Inhomogeneous broadening of optical spectra in mixed crystals: Basic
model and its application to Sm 2 + in SrFClxBr1_x
R. Jaaniso,a) H. Hagemann, and H. Bill
Department of Physical Chemistry, University of Geneva, 30 quai Ernest-Ansermet, CH-1211 Geneva 4,
Switzerland
(Received 12 July 1994; accepted 14 September 1994)
We have developed a model to describe the inhomogeneous broadening of optical spectra in the
substitutionally disordered crystals. The comparison with the experimental f - f fluorescence spectra
of SrFClxBrl_x:Sm2+ (O~x~ I) allowed to establish, in a very detailed manner, the relationship
between the inhomogeneous spectral distribution and the crystal structure around the Sm2 +
impurity. © 1994 American Institute of Physics.
I. INTRODUCTION
Inhomogeneous broadening (IHB) of spectral lines due
to imperfections or disordered structure is a general intrinsic
property of solids, which comprises the information about
the nature and distribution of the structural irregularities, as
well as about the interactions between the optical center and
the host. I The relationship between the inhomogeneous spectral distribution (ISD) and the structure of the solid plays
also a crucial role in the physical phenomena which were
previously essentially hidden by the inhomogeneous spectral
"envelope" (excitation energy transfer within an ensemble
of centers, correlations between the transition frequencies
and other measurable quantities characterizing the optical
centers; see, for example, Refs. 2-4, and references therein)
and which have become accessible after introduction of tunable lasers and spectrally selective techniques. 5- 8 In rather
general terms, IHB can be also viewed as a useful property
for the potential technological applications of spectral hole
burning technique. 9
In the crystals, the variations in the local environments
of the impurity centers are caused by a relatively small number of point defects (below I at. %) or dislocations, which
give rise to the inhomogeneous spectral Iinewidths of the
order of 0.1-1 cm -I. The theory of IHB in crystals, comprehensively reviewed by Stoneham,l is based on two main assumptions; (i) superposition of pairwise impurity centerdefect interactions; and (ii) independent (uncorrelated)
spatial distributions of the defects. The latter assumption is
rigorously valid at low defect concentrations (small x or I-x
values in the title compound), when the line shape resulting
from the (locally neutral) point defects is predicted to be a
Lorentzian. The experiment has verified the linear dependence of the linewidth and shift with the defect
concentration. lo - 12 Theory has been generalized for the correlated distributions of different transitions,I3 for degenerate
transitions and correlated strains,14 and for two-dimensional
solids. IS
Amorphous solids with large IHB of spectral lines (I 00
cm- I and more) represent the other extreme in the given
problem. Two approaches have been applied to describe the
·)Permanent address: Institute of Physics, Estonian Academy of Sciences,
Riia 142. EE-2400, Estonia.
IHB in these systems. First, essentially the same statistical
model, as for dilute defect systems, has been explored under
the assumptions that it can be extrapolated to higher defect
concentrations, and the so called Gaussian approximation is
justified. 16 By using continuous spatial distributions with an
impurity (solute) represented by a small sphere, a turnover
from the low-concentration limit to the Gaussian line shape
was also modeled within this approach.17 The second
method, initially proposed for the fluids of rare gas
elements,18 and thoroughly elaborated for both liquid and
glassy environments,I9 allows us to overcome the assumption about the independent particle distributions by taking
into the account the three-particle spatial correlations. The
examples of experimentally determined ISD in glassy solids
can be found in Refs. 20 and 21. Comparison with the experiment has been most successful for the spectral holes,
inhomogeneously
broadened
under
the
external
pressure. 16,22,23
The description of mixed crystals is usually considered
as being of intermediate complexity between of the one of
the crystals and of the glasses, respectively.24 The title compound represents the simplest subclass in the family of the
mixed crystals 25-the one with binary substitutional disorder
and with full range miscibility (Le., O~x~ I). The composition of this type of compounds, and hence, their degree of
disorder, can be varied continuously within wide limits,
which makes up for their advantage as a model system. The
IHB can be studied starting from the low-concentration limit
up to maximum disorder at x=0.5.
In order to describe the IHB for the full range of x values, two essential modifications of the theory I seem to be
unavoidable. The first point follows from the fact that at a
large number of substitutions the major effect to the spectral
distribution will be determined by the lattice sites closest to
the (Sm) impurity. Obviously, the description of the defectimpurity interactions via long range defect fields or field gradients should be replaced by an appropriate crystal field
model for the first coordination sphere(s). The second generalization concerns the proper accounting of different configurations in a binary compound AxB l_x. 24 The assumption
(ii), cited above, becomes inadequate because it allows multiple occupancy of the lattice sites. In connection with the
first point, it is convenient to separate the near neighbor clus-
ter from the outer crystal region in the statistical averaging
J. Chem. Phys. 101 (12), 15 December 1994
0021-9606/94/101 (12)/1 0323/15/$6.00
© 1994 American Institute of Physics
10323
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
10324
TABLE I. Crystallographic data for SrFX compounds.
Compound
a (A)
c (A)
Vel (A3)
r(Sr-Xl)
(A)
r(Sr-X4)
(A)
SrFC1'
SrFB(>
4.126
4.218
6.958
7.337
59.22
65.27
3.072
3.389
3.112
3.222
"Reference 29.
bReference 30.
FIG. 1. Unit cell for the SrFX compound (1) and the projection of a portion
of the crystal on an (a,c) plane (2). The labeled atoms on (2) pertain to the
same plane (except of X4 in parentheses). The blank atoms are situated in a
parallel plain displaced by the distance a12. Cl and Br sites are generally
labeled by X; Sm2+ ligands by (Xl,X4); and the second type of actual Sm2+
near neighbors by (Yl,Y4).
problem. This will allow us to avoid an a priori application
of the superposition principle [assumption (i)] for the near
neighbor substitutions, and, in addition, to describe the clustering effects around the optical center. Such separation of
two crystal regions is, in fact, an extension of the strategy
used in the Mott-Littleton method26 for computing the defect structures in ionic materials. It has been also discussed
by Stoneham 1 in connection with the satellite line spectra at
low defect concentrations. The basic model with mentioned
generalizations was first briefly described in Ref. 27 together
with its simplified application to the title compound experimental spectra. This paper presents the results of a comprehensive study, where the parameters for Sm near neighbor
substitutions are determined from the experiment. Recently a
model, similar to the one we applied to the outer crystal
region was published together with the calculations for a
cubic crystal structure.2s
The paper is organized as follows. The experimental details are given in Sec. II. The basic model is described and
developed to a fully analytical result in Sec. III. Finally Sec.
IV applies the model to the SrFClxBrl_x structure, compares
its results with the fluorescence spectra of two Sm2 + I-I
transitions D 1- 7 F 0 and 5 Do- 7 F 0) and discusses the results.
cooled to 900°C before switching the furnace off or grown
by pulling. All measurements were performed on single crystals with minimum dimensions of 2· I· 0.1 mm 3. The concentration of dopant was chosen to be one order smaller than the
smallest value of x (0.5%) in order to avoid extra effects
from Sm pairs.
Fluorescence was excited with 476.5 nm Ar+ laser light
and was recorded through a Spex 1403 based spectrometer
with 0.2 cm -I spectral resolution. Fluorescence spectra of
7
5 D 1- 7 F 0 and 5 D 0- F 0 transitions, excited via 1- d absorption, were measured for different values of x. An overview of
the 5Do- 7 F 0 transition spectra is presented in Fig. 2. One
obtains a number of subbands which are broadening with
increasing disorder, and are shifting towards the low frequency side with increasing x-value. Similar spectra (at x
equal to 0.2, 0.5, and 0.8) have been published for
BaFClxBrl_x:Sm2+ together with a short interpretation [the
sub bands were ascribed to six different compositions of the
clusters CI 5 - n Bf" (n =0, ... ,5) around Sm ion, but without
taking into the account the general shift of the spectra].33
Our final analysis was performed on the spectra measured at 30 K in a He flow cryostat. This temperature was
sufficiently low to avoid any measurable contributions from
the homogeneous linewidths of the order of 0.0 I cm -1.27 As
a result, the experimental spectra could be directly compared
with the model lSD, even for the crystals with minor mixing.
