2 [6]. An independent determination of the sign and the magnitude of R was obtained by Dubiel and Le Caer [7] from a ll9Sn M6ssbauer-effect experiment on a single-crystal sample of chromium. Their value for R is 2.5(8) × 10 -2 at room

temperature. The method based on the M6ssbauer effect seems to be

especially well-suited for the investigation of SDWs because

* Corresponding author. Tel. +48 12 333740, fax +48 12 340010.

0168-583X/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO168-583X(94)O0422-6

the spectral parameters of this method, i.e. the hyperfine field H and the isomer shift I, are directly proportional to the spin- and to the charge-density, respectively. Recent model calcu- lations have demonstrated [ 8 ] that 119 Sn M6ssbauer spectra are sensitive both to the amplitude and the sign of the higher- order harmonics of the SDW. In addition, this method is the only one which simultaneously gives information on the spin- and the charge-density, hence it is also suitable to be applied in the investigation of charge-density waves (CDW) that are inevitably induced in the conduction-electron sys- tem by the incommensurate SDWs [3]. The CDW can be described in terms of the even-harmonics as follows:

O<3

CDW = Z p2i sin(2ia + ~), i--0

(2)

where p2i is the amplitude of the ith harmonic of the CDW. Through an electron-phonon interaction [9] CDWs lead

to a periodic modulation of the lattice constant, which is known as a strain-wave (SW). From neutron-diffraction [ 5] and X-ray diffraction experiments [ 10] comes the evidence that the displacement of the lattice associated with the 2Q- vector amounts to 1.6 x 10 -3 or 2.5 x 10 -3, respectively. These results may indicate that both CDWs and SWs are present in metal chromium.

In the present paper we describe a new method for the iter- ative numerical analysis of the M6ssbauer spectra in terms of higher-order harmonics of SDWs and CDWs. The arrange- ment of the paper is as follows: We describe the method in Section 2. In Section 3 we calculate model spectra for pure CDWs and for SDWs and concomitant CDWs. We test the method on model spectra with various statistics in Section 4 and finally in Section 5 we apply the method to analyze 119Sn spectra registered on real samples of chromium.

132 J. Cieglak, S.M. Dubiel/Nucl. Instr. and Meth..in Phys. Res. B 95 (1995)131-140

2 . D e s c r i p t i o n o f t h e m e t h o d

We consider the problem of fitting an N-parameter func- tion F to a measured M-point spectrum Y. We denote:

(iii) The overall spectrum is calculated by adding all the partial sextets obtained as described above.

The algorithm of the present fitting procedure has been based on a method described elsewhere [ 11-13].

F = F(xi, p i , ' " , p N ) , (3) 2.1. Determination of the correction vector

Y = Y(xi), (4)

with i = 1 , 2 , . . . , M . In our particular case we want to fit a measured spectrum

with a finite number of sextets whose hyperfine magnetic splitting H follows the amplitudes of the SDW given by Eq. (1), and the isomer shift I follows the amplitudes of the CDW given by Eq. (2)• Consequently, the function F has in this case the following form:

LM 6

G (5) F=z ' -~ i ~--~-'~ ( ( x i - Io - Fc + FOIC;)2 + l ' /=l j=l

with

KM

Fc = Z 12k sin(2ko~ + ~), (6a) k=l

NM

Fs = W j ~ H 2 n - X s in[ (2n - 1 )a ] , (6b) n=l

- LM - number of sextets taken into account, - Cj - amplitude of the j th line, - Gj - width at the half maximum of the j th line, - Io - average isomer shift of the spectrum, - Wj - position of the j th line, - KM - number of even-harmonics of the CDW taken into

account, - 12k - amplitude of the (2k)th harmonic of the CDW, - NM - number of odd-harmonics of the SDW taken into

account, - Hzn- l - amplitude of the (2n- 1 ) th harmonic of the SDW.

In other words, our procedure is as follows: (i) For a chosen number of higher-order harmonics of

the SDW the resultant shape of the SDW is determined for 0 ° < a < 180 °.

(ii) The range of [0 °, 180 °] is divided into LM equidis- tant intervals and for each of them a sextet is constructed assuming it consists of Lorentzian-shaped lines with ampli- tudes Cj and width Gj at half-maximum. The splitting of a particular sextet is proportional to the amplitude of the SDW in the given interval and its isomer shift is proportional to the amplitude of the CDW in that interval. The value of LM should be large enough if one wants to fit spectra due to in- commensurate SDWs. The actual value of LM depends on the maximum amplitude of the SDW, e.g. for Hmax = 60 kOe, LM_> 15.

The determination of the correction vector can be de- scribed as follows: We want to minimize the value of S 2 given by

M

S 2 = Z [ Y ( x i ) - F ( x i , p l , . . . , p N ) ] 2 l Y ( x i )

i=1

(7)

Let p~ be the value of the j th parameter after the kth iteration (pO being its starting value). The function F can be approximated by the following formula:

l--O

(8)

Taking into account Eqs. (7) and (8) and the condition for the minimum of S 2,

O( SZ) k - - = 0 ; l = 1 , . - - , N , ( 9 )

0pl

one can determine values of the correction vector, AP by solving the following matrix equation:

A . A P = C , 10)

where

a~= ~ (OF(x,) ~k ( a F ( x / ) ' ~ k 1

,:l t-D-Y-p,/ r(x,--7 11)

are elements of the matrix A, and

k = cj [Y(xD - F(xt;pkl, ' ' ' ,pku)] OF 1 Y(xD

/=1

12)

are elements of the matrix C.

2.2. Determination of the correction coefficient, fl

As only the first term in the Taylor series - Eq. (8) - was taken into account, we introduce a correction coefficient,/3 as follows:

p)+' = p) +/3Apy. (13)

It accelerates the minimization time as well as improves the convergence of the procedure. Upon termination of the

7. CieaTak, S.M. Dubiel/Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 131-140 133

-3 -2 -1 0 1 2 V (mm/s)

Fig. I. Simulated 119Sn M6ssbauer spectra for pure CDWs, CDW=I 2 cos2a. The value of 12 changes from 0 (top spectrum ) to 1 (bottom spectrum) with a step of 0.2.

H-0 (single line) 12=0.0- 1.0mm/s

(downwards) with 0.2 step

Vmax:3 mm/s

iteration procedure, an error analysis is performed. As out- lined in Ref. [ 13 ], a good approximation of the error, o'z, of the/th parameter is given by

1

0"1 = I1 (14)

where u is the number of degrees of freedom. In the following sections we apply this procedure to fit

]]9Sn simulated spectra with various statistics generated by a random number generator as well as those measured on real samples of metal chromium.

3. Examples of simulated spectra

Le CaEr and Dubiel [8] were already studying the influ- ence of the sign and the amplitude of the third-order, H3, and of the fifth-order, Hs, harmonics of the SDW on the shape of 119Sn MOssbauer spectra. Consequently, we limit here ourselves to show how such spectra would look like if (Section 3.1) only CDWs and (Section 3.2) both CDWs and SDWs were present.

3.1. The case o f pure CDWs

We set H2n-I = 0 in Eq. (5) and generate llgsn M6ssbauer spectra for various values of the CDW param- eters. Fig. 1 illustrates, as example, such spectra obtained for the simplest case i.e. where only the second-order har- monic of the CDW, 12, is present. One can see that the spectrum is sensitive to the value of 12 and on increasing its value it changes from a single-line spectrum, for 12 = O,

into a quite well-resolved double-line spectrum, for 12 = 1. Such behaviour proves that ILgsn M6ssbauer spectroscopy is suitable to detect and study CDWs.

3.2. The case o f SDWs and concomitant CDWs

3.2.1. SDW = Hls in a and CDW = ~sin(2c~ + 8)

Let us start with the case of the fundamental SDW, i.e. SDW=H1 sin a, and the second-order term of the CDW, i.e. CDW=I2 s in(2a+a) . Let Ht = 90 kOe (which corresponds to the case of a 4.2 K n9sn spectrum of a single-crystal sample of chromium) and a = o. Having assumed that the relative intensities of the lines in the partial sextets are like 3:2:1, we generate lt9Sn spectra for various values of 12 ranging from 0 to 1. Some chosen examples of the spectra obtained are shown in Fig.2a. One can readily see that the main effect of/2 on the shape of the spectra manifests itself in their central part. We want next to see what is the influence of the phase ~ on the shape of the spectrum. To illustrate this we keep the same SDW and CDW and simulate the spectra for various a-values. Some chosen examples are displayed in Figs. 2b-2g. We note that the a-parameter determines the symmetry of the spectrum.

Fig. 3 illustrates a similar case as above but for HL = 60 kOe (which corresponds to a real llgsn spectrum of a single-crystal sample of chromium measured at 295 K). The sensitivity of the spectra both to I2 and 6 is obvious.

Finally, in Fig. 4 we present the simulated spectra for H1 = 30 kOe and various values of Iz and a indicated.

3.2.2. SDW = H l s i n a :]: H3s in3a and

CDW = 12sin( 2a + ~)

The ll9Sn spectra obtained for H1 = 90 kOe, H3 = 5 kOe and a = 0 ° for various/:-values ranging from 0 to 0.5 are presented in Fig. 5a. They illustrate the influence of 12 on their shape. The spectra displayed in Figs. 5b-5g show how the shape of the spectra depends on the phase parameter 6.

Fig. 6 illustrates the shapes of the spectra obtained for the same parameters as those in Fig. 5 except the sign of H3 being negative. The reversal of the sign reflects itself in the changed shape of the spectra, which proves that the spectra are also sensitive to the sign of the higher-order harmonics of the SDW.

3.2.3. SDW = H l s i n a and CDW = 14sin(4ce + ~)

Finally, in this section we would like to illustrate the in- fluence of the fourth-order harmonic of the CDW, 14, on the shape of the la9Sn spectra. To this end we set H~ = 90 kOe and simulate the spectra for various values of 14 and & A selection of the spectra obtained in this way is displayed in Figs. 7a-7g.

4. Test for the limit of detection of the higher-order harmonics

In single-crystal samples of chromium 1.5-2% contribu- tion of H3 and ca. 0.2% contribution of I2 was already de- tected. It is of interest to test what is the limit of the detection of the higher-order harmonics of SDWs and CDWs from the

134 J. Ciedlak, S.M. Dubiel/Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 131-140

6=0 °

Hl:gOkOe

12:0.0 - 1.0 mm/s (downwards) with 0.2 step

Vmax:14 mm/s

6=_30 °

b=30 °

o

r

. r

c----

C'-- C " - -

, . . . . - - -

r

Fig. 2. Simulated ll9Sn M6ssbauer spectra for SDWs and concomitant CDWs in case of SDW=90 sin a and CDW=I 2 sin(2a + 6). The figure illustrates the influence of the amplitude of 12, which changes between 0 and 1 with a step of 0.2, and that of the phase 6 on the shape of the spectra. The maximum velocity is denoted by Vmax.

6 _ 0 °

Hl=60kOe 12--0.0 - 1.0 mm/s

(downwards) with 0.2 step

Vmax=lO mm/s

~--~ (b)p

6=_30 °

~(c) p

~(~) p

6_30 ° 6O

Fig. 3. The same as in Fig. 2 but for SDW=60 sin a.

----S~ 6=_90o

~ (g)

6 90 ° l

J. Cieglak, S.M. Dubiel/Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 131-140 135

~--~(a) /~

b = O °

H1-30kOe 12-0.0 - 0.5mm/s

(downwards) with 0.1 step

Vmax--6 mm/s

6 -30

I

(~) f ~ / #

J/¢

J ~=-60 °

b 3(

(e)

, , , - , j , - - , j

f

f ~

6=_90

~ \ (g)

DO 6=90 °

Fig. 4. The same as in Fig. 2 but for SDW=30 s ina and 12--0 to 0.5 with a step of 0.1.

6 0 ° 6=-30 z -

Hl=g0kOe H3=+5kOe 12--0.0 - 0.Smm/s

(downwards) with 0.1 step

Vm~×-14 mm/s

, r

C---

c----- c-----

Fig. 5. The same as in Fig. 4 but for SDW=90 sin ,~ + 5 sin 3o~.

136 J. Cieglak, S.M. Dubiel/NucL Instr. and Meth. in Phys. Res. B 95 (1995) 131-140

6 -30 °

H 1 90kOe H3--5kOe 12--0.0 - 0.5mm/s

(downwards) with 0.1 step

Vmax-14 mm/s

r

- -~ (d)

90 °

Fig. 6. The same as in Fig. 4 but for SDW=90 sin a - 5 sin 3a.

L 5=-60°

Hl__90kO e (e) , ~ ~ 14=0.0 - 1.0 mm/s

(downwards) with 0.2 s t e p

Vma x 14 mm/s

(~

A,/

6 = 9 0 ° I

Fig. 7. Simulated l l9Sn M6ssbauer spectra for coexisting SDWs and CDWs in the case of SDW= )sin o~ and CDW=I 4 sin(2a + 6). The figure illustrates the influence of the amplitude of I 4, which changes from 0 to 1 with a step of 0.2, and that of the phase 6 on the shape of the spectra. The maximum velocity is denoted by t~max.

J. Cie,(lak, S.M. Dubiel/Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 131-140 137

M6ssbauer spectra using the method outlined in Section 2. It is obvious that the limit will depend on the amplitude of the fundamental term of the SDW, Ht, and the quality of the spectra. In our test we regard a given higher-order har- monic for detectable if its absolute value is greater than the error of its determination, o', given by Eq. (14). If the er- ror is greater or the value of the harmonic is not correct, we regard this case as undetectable.

In order to cheque the influence of the quality of the spectrum, in terms of its statistics, on the detectability, the spectra of known composition, in terms of SDWs and CDWs, were simulated with the use of a random number generator. As a measure of the statistical quality of the spectrum, we take the following quantity:

N - ~ K = - - (15)

where N is the number of counts in the background of the spectrum and No that in the resonance line.

4.1. Detectabili~ of H 3 and H 5

Figs. 8a-8f illustrate diagrams from which limits of the detection of H3 and//5 as a function of the statistical quantity of the spectra, K, can be deduced. Full squares stand for the detectable cases, and open squares for undetectable cases. In particular, from Fig. 8a the limit of the detection of H3 > 0 and from Fig. 8b that of H3 < 0 can be estimated for HI = 60 kOe, while Figs. 8c and 8d show the similar cases for H1 = 30 kOe. Figs. 8e and 8f stand for the case of the detectability of H5 in a spectrum containing H1 = 60 kOe and H3 = +1.5 kOe, respectively. Fig. 9 illustrates similar diagrams for the detectability of the second-order harmonic of the CDW, 12, coexisting with the fundamental SDW with HI = 90, 60 and 30 kOe.

From the diagrams displayed in Figs. 8 and 9 the following conclusions can be drawn:

(a) The limit of the detection of the higher-order harmon- ics clearly depends on the value of K: the larger the K-value (better statistics) the smaller the values of H3, Hs and 12 that can be detected.

(b) It seems easier to detect H3 < 0 than H3 > 0 for SDWs consisting of H~ and H3.

(c) It seems easier to detect//5 < 0 than //5 > 0 for SDWs consisting of H1, H3 and Hs.

(d) The detectability of the second-order harmonic of the CDW,/2, concomitant with the fundamental SDW, H~, depends on the value of Hi: the smaller the Hi-value the lower the detection limit of I2.

5. Applicat ion of the method to real spectra

We applied the method outlined in Section 2 to analyze U9Sn M6ssbauer spectra measured at room temperature of:

- a (110) face single-crystal sample of chromium doped with 0.15 at.% Sn enriched to ca. 93% in the l l9Sn isotope,

- a polycrystalline sample of Cr containing 0.1 at.% of the enriched tin. The spectra were analyzed assuming the ratio of the am-

plitudes in the partial sextets is like 3:2:1 and all the lines have the same width at half maximum.

The former spectrum is shown in Fig. 10a. Its quality parameter K ~ 43. In the fitting procedure it was assumed that both SDWs and CDWs are present, the latter having the phase 6 = 90 ° with respect to the SDW [14]. The best-fit spectrum obtained is marked by a full line. It has turned out that the first-five odd-harmonics of the SDW have meaningful contributions with the following amplitudes: HI = 57.1(7) kOe, H3 = 0.8(4) kOe, //5 = 0.6(3) kOe and H7 = - 0 . 5 ( 4 ) kOe. The shape of the resulting SDW can then be described by the following expression:

SDW = 57.1 sin a + 0.8 sin 3a + 0.6 sin 5a - 0.5 sin 7o~.

(161)

It is visualized in Fig. 10b, and the histogram of the cor- responding hyperfine field distribution (HFD) in Fig. 10c with the average value H~v = 37.0 kOe.

The fitting procedure yielded for the amplitude of CDWs the following values (we assumed that their number was the same as that of SDWs): 12 = 0,004(26) ram/s, 14 = 0.030(25)ram/s, 16 = 0.017(24)mm/s and & = 0,008(24)mm/s. The shape of the resulting CDW can be described by

CDW = 0.004 cos 2a + 0.030 cos 4a

+0.017 cos 6a + 0.008 cos 8o~ (17)

and it is illustrated in Fig. 10d. In order to discuss these results we recall our up-to-date

experimental knowledge on the harmonics of SDWs and CDWs in chromium.

Concerning the former only the evidence of H3 was found from neutron-diffraction measurements on a single- crystal sample [5]. Its relative amplitude was estimated as 0.0165(5) and it was found to be in phase with the fun- damental term. From Eq. (16) it follows that H~ > 0, i.e. is in phase with H1, and its relative amplitude H3/Hj = 0.014(7), both in good agreement with the neutron data. A novel feature revealed by the present analysis is the exis- tence of the 5th-order, Hs, and the 7th-order, H7, harmonics of the SDW. Their existence was theoretically predicted

[2,31. Finally, it should be mentioned that the HFD-histogram

obtained by the present analysis of the spectrum agrees well with that obtained by the model-independent method [ 15 ]

- see Fig. 10c. Concerning now CDWs it should be recalled that accord-

ing to theory [3] they are inherently associated with the in- commensurate SDWs. However, their existence in chromium

138 J. Cie~lak, S.M. Dubiel/Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 131-140

50

40

(7 30

~ m m m m m m m m

(a)

~ m l m m mm m m m

~ m m m m mmm m

~ m m m m m m 20

~ a ~ m ~ _ m _ m nn ~ nn m ~ m m • l

10 ~ ~ ~ ~ m ~ ~ m mm m

50

40

(7 30

20

10

~ m i i i i m m m

[~ [~ 1 i i i 1 n I

[] [~ 1 m 1 m m 1 i

K] ~ m n I i 1 m •

c] ~ n i i • D,E] [~ El m n ~ m,m

50

40

(7 30

2O

10

D m m m m m m m m m

~ n m m m m m m ,

o m l m l I I : : : ~ s m ~ m | m ~ m

0.0 0.2 0.4 0,6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 HI (kOe) H 3 (kOe) H 5 (kOe)

50 ~ m m m m m m m m m 50 D m m m mm m m m m 50 D ~ m m m m m mm m m

40 ( b ) 40 ( d ) 40 ( f )

o ' El D El m m n 1 1 1 1 (7 El m ml 1 iN 1 1 l 1 O" E l l ml m mlm m l n Hi

30 D D ~ m m m m m m m 30 D m mm m m m m m m 30 D D m m m m m m m m m

~ mm m m 1 m m m m m m mm ml • m m m m m ~ m ~ m m m m m m mm 20 20 20

,_. . . . . ~m l mmm ~ m • • m m i ~ m m • •

I0 ~ ~ ~ G ~ m m m m m m,1 m m m m m m,m 10 10 m ~ m ~z m m l ) m m

0.0 0.2 04 0,6 08 1.0 0.0 0,5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 -H 3 (kOe) -H 3 (kOe) -H 5 (kOe)

Fig. 8. Diagrams for the detectability of /13 and /15 for various cases of SDWs: (a) SDW= 60sin o~ + / t 3 sin 3a, (b) SDW= 60sin a - /13 sin 3a, (c) SDW= 30 sin o~ +/13 sin 3a, (d) SDW= 30 sin a - / 13 sin 3a, (e) SDW= 60 sin a + 1.5 sin 3o~ +/15 sin 50~ and ( f) SDW= 60 sin a + 1.5 sin 3a - / 1 5 sin 50~. Full symbols stand for the detectable cases and the open symbols for the undetectable cases.

50

40

(7 3O

~mmmmmmmmmmmmmm

(4 ~mmmmmmmmmmlmmm

~Dmmmmmmmmmmmmm

Dmmmmmmmmmmuammmmm 20

~ l S l m l U l l m l m m l l ~ l ~ l m m l m m l m m l m

10 ~slm 'mmmmmmmmmmmu

0.0 0,05 0.1 0.15 12 (mm/s)

5O

40i

(3' 30

~Dmnmmmnmnmmmmm 50 DGmmmmmmmmnmmmm

(b) 40 (c) ~m.mmnmlmnmlnmm (7 []mm[]~lmlmm~mmmm

~Dmummmmmmmmmmm 30 ~Dmmmmmmmmmmmmm

DGNNNDNNNNNNNNN D~mm~mmmmmmmmmm 20 ' 20

o [ ] [ ] D o m l U l n l m l m u [ ]DoD[ ]smmml lmlmB

. . . . . . . : : : : i . . . . . . . . . . . i0 ~ D ~ D D m m N m I0 ~O~m~lm@mmlmm

0.0 0025 0.05 0.07 0,0 0.015 0.03 0.045 12 (mm/s) [2 (mm/s)

Fig. 9. Diagrams for the detectability of the second-order harmonic of the CDW, 12 coexisting with the following fundamental SDWs: (a) SDW= 90 sin a,

(b) SDW= 60 sin o~ and (c) SDW= 30 sin a. The full symbols represent the detectable cases and the open symbols the undetectable ones.

has not yet been directly proved. There exist, however, an indirect indication from neutron- and X-ray-diffraction mea- surements that the second-order harmonic of the CDW, 12, with the relative amplitude of the order of 0.002, does occur in chromium. Also a theoretical estimation for the relative value of 12 lies between 10 -2 and 10 -3 [10]. In addition,

one expects the maximum of the charge-density in the re- gion of the loop of the SDW [ 14].

From our present analysis of the single-crystal spectrum it follows that among the four harmonics of the CDW only the amplitude of/4 is statistically meaningful and has the relative value of 14/h) ~ 0.02, in accordance with the theoretical estimation, Io being the average isomer shift.

Concerning now the spectrum of the polycrystalline sam- ple of chromium, which has been recorded with the quality parameter K ~ 133 and which is shown in Fig. 11 a, one has

to note its remarkable difference from that registered for the single-crystal sample. Its shape suggests that contributions of higher-order harmonics are substantial [8]. Its analysis by means of the presently outlined method confirmed this expectation. The resulting SDW can be described by the fol- lowing expression:

SDW = 61.6sin a - 23.2 sin 3or + 6.0 sin 5o~ - 3.2 sin 7a.

(18)

The shape of this wave is illustrated in Fig. 1 lb and the corresponding HFD-histogram in Fig. 1 lc.

For the CDW the analysis yielded the following result:

CDW = 0.033 cos 2a - O.O03cos4a

-0.061 cos 6oL + 0.043 cos 8a. (19)

J. Cie,@ak, S.M. Dubiel/Nucl. Instr• and Meth. in Phys. Res. B 95 (1995) 131-140 139

1000 998

~ 996

E 994 m .b 992 ~ 9 9 0

98.8

98 6

. . . . '=-' " " ~ " " " ";~ '7"" a :' " ' ' "', , • . , .:-. ( ) ..," ,~.,, •

-8 -6 -4 2 0 2 4 6 8 30 60 90 120 150 180 v (m~/~) ~ (o)

H 1 57.11 kOe H3-0.82 H5=0.65 H7--0.50

(c)

12=0.004 mm/s 14=0030 16 0.017 18=0.008

0 10 20 30 40 50 60 30 60 90 120 150 180 H (kOe) *~ (o)

Fig. 10. (a) Room temperature l l9Sn M6ssbauer spectrum of a (1 I0) face of a single-crystal sample of chromium. The solid line stands for the best-fit spectrum obtained by means of the method outlined in Section 2; (b) The shape of the resulting SDW= 57.1 sin c~ + 0.8 sin 3c~ + 0.65 sin 5a' - 0.5 sin 7a: (c) The HDF-histogram corresponding to the found SDW (bars) and the one yielded by the model-independent method [ 151 (solid line), and (d) the shape of the resulting CDW= 0 .004cos2a + 0 .030cos4a + 0 .017cos6a + 0.008cosSm The dotted line stands for the average isomer shift /0 = 1.555 mm/s.

100

99 g

:~ 98

~97

96

8 -6 -4 2 0 2 4 6 8 V (mm/s)

H1=61.58 kOe H3=-23.21 H5:6.01 H7:-3.19

12:0.033 mm/s 14:-0.003 16---0,061 18:0.043

(c)

~'o 6o 9o 12o ~o ~8o ~ (°)

010203040s060708090100 30 6o 90 120 ISO 180 H (,Oe) ~ (o)

Fig. 11. (a) Room temperature 119Sn M6ssbauer spectrum of a polycrystalline sample of chromium. The solid line represents the best-fit spectrum obtained by means of the method outlined in Section 2; (b) The shape of the resulting SDW= 61.6 sin a - 23.2 sin 3a + 6.0 sin 5a - 3.2 sin 7a; (c) The HFD-histogram corresponding to the found SDW (bars) and the one yielded by the model-independent method [15] (solid line), and (d) the shape of the resulting CDW= 0,033 cos 2o~ - 0.003 cos 4 a - 0.61 cos 6a + 0.043 cos 8a. The dotted line shows the average isomer shift I0 = 1.553 minis.

140 J. Cieglak, S.M. Dubiel/Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 131-140

The shape of this wave is presented in Fig. 1 ld. As the only difference be tween the two samples is the ex-

is tence of grains in the latter, we conclude that it must be

an interact ion be tween SDWs and CDWs with grain bound- aries which deforms the shape of the SDW and the C D W

and enhances thereby the ampl i tudes of the higher-order har-

monics as observed for the polycrystal l ine sample, A more detailed presenta t ion and discuss ion of this problem will be

given elsewere [ 16].

6. Summary

We have out l ined a new iterative method for the numeri- cal analysis of MOssbauer spectra in terms of h igher-order

ha rmonics of SDWs and CDWs. We have used this method

to show how the shape of l l9Sn M6ssbaue r spectra is influ- enced by the presence of C D W s or both SDWs and CDWs

having various relevant parameters . We tested the method by apply ing it to analyze s imulated spectra of a known com- posit ion and different statistics as well as to real spectra

measured at room temperature on a single-crystal and poly- crystal l ine sample of chromium. We found for the first t ime

the ev idence of the 5 th-order and the 7th-order harmonic of

the SD W in c h r o m i u m and measured for the first t ime the C D W directly.

Acknowledgment

The Alexander yon Humboldt -St i f tung is acknowledged

for the donat ion of a PC at which the presented calculat ions have been carried out.

References

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