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Theory of conversion electron Mössbauer spectroscopy (CEMS)
Nuclear
70
Instruments
and Methods
in Physics Research
Bl (1984) 70-84
North-Holland,
THEORY OF CONVERSION Francesc Faculrad
Received
SALVAT
MijSSBAUER
SPECTROSCOPY
(CEMS)
and Joan PARELLADA
de Fisrca, Universidad
7 September
ELECTRON
Amsterdam
de Barcelona,
Diagonal
645, Barcelona - 28, Spain
1982 and in revised form 6 June 1983
A theory of conversion electron MGssbauer spectroscopy (CEMS) including second order effects, i.e. secondary electron emission, and detection coincidence corrections has been derived. The theory is applicable to surface films containing any number of distinct layers. The partial spectra are given in terms of analytical functions of the Miissbauer parameters and the physical characteristics of each layer in the sample. Numerical results are compared to available experimental data.
1. Introduction At the end of the sixties, Bonchev et al. [l] and Swanson and Spijkerman [2] showed that the electrons emitted, following the nuclear resonant absorption, in the MGssbauer absorber could be used to characterize surface layers in the sample. Since then, different methods and theories have been developed to obtain and interpret conversion electron Massbauer (CEM) spectra. The CEM spectrum can be obtained either ‘by collecting essentially all the electrons leaving the surface (integral technique), or by selecting the ones in a given energy range by means of a beta-ray spectrometer (differential or depth selective CEMS). Due to the relative sophistication of the differential techniques [1,3-61, the largest number of papers dealing with CEMS use the integral method. This method allows the use of simple and in expensive detecting equipment, mainly flow type proportional detectors [2,7] in which large counting rates can be obtained. This last characteristic makes possible the study of samples with the natural abundance of the Mijssbauer isotope. The information furnished by the integral measurements can be increased by using various angles of incidence [7] or by depositing thin layers of inert material on the sample [S]. Krakowski and Miller [9], Bainbridge [lo] and Huffman [ll] have developed theories hor the interpretation of integral CEMS measurements. In all of them the attenuation of the emerging electrons flux is described by the empirical formula of Cosslett and Thomas [12]. These theories can be considered to be of first order, because they neglect the contribution to the CEM signal of the electrons generated by X and/or gamma radiation which comes from the de-excitations of resonant nuclei deeply placed in the absorber. This contribution, which was experimentally observed by Tricker et al. [13], has been introduced in the theory of Liljequist et al. [14], hereafter referred to as LEB, who use a more realistic attenuation law derived from Monte Carlo calculations. Latterly, Deeney and McCarthy [15] take into account the coincidence effect in the detection and remove some excessive simplifications made by LEB. However, the spectral properties of the resonant absorption and the depth dependence of the secondary electron production rate have not been taken into account with enough detail. The coincidence or pairing effect has also been introduced by Liljequist, together with a more accurate attenuation law, in an approximate theory for two phases (duplex) non-enriched 57Fe absorbers [16,17]. In this work, a second order theory, including coincidence effects, is established under the usual assumption of isotropic emission of the electrons and radiations involved in the Miissbauer decay. The theory is formulated for the general case of a multilayer sample. When an analytical fit to the actual electron attenuation law is used, simple expressions for the spectral functions of each one of the sample layers are obtained. These functions, directly related to the weight functions of B&verstam et al. [4], give 0168-583X/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
F. Salwi,
J. Purelldo
/ Theory of CEMS
11
rise to the CEM spectrum when they are convoluted with the source emission spectrum. In this way, using a minicomputer, fast numerical simulations of CEM spectra can be done, thus it is possible to quantitatively relate the characteristics of the different layers with the observed CEM signal.
2. Theory 2.1. Overview The problem we deal with is shown schematically in fig. 1. A collimated beam of resonant gamma radiation coming from a Mossbauer source impinges on the surface of an ideal multilayer absorber with NL different layers of thickness X,, X,, . . . , X,,. The materials and/or the concentrations of resonant nuclei can be different for each layer. In the energy range used in CEMS, the incident radiation can interact with the absorber through two kinds of processes: (a) conventional interactions - photoelectric and Compton effects, and (b) nuclear resonant absorption - Mossbauer effect. Due to conventional interactions the beam is attenuated and electrons are emitted from the sample. The nuclear de-excitation following the resonant absorption takes place by emission of either a gamma ray or an internal conversion (1C) electron. In the latter case, the atom is left in an ‘excited’ state with a hole in an inner shell; the energy excess is given away with emission of Auger electrons and/or X-rays. Thus, the electrons emitted from the sample as a consequence of the Mossbauer absorptions are: (a) primary (IC or Auger) electrons originated in the de-excitations of the nuclei excited by the incident beam, and (b) secondary electrons originated by conventional interactions of photons (or resonant absorption of gamma rays) emitted after resonant absorptions. We wish to calculate the total number of electrons leaving the surface of the sample per unit time when the relative velocity of the source is v. Let Z, be the flux of resonant gamma radiation incident on SO, A the area of the sample surface and 8 the angle of incidence. The energy distribution of the recoilless gamma radiation is given by RZ”‘( E, v)dE
= &A cos df,L( E, v)dE,
f; being the source recoilless 2 L(E’v)=q
fraction
(1)
and
(r,/2)2
(2)
(Q’2)2+[E-EO(l+v/c)]2
-Y-
-.IJ
xi==-
* XN
Fig. 1. Schematic representation of the geometry in an integral CEMS measurement. secondary (e,) electrons are shown. Note the way of defining the depths x and x’.
The generation
mechanisms
of primary
(e,)
and
72
F. Sulwa, .I. Parellrdu
/ Theorv of CEMS
a Lorentzian distribution. E, is the excitation energy of the source nuclei, r, is the width of the emission lineshape and c is the speed of light. It is convenient to write eq. (1) in terms of the reduced energy y = 2( E - &)/F and the reduced velocity s = 2u/r,, where F is the natural level width and r, = CT/E,. Thus L(s,y)dY
dY = 3 = l+y,i(.Y-,F)*’
with y, = r/c,. In the following we shall use upper indices between brackets to distinguish among the different layers in the sample. The gamma ray absorption at the Nth layer due to conventional processes can be represented by a linear absorption coefficient p 6”’ which, in the energy range swept by the Doppler effect, is independent of the energy and practically coincides with the photoelectric linear absorption coefficient. The nuclear resonant absorption cross section at the Nth layer is given by the Breit-Wigner formula. In the most general case: the absorber nuclei can show hyperfine splittings. Thus, if E,(N0 and r!‘l are the position and width of the jth resonance, the resonant absorption can be characterized by a linear absorption coefficient [9]
to thejth peak and /3,(“) vJ(N) being the velocity corresponding where Y/(~)= T/T:N) and sCN’= 2ujN)/li,, being the associate statistic;1 weight. The effective thickness per unit length, thN’, 1s ’ given by r(N) = nf~) 0
f(N+, 0’
(5)
nCN) and fcN)are the resonant nuclei respectively. a,, is the total resonant cross The flux of recoilless gamma radiation Y and y + dy, when the reduced velocity
concentration and the Debye-Waller factor of the Nth layer section. that reaches a depth x inside the Nth layer with energies between is s, is given by _
(6)
RI’N’(s,Y,x)dY=Rl’*‘(s,?,)Q(N’(~)R(N)(Y,x)dY, where the factors N-l
c
pLL.‘XL+pbN)x
I.= 1
Ii ,
(7)
and N-l
R
(N)
-secB
(y,x)=exp
2
~~)(y)XL~~~N){y)x
i L=l
i
1~
(8)
interactions and resonant absorption reproduce the attenuation of the incident beam due to conventional respectively. The rate of recoilless gamma rays resonantly absorbed at depths between x and x + dx in the Nth layer can be written as RIACN)( S, x)dx
= set 8
I
(9)
M RI’N’(s,y,x)ELaN’(y)dydx.
--m
From now on, we will drop the argument
in the attenuation
function
when it takes its maximum
value,
Table 1 Radiations
emitted
as a result of the 57Fe* de-excitations.
(kev)
Emission probability
E, = 14.4 e, = 7.3 e, =13.6 e3 = 14.3 E, = 6.3 e4 = 5.4 < 0.85
w,” = 0.09 w; = 0.81 w; = 0.09 w; = 0.01 w,’ = 0.24 w4’ = 0.57 = 0.91
Energy
Kind of radiation
Resonant gamma ray K shell IC electron L shell IC electron M shell IC electron K ~ X-ray K-LL Auger electron Other electrons and X-rays
i.e. Q’“‘(
&,)
z QcN).
2.2. Electron generation As a result of the nuclear de-excitations, primary electrons with (discrete) energies e,, e,, . . . ,e,, gamma rays of energy E, and X-rays with characteristic energies E,, E,. . . . . Em are emitted. Let wt, . . . , w,‘, w,“, ’ be the corresponding emission probabilities. All these quantities can be obtained from the q, ‘. ., wn, knowledge of the binding energy, internal conversion coefficient and fluorescence yield for each shell of the Mossbauer atom [9]. Table 1 shows these data for 57Fe*. The primary electron production rate at depths between x and x + dx inside the Nth layer is given by NPE,‘N’(s,
x)dx
= w,“RIACN+,
x)dx.
(IO)
Both the X and gamma radiation emitted in the nuclear de-excitation can be absorbed by the surrounding material by conventional interactions, while the gamma radiation can also be resonantly absorbed. In any case secondary electrons are generated. This mechanism, ignored in the first theoretical papers on this subject, cause layers at large depth in the sample to contribute to the effect. (N) be the conventional absorption coefficient of the material at the Nth layer for radiation of Let EL, energy E, [18]. The Lth layer absorption coefficient for secondary gamma radiation generated in the Nth layer is given by the conventional contribution, pL(dy),plus the resonant one. The latter is
01) where the factor inside brackets normalizes the emission spectrum. After the photoelectric absorption of an X or gamma ray of energy E, in the Nth layer, a photoelectron and, possibly some Auger electrons are emitted. Let E:/) and w/$v) be the possible emission energies and corresponding probabilities. For each two layers, N and L, we shall define the linear absorption functions
+O(N,L)[I_~~~'+~~~~)]X-~(N,L)[~U(OL)+~L(RNS.~)]X',
AjN’L)(x,x’) = &;"'X, J
+ 0( N, L)pJN)x - 0( N,
L)@‘x’,
(12) (13)
14
F. Salwt, J. Parelludcc / Theop
of CEMS
where O(N,L)=
1 = -1
and the summation
ifN>L
orN=Landxrx’,
ifN
orN=Landx
extends
(14)
to all indices J that fulfill the condition
min( N, L) I J < max( N, L).
(15)
Although the generation of primary electrons is localized in the resonant layer, it is not true for the secondary electrons and, in fact, the whole absorber acts as an extended secondary source. The number of secondary electrons generated per unit time at depths between x’ and x’ + dx’ in the layer L correlated with nuclear de-excitations at depths between x and x + dx in the Nth layer can be computed as the sum of the possible contributions: _ electrons generated by photoelectric effect with energy E.$-) N.!&?‘(s, - IC electrons
x, x’)dxdx’=
w:~L)~.:~)~w,~RIA(~)(~,
x)E,
(A;=‘(x,
x’)}dxdx’;
(16)
with energy e,
NSE,‘N.L’(~,
x, x’)dxdx’
= w,~~(~~~)~wOXRIA(~)(S, x)E,{
A~~,‘~‘(x, x’)}dxdx’,
(17)
where E, is the exponential integral function [19]. These expressions are obtained by integrating over all directions the secondary radiation flux that, generated at layer N, crosses the Lth layer and is absorbed there. We have assumed that the effective thickness of the sample is very small compared with the linear dimensions of its surface, thus the integration over the polar angle can be extended to the whole interval (0, r/2). Following the existing theories [9-11,14-171, the electrons are assumed to be emitted isotropically. This hypothesis is essentially correct for Auger and IC electrons, however is not a very good one for photoelectrons because it is well known that the photoelectric differential cross section depends markedly on the emission angle [20]. Although, considering that the X and gamma secondary radiation is emitted isotropically, we can conclude that, in average, our assumption should also be true for the photoelectric emission. 2.3. Electron transport theory Due to the hypothesis of isotropic emission, the escape probability, P( e, x), for the electrons emitted at a depth x in a homogeneous sample with initial energy e will depend only upon x and e. In the first theoretical papers on integral CEMS [9-111 the electron transport was described by using the usual exponential attenuation law P(e,
x) =pO exp[-~(0x1,
08)
where q(e) is the attenuation coefficient for electrons of energy e. Electron transmission measurements of Cosslett and Thomas [12] have shown that, for thick films, exponential attenuation agrees well with experiment. In these early theories, the empirical attenuation coefficients found by Cosslett and Thomas, by inspection of their experimental transmission curves for thick films, were used. Although the exponential behaviour of the attenuation law (18) is qualitatively correct, it is not clear if, and how, the Cosslett and Thomas empirical attenuation coefficients can be used with CEMS geometry. Krakowski and Miller [9] and Bainbridge [lo] assume x to be the length of the electron trajectory considered as a straight line defined by the exit direction (i.e. our x/cos 0,) while Huffman [ll] uses eq. (18) taking x to be the depth in the absorber. In fact, as the energy degradation suffered by the incident electrons before diffusion conditions are achieved is not taken into account by Cosslett and Thomas in
F. Sdvcrr, .I. Purellrdu / Theo?
of CEMS
75
finding their attenuation coefficients, it is apparent that these cannot work well when used with CEMS geometry. In any case, there is no reason to use eq. (18) when the electron attenuation is treated on a purely individual basis as in the works of Krakowski and Miller [9] and Bainbridge [lo]. It seems clear that the actual attenuation law must be derived either from experimental measurements or from Monte Carlo calculations together with suitable scaling rules. For a given energy the electron attenuation law in absorbers of different composition can be reduced to the same function by changing the scale of the x axis. The escape probabilities for plane isotropic electron sources (of the same energies) embedded in materials of different atomic numbers are approximately the same if the external layers have equal mass thickness. LEB [14], using simple scattering models and the Monte Carlo method, have shown that, for the energies involved in CEMS, the escaping probability in iron can be given as a single function of the reduced depth PcFe)(e,x)
=O.74-2.7x+2.5 G(e) = 0
e 1
1 *
B
ifxIOS5r,(e), (19) if x > 0.55 r,(e),
where rB( e) is the Bethe range (in iron) for electrons of energy e. Furthermore, for two other materials - Al and Fe,O, - and shown that the scaling rule p’x’
Pcx)( e, x) = PcFe) e,xx i
P
LEB has computed
1
P( e, x)
(20)
holds with good precision for these materials. Recently, Liljequist has carried out a similar but more accurate Monte Carlo calculation and shown that eq. (20) also holds for Fe,O,. Following the usual practice, we shall take iron as the pattern material and use eq. (20) to derive the escape probabilities for other absorbers. To make the theory as general as possible, we will assume the actual attenuation law (in iron) to be given by
pfFe)(e3 x) =f[*). where f is known to be a continuous function strictly positive, which is non-zero decreasing for large values of the variable. Therefore it is possible to approximate decreasing exponentials
(21) for x = 0 and rapidly f by a finite series of
(22) The choice of this expression for the attenuation law, which although certainly arbitrary is sufficiently general, would allow us (1) to dispose of an analytical expression valid for all x and (2) to solve analytically the integrals appearing in the theory. In particular, Liljequist’s results fit to eq. (22) with pi = 0.203851,
q1 = 31.4112,
p2 = 147.585,
q2 = 5.34677,
p3 = -42.1327,
q3 = 6.02607,
p4 = - 106.307,
q4 = 5.08245,
ps = 1.41436,
q5 = 3.12760,
(23)
the difference between his data and our fit being lower than the statistical uncertainty. Hereafter we shall use eq. (22) with the parameters (23) as the attenuation law in iron because, in our opinion, it is the most accurate available at this time. Possible improvements in the attenuation law will
affect the theory only by changing the values of the parameters pt and ql. For a multilayer sample, the escape probability of an electron emitted at a depth x in the N th layer with initial energy e is given by P)(e,
x) =CP,‘“‘(e,
x)
(24) N-1
-4,
=CPrexp f
I
C ,r~)(e)X~+~(~)~e~x~~. i L==1
(25)
where py
e)
pf E
-
1
(26)
-
p(Fe) rg(e)
’
We point out that the validity of eq. (25) is only limited by the accuracy of the scaling rule (20), which can be used with good confidence for materials within a limited atomic number range as shown throughout Monte Carlo calculations. 2.4. Miissbauer spectrum An integral CEM spectrum is recorded detecting all the electrons leaving the surface of the sample during some time that we shall take as the unit time. As the depth dependence of the emission rate is at our disposal, we can now calculate the cont~but~on of each layer to the total CEM signal i.e., the partial spectra. 2.4.1. Primary signal The Nth layer contribution NPE’“‘(s)
= ix’
i
to the effect due to primary NPEjN’(s,
which, after an easy calculation, NPE(N’(s)
= set 8
x)PCN)(e,,
can be written
electrons
- first order theory - is given by
x)dx,
(27)
as
J-mmlil’“‘(s,y)~.kN)(y)WP(N)(y)dy,
(28)
where.
VT(~)(Y)
=
2 w~~KvP(“)(ei,
x)Q(N’(x)R’N’(_y, x)dx
(29)
i=l is the primary electron contribution to the Nth layer weight function. attenuation law, it can be shown that WPCN’( y) = R (hi-‘)(y)C
i
a,(;Y’I;“‘(y),
Using
the expression
(25) of the
(30)
f I=1
(31) and I,l”‘( y) =JoX”exp{
- [ sec8{&v)(y)+&“)}
+q,$“‘)(e;)]x}dx.
(32)
The secondary
electron
contribution
to the Nth layer signal is
to avoid unnecessary corn~~~ta~~o~ time, & is taken to be the smallest index such that P(-Wcr’t’(&) becomes negligibte. After a caicufation similar to that of the primary case, an expression similar to eq. (2X> is obtained where,
N,SE’“‘{s) ‘The secondary
(34)
= set B O” RI’ai(,,5’)~~‘(y)WS’N!(y)dy, I ---Do electron
x
co~t~ib~~i~~~
WS(“‘(y),
to the Nth layer weight function
fo”=.xp( -qt~‘L~x’)El(A;N*L)(x, xl)jdx’d.w,
is given by
(37)
snd qctt=, q it1 (e,) or qCL’($-)) as corresponding. The integrals in eq. (37) follow the pattern (71) analyzed in the appendix, thus they can be a~g~b~~ieal~~ solved, For N = L special care has to be taken due to the change nf sign in U(N, t): in this case the involved integrals hft5w the pattern (72).
In an actual experiment the GEM signal is superimposed on the background, rate in the reduced velocity interval between s and s + ds is
NB( s), thus the total count
An important contribution to the background is due to the photoelectrons generated by the incident beam because their creation rate is independent of the source velocity. Unfortunately there are also large contributions due to other X and ~~rn~~~ rays with energies far away from the resonance, whose t~~~trn~nt
is prohibitively difficult, therefore contribution that can be calculated can be written as NRB(s)
= set 6’ c
the background can only be experimentally determined. is the resonant one which, under the hypothesis of isotropic
The only emission,
p’;“’
N=l
1
In fact, this value is a lower limit to the experimental background. Because of the resonant absorption of the recoilless gamma radiation, the background shows a small decrease under the spectral peaks. To show this, eq. (40) can be rearranged and written in the form
142)
and (43) where
,
x)
1
Q'"'(x)[
1 - RCN’( y, x)ldx
is the background (negative) contribution to the Nth layer weight function. In peak intensity is, in generaI, small because the attenuation of the incident beam at a depth of the order of the Bethe ranges is also small. Although this is true abundance of the ~~ssbauer isotope, this effect becomes more important ignoring it could lead to appreciable errors. After the routine integration of eqs. (42) and (44) we get
any due for for
(44) case, this effect on the to resonant absorption sampIes with a natural enriched samples and
(45) N=l
and
F. S&ut.
J. Parelkdrr
/ Theor?, of CEMS
79
and (48)
2.4.4. Spectral functions With these results eq. (38) can be written NE(s)ds=
NB(oo)
+ 2
NE(N)(s)
ds,
N=l
[
where NB(oa) is the experimental NEtN’(s)=
as (49)
I
background
and
6~~)NPE'N'(s)+NSE("'(s)-NBD'N'(s)
is the total signal due to resonant absorption Using eqs. (28) (34) and (43) we obtain
(50)
in the Nth layer,
e m Rl’“‘(s,y)Cl~)(y)W’N’(,~)dy, i --oo
NE”V’(~)
= see
W(N)(y)
= S~~)HT@)(
(51)
where y) + w.s(“)(
y) - IVB’N’( y)
(52)
is the Nth layer weight function. We should point out that this weight function is the generalization second order of the one defined by Baverstam et al. [4] for the analysis of CEM spectra. We shall define the Nth layer spectral function as SPC’N’( y) = pkN’( y )W’N’( y ). Let us notice that, if A(s -y) NE’N’(s)
= I,Af,[n(s)
to
(53)
= L(s, y), * SP@‘(s)j,
(54)
product of the source emission spectrum A i.e., the partial spectra NEcN) can be written as the convolution and the spectral function SPC (N) .Evidently, the same result holds for multiline sources. In order to obtain a more manageable expression for the spectral functions, we have used the series expansion exp( -set
e~~)(
y)x}
= f t-s~lBX)r r=O
~~~~~~~)~ +,
(55)
and the relationship
I yx~exp(-ax)f(x)dx=j-l)“~~Yexp(-ax)f(x)dx X
(56) X
civ)( y). Introducing to write each integral in eqs. (32), (37) and (48) as a series in pR (35) and (46) and using eq. (52) a series expansion SPC’N’( y) = R
these series into eqs. (30),
(57) r-o
is obtained after some simple but tedious manipulation. Using the results of the appendix, the coefficients BtN) can be calculated analytically - at least for the low order terms. It is clear that these coefficients do r&t depend on the reduced energy y. We shall say that the Nth layer is of low absorption if thN’
80
F. Sulwi,
set 8X, < 0.35, in such a case expansion SPC’N’( the truncation convenient to (Y = 0, 1, 2, 3) corresponding
y)
z R ‘“-l’(y)
J. Parelludu / Theon;
of CEMS
(57) can be approximated
t: Byqpyyy)]r-cp?
by its four first terms
(58)
error being lower than 0.01%. In order to optimize the computation work, it will be split each layer of the sample into low absorption slices. Thus the coefficients BJN) resume all the non-resonant transport properties of the Nth low absorption layer, while the Mbssbauer information is included in the resonant absorption coefficient ~a”)( y ).
2.5. Detection coincidence effect We have just shown how an integral CEM spectrum can be calculated assuming that all the electrons leaving the surface of the sample are detected. However, when several electrons reach the detector simultaneously or, to be more precise, in a time interval that is smaller than the detector dead time, a single pulse is obtained from the detector. When this effect is ignored, the values of the scattering peak intensities are exaggerated. To fix some ideas, let us consider the case of 57Fe* (table 1). It should be clear that the main contribution to the coincidence effect comes from the detection of an IC K electron or a K photoelectron plus its subsidiary L Auger electron, because the emission of both electrons is practically simultaneous. For a pair (K, L Auger) of energies e, and e, and emissioqprobabilities wi and w,, the probability of obtaining a single count after the absorption of the appropriate photon at a depth x inside the Nth layer is given by (59) Thus, to take into account the coincidence effect, additional terms of the form - w2P(N)(e,, x)PcN)(e,, X) should be included in the definition of the different contributions to the weight function. For primary electrons and background correction these terms lead to corrections of the characteristic coefficients B,! N, which can be calculated easily. The computation of coincidence effects for secondary electrons is much more complicated, although it becomes reasonably easy if one assumes that the secondary electron emission rate from layer L due to resonant processes taking place at the Nth layer is constant in depth. Under this assumption, the secondary coincidence correction can finally be computed in terms of integrals of the type of eq. (67). It is important to notice that the coincidence correction just changes the values of the characteristic coefficients BCN), but the expression (57) defining the spectral function, remains unchanged. r
3. Numerical results sample A computer code to obtain the characteristic coefficients BJN) (r = 0, 1, 2, 3) for a multilayer with 57Fe has been written following the theory explained in the last section. According to LEB, the energy spectrum of secondary photoelectrons is simplified by assuming they have an initial energy equal to that of the absorber photon [14,15] and ignoring the subsidiary Auger electrons. Such a simple model allows us to establish quantitatively the details of integral CEM spectra from a small number of sample characteristics. In fact the only non-trivial parameters are the attenuation coefficients [18] and the Debye-Waller factors for each one of the layers in the sample. Of course, one can employ a more accurate description of the secondary spectrum if binding energies and fluorescence yields for all the atoms in each layer are considered. For integral CEMS, this procedure will not appreciably improve the accuracy of the theoretical results and furthermore it will largely increase the complexity of the program. The CEM spectrum area, given by -L/m Amt - NB(m)
[NE(s)-NB(co)]ds, PC0
(60)
F. Sohut, J. PureNudcr / Themy of CEMS
81
10
“r
0 7:
’
/
OF
02:
I 100
200
MO
400 nm
Fig. 2. U,(d)
function
derived
from the present
50
Fe
lco nm
theory for a homogeneous
non-enriched
Fe
iron absorber.
Fig. 3. Stainless steel area percentage vs. the iron layer thickness from the integral measurements of Thomas et al. [21]. 0 Experimental results, a - present theory, b - first order theory, c - second order theory without coincidence effect and d - derived from the scaling rule (65).
can be written A to, =
as the sum of the areas contributed
by each layer in the sample,
F AcN),
i.e.
(61)
N=l
with
The most basic quantities
experimentally
APCN’ = 100 AcN)/A,,, , and relative
determined
are the area percentages (63)
signals
RS’N.L’ 3 /f(N) AcL) / .
(64)
These quantities, being independent of the spectral background NB(cc), can be easily computed from the spectral functions. As suggested by Liljequist, an homogeneous non-enriched iron absorber can be thought of as a duplex absorber; the partial signal coming from the external layer (x < d) defines a ‘universal’ function, U,(d), which should be useful in practical estimates of thicknesses. The area contributions for non-enriched 57Fe multilayer absorbers can be approximated from U,(d) by re-scaling the thicknesses [16]:
XPC 0
------
--.-_..Iz---_.__ ------
,
,
50
100 nm
Fe
Fig. 4. contributions to the stainle.ss steel signal: XPE = secondary photoelectrons due to X rays. GPE = secondary gamma rays. CD -coincidence defect in the count rate. The contributions due to IC and Auger primary electrons Fig. 5. Iron to stainless steel relative signal continuous curve is the theoretical prediction, Tricker et al. is also shown.
nm Al electrons due to are also shown.
from the integral measurements of Tricker et al. [13]. l Experimental results. The and the broken one has been calculated using the scaling rule (65). The linear fit of
the constant C being cancelled when ratios of signals are considered. The function U,(d) as computed from the present theory (fig. 2) agrees with Liljequist’s one to within a few percent. Likely, these small differences are more a consequence of the diverse sets of parameters used than of approximations in either theory. The experimental results of Thomas et al. [21] have been reproduced numerically. These workers had performed a set of integral CEMS measurements on a stainless steel sample covered with layers of natural iron (2% of 57Fe) of different thickness. An ideal stainless steel composition of 71% Fe, 19% Cr and 10% Ni was used in the numerical simulation. The continuous curve in fig, 3 corresponds to the numerical. prediction when the Debye-Waller factors of iron and stainless steel are 0.7 and OS respectively. In the same figure, simulations according to the first order theory and the second order theory without the coincidence effect are also shown. Actually this second order theory without coincidence corrections is merely that of LEB free of their assumption on the homogeneous activity of each layer. In the LEB approximation it is assumed that the de-excitation rate does not depend on depth. Due to the exponential decrease of the incident gamma radiation flux, this approximation is obviously wrong. It is clear that both the secondary contributions and the coincidence corrections lead to an increase of the stainless steel area percentage. The chained curve in fig 3 corresponds to Liljequist’s scaling prescription eq. (65). In fig. 4 the different contributions to the stainless steel signal are shown. For a homogeneous sample, the total number of electrons emerging from the surface can be classified into secondary X-ray photoelectrons, 5%, secondary electrons generated by secondary gamma rays, 58, Auger electrons, 21.7%, K-IC electrons, 50.3% and (L + M)-IC electrons, 18%. Due to the detection coincidence effect there is a 8.7% decrease in the count rate, which is of the same order of magnitude as the secondary contribution (IO%). The coincidence correction to the total signal gets smaller for deeper layers, being negligible for layers buried at more than about 70 nm. As proved by Deeney and McCarthy, LEB obtained a greater secondary contribution due to the homogeneous activity approximation. However, Deeney and McCarthy obtained a value that was too low (3.5%) for the secondary gamma contribution because:
I) the secondary electrons generated by secondary radiation originated at depths lower than the secondary pb~to~~~tron range are not taken into account, and 2) the resonant attenuation of the secondary gamma radiation is ignored. fn general trends our results agree fairly well with those obtained by Liljequist using his approximate theory for eon-enriched 57Fe duplex absorbers, Tricker et al. f13] showed the contribution of the secondary electrons in a set of measurements where the sample was a 90 nm Layer of natural iron deposited in a SS substrate while covered with a~uminium~ fn fig. 5 the iron to SS relative signaf as a function of the ahnninium Iayer width is shown. The continuous fine is our numerical result obtained assuming that the active materials are the same as before. The qualitative behaviour of the theoretical curve can easily be explained. When the alurninium layer is thin, up to 30 nm, only the electrons created in deep layers are absorbed in the Al film, so the relative signal is mainly constant, For layer widths larger than 30 nrn and up to 200 nm this attenuation effect still persists while the emission of secondary electrons related to resonant de-excitation in the steel becomes larger, then the line shows a ~onti~~~~~s decrease of the siope. From 200 nm on, the slope gets s~~~~t~~ Larger because of the greater probability in generating secondary electrons in the Af layer related to nuclear de-excitations in the iron layer. For larger widths, 500 nm and more, the relative signal is practically exponential (linear in the figure). The linear fit of Tricker et al. f13] would correspond to the theoretical behaviour for larger widths, but clearly is not correct for thin layers. It is possible that the experimental error, obviously a large one from the large dispersion of the experiments data, could hide the correct behaviour. Furthermore, as has been said above, we have assumed the active material to have the same physical characteristics as the ones used in the simulation of fig. 3, due to a lack of information. One can check that the relative iron to SS signal for width zero of aluminium does not agree with the value from fig. 3, so maybe both sets of experiments were performed under different experimental conditions which have not been inchrded in our numerical procedure.
A second order theory, in&ding
coincidence
effects, has been established
for the general case of CEMS functions have been
m~as~~~rne~ts cm muttitayer samples. Simple anafytkzd expressions for the spectral
derived. Although only spectral areas are considered in this paper, the contribution to the actual spectrum from each layer in the sample can easily be obtained from the convolution of its spectral function with the source lineshape. A program to simulate CEM spectra has been written and will be published elsewhere. The authors
wish to thank
Dr. D. Li&quist
for many valuable
and ~~Ii~hten~n~
comments.
Appendix We shall estabhsh some integration theorems, related to the exponential integral function f19f_ which are useful for the anatyticaf evaluation of the integrals appearing in the theory. Looking for computational facilities, the final results will be expressed in terms of the function E;(x)
3 eXEl(x),
instead of the exponential integral E,(x), because I?;(x) fntegrating by parts, the following identity is found
(65) remains
finite for large values of the variable.
where
Fl(@,P)= ~{E;([a+l]p)-E’(p)). It is just a matter
of direct calculation
(68)
to establish
9
sP
e-““F,(b,x)dx=F,(a,b,g)-F,(a,b,p),
(69)
with
&(a, b,p) = ;;;I”;;‘::
(70)
{ bE;([ a+b+l]p)-(a+b)E~([b+I]p)-taE~(p)}.
The integrals J, -ixxn
e “fe
(71)
-h’E1(~+f~+~)dydx,
and (72) can be analytically J,= (+&Jo.
solved in terms of the derivatives
of the function
F2 by noting
that (73)
References (11 [2] [3] [4] [5] [6] [7] [8) [9] [lo] Ill] [12] [13] [14] [I51 1161 [17] [18] [19] [20] [Zl]
Zw. Bonchev, A. Jordanov and A. Minkova, Nucl. Instr. and Meth. 70 (1969) 36. K.R. Swanson and J.J. Spijkerman, J. Appl. Phys. 41 (1970) 3155. T. Shigematsu, H.D. Pfannes and W. Keune, Phys. Rev. Lett. 45 (1980) 1206. U. Baverstam, T. Ekdahl, Ch. Bohm, B. Ringstrbm. V. Stefansson and D. LiIjequist, Nucl. Ins&. and Meth. 11s (1974) 373. U. Baverstam, C. Bohm, T. Ekdahi, D. Liljequist and B. Ringstrbm, Miissbauer effect methodology, vol. 9. eds., J.J. Gruverman, C.W. Seidel and D.K. Dieterly (Plenum, New York, 1974) p. 259. D. Liljequist, C. Bohm and T. Ekdahl, Nucl. Instr. and Mcth. 177 (1980) 495. M.J. Tricker, A.G. Freeman, A.P. Winterbottom and J.M. Thomas, Nucl. Instr. and Meth. 135 (1976) 117. M.J. Tricker, L. Ash and W. Jones, Surf. Sci. 79 (1979) L333. R.A. Krakow&i and R.B. Miller, Nucl. Instr. and Meth. 100 (1972) 93. J. Bainbridge, Nucl. Instr. and Meth. 128 (1975) 531. G.P. Huffman, Nucl. Instr. and Meth. 137 (1976) 267. V.E. Cosslett and R.N. Thomas, Brit. J. Appl. Phys. 15 (1964) 883. M.J. Tricker, L.A. Ash and T.E. Cranshaw. Nucl. Instr. and Meth. 143 (1977) 307. D. Liljequist, T. Ekdahl and U. Baverstam, Nucl. Instr. and Meth. 155 (1978) 529. F.A. Deeney and P.J. McCarthy, Nucl. Ins&. and Meth. 166 (1979) 491. D. Liljequist, USIP Report 80-07 (1980). D. Liljequist, Nucl. Instr. and Meth. 179 (1981) 617. International tables for X-ray crystallography, ~01s. 3 and 4, eds., J.A. Ibers and W.C. Hamilton (The Kynoch Press, Birmingham, England, 1974). Handbook of mathematical functions, eds., M. Abramowitz and LA. Stegun (Dover, New York, 1972). C.M. Davisson and R.D. Evans, Rev. Mod. Phys. 24 (1952) 79. J.M. Thomas, M.J. Tricker and A.P. Winterbottom, J. Chem. Sot,, Faraday II, 71 (1975) 1708.
70
Instruments
and Methods
in Physics Research
Bl (1984) 70-84
North-Holland,
THEORY OF CONVERSION Francesc Faculrad
Received
SALVAT
MijSSBAUER
SPECTROSCOPY
(CEMS)
and Joan PARELLADA
de Fisrca, Universidad
7 September
ELECTRON
Amsterdam
de Barcelona,
Diagonal
645, Barcelona - 28, Spain
1982 and in revised form 6 June 1983
A theory of conversion electron MGssbauer spectroscopy (CEMS) including second order effects, i.e. secondary electron emission, and detection coincidence corrections has been derived. The theory is applicable to surface films containing any number of distinct layers. The partial spectra are given in terms of analytical functions of the Miissbauer parameters and the physical characteristics of each layer in the sample. Numerical results are compared to available experimental data.
1. Introduction At the end of the sixties, Bonchev et al. [l] and Swanson and Spijkerman [2] showed that the electrons emitted, following the nuclear resonant absorption, in the MGssbauer absorber could be used to characterize surface layers in the sample. Since then, different methods and theories have been developed to obtain and interpret conversion electron Massbauer (CEM) spectra. The CEM spectrum can be obtained either ‘by collecting essentially all the electrons leaving the surface (integral technique), or by selecting the ones in a given energy range by means of a beta-ray spectrometer (differential or depth selective CEMS). Due to the relative sophistication of the differential techniques [1,3-61, the largest number of papers dealing with CEMS use the integral method. This method allows the use of simple and in expensive detecting equipment, mainly flow type proportional detectors [2,7] in which large counting rates can be obtained. This last characteristic makes possible the study of samples with the natural abundance of the Mijssbauer isotope. The information furnished by the integral measurements can be increased by using various angles of incidence [7] or by depositing thin layers of inert material on the sample [S]. Krakowski and Miller [9], Bainbridge [lo] and Huffman [ll] have developed theories hor the interpretation of integral CEMS measurements. In all of them the attenuation of the emerging electrons flux is described by the empirical formula of Cosslett and Thomas [12]. These theories can be considered to be of first order, because they neglect the contribution to the CEM signal of the electrons generated by X and/or gamma radiation which comes from the de-excitations of resonant nuclei deeply placed in the absorber. This contribution, which was experimentally observed by Tricker et al. [13], has been introduced in the theory of Liljequist et al. [14], hereafter referred to as LEB, who use a more realistic attenuation law derived from Monte Carlo calculations. Latterly, Deeney and McCarthy [15] take into account the coincidence effect in the detection and remove some excessive simplifications made by LEB. However, the spectral properties of the resonant absorption and the depth dependence of the secondary electron production rate have not been taken into account with enough detail. The coincidence or pairing effect has also been introduced by Liljequist, together with a more accurate attenuation law, in an approximate theory for two phases (duplex) non-enriched 57Fe absorbers [16,17]. In this work, a second order theory, including coincidence effects, is established under the usual assumption of isotropic emission of the electrons and radiations involved in the Miissbauer decay. The theory is formulated for the general case of a multilayer sample. When an analytical fit to the actual electron attenuation law is used, simple expressions for the spectral functions of each one of the sample layers are obtained. These functions, directly related to the weight functions of B&verstam et al. [4], give 0168-583X/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
F. Salwi,
J. Purelldo
/ Theory of CEMS
11
rise to the CEM spectrum when they are convoluted with the source emission spectrum. In this way, using a minicomputer, fast numerical simulations of CEM spectra can be done, thus it is possible to quantitatively relate the characteristics of the different layers with the observed CEM signal.
2. Theory 2.1. Overview The problem we deal with is shown schematically in fig. 1. A collimated beam of resonant gamma radiation coming from a Mossbauer source impinges on the surface of an ideal multilayer absorber with NL different layers of thickness X,, X,, . . . , X,,. The materials and/or the concentrations of resonant nuclei can be different for each layer. In the energy range used in CEMS, the incident radiation can interact with the absorber through two kinds of processes: (a) conventional interactions - photoelectric and Compton effects, and (b) nuclear resonant absorption - Mossbauer effect. Due to conventional interactions the beam is attenuated and electrons are emitted from the sample. The nuclear de-excitation following the resonant absorption takes place by emission of either a gamma ray or an internal conversion (1C) electron. In the latter case, the atom is left in an ‘excited’ state with a hole in an inner shell; the energy excess is given away with emission of Auger electrons and/or X-rays. Thus, the electrons emitted from the sample as a consequence of the Mossbauer absorptions are: (a) primary (IC or Auger) electrons originated in the de-excitations of the nuclei excited by the incident beam, and (b) secondary electrons originated by conventional interactions of photons (or resonant absorption of gamma rays) emitted after resonant absorptions. We wish to calculate the total number of electrons leaving the surface of the sample per unit time when the relative velocity of the source is v. Let Z, be the flux of resonant gamma radiation incident on SO, A the area of the sample surface and 8 the angle of incidence. The energy distribution of the recoilless gamma radiation is given by RZ”‘( E, v)dE
= &A cos df,L( E, v)dE,
f; being the source recoilless 2 L(E’v)=q
fraction
(1)
and
(r,/2)2
(2)
(Q’2)2+[E-EO(l+v/c)]2
-Y-
-.IJ
xi==-
* XN
Fig. 1. Schematic representation of the geometry in an integral CEMS measurement. secondary (e,) electrons are shown. Note the way of defining the depths x and x’.
The generation
mechanisms
of primary
(e,)
and
72
F. Sulwa, .I. Parellrdu
/ Theorv of CEMS
a Lorentzian distribution. E, is the excitation energy of the source nuclei, r, is the width of the emission lineshape and c is the speed of light. It is convenient to write eq. (1) in terms of the reduced energy y = 2( E - &)/F and the reduced velocity s = 2u/r,, where F is the natural level width and r, = CT/E,. Thus L(s,y)dY
dY = 3 = l+y,i(.Y-,F)*’
with y, = r/c,. In the following we shall use upper indices between brackets to distinguish among the different layers in the sample. The gamma ray absorption at the Nth layer due to conventional processes can be represented by a linear absorption coefficient p 6”’ which, in the energy range swept by the Doppler effect, is independent of the energy and practically coincides with the photoelectric linear absorption coefficient. The nuclear resonant absorption cross section at the Nth layer is given by the Breit-Wigner formula. In the most general case: the absorber nuclei can show hyperfine splittings. Thus, if E,(N0 and r!‘l are the position and width of the jth resonance, the resonant absorption can be characterized by a linear absorption coefficient [9]
to thejth peak and /3,(“) vJ(N) being the velocity corresponding where Y/(~)= T/T:N) and sCN’= 2ujN)/li,, being the associate statistic;1 weight. The effective thickness per unit length, thN’, 1s ’ given by r(N) = nf~) 0
f(N+, 0’
(5)
nCN) and fcN)are the resonant nuclei respectively. a,, is the total resonant cross The flux of recoilless gamma radiation Y and y + dy, when the reduced velocity
concentration and the Debye-Waller factor of the Nth layer section. that reaches a depth x inside the Nth layer with energies between is s, is given by _
(6)
RI’N’(s,Y,x)dY=Rl’*‘(s,?,)Q(N’(~)R(N)(Y,x)dY, where the factors N-l
c
pLL.‘XL+pbN)x
I.= 1
Ii ,
(7)
and N-l
R
(N)
-secB
(y,x)=exp
2
~~)(y)XL~~~N){y)x
i L=l
i
1~
(8)
interactions and resonant absorption reproduce the attenuation of the incident beam due to conventional respectively. The rate of recoilless gamma rays resonantly absorbed at depths between x and x + dx in the Nth layer can be written as RIACN)( S, x)dx
= set 8
I
(9)
M RI’N’(s,y,x)ELaN’(y)dydx.
--m
From now on, we will drop the argument
in the attenuation
function
when it takes its maximum
value,
Table 1 Radiations
emitted
as a result of the 57Fe* de-excitations.
(kev)
Emission probability
E, = 14.4 e, = 7.3 e, =13.6 e3 = 14.3 E, = 6.3 e4 = 5.4 < 0.85
w,” = 0.09 w; = 0.81 w; = 0.09 w; = 0.01 w,’ = 0.24 w4’ = 0.57 = 0.91
Energy
Kind of radiation
Resonant gamma ray K shell IC electron L shell IC electron M shell IC electron K ~ X-ray K-LL Auger electron Other electrons and X-rays
i.e. Q’“‘(
&,)
z QcN).
2.2. Electron generation As a result of the nuclear de-excitations, primary electrons with (discrete) energies e,, e,, . . . ,e,, gamma rays of energy E, and X-rays with characteristic energies E,, E,. . . . . Em are emitted. Let wt, . . . , w,‘, w,“, ’ be the corresponding emission probabilities. All these quantities can be obtained from the q, ‘. ., wn, knowledge of the binding energy, internal conversion coefficient and fluorescence yield for each shell of the Mossbauer atom [9]. Table 1 shows these data for 57Fe*. The primary electron production rate at depths between x and x + dx inside the Nth layer is given by NPE,‘N’(s,
x)dx
= w,“RIACN+,
x)dx.
(IO)
Both the X and gamma radiation emitted in the nuclear de-excitation can be absorbed by the surrounding material by conventional interactions, while the gamma radiation can also be resonantly absorbed. In any case secondary electrons are generated. This mechanism, ignored in the first theoretical papers on this subject, cause layers at large depth in the sample to contribute to the effect. (N) be the conventional absorption coefficient of the material at the Nth layer for radiation of Let EL, energy E, [18]. The Lth layer absorption coefficient for secondary gamma radiation generated in the Nth layer is given by the conventional contribution, pL(dy),plus the resonant one. The latter is
01) where the factor inside brackets normalizes the emission spectrum. After the photoelectric absorption of an X or gamma ray of energy E, in the Nth layer, a photoelectron and, possibly some Auger electrons are emitted. Let E:/) and w/$v) be the possible emission energies and corresponding probabilities. For each two layers, N and L, we shall define the linear absorption functions
+O(N,L)[I_~~~'+~~~~)]X-~(N,L)[~U(OL)+~L(RNS.~)]X',
AjN’L)(x,x’) = &;"'X, J
+ 0( N, L)pJN)x - 0( N,
L)@‘x’,
(12) (13)
14
F. Salwt, J. Parelludcc / Theop
of CEMS
where O(N,L)=
1 = -1
and the summation
ifN>L
orN=Landxrx’,
ifN
orN=Landx
extends
(14)
to all indices J that fulfill the condition
min( N, L) I J < max( N, L).
(15)
Although the generation of primary electrons is localized in the resonant layer, it is not true for the secondary electrons and, in fact, the whole absorber acts as an extended secondary source. The number of secondary electrons generated per unit time at depths between x’ and x’ + dx’ in the layer L correlated with nuclear de-excitations at depths between x and x + dx in the Nth layer can be computed as the sum of the possible contributions: _ electrons generated by photoelectric effect with energy E.$-) N.!&?‘(s, - IC electrons
x, x’)dxdx’=
w:~L)~.:~)~w,~RIA(~)(~,
x)E,
(A;=‘(x,
x’)}dxdx’;
(16)
with energy e,
NSE,‘N.L’(~,
x, x’)dxdx’
= w,~~(~~~)~wOXRIA(~)(S, x)E,{
A~~,‘~‘(x, x’)}dxdx’,
(17)
where E, is the exponential integral function [19]. These expressions are obtained by integrating over all directions the secondary radiation flux that, generated at layer N, crosses the Lth layer and is absorbed there. We have assumed that the effective thickness of the sample is very small compared with the linear dimensions of its surface, thus the integration over the polar angle can be extended to the whole interval (0, r/2). Following the existing theories [9-11,14-171, the electrons are assumed to be emitted isotropically. This hypothesis is essentially correct for Auger and IC electrons, however is not a very good one for photoelectrons because it is well known that the photoelectric differential cross section depends markedly on the emission angle [20]. Although, considering that the X and gamma secondary radiation is emitted isotropically, we can conclude that, in average, our assumption should also be true for the photoelectric emission. 2.3. Electron transport theory Due to the hypothesis of isotropic emission, the escape probability, P( e, x), for the electrons emitted at a depth x in a homogeneous sample with initial energy e will depend only upon x and e. In the first theoretical papers on integral CEMS [9-111 the electron transport was described by using the usual exponential attenuation law P(e,
x) =pO exp[-~(0x1,
08)
where q(e) is the attenuation coefficient for electrons of energy e. Electron transmission measurements of Cosslett and Thomas [12] have shown that, for thick films, exponential attenuation agrees well with experiment. In these early theories, the empirical attenuation coefficients found by Cosslett and Thomas, by inspection of their experimental transmission curves for thick films, were used. Although the exponential behaviour of the attenuation law (18) is qualitatively correct, it is not clear if, and how, the Cosslett and Thomas empirical attenuation coefficients can be used with CEMS geometry. Krakowski and Miller [9] and Bainbridge [lo] assume x to be the length of the electron trajectory considered as a straight line defined by the exit direction (i.e. our x/cos 0,) while Huffman [ll] uses eq. (18) taking x to be the depth in the absorber. In fact, as the energy degradation suffered by the incident electrons before diffusion conditions are achieved is not taken into account by Cosslett and Thomas in
F. Sdvcrr, .I. Purellrdu / Theo?
of CEMS
75
finding their attenuation coefficients, it is apparent that these cannot work well when used with CEMS geometry. In any case, there is no reason to use eq. (18) when the electron attenuation is treated on a purely individual basis as in the works of Krakowski and Miller [9] and Bainbridge [lo]. It seems clear that the actual attenuation law must be derived either from experimental measurements or from Monte Carlo calculations together with suitable scaling rules. For a given energy the electron attenuation law in absorbers of different composition can be reduced to the same function by changing the scale of the x axis. The escape probabilities for plane isotropic electron sources (of the same energies) embedded in materials of different atomic numbers are approximately the same if the external layers have equal mass thickness. LEB [14], using simple scattering models and the Monte Carlo method, have shown that, for the energies involved in CEMS, the escaping probability in iron can be given as a single function of the reduced depth PcFe)(e,x)
=O.74-2.7x+2.5 G(e) = 0
e 1
1 *
B
ifxIOS5r,(e), (19) if x > 0.55 r,(e),
where rB( e) is the Bethe range (in iron) for electrons of energy e. Furthermore, for two other materials - Al and Fe,O, - and shown that the scaling rule p’x’
Pcx)( e, x) = PcFe) e,xx i
P
LEB has computed
1
P( e, x)
(20)
holds with good precision for these materials. Recently, Liljequist has carried out a similar but more accurate Monte Carlo calculation and shown that eq. (20) also holds for Fe,O,. Following the usual practice, we shall take iron as the pattern material and use eq. (20) to derive the escape probabilities for other absorbers. To make the theory as general as possible, we will assume the actual attenuation law (in iron) to be given by
pfFe)(e3 x) =f[*). where f is known to be a continuous function strictly positive, which is non-zero decreasing for large values of the variable. Therefore it is possible to approximate decreasing exponentials
(21) for x = 0 and rapidly f by a finite series of
(22) The choice of this expression for the attenuation law, which although certainly arbitrary is sufficiently general, would allow us (1) to dispose of an analytical expression valid for all x and (2) to solve analytically the integrals appearing in the theory. In particular, Liljequist’s results fit to eq. (22) with pi = 0.203851,
q1 = 31.4112,
p2 = 147.585,
q2 = 5.34677,
p3 = -42.1327,
q3 = 6.02607,
p4 = - 106.307,
q4 = 5.08245,
ps = 1.41436,
q5 = 3.12760,
(23)
the difference between his data and our fit being lower than the statistical uncertainty. Hereafter we shall use eq. (22) with the parameters (23) as the attenuation law in iron because, in our opinion, it is the most accurate available at this time. Possible improvements in the attenuation law will
affect the theory only by changing the values of the parameters pt and ql. For a multilayer sample, the escape probability of an electron emitted at a depth x in the N th layer with initial energy e is given by P)(e,
x) =CP,‘“‘(e,
x)
(24) N-1
-4,
=CPrexp f
I
C ,r~)(e)X~+~(~)~e~x~~. i L==1
(25)
where py
e)
pf E
-
1
(26)
-
p(Fe) rg(e)
’
We point out that the validity of eq. (25) is only limited by the accuracy of the scaling rule (20), which can be used with good confidence for materials within a limited atomic number range as shown throughout Monte Carlo calculations. 2.4. Miissbauer spectrum An integral CEM spectrum is recorded detecting all the electrons leaving the surface of the sample during some time that we shall take as the unit time. As the depth dependence of the emission rate is at our disposal, we can now calculate the cont~but~on of each layer to the total CEM signal i.e., the partial spectra. 2.4.1. Primary signal The Nth layer contribution NPE’“‘(s)
= ix’
i
to the effect due to primary NPEjN’(s,
which, after an easy calculation, NPE(N’(s)
= set 8
x)PCN)(e,,
can be written
electrons
- first order theory - is given by
x)dx,
(27)
as
J-mmlil’“‘(s,y)~.kN)(y)WP(N)(y)dy,
(28)
where.
VT(~)(Y)
=
2 w~~KvP(“)(ei,
x)Q(N’(x)R’N’(_y, x)dx
(29)
i=l is the primary electron contribution to the Nth layer weight function. attenuation law, it can be shown that WPCN’( y) = R (hi-‘)(y)C
i
a,(;Y’I;“‘(y),
Using
the expression
(25) of the
(30)
f I=1
(31) and I,l”‘( y) =JoX”exp{
- [ sec8{&v)(y)+&“)}
+q,$“‘)(e;)]x}dx.
(32)
The secondary
electron
contribution
to the Nth layer signal is
to avoid unnecessary corn~~~ta~~o~ time, & is taken to be the smallest index such that P(-Wcr’t’(&) becomes negligibte. After a caicufation similar to that of the primary case, an expression similar to eq. (2X> is obtained where,
N,SE’“‘{s) ‘The secondary
(34)
= set B O” RI’ai(,,5’)~~‘(y)WS’N!(y)dy, I ---Do electron
x
co~t~ib~~i~~~
WS(“‘(y),
to the Nth layer weight function
fo”=.xp( -qt~‘L~x’)El(A;N*L)(x, xl)jdx’d.w,
is given by
(37)
snd qctt=, q it1 (e,) or qCL’($-)) as corresponding. The integrals in eq. (37) follow the pattern (71) analyzed in the appendix, thus they can be a~g~b~~ieal~~ solved, For N = L special care has to be taken due to the change nf sign in U(N, t): in this case the involved integrals hft5w the pattern (72).
In an actual experiment the GEM signal is superimposed on the background, rate in the reduced velocity interval between s and s + ds is
NB( s), thus the total count
An important contribution to the background is due to the photoelectrons generated by the incident beam because their creation rate is independent of the source velocity. Unfortunately there are also large contributions due to other X and ~~rn~~~ rays with energies far away from the resonance, whose t~~~trn~nt
is prohibitively difficult, therefore contribution that can be calculated can be written as NRB(s)
= set 6’ c
the background can only be experimentally determined. is the resonant one which, under the hypothesis of isotropic
The only emission,
p’;“’
N=l
1
In fact, this value is a lower limit to the experimental background. Because of the resonant absorption of the recoilless gamma radiation, the background shows a small decrease under the spectral peaks. To show this, eq. (40) can be rearranged and written in the form
142)
and (43) where
,
x)
1
Q'"'(x)[
1 - RCN’( y, x)ldx
is the background (negative) contribution to the Nth layer weight function. In peak intensity is, in generaI, small because the attenuation of the incident beam at a depth of the order of the Bethe ranges is also small. Although this is true abundance of the ~~ssbauer isotope, this effect becomes more important ignoring it could lead to appreciable errors. After the routine integration of eqs. (42) and (44) we get
any due for for
(44) case, this effect on the to resonant absorption sampIes with a natural enriched samples and
(45) N=l
and
F. S&ut.
J. Parelkdrr
/ Theor?, of CEMS
79
and (48)
2.4.4. Spectral functions With these results eq. (38) can be written NE(s)ds=
NB(oo)
+ 2
NE(N)(s)
ds,
N=l
[
where NB(oa) is the experimental NEtN’(s)=
as (49)
I
background
and
6~~)NPE'N'(s)+NSE("'(s)-NBD'N'(s)
is the total signal due to resonant absorption Using eqs. (28) (34) and (43) we obtain
(50)
in the Nth layer,
e m Rl’“‘(s,y)Cl~)(y)W’N’(,~)dy, i --oo
NE”V’(~)
= see
W(N)(y)
= S~~)HT@)(
(51)
where y) + w.s(“)(
y) - IVB’N’( y)
(52)
is the Nth layer weight function. We should point out that this weight function is the generalization second order of the one defined by Baverstam et al. [4] for the analysis of CEM spectra. We shall define the Nth layer spectral function as SPC’N’( y) = pkN’( y )W’N’( y ). Let us notice that, if A(s -y) NE’N’(s)
= I,Af,[n(s)
to
(53)
= L(s, y), * SP@‘(s)j,
(54)
product of the source emission spectrum A i.e., the partial spectra NEcN) can be written as the convolution and the spectral function SPC (N) .Evidently, the same result holds for multiline sources. In order to obtain a more manageable expression for the spectral functions, we have used the series expansion exp( -set
e~~)(
y)x}
= f t-s~lBX)r r=O
~~~~~~~)~ +,
(55)
and the relationship
I yx~exp(-ax)f(x)dx=j-l)“~~Yexp(-ax)f(x)dx X
(56) X
civ)( y). Introducing to write each integral in eqs. (32), (37) and (48) as a series in pR (35) and (46) and using eq. (52) a series expansion SPC’N’( y) = R
these series into eqs. (30),
(57) r-o
is obtained after some simple but tedious manipulation. Using the results of the appendix, the coefficients BtN) can be calculated analytically - at least for the low order terms. It is clear that these coefficients do r&t depend on the reduced energy y. We shall say that the Nth layer is of low absorption if thN’
80
F. Sulwi,
set 8X, < 0.35, in such a case expansion SPC’N’( the truncation convenient to (Y = 0, 1, 2, 3) corresponding
y)
z R ‘“-l’(y)
J. Parelludu / Theon;
of CEMS
(57) can be approximated
t: Byqpyyy)]r-cp?
by its four first terms
(58)
error being lower than 0.01%. In order to optimize the computation work, it will be split each layer of the sample into low absorption slices. Thus the coefficients BJN) resume all the non-resonant transport properties of the Nth low absorption layer, while the Mbssbauer information is included in the resonant absorption coefficient ~a”)( y ).
2.5. Detection coincidence effect We have just shown how an integral CEM spectrum can be calculated assuming that all the electrons leaving the surface of the sample are detected. However, when several electrons reach the detector simultaneously or, to be more precise, in a time interval that is smaller than the detector dead time, a single pulse is obtained from the detector. When this effect is ignored, the values of the scattering peak intensities are exaggerated. To fix some ideas, let us consider the case of 57Fe* (table 1). It should be clear that the main contribution to the coincidence effect comes from the detection of an IC K electron or a K photoelectron plus its subsidiary L Auger electron, because the emission of both electrons is practically simultaneous. For a pair (K, L Auger) of energies e, and e, and emissioqprobabilities wi and w,, the probability of obtaining a single count after the absorption of the appropriate photon at a depth x inside the Nth layer is given by (59) Thus, to take into account the coincidence effect, additional terms of the form - w2P(N)(e,, x)PcN)(e,, X) should be included in the definition of the different contributions to the weight function. For primary electrons and background correction these terms lead to corrections of the characteristic coefficients B,! N, which can be calculated easily. The computation of coincidence effects for secondary electrons is much more complicated, although it becomes reasonably easy if one assumes that the secondary electron emission rate from layer L due to resonant processes taking place at the Nth layer is constant in depth. Under this assumption, the secondary coincidence correction can finally be computed in terms of integrals of the type of eq. (67). It is important to notice that the coincidence correction just changes the values of the characteristic coefficients BCN), but the expression (57) defining the spectral function, remains unchanged. r
3. Numerical results sample A computer code to obtain the characteristic coefficients BJN) (r = 0, 1, 2, 3) for a multilayer with 57Fe has been written following the theory explained in the last section. According to LEB, the energy spectrum of secondary photoelectrons is simplified by assuming they have an initial energy equal to that of the absorber photon [14,15] and ignoring the subsidiary Auger electrons. Such a simple model allows us to establish quantitatively the details of integral CEM spectra from a small number of sample characteristics. In fact the only non-trivial parameters are the attenuation coefficients [18] and the Debye-Waller factors for each one of the layers in the sample. Of course, one can employ a more accurate description of the secondary spectrum if binding energies and fluorescence yields for all the atoms in each layer are considered. For integral CEMS, this procedure will not appreciably improve the accuracy of the theoretical results and furthermore it will largely increase the complexity of the program. The CEM spectrum area, given by -L/m Amt - NB(m)
[NE(s)-NB(co)]ds, PC0
(60)
F. Sohut, J. PureNudcr / Themy of CEMS
81
10
“r
0 7:
’
/
OF
02:
I 100
200
MO
400 nm
Fig. 2. U,(d)
function
derived
from the present
50
Fe
lco nm
theory for a homogeneous
non-enriched
Fe
iron absorber.
Fig. 3. Stainless steel area percentage vs. the iron layer thickness from the integral measurements of Thomas et al. [21]. 0 Experimental results, a - present theory, b - first order theory, c - second order theory without coincidence effect and d - derived from the scaling rule (65).
can be written A to, =
as the sum of the areas contributed
by each layer in the sample,
F AcN),
i.e.
(61)
N=l
with
The most basic quantities
experimentally
APCN’ = 100 AcN)/A,,, , and relative
determined
are the area percentages (63)
signals
RS’N.L’ 3 /f(N) AcL) / .
(64)
These quantities, being independent of the spectral background NB(cc), can be easily computed from the spectral functions. As suggested by Liljequist, an homogeneous non-enriched iron absorber can be thought of as a duplex absorber; the partial signal coming from the external layer (x < d) defines a ‘universal’ function, U,(d), which should be useful in practical estimates of thicknesses. The area contributions for non-enriched 57Fe multilayer absorbers can be approximated from U,(d) by re-scaling the thicknesses [16]:
XPC 0
------
--.-_..Iz---_.__ ------
,
,
50
100 nm
Fe
Fig. 4. contributions to the stainle.ss steel signal: XPE = secondary photoelectrons due to X rays. GPE = secondary gamma rays. CD -coincidence defect in the count rate. The contributions due to IC and Auger primary electrons Fig. 5. Iron to stainless steel relative signal continuous curve is the theoretical prediction, Tricker et al. is also shown.
nm Al electrons due to are also shown.
from the integral measurements of Tricker et al. [13]. l Experimental results. The and the broken one has been calculated using the scaling rule (65). The linear fit of
the constant C being cancelled when ratios of signals are considered. The function U,(d) as computed from the present theory (fig. 2) agrees with Liljequist’s one to within a few percent. Likely, these small differences are more a consequence of the diverse sets of parameters used than of approximations in either theory. The experimental results of Thomas et al. [21] have been reproduced numerically. These workers had performed a set of integral CEMS measurements on a stainless steel sample covered with layers of natural iron (2% of 57Fe) of different thickness. An ideal stainless steel composition of 71% Fe, 19% Cr and 10% Ni was used in the numerical simulation. The continuous curve in fig, 3 corresponds to the numerical. prediction when the Debye-Waller factors of iron and stainless steel are 0.7 and OS respectively. In the same figure, simulations according to the first order theory and the second order theory without the coincidence effect are also shown. Actually this second order theory without coincidence corrections is merely that of LEB free of their assumption on the homogeneous activity of each layer. In the LEB approximation it is assumed that the de-excitation rate does not depend on depth. Due to the exponential decrease of the incident gamma radiation flux, this approximation is obviously wrong. It is clear that both the secondary contributions and the coincidence corrections lead to an increase of the stainless steel area percentage. The chained curve in fig 3 corresponds to Liljequist’s scaling prescription eq. (65). In fig. 4 the different contributions to the stainless steel signal are shown. For a homogeneous sample, the total number of electrons emerging from the surface can be classified into secondary X-ray photoelectrons, 5%, secondary electrons generated by secondary gamma rays, 58, Auger electrons, 21.7%, K-IC electrons, 50.3% and (L + M)-IC electrons, 18%. Due to the detection coincidence effect there is a 8.7% decrease in the count rate, which is of the same order of magnitude as the secondary contribution (IO%). The coincidence correction to the total signal gets smaller for deeper layers, being negligible for layers buried at more than about 70 nm. As proved by Deeney and McCarthy, LEB obtained a greater secondary contribution due to the homogeneous activity approximation. However, Deeney and McCarthy obtained a value that was too low (3.5%) for the secondary gamma contribution because:
I) the secondary electrons generated by secondary radiation originated at depths lower than the secondary pb~to~~~tron range are not taken into account, and 2) the resonant attenuation of the secondary gamma radiation is ignored. fn general trends our results agree fairly well with those obtained by Liljequist using his approximate theory for eon-enriched 57Fe duplex absorbers, Tricker et al. f13] showed the contribution of the secondary electrons in a set of measurements where the sample was a 90 nm Layer of natural iron deposited in a SS substrate while covered with a~uminium~ fn fig. 5 the iron to SS relative signaf as a function of the ahnninium Iayer width is shown. The continuous fine is our numerical result obtained assuming that the active materials are the same as before. The qualitative behaviour of the theoretical curve can easily be explained. When the alurninium layer is thin, up to 30 nm, only the electrons created in deep layers are absorbed in the Al film, so the relative signal is mainly constant, For layer widths larger than 30 nrn and up to 200 nm this attenuation effect still persists while the emission of secondary electrons related to resonant de-excitation in the steel becomes larger, then the line shows a ~onti~~~~~s decrease of the siope. From 200 nm on, the slope gets s~~~~t~~ Larger because of the greater probability in generating secondary electrons in the Af layer related to nuclear de-excitations in the iron layer. For larger widths, 500 nm and more, the relative signal is practically exponential (linear in the figure). The linear fit of Tricker et al. f13] would correspond to the theoretical behaviour for larger widths, but clearly is not correct for thin layers. It is possible that the experimental error, obviously a large one from the large dispersion of the experiments data, could hide the correct behaviour. Furthermore, as has been said above, we have assumed the active material to have the same physical characteristics as the ones used in the simulation of fig. 3, due to a lack of information. One can check that the relative iron to SS signal for width zero of aluminium does not agree with the value from fig. 3, so maybe both sets of experiments were performed under different experimental conditions which have not been inchrded in our numerical procedure.
A second order theory, in&ding
coincidence
effects, has been established
for the general case of CEMS functions have been
m~as~~~rne~ts cm muttitayer samples. Simple anafytkzd expressions for the spectral
derived. Although only spectral areas are considered in this paper, the contribution to the actual spectrum from each layer in the sample can easily be obtained from the convolution of its spectral function with the source lineshape. A program to simulate CEM spectra has been written and will be published elsewhere. The authors
wish to thank
Dr. D. Li&quist
for many valuable
and ~~Ii~hten~n~
comments.
Appendix We shall estabhsh some integration theorems, related to the exponential integral function f19f_ which are useful for the anatyticaf evaluation of the integrals appearing in the theory. Looking for computational facilities, the final results will be expressed in terms of the function E;(x)
3 eXEl(x),
instead of the exponential integral E,(x), because I?;(x) fntegrating by parts, the following identity is found
(65) remains
finite for large values of the variable.
where
Fl(@,P)= ~{E;([a+l]p)-E’(p)). It is just a matter
of direct calculation
(68)
to establish
9
sP
e-““F,(b,x)dx=F,(a,b,g)-F,(a,b,p),
(69)
with
&(a, b,p) = ;;;I”;;‘::
(70)
{ bE;([ a+b+l]p)-(a+b)E~([b+I]p)-taE~(p)}.
The integrals J, -ixxn
e “fe
(71)
-h’E1(~+f~+~)dydx,
and (72) can be analytically J,= (+&Jo.
solved in terms of the derivatives
of the function
F2 by noting
that (73)
References (11 [2] [3] [4] [5] [6] [7] [8) [9] [lo] Ill] [12] [13] [14] [I51 1161 [17] [18] [19] [20] [Zl]
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