In order to see the gradual change in the spectra and to obtain
a reasonable basis for comparison with the model, altogether
e
II. EXPERIMENT (SAMPLE PREPARATION AND
FLUORESCENCE MEASUREMENTS)
The parent compounds, SrFCI and SrFBr, crystallize in
the P4/nmm structure (point group D 4h) .29 ,30 The tetragonal
structure persists for all intermediate compounds (values of x
between 0 and 1), whereas the CI and Br ions are located in
the double layers (X) between Sr-F-Sr sheets (Fig. 1).31,32
Sm ions enter substitutionally into Sr positions (C 4v point
symmetry in ordered crystals), and have five nearest neighbor heavy halogens (one axial, labeled as Xl in Fig, 1, and
four nonaxial, labeled as X4). Selected crystallographic parameters are given in Table I.
Samples were made by mixing stoichiometric amounts
of previously prepared SrFCI and SrFBr, both doped nominally with 0.05% SmF3. The ground mixtures were either
molten under an Ar atmosphere at llOO °C and slowly
x - value
1
O.5~--
O~=====14*500~=====1~M~5~O~Emission Frequency [cm- I ]
FIG. 2. Fluorescence spectra of SrFClxBrl_x:Sm2+
T=lOO K.
5Do - 7 F 0
transition;
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
10325
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
21 different compositIOns (x=O, 0.005, 0.01, 0.02, 0.03,
0.04.0.1,0.2.0.3.0.4,0.5,0.6,0.7,0.8,0.9,0.96,0.97,0.98,
0.99. 0.995, I) were prepared.
(4)
Gx{V)= ~
III. BASIC MODEL
±P~{{TJk})8(V-~
~
'7N=O
'71 =0 '72=0
The ISD of a nondegenerate impurity transition in a binary compound A" B I _ x (which may also be a sublattice of a
more complex compound) can be written as a configurational
average,
(1)
where P~(j) is the realization probability for a certain configuration (j) of the particles A and B around the impurity at
the molar ratio x. and Vj + Vx is the transition frequency in
this configuration. The latter quantity is divided into two
parts, where Vx is a common frequency shift caused by the
uniform dilatation in the finite crystal (usually called the Eshclby shift).34 The Vx is assumed to depend only on the molar ratio. For sake of generality we mention that if the actual
Eshelby shifts are slightly different for different configurations, then these differences are included into Vj'
Following the comments made in the Introduction we
will separate the configurations of a small cluster around the
impurity center from the surrounding lattice. The full advantage of this separation becomes clear later when we apply the
model to the real system. At this stage one more reason can
be noticed-the interaction of the impurity center with its
nearest neighbors is generally specific-it does include overlap and covalency contributions even for rare earth impurities in ionic crystals. 2 .35 In what follows we will make the
difference between the crystal region I (the first, and if applicable. the next nearest neighbors) and crystal region II (the
outer lattice).
For region II we suppose that the substitution of B by A
at a lattice site k induces the shift Vk of the actual transition
frequency, and, that these shifts are additive. We also suppose the additivity of the total spectral shifts from two crystal regions, and, in addition, that the probability factors for
the region I do not depend on the actual configuration of the
surrounding lattice. Now the overall configuration probability factorizes into two parts,
VkTJk),
k
(5)
where the last expression gives the ISD due to the interactions with crystal region II.
Further factorization follows if we assume that the distribution of the particles A and B in region II is completely
random,
P~( {TJd) =
II bx{l, TJk)'
(6)
k
If the selections {TJk ; k = 1, ... ,N} are taken from a consider-
ably larger ensemble, then the probabilities that a certain
lattice site is occupied by A or B, are given by bx( I , I) = x or
b x (1,O)=(1-x), respectively.
It follows from the assumptions made above for region
II that the spectral shifts Vk should be equal for symmetrically located sites. This allows, in principle, to reduce the
number of products in Eq. (6) by using the joint occupation
probability function for each subgroup of equivalent sites.
This function is given by the binomial law,
(7)
where m is the number of equivalent sites (for example,
m =4 for nonaxial sites in C 4v symmetry), and n is the actual
number of particles A in the subgroup. This approach will be
advantageous in the numerical calculations (Secs. IV Band
IV C). We will proceed here with the general case with each
site treated separately.
Formula (5) can be further simplified by writing the
8-function in its spectral representation,
G x ( v) = (1I27T)
X
f:",
~ [II
{'7k}
dr exp{ - ivr)
bAI,TJk).exP(ivkTJkr)].
(8)
k
(2)
Here the sum over the configurations is written in a simplified form and the exponent of i r~ Vk TJk has been transformed
into a product. After performing the summation one obtains
and the transition frequency is given by
N
Vj = Vi + ~ vk TJk .
(3)
k=1
Configuration of region II is given by the set of occupation numbers {TJk ; k = 1, ... , N}, where k accounts for the different lattice sites, and TJk is equal to I or 0, depending on
whether there is A or B in the lattice site k, respectively.
Label i accounts for the relatively small number of region I
configurations.
Under these assumptions formula (1) can be rewritten as
(9)
Fx{ r)=
II [(l-x)+x exp(ivkr)].
(10)
k
At this point we note that at x ~ I the characteristic func-
tion F) 7) can be reduced to a form familiar from Ref. 1,
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
10326
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
n=xNIV
ex p ( -n !vdP(r){l-eXP[iV(r)r]}).
(11)
Here dp(r) is the probability to find a defect in the vicinity
of point r in continuous medium and v(r) can be related to
the long range field of a lattice defect.
The Fourier transform (9) can be performed analytically
by using the cumulant expansion of the function Fx(r). The
result can be interpreted as a special case of a general result
in the statistical analysis (see, for example Ref. 36), which
allows us to correct the probability distribution through the
cumulants of the random variables. Corresponding formula
have been derived for the generating function (11) in Ref. 37.
We will give below the explicit results for the arbitrary values of x.
The general formula for the cumulant expansion is
In[FA r)]=
2:
(12)
Kj(X)' (ir)jlj!.
j=1
The cumulant coefficients K/X) may be calculated directly
from the definition,
j
(13)
K/X) = i-j[d In Fx(r)ldri]IT=O,
(14)
K/X)=q/X)2: (Vk)j,
k
where q/x) are the polynomials of x,
ql(X)=X,
j
2:
[x(l-x)]i( -1)i-I(2i- 1)!aj;'
(IS)
;=1
j
q2j+I(X)=(l-2x)2: [x(l-X)]i( - 1);-1
;=1
X(2i-l)!iaji'
The expressions for the numerical coefficients a ji are given
in the Appendix.
Formula (12) can be further rewritten as
j=3
(16)
which becomes
FA r) = exp[i rvx - (rO'xf/2][ I +
[(V- iix)2]
2
I
G( v)x= ----,,::-- exp ,,2'TT'ux
2ux
In conclusion, we have reached to a fully analytical expressions for the lSD, Eqs. (4) and (I8). In the simplest case,
the coefficients P~(i) in formula (4) are given by the binomial distributions of the type (7), but in general they depend
on the chemical potentials <I>(i) of different near neighbor
clusters, P~( i) - exp[ - <I>(i)lkTm]. 1,38 The series in formula
(18) is, in fact, semiconvergent and is thereby given in a
truncated form. It becomes totally divergent for the small
x-values (or for the x-values close to unity), but gives reasonable approximation at the x-values which are of primary
interest in the present work (see Sec. IV B).
IV. RESULTS AND DISCUSSION
by using Eq. (10). The final result is
q2/X)=
after expanding each exponent under the infinite product into
the Taylor series and by multiplying all these series. The new
coefficients d/x) are determined by the combinations of
normalized cumulants Ki(X)/(uX>' (see Appendix). We have
used here the notations iix=KI(X) and U;=K2(X),
After the substitution of Eq. (17) into Eq. (9) each term
in the series can be directly Fourier transformed. The final
result is a Gaussian multiplied by a functional series with
Hermite polynomials He/z),
i
d/x)· (i rO'x)j]
J=3
(17)
A. Effect of near neighbors: Satellite line structures
from region I
The spectral shifts due to substitutions in the nearest
neighbor sites can be calculated, in principle, by ab initio
methods or, perhaps, with the help of the crystal field superposition mode1. 35 In the situation of a solid solution with a
full range solubility there is, however, a possibility to determine these shifts from the experiment by analyzing the spectra at a small number of substitutions. At x<0.05 and
x>0.95 the IHB due to the outer crystal region is small and
the satellite line structure due to near neighbor substitutions
is clearly resolved. Satellite lines have been a frequent subject of studies, although their assignment to precise atomic
configurations around the optical center has been less common (for a recent example, see Ref. 39). The main aim of
this section will be the identification of all of the resolved
satellite lines emerging from Cl+-... Br substitutions, in order
to have an empirical set of parameters for the crystal region
I.
With this purpose we analyzed the spectra of 12 compounds at x<O.1 and x>0.9. An example for x= 1% is given
in Fig. 3. The intensity of the satellite lines shown on this
figure increased linearly with the Cl concentration, which
indicates that they are induced by single Cl substitution in
Sm neighborhood. There are two inequivalent such substitutions in the cluster of five nearest neighbors; one with axial
Cl (XI position in Fig. 1), and the other with non axial Cl
(X4 position). Consequently, two satellite lines are expected
with the ratio of the intensities of I :4. The assignment shown
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
10327
TABLE II. Assignment of first neighbor satellite lines (5 Do - 7 F0)'
line No.(i) v(i) [em-I]
n(i)
m(i)
configuration
0
1
0
0
1
2
7.0
1
4
0
0
0
0
3
14.0
2
4
4
14.0
2
2
5
20.0
3
4
6
27.0
4
1
7
8.0
1
1
14495
14490
14485
frequency [cm- 1]
FIG, 3. Fluorescence spectra of ~Do-7Fo and 5D)-7F o transitions at
x=O,OI. Three configurations of the X-cluster are shown by their projections onto the plane perpendicular to the crystal axis (filled circle Cl; empty
circle. Bd: T=30 K.
in Fig. 3 is further confirmed by the splitting of the doubly
degenerate 5 D 1(£) - 7 F O(A 1) line in the configuration of
lower symmetry. The two satellites labeled as Y 1 and Y 4 on
the other side of the main line are attributed to substitutions
at the next nearest neighbor sites and will be discussed later.
There are 12 different configurations in the X-cluster
(4· X4+ X I). which are schematically shown in the last column of Table II (the symbols are the same as used in Fig. 3).
In what follows. the configurations will be numbered according to this table. The satellite lines, corresponding to different configurations can be identified, first of all, by analyzing
the concentration dependence of the line intensities (which
gives the composition of the cluster, ClnB~ -n, n =0, ... ,5),
and by comparing the relative intensities of the lines corresponding to the same composition. If the occurrence of different configurations is completely random, then the intensities of the lines are governed by the formula,
px(i)=m(i)x n(i){I-x}[5-n(i)],
(i= 1, ... ,12).
(19)
The numbers of Cl ions in the X-cluster n(i) and the
combinatorial weights m(i) are given in Table II.
In order to illustrate the identification of all the satellite
lines, the spectra in Fig. 4 are shifted from their original
position in the frequency scale. First, the maxima of the main
lines are shifted to the same frequency within one subset
(x=O.OI, 0.03, 0.1 for the upper subset; x=0.96, 0.97, 0.99
for the lower subset), and second, the lower subset is shifted
so that the maxima of the satellite lines, resolved and identified in both subsets, do coincide. The spectral shifts introduced in this way follow approximately a linear dependence
8
18.0
2
4
9
26.5
3
4
10
26.5
3
2
11
35.9
4
4
12
43.9
5
1
0
0
0
0
0
•
•
•
•
•
••
•
••
••
•
••
•••
••
•
•••
•
•••
0
0
0
0
0
0
0
0
14500
0
0
0
0
0
0
0
0
0
0
0
0
••
with the x-value, with the proportionality coefficient being
equal to -57 cm- I . The frequencies v(i) of the satellite lines
as measured from the position of the line No. I are collected
into Table II.
The strongest satellite lines are associated with n = I and
n =4. The assignment of the lines No.2 and No.7 was already demonstrated in Fig. 3. Corresponding lines at x>0.9
(No.6 and 11) were identified in a similar way. Note here
the difference for the substitutions in X4 and Xl sites. For
X4 the spectral shifts are practically equal for the substitutions CI 5 -+CI 4 Br [1I(l2)-1I(l1)=8 cm- I ] and B~
-+Br4 Cl[1I(2)-lI(l)=7 cm- I ]; for Xl the corresponding
shifts are essentially different; 1I(l2)-v(6)=16.9 cm- I and
1I(7)- v(1)=8 cm- I .
For n = 2 (or 3) and m =4 two lines with identical intensity dependence are predicted. In this case the shift due to the
axial (X 1) substitution was assumed to be bigger, as it was
established for n = 1 and n =4 lines. Only for this assignment
a regular behavior could be established for the frequencies
v( i). This regularity is characterized, first of all, by the additive spectral shifts for X4 sites. This can be directly seen
from Table II, the frequencies are increasing by =7 cm -I per
substitution for i= 1, ... ,6, and by =8.5 cm- I per substitution
for i =7,. .. 12. For an axial substitution the frequency shift is
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
10328
o
X
x=0.1
,
I
'x=0.03:
,,
,,
,
12
11
9,10
6
,,
I
" ,
I
""
~ 3'4 72
5'
I I
,
I
I
I
I
I
I
I
I I
, I
,,
I
x=O.96
x=O.97
50
40
30
20
o
10
relative frequency [cm- 1]
FIG. 4. Identification of 12 satellite lines of the X-cluster in the fluorescence
spectra of SD 0 - 7 F 0 transition.
not constant but depends on the number of CI ions in nonaxial sites. However, the frequency shifts [v(i + 6) - v(i);
i= 1, ... ,5] are gradually increasing, which can be associated
with the decreasing metal-ligand distance r(Sr-XI) between
SrFCI and SrFBr (see Table I). Note that the corresponding
change is much smaller for r(Sr-X4), and, that the values of
r in Table I are probably well suited for Sm-ligand distances
because of the very close ionic radii of sr2+ and Sm2+.
The lines with m =2 (No.4 and 10) were not clearly
resolved and, at this stage, corresponding frequency shifts
were set equal to the one due to stronger line (m =4) with the
same axial ligand.
The upper and lower subsets in Fig. 4 are adjusted in the
frequency scale by coinciding the lines No. 3/4, which are
clearly visible in all the spectra at x>O.OI and x<0.99.
Close inspection of the line No.5 indicated the presence of a
shoulder at the low frequency side, of increasing intensity
with the concentration. The position of this shoulder coincides with the maximum No. 8 in the upper subset. Finally,
the weak line on the left side of the x=O.1 spectrum can be
connected with the line No. 11 in the lower subset.
The preliminary assignment of the first neighbor satellite
lines can be summarized as follows: each configuration has
probably a well defined frequency over the whole range of x;
for non axial ligands the frequencies follow closely the superposition principle; the deviation from the superposition principle for an axial ligand is continuous with x and can be
associated with the change in the crystal structure.
The satellite lines appearing on the opposite side of the
main line are ascribed to the substitutions in next nearest
neighbor positions (YI and Y4). For higher concentrations
these lines are not clearly resolved, but are still visible as
shoulders. Similar shoulders are visible at the first neighbor
satellites (see lines No. 2,5,6,11 in Fig. 4), indicating that, at
least approximately, additivity holds for the frequencies of X
and Y substitutions. The intensities of the lines Y I and Y4 in
Fig. 3 are again in the ratio close to I :4. Hence, the smaller
line can be assigned to the substitution in the next closest
axial site (Yl in Fig. 1). The assignment of the nonaxial site
(Y4) is not final at the present stage and will be discussed in
Sec. IV. The values of the spectral shifts, measured from the
position of the main line, are as follows: -1.2 (Y4) and
-2.2 (Y1) cm- I at x=O, and -2.7 (Y4) and -4.0 (Yl)
cm- I at x=l.
At the lowest concentrations (x<O.Ol and x>0.99) some
additional weak lines, probably induced by chemical impurities or internal defects, were observed. All these lines were
shown not to originate from Cl+-+Br substitutions by investigating the x-dependence of the spectra. In the worst case
such a line was situated at the same frequency as the "true"
satellite. For example, the intensities of the Y 1 and Y4 lines
were not in the 1:4 ratio for x>98% samples, and only comparison of the samples with different concentrations clarified
the situation (there was a weak "false" line at the Yl position, present even in nominally pure SrFCl). The presence of
foreign impurities was not, however, a major problem, and
the purity of the starting materials was estimated to be better
than 0.2%.
5 D I - 7 F 0 transition spectra were analyzed in a similar
manner. One-to-one correspondence with 5 Do- 7 F 0 satellite
lines was observed, but, all the spectral shifts v( i) were
somewhat (5%-10%) bigger. The parameters for the
5D I - 7 F 0 transition wiII be given after the final fit of mixed
crystal spectra in Sec. IV C.
B. Influence of the outer lattice (region II)
The information about the static interactions between the
impurity and the outer crystal region remains hidden in the
line shape, even at a low number of substitutions.
In order to evaluate the line shape on the basis of the
results of Sec. II, one has to calculate the spectral shifts Vk
induced by the substitutions at all the lattice sites of region
II. For distant substitutions these shifts can be calculated if
the spatial dependence of the point defect induced strain,
and, the strain coupling constants for a given transition, are
known. Unfortunately, the long range strain cannot be given
by an explicit analytical formula for a tetragonal crysta1. 34 In
order to analyze the applicability of Eq. (18) and to obtain
some qualitative information about the line shape one may
use the expressions for the continuous medium. I As a crude
approximation we will use the formula
vk=~(Vel/ri)(1-3 cos 2
ek),
(20)
where ~ is the coupling strength and Vel the elementary cell
volume.
The lattice product (10) was calculated by taking into the
account the influences from 777 CIlBr sites in the rectangular
crystal around the Sm. The 10 near neighbor sites (4·X4
+ Xl +4· Y4+ Yl) were excluded. The positions of the ions
[given in Eq. (20) by the distance from Sm, rk, and by the
equatorial angle, ed, were calculated from the averaged el-
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
FIG. 5. Theoretical ISO for the crystal region II [solid curve, exact result;
dotted curve. approximation with formula (18); dashed curve, Gaussian approximation], The values of x are indicated at the curves. The number of
terms in the ,eries (18) was given by M =4 (x =0.5), M = 12 (x =0.1, x =0.9)
or M=28 (x =0.04. x=0.96); ~=-5.9 em-I.
ementary cell parameters (see Ref. 32). The accuracy of the
final results was checked by comparing the moments of the
calculated Fourier transform (9) with the moments directly
calculated from the cumulant coefficients (14). We will give
below the explicit expressions for the average frequency,
variance, scewness, and excess coefficients of the distribution Gx(l'),
I'I=K3/a3=~_~(~~XX) (L v~)
4
1'2= K4/a =
I
/(L v~rl2,
~t~~~~X) (L vi) / (L v~
(21)
r
On the other hand, the spectra were calculated with formula (18). The results, obtained within different approaches,
are compared in Fig. 5. One obtains that the spectrum is
closely Gaussian at x =0.5. For a realistic distribution of the
frequency shifts Ilk the first order corrections to the Gaussian
in formula (18) are coming from the 3rd and 4th order cumulants (more precisely, from the dimensionless parameters
1'1 and 1'2)' The x-dependence of 1'1 and 1'2 is shown in Fig.
6. For x=O.5 the 1'1 [as well as all other odd cumulants, see
formula (15)] vanish and the spectrum is symmetric. There is
always some negative excess factor at x =0.5 but, as one can
see from Fig. 5, this has a negligible effect on the spectrum
when near neighbor sites had been excluded.
In the range of 0.2:'Sx:'S0.8 (range A in Fig. 6) the deviations from the Gaussian curve are still relatively small and
can be neglected in the first approximation. If the x-region is
extended to A + B, the inclusion of only the 3rd and 4th
cumulants [i.e., only K3 and K4 are nonzero in formula (A2)]
will provide sufficient accuracy. This region can be approximately given by the inequality 11'1112+11'21<0.2. FinaIly, formula (18) is still applicable in the range C in Fig. 6 but in
10329
FIG. 6. Dependence of the scewness (YI) and excess (Y2) factors on the
x-value. A, B, and C indicate different regions for the applicability of the
formula (I 8).
this case up to 30 higher order cumulants should be included.
At smaller (or bigger) x-values the series becomes divergent;
in other words, the cumulant expansion becomes unusable.
The rules sketched above for the applicability of the formula (18) should be quite general, provided that the closest
sites to the impurity (with spectral shifts bigger than = I
cm -1) are incorporated into the crystal region I. The convergency at the edges of region C is, indeed, somewhat dependent on the exact borderline between two crystal regions and
should be checked in each particular case. We also note that
the change in the coupling strength g does not alter the shape
of the line; g acts as a scaling factor determining the central
position Vx and the variance ~. By changing the sign of g
the spectrum will be reflected with respect to point v=O. In
the model calculations we used the negative value of g, as in
this case the asymmetry coincided with that observed in the
experimental spectra (see the main line in Fig. 3).
Similar calculations basing on the direct Fourier transform of the lattice product (10) or its cumulant expansion
(12) have been made by Orth et al. for a simple cubic
lattice?8 Although all near neighbor sites were included at
x=0.5, the calculated spectrum was not so different from a
Gaussian. In a cubic (or hexagonal) crystal there are less
inequivalent sites around the substitutional impurity, and, if
the superposition principle applies, a more uniform distribution of the first neighbor lines is expected, indeed. It was also
shown that a sufficient correction to the Gaussian distribution could be obtained by using the direct Fourier transform
of the cumulant series truncated at K4' This is, in essence,
equivalent to the application of the formula (18) with only K3
and K4 included.
The analysis given above determines also quantitative
limits for the so-called Gaussian approximation. The result
that the ISD induced by point defects tends to a Gaussian if
the defects are excluded from an increasing volume around
the impurity has been derived in continuum approximation in
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
10330
Ref. 40. Within the present formalism, the same result will
be obtained when the spectral shifts vk--+O, and the number
of sites N --+00. This can be interpreted as a special case of the
central limit theorem, indeed. The calculation scheme, where
enough sites around the optical center are treated exactly so
that the residual line shape G xCv) is approximately Gaussian,
was also suggested in Ref. 28. In order to describe the overall ISD within this approach the influence of only a limited
number of near neighbors should be known exactly. This
scheme is clearly advantageous if the long range fields cannot be treated explicitly but enough information about the
near neighbor sites can be obtained from the satellite lines
and/or atomistic simulations. In our example the satellite
lines provided enough data for using the Gaussian G x( v) at
the x-values between 0.2 and 0.8 but additional corrections
were needed for the outer x-range.
c.
Having now the two crystal regions concretely determined, and armed with the empirical spectral shifts for the
near neighbor substitutions, the final formula for fitting the
Sm2+:SrFClxBrl_x spectra can be written as a sum over region I configurations,
12
4
2: 2: 2:
(22)
where the product of the first four factors under the sum
equals to P~ in Eq. (4). Label i accounts for the first neighbor
configurations (see Table II), j for the number of Cl ions in
Y4 sites and j' for the number of CI ions in Yl site. p xU) is
given by Eq. (19), and bx(m,n) by Eq. (7). By assuming
spectrum (22) to be normalized to unity, we have
12
w(i)pAi)= 1.
v
hence, one should introduce an interpolation for the intermediate values. The simplest ad hoc assumption is the linear
dependence of these parameters with x,
vytCx) = VYI(O)+X[VYI(1)- VYI(O)],
w(i)pAOb x (4,j)b x (1,j')
i= 1 j=O j' =0
2:
o
FIG. 7. Schematic representation of the satellite lines and their spectral
shifts in the model. The positions of Sm2+ lines in the parent compounds are
labeled as SrFCI and SrFBr, for other explanations see text.
Fitting the mixed crystal spectra
IAv)=
x
(23)
i=1
The additional weights wU) measure the deviations of
the first neighbor line intensities from the binomial law.
When comparing the results of this model to the experimental spectra, one has to take into account that these deviations
may also be caused by the parameters relating the ISD to the
emission or absorption spectrum (absorption strengths, fluorescence yields of different clusters).
As not all the satellite lines emerging from Y I and Y4
substitutions could be resolved, we will use for the corresponding frequency shifts a similar additivity hypothesis as
for the outer crystal region. The frequencies for the region I
configurations are then given by
vA i,j ,j') = v( SrFBr) + v(i) + jVY4(X) + j' VYI (x) + x
(24)
where v(SrFBr) is the transition frequency in pure SrFBr
compound. VYl and VY4 are the frequency shifts for a single
substitution in Y I or Y4 site, respectively. The latter quantities were determined to be different at x =0 and x = 1, and
(1= 1,4).
(25)
The quantity A Vo is equal to the frequency change of
region I configurations when the host composition is
changed from SrFBr to SrFCl. It is defined by the relation
(26)
which states that the frequencies of region I configurations
change linearly with the x-value. For iix the linear dependence can be seen from Eq. (21). The average Eshelby shift
Vx should also change linearly with mole fraction, as long as
the strain coupling is linear and the lattice parameters change
linearly with x. The x-ray investigations confirm the latter
assumption for the title compound?I,32 High-pressure experiments on SrFCI:Sm2+ have also shown that the frequencies
of f - f transitions are changing proportionally to the lattice
parameters41 (at least up to the changes which correspond to
the differences between SrFCI and SrFBr lattices).
In Fig. 7 the spectral shifts given by formulas (24) and
(25) are shown schematically. The satellites near x =0 and
x = I are represented by vertical lines. The labels in the
circles indicate the configuration of the X-cluster (i=I,12).
For simplicity, only one line is shown for intermediate cluster compositions (i =2, ... , 11). The thick (thin) lines at the
lower frequency axis correspond to, the Y-cluster with only
Br (Cl) ions. Again, the scheme is simplified as the intermediate compositions are not shown. At x = 1, the Y-cluster with
only Cl ions is more probable and corresponding satellites
are presented by thick lines. The'satellites corresponding to
the same configuration are connected with the dashed lines
which give the frequency positions 1Jx (i,j,j') at the intermediate x-values. As one can see from Fig. 7, the slope of the
function vxCi ,j ,j') depends on the composition of the
Y-cluster [the extreme cases being represented by Avo (j=0,
j' =0) and AVI (j= 1, j' =4)].
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
0,1
TABLE ill, Final spectral parameters (in cm-),
0,15
0,15
X=0,9
0,1
X-o,1
p(SrFBr)
p(SrFCI)
0,05
0,05
J VI~
°
aPn.c,
°
dvo
dv)
lIyiO)
lIy4(1)
lIy,(O)
lIy)(I)
(V2- V)
(V3- v,)
(v4- v ,)
(vs-v,)
(V6- V)
(V7- V,)
(vs-v,)
(v9- v)
(vJO-v)
(VII-V,)
(v'2- v )
0,1
0,1
X-o,2
0,05
0,05
a
0,1
1«50
14500
° 14450
0,1
X.O,7
X-o,3
0,05
0,05
a
a
0,1
0,1
x.o,a
X-o,4
J!,(Vk?
lIx4
lIx'
dllx4X)
0,05
0,05
10331
sDo- 7FO
sD)- 7FO
14489.4
14475,8
[3,6
-41.1
-49,2
-1.2
-2,7
-2,2
-4,0
7,0(2)
12,6(4)
[3,9(9)
19,3(7)
25,1(5)
8,1(2)
17,5(3)
26.4(9)
27,8(9)
34,2(3)
42,0(5)
6,1
6,3
8,5
2,2
15829,6
15811.4
18,2
-46,8
-55,0
-1.4
-2,9
-2,6
-4,8
7,3(2)
13,5(3)
15,8(10)
20,8(8)
26,6(8)
9.4(3)
19,1(4)
28,0(10)
29,0(10)
35,8(3)
43,3(3)
6,6
6,5
9,0
1.9
OL-~JUUL~~~~
o 14450
14500
frequency [em- 11
14450
14500
frequency [em- 11
wU) = I, but it was immediately clear that the subband in-
It is important to note here, that in spite of the relatively big
number of the weight parameters, it was possible to determine them uniquely for the most cases because of the existence of the clearly resolved subbands in the spectra, This is
particularly true for x<O.4 and x>O,g, Some barely resolved
maxima can also be seen for the central x-values between 0.4
and 0,8.
Fitting was performed step by step, starting with the
spectra at x=0.9 (or x=O, 1) and moving to the smaller (bigger) x-values, After each step the adjustable parameters were
extrapolated to the next x-value, and then, were adjusted for
best fit by comparing visually the experimental and model
spectra,
In a second stage the starting parameters v(i) were also
slightly adjusted for better coincidence with the maxima in
the experimental spectra, In these cases where the maxima
were not clearly resolved (x =0.5,0,6,0,7), the differences
v(i) - v(I) were held constant and only the common shift for
all the frequencies was adjusted,
Figure 8 presents the results for 0, 1~x~0,9, Two sets of
model spectra are shown, Those weighted by wei) are practically indistinguishable from the experimental curves (except for a slight discrepancy at the strongest line for x =0,9),
For the simplified application of the model [we i) = 1 for all i]
clear deviations can be seen, even for the spectra with well
resolved subbands, We note that the simplified model spectra
shown in Fig, g are calculated with the final frequency values
(Table III) and that the discrepancies were even bigger if the
initial values from Table II were used,
The 8-lines in Fig. 8 correspond to different X-cluster
configurations, The positions of the lines are obtained by
averaging Eq, (24) over the binomial distributions of Yl and
tensities start to deviate from the distribution given by p x( i).
Y4 sites,
FIG, 8, Fitting results for 0.1 <x<0,9 (experimental spectra, solid curves;
simplified model, dashed curves; final model, dotted curves),
In order to complete the description of the model we will
list all the parameters which have to be used in fitting the
experimental spectra, The values of spectral parameters in
formulas (24) and (25) [v(2), ... ,v(12), VYl(O), VY4(0), VYl(I),
z/Y4( I)] were determined in Sec, IV A, If one adds the transition frequencies in the pure compounds, v(SrFCl) and
z1('SrFBr), then there will be altogether 17 fixed input parameters. Note that Llvo is not an independent parameter, as it
follows from Eq. (24) that
- Ll vo= v(SrFBr)- v(SrFCI)+ v(12)+ Vy(I).
(27)
We used here the definition Vy(x) =4vY4(x) + vYl(x) for the
total spectral shift of Y-cluster, and the conventions v(l)=O,
v,(I2,1,4)=v(SrFCl). Relation (27) can be seen in graphical
form in Fig, 7,
The parameters to be adjusted are the weights wei) (i
= 1,.,.,12) and those determining the function Gx(v) [this is
actually the coupling strength g in formula (20)], In fact,
only nine independent weight parameters were used as the
values of wU) were set equal for the lines with similar concentration dependence, and which could not be resolved in
the spectra at 0,}<x<0.9: w(2)=w(7), w(3)=w(4), and
w(9)=w(10).
At the beginning we tried to fit the spectra by putting all
J, Chem, Phys" Vol. 101, No, 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
10332
~
+
.,.
v
4
3
1
2,7
3,4
8
0
5
•
9,10
6
11
12
.
0
x
2
x
FIG. 10. Weight factors w(i) of X-cluster configurations i= 1,... ,12.
14460
14480
14500
14520
frequency [em- 1]
FIG. 9. Modeling of IHB at x=0.5 (solid line, experimental spectrum;
dashed line, final model spectrum). The 8-line structures correspond to the
clusters shown in the right-hand comers [in the upper part, X-cluster; in the
lower part, (X + Y}-cluster]'
iixCi) = <vx(i,j ,j') )j,j' = v(SrFBr) + v(i) + X· [Vy(x)
+a VO].
(28)
The intensities of the lines are given by w( i) . P x( i). The
x-dependencies of the X-cluster average frequencies iix(1)
and vx(12) were shown also in Fig. 7 (by the solid lines
between the dashed ones).
The final result for x=0.5 is shown separately in Fig. 9.
The line spectrum in the upper part of the figure is similar to
the ones given in Fig. 8. The line structure in the lower part
is due to the bigger (X + Y) cluster and is given by Eq. (22)
when one replaces Gx(v) by a 8-function.
The weights wei), as determined by the fit, are shown in
Fig. 10. Note that the lines No.5 and No.8 have almost the
same frequency and do form one subband at 0.I<x<0.9.
The
intensity
of this
subband
is
given
by
w(5)Px(5)+w(8)pX<8), from which the values ofw(5) and
w(8) cannot be determined uniquely. Thereby we used here
the additional restriction that the dependence of w(5) and
w(8) on the x-value should be as smooth as possible. The
same rule was applied to the lines No.6 and No. 9/10 which
also do form a single subband.
We see from Figs. 8 and 10 that a quantitatively perfect
fit was only possible if the weights wei) were allowed not
only to depart from unity but were also allowed to depend on
the x-value. This dependence is generally smooth and monotonic, except for the x-values between 0.6 and 0.8. By investigating samples from different batches grown with this
range of x-values, we observed slight spectral variations for
samples with nominally the same compositIOn. The phase
diagram of SrFClxBrl_x system is unsymmetrical with respect to the x =0.5 point, the minimum melting temperature
occurring at x=O.3Y Correspondingly, the equilibrium has
the widest x-splitting between the solidus and the liquidus
lines on the other side of the central x value, approximately
at x=0.7. One may conclude that 0.6<x<0.8 is exactly the
region where the largest compositional variations may be
expected. From the differences in the spectra we estimated
these variations to be smaller than 5%.
The physical meaning of the weight factors remains
somewhat unclear at the present stage. One interpretation is
obviously, that some of the Sm-ligand configurations are
thermodynamically more favorable. Yet, not very large differences may be expected, because the lattice distortions
around the Sm2+ ion are small, and because the compositional stability of the crystals is generally good. The other
factor to be considered is that the nonradiative relaxation
rates42 may be somewhat sensitive to the local structure as
well as to the mean x-value (through the nonlocal interactions). Additional studies of the lifetimes, pressure, and temperature dependencies are probably useful for making the
distinction between these mechanisms.
As it was mentioned above, the presence of clearly resolved subbands in the spectra allowed us also to adjust the
frequencies v(i). With this we relaxed the model constrain
that the values of v(i) are exactly constant over the whole
range of x. The results for v( i) are presented in Fig. 11. We
also tried to locate more precisely the weaker lines No.4 and
No. 10. This was possible only at the edges of the x-scale;
for the central x-values the differences v(4)-v(3) and v(lO)
- v(9) were held constant. In Table III we presented the
quantities (v(i) - v(l), which are the averages over all values shown in Fig. 11. By subtracting the value of v(l), one
cancels the possible error in spectrometer repeatability or the
deviation because of the compositional variations. To illustrate the latter effect, we used for x=0.6 the spectra of the
sample where the actual composition can be predicted to be
x=0.63.
Regarding the broadening by the outer crystal region, the
spectra could be fitted with a constant value of the coupling
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso. Hagemann. and Bill: Inhomogeneous broadening in mixed crystals
10333
D. Discussion
0.8
0.6
x
0.4
0.2
0L-~~--~~i~0------2~0------~30------~4~0~
relative frequency [em' i )
FIG. II. Frequency shifts v( i) (i = 1..... 12) of X·cluster configurations.
parameter g=-5.9±O.6 cm- I . During the fitting procedure,
in fact, we tried to adjust g for each value of x; the resulting
values of the variance (which is uniquely determined by g
and characterizes the subband width) are shown in Fig. 12.
We see that the theoretical dependence, ax - [xC 1 - x)] 1/2
[see formula (21)], which is a direct consequence of the superposition principle, is nicely followed.
Although the model is rigorously valid only for the transitions between non degenerate states, it appeared to describe
well the 5D I(E) - 7 F o(A I) transition spectra because of the
relatively small splitting between the 5D I (E)-states. The latter spectra were in a good one-to-one correspondence with
the 5 Do(A I) - 7 F o(A I) transition spectra. The fitting could
be done exactly in the same manner and all the remarks
made above on the behavior of the weights and spectral parameters apply. The final values of the parameters are given
in Table III. We note that the nearby 5D I(A 2) - 7 F o(A I)
transition was neglected in the fitting procedure because it
contributed less than 1% to the total area under the curve.
... 0 ..
3
1
25
.
~
2
S
b
%~~O.~1--~O.~2--~O.~3--~O.~4--~O.-5--0~.6---0~.7---0~.8---0~.9--~
x
FIG. 12. Broadening parameters for crystal region II as the functions of
composition (rings represent the 0', and crosses the half-width r ih l2). Theo·
retical dependence of
0',
is shown by a solid line.
By working out the model in detail in Secs. IV A-IV C
we achieved finally a rather precise description of the mixed
crystal spectra. At the same time, a set of well defined parameters was determined for the Sm2+ :SrFClxBrl_x system.
The crucial factor for the successful application of the model
was the separation, from the very beginning, of the nearest
neighbor crystal sites, and the use of different approaches for
the two crystal regions. In the simplest form the final model
contained a single parameter W which had to be adjusted by
comparison with the experimental spectra. As we see from
Fig. 8 this basically allowed us to predict the position of the
spectrum in the frequency scale (and its width) but failed to
reproduce accurately the line shapes. After inclusion of the
weight parameters and slight adjustment of region I frequencies the line shapes could be described very precisely. The
accuracy of the data determined at the final fitting with numerous parameters, depends, in tum, on the rightness of the
underlying model assumptions. In particular, the assumption
postulated by formula (25) will need verifying arguments.
On the other side, the quasifree treatment of the frequencies vCi) in the previous section allows us to check the validity of the model assumptions summarized schematically in
Fig. 7. Note that the scheme in Fig. 7 can be compared to the
one given for Nd3+ in LaClxBr3_x in Ref. 43, where the
difference between the transition frequencies in LaCl 3 and
LaBr3 was described as a sum of "ionicity shift" and "volume shift." The first one corresponds to v(l2) in our notation, and the second one to the average (Avo+Avl)l2. The
term "ionicity shift" keeps only a qualitative meaning (for a
general discussion on this matter, see Ref. 44) because of the
lack of exact additivity for Xl-sites and because of the relatively strong influence from the second nearest neighbors.
Let us tum to the matter of the "volume shifts" (spectral
shifts of nonlocal origin). Provided that these shifts are equal
for all configurations one should observe 12 strictly vertical
lines in Fig. 11. We see that this is quite the case--the deviations from the vertical line are smaller than 2 cm -I which
makes about 5% from the average shift (Avo+Avl)/2. One
may conclude that the local strain sensitivities do not differ
by more than 5% for different X-clusters. Recalling now that
the corresponding difference for the Y-clusters is equal to
AVI-Avo=8 cm- I , we find, perhaps surprisingly, that the
changes in the more remote Y-cluster have somewhat bigger
influence on the strain sensitivity.
However, one should make clear at this point that the
last conclusion is directly related to the assumption made
with Eq. (25); the Vy depend on x-value (but not on the
composition of X-cluster). In order to check the accuracy of
this assumption, we tried to fit the spectra also under the
opposite assumption, i.e., the Vy depend only on the composition of X-cluster (and not on x-value). More exactly, we
supposed that the shifts Vy are increasing proportionally to
the number of Cl ions in the X-cluster [with the values of
Vy(O) and Vy(l) in Table II linked to i = I and i = 12, respectively J. The results of the fit, analogous to the ones given in
Fig. 11, are shown in Fig. 13. One can conclude from Fig. 13
that the average "volume shifts" are decreasing from i = 12
to i = 1. This is in accordance with our alternate postulate.
J. Chem. Phys .• Vol. 101. No. 12. 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
10334
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
18
16
14
)(
12
0
-
0.4
'E
is
t:
0L-~0~--~-1~0------2~0------3~0------~~~~
relative frequency [cm- 1]
CD
:::J
I
~
E
~
FIG. 13. Frequency shifts v(i) (i= 1, ... ,12) for the alternate model. The
difference with Fig. II is explained in the text.
What is no longer in accordance with the model is the nonlinearity of the "volume shifts" [see comments following
formulas (26) and (32)]. In addition, the spectra could not be
fitted with a constant value of g but we had to choose smaller
g for the clusters with bigger vy-values (which seems to be
artificial as Vy is related to crystal region I and g to region
II). As a conclusion, we see that formula (25) provides a
more reasonable interpretation of the data. A further conclusion one can draw from the insensitivity of the vy-shifts on
the composition of the X-cluster is that the Y- and X-sites are
probably not in direct contact. For that reason we are eager to
identify the Y 4-sites as shown in Fig. 1.
As it follows from Eq. (26), the "volume shift" arises
from two components, vx and vx' The Eshelby shift can be
estimated as follows. By assuming that the strength of the
image forces at the surface 45 builds up linearly with the number of substitutions in a solid solution,34 one may write
xLl VI
Ll vp Ll V .
(29)
The Ll vp is the frequency shift induced by such hydrostatic
pressure which changes the elementary cell volume by the
value equal to Ll V between SrFCI and SrFBr. The volume
change because of image forces at the surface, Ll VI' is a
fraction of the total volume change which occurs when the
composition changes from x =0 to x = 1. The corresponding
ratio can be estimated by using the result of the continuum
elasticity theory, 34
Ll VI _ 2 ( 1 - 2 P )
LlV-3
!"=p'
(30)
where P is Poisson's ratio. By assuming the average value of
P to be around 0.2, the ratio (30) will be close to 0.5. Consequently, the total Eshelby shift VI is about half of the spectral shift measured at such a hydrostatic pressure which produces a fractional volume change equal to (Ll V/VkIBr' From
the high-pressure data of SrFCI:Sm2+ (Ref. 41) one can obtain that Llvp =-155 cm- l for 5DO-7FO transition, and
hence x = - 78x cm -I. The absolute value of Ll vp is about
7% bigger for 5D 1- 7F 0 transition, but this difference is
within the precision of the estimate.
v
4
45
35
,-<5
,,
o , ,
30
j;Y
25
X1=C1
20
, ,Cf'
15
,,
10,..'
_
3
2
.2. 40
0.2
vx =
-.-
_._.->t-
10
x
---
_--x-
--}-
-
-""
X1=Br
...,...-
5
'-0
2
n(X4)
3
FIG. 14. Frequency shifts of the X-cluster as the functions of the number of
Cl ions in X4 positions CDo-7FO transition). The upper part shows the
spectral shift due to substitution in X I site. The lower part shows the total
shift due to substitutions in X4 sites for two types of X I ligand.
Within the same approximation (isotropic continuum),
the coupling strength in formula (20) can be shown to be
proportional to (Ll V/V)CIBf' and consequently, the spectral
shift Vx to be proportional to the volume change. However,
for vx the relation is not as direct as for vx and is more
sensitive to the actual structure of the crystal. In order to
estimate the coupling constant g one needs, in addition, the
spectral responses of the given transition with respect to individual components of the strain tensor. Whereas the latter
estimate is difficult to make at the present stage, one can
obtain an independent experimental value of g by using the
inhomogeneous line widths f ih measured at small x (or 1- x)
values. For the spatial dependence given by expression (20)
the following relation holds: f ih = 1O.11g\x .46 From the values
of the half-widths shown in Fig. 12 one can obtain the derivative Idf mldxl=40 cm- I (both for x--+O and x--+ 1), which
leads to 19\=4 cm- I . The difference with the previously determined value (Ig\ =5.9 cm -I) is rather reasonable if we consider the crude character of Eq. (20). Regarding the sign of g,
we note that the proposed negative value gives the right sign
for both, the asymmetry ('YI>O at x<0.5) and the frequency
shift vx(>O). The latter quantity should be positive, indeed,
in order to cover the difference between the experimental
volume shift and the estimated Eshelby shift [see formula
(26)].
Let us tum now to the spectral shifts of local origin. In
Fig. 14, the quantities <v( i) - v(l) are shown as the functions of the number of CI ions in X4 positions. We see that
the superposition principle is rather precisely fulfilled for
X4-sites; by adding one Cl to an X4-site the frequency increases by 6.3 cm- I (Xl=Br) or by 8.5 cm- I (Xl=CI). On
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
the upper part of the figure not the total cluster frequency but
the shift resulting from a single Cl-+Br substitution in Xl
position is shown. In this case the elementary shifts are not
equal but depend on n(X4). The shift is about two times
bigger for n(X4)=0 (all Br in X4 positions) than for
n(X4)=4 (all Cl in X4 positions), and increases approximately linearly between these extremes. All the linear approximations shown in Fig. 14 can be given by the equation
vU) = vX4n(X4) + vXln(XI) +
a vX4Xln(X4 )n(XI),
(31)
where fl(X!) and II (X4) are the numbers ofCI ions in Xl and
X4 positions, respectively, and the values of spectral parameters (vX4' VXI' aVX4XI) are given in Table III. Consequently,
the description of the frequencies of 12 different configurations can be reduced, in a good approximation, to three parameters. One may expect this type of reduction to occur,
keeping in mind the successful application of the crystal field
superposition model (CFSM) to the lanthanide crystal field
spectra. 35 In the frame of CFSM the third term in Eq. (31) is
to be related to the distortions in the cluster geometry at the
substitution. The presence of the cross-term may also reflect
the fact that the problem cannot be totally reduced to pairwise interactions. Note that formula (31) is a direct representation of the experimental data. This implies that the method
used in the present work may be helpful for testing the
CFSM postulates.
The ions in the Y-sites are not in contact with Sm and
hence their influence may be thought to be essentially Coulombic. We estimate here only the first order contribution
which can be effectively described as a coupling between
all(Sm2 +) (difference between the ground state and excited
state dipole moments of Sm2 +) and all(CllBr) (effective
dipole moment induced by substituting Cl-+Br). The spatial
dependence of this dipolar contribution is given by Eq. (20).
Direct calculation shows that the ratio VytfVY4 should be
equal to 15 (actually is equal to 1.6), and the values of Vy at
x =0 and x = 1 should differ by 4%-8% (the values at x = 1
are actually two times bigger). The large discrepancy between the calculated and experimental ratios shows that the
influence of the Y-sites should be essentiaIly mediated by the
F- ligands. The relatively strong dependence of the shifts Vy
on the x-value can be associated with the detailed balance of
the contributions from different types of ligands in the
present compounds (which, in particular, makes the crystal
field parameters sensitive with respect to X4-Sm angle).47,48
A direct electrostatic contribution should, in fact, be considered also for the crystal region II. Although the comparative calculations for Y-sites seem to agree with an interpretation where the strain coupling is dominating, an
independent estimate would be useful for Coulombic interactions. For the remote sites only the dipolar term is of importance, whereas the coupling constant in formula (20) can
be given through the coefficient of the linear pseudo-Stark
effect aVE:g = ±iall(CllBr)i . aVE/2Vel • Note that
the sign of g will be opposite for the "lower" and "upper"
layer in the double layer of ClIBr ions. The induced dipole
moment can be estimated as foIlows. If we assume the
change in the position of the point charge to be between
10335
a r!2 and a r (a r =0.15 A is the difference in the ionic radii
of Cl- and Br-), then the corresponding dipole strength values are between 0.36 and 0.72 D. The other contribution,
arising from the change in the ionic polarization, can be estimated to be 0.5 D on the basis of BaFCI and BaFBr data. 49
Taking into the account that the two mentioned dipoles are of
opposite direction, one can write for the resulting value,
iall(CIlBr)i ~ 0.2 D. By using the Stark-coefficients of
Sm2 +:BaFCI (avE=0.07 MHzN cm- I) (Ref. 50) one finally
obtains iel~0.6 cm- I.
The effect of the deviations from the superposition principle may also be considered for crystal region II. The first
order generalization of formula (3) has similar structure as
Eq. (31) and is given by
Vj=V;+L Vk1lk+L aVik1lk+L 'aVkk'1lk1lk' (32)
k
k
k,k'
The third term in Eq. (32) takes into the account the fact
that the coupling to region II may be slightly different for
different near neighbor clusters. This generalization can be
simply included into the model; the result will be that each
region I configuration will have its own residual line shape
G~(v-
iJ,,).
The last term in Eq. (32) may be interpreted as resulting
from pairwise interactions between the substitutions. The
main contribution to this term may be thought to arise from
the nearest sites to the given substitution (i.e., k' runs over a
smaIl number of sites around k). For example, if we consider
the substitution in site k to be the source of the strain field,
then the strength of the elastic field can be evaluated through
the force constants of pairwise interactions with nearest
neighbors. 51 These strengths will be generally different for
different neighbors which results in the variations of the frequency shift associated with site k.
One result arising from the last term in Eq. (32) can be
seen relatively easily; the average frequency jjx will be a
nonlinear function of x (an x 2 term wiIl appear). Note that
this nonlinearity should be common for all the configurations
of crystal region I and cannot explain the behavior we found
when testing an alternate dependence to the one given by
formula (25) (see Fig. 13). The absence of significant nonlinearity in the volume shifts may serve as a supporting argument for the use of the linear approximation (3) with a
fixed set of parameters Vk' The relatively small effect of the
second order terms in Eq. (32) is also in accordance with the
estimate we obtained for the strain magnitude €; by using the
value of the fractional volume change (a V/V)C1Br=0.097
and formulas of continuum elasticity theory 34 we estimated
€<2X 10- 3 , which is five times smaIler than the commonly
used limiting value for the linear elasticity.
Finally, we would like to point out that during the course
we foIl owed in clarifying the formation of inhomogeneous
broadening, it became evident, in parallel, that the mixed
crystal spectra can provide rich information for a detailed
crystal field analysis (Table III). Figure 15 gives an additional illustration to this point. Here the model is applied to a
satellite line spectrum. One can see that a rather fine spectral
selection of numerous clusters is possible. Close inspection
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
10336
ACKNOWLEDGMENTS
.~.
This work has been supported by the Swiss National
Science Foundation and (as a part of optical characterization
of rare earth doped PbFCI family mixed crystals) by the
Swiss federal program "Optique."
APPENDIX
The numerical coefficients a ji in Eq. (15) obey the recurrent relations,
(Al)
14450
14460
14470
aji=i2.aj-li+aj-li-l
14480
i=2, ... ,j-l.
frequency [cm- 1]
Coefficients d j in Eqs. (17) and (18) are given by the
relations
FIG. 15. Modeling of the satellite line spectrum at x=0.98. At each line the
composition of crystal region I is shown by three indexes (i ,j ,j') as in
formula (22).
dj=lj
j=3,4,5,
d 6 = 16+(h)2/2!,
of the line shifts shows that for the Y-sites the superposition
principle is fulfilled approximately with the same accuracy
as given by Eq. (31) and that the shifts Vy have (as it has
been assumed) approximately the same value for different
X-clusters (i=5,6,1l,12). By improving the purity of the
samples and the dynamic range of the measurement one may
probably have even a more detailed description of the rareearth-near-neighbor clusters. In order to check further the
generality of the model and the accuracy of the interpretation, it will be also useful to have experimental data for the
analogous
related compounds. Work on closely
Sm2+:BaFClxBrl_x and on the compounds with cationic disorder (like Sm2+:SryBal_yFC1) will be completed in near
future. Another interesting test of the model assumptions
could be a study of the correlation between the distributions
of two different transitions. 52
V. CONCLUSIONS
We developed a model to describe the inhomogeneous
broadening of optical spectra in the substitutionally disordered crystals, which is compared to the experimental I-I
fluorescence spectra of Sm2 + impurity in SrFClxBrl_x'
The model, in its final form, was applied in a semiempirical way so that its basic parameters were determined from
the experiment. The success of such an approach is based on
two features; (i) the initial parameters for the near neighbors
can be found from the satellite line spectra; (ii) the line shape
due to the interactions with the outer lattice is relatively insensitive to the exact form of these interactions in the range
ofO.l<x<O.9.
We believe that the approach used in the present work is
generally applicable to the compounds with substitutional
disorder, especially to those having the full range miscibility.
It allows not only to understand the formation of IHB and to
relate the structural variations to the spectra, but also provides a sensitive tool for probing the impurity-ligand interactions (through the possibility to compare different ligand
clusters eventually in the same solid).
d 7 =h +1314 ,
d 8=18+ 1315+ (f4)2/2!,
(A2)
d 9= 19+ 1316+ 1415 + (f3)3/3 !,
dlO= 110+ 1317+ 1416 + (f5)2/2!,
where we used the notation Ij =
K/ (
~ j!). In general
d.= ~
li(l-!i(2)" "Ii(n)
,
k j (l)!k j (2)!" 'kj(m)!'
(A3)
where the sum should be taken over all combinations such
that i(1)+i(2)+···+i(n)=i. The factorial k j ! appears in
the denominator each time when some of the indexes among
i (1 ), ... , i (n) are equal, kj being the number of equal indexes.
The Hermite polynomials in Eq. (18) can be most conveniently evaluated from the recurrent relations,36
Hel(Z)=Z,
He2(z)=z2-1,
(A4)
Hen + 1(Z) = zHen(z) -nHen-l (Z).
A. M. Stoneham, Rev. Mod. Phys. 41, 82 (1969).
of Solids Containing Rare Earth Ions, edited by A. A. Kapljanskii and R. M. Macfarlane (North-Holland, Amsterdam, 1987).
3J. L. Skinner, B. B. Laird, and L. Root, J. Lumin. 45, 6 (1990).
4R. M. Macfarlane, J. Lumin. 45, 1 (1990).
5N. A. Kurnit, I. D. Abella, and S. R. Hartmann, Phys. Rev. Lett. 13, 567
(1964).
6 A. Szabo, Phys. Rev. Lett. 27, 323 (1971).
7 A. A. Gorokhovskii, R. K. Kaarli, and L. A. Rebane, JETP Lett. 20, 216
(1974).
8B. M. Kharlamov, R. I. Personov, and L. A. Bykovskaya, Opt. Commun.
12, 191 (1974).
9 Persistent Spectral Hole Burning: Science and Applications, edited by E.
W. Moerner (Springer, Berlin, 1988).
10 A. E. Hughes, J. Phys. Chern. Solids 29, 1461 (1968).
II A. A. Kapljanskii and R. B. Rozenbaum, SOY. Phys. Solid State. 13, 2200
(1972).
12 A. M. Freiberg and L. A. Rebane, SOY. Phys. Solid State 16, 1704 (1975).
13 J. Kikas and M. Ratsep, Phys. Status Solidi B 112, 409 (1982).
I
2 Spectroscopy
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Jaaniso, Hagemann, and Bill: Inhomogeneous broadening in mixed crystals
14H. Bill and D. Lovy, J. Phys. Condensed Matter 1, 10 265 (1989).
I~N. S. Averkiev, V. S. Vikhnin, N. S. Sokolov, and N. L. Yakovlev, JETP
Lett. 56,9 (1992).
16B. B. Laird and J. L. Skinner, J. Chern. Phys. 90, 3274, 3880 (1989).
17L. Kador, J. Chern. Phys. 95, 5574 (1991).
IRI. Messing. B. Raz, and J. Jortner, I. Chern. Phys. 66, 2239,4577 (1977).
19H. M. Sevian and J. L. Skinner, Theor. Chim. Acta 82,29 (1992).
20J. Fiinfschilling, I. Zschokke-Granacher, and D. F. Williams, J. Chern.
Phys. 75, 3669 (1981).
~I R. Jaaniso and J. Kikas, Chern. Phys. Lett. 123, 169 (1986); R. Yaaniso
and R. Avarmaa, Sov. J. App!. Spectrosc. 44, 365 (1987).
22Th. Sesselmann, W. Richter, D. Haarer, and H. Morawitz, Phys. Rev. B 36,
760 I (1987).
2'L. Kador, S. Jahn. D. Haarer. and R. Silbey, Phys. Rev. B 41, 12215
(1990).
J. Elliott, I. A. Krumhansl. and P. L. Leath, Rev. Mod. Phys. 46, 465
(1974).
"~A. I. Kitaigorodsky, Mixed Crystals (Springer, Berlin, 1984).
16N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34,485 (1938).
17R. Jaaniso, H. Hagemann, and H. Bill, Opt. Soc. Am. Tech. Digest 22, 79
(1992).
~8D. L. Orth, R. J. Mashl, and J. L. Skinner, J. Phys. Condensed Matter 5,
2533 (1993).
,QM. Sauvage. Acta Crystallogr. B 30, 2786 (1974).
1QH. P. Beck. J. Solid State Chern. 17,275 (1976).
.\1 L. Jianhua and S. Mianzeng, Gaodeng Xuexiao Huaxue Xuebao 6, 957
24R.
(1985).
'~H. Hagemann. F. KubeI. and H. Bill, Mater. Res. Bull. 28, 353 (1993).
)'L. Zhang, J. Yu, and S. Huang, J. Lumin. 45, 301 (1990).
\.II. D. Eshelby, Solid State Phys. 3, 79 (1956).
·\~D. I. Newman, Adv. Phys. 20,197 (1971); D. I. Newman and Betty Ng.
Rep. Prog. Phys. 52, 699 (1989).
10337
of Mathematical Functions, edited by M. Abramowitz and I. A.
Stegun (National Bureau of Standards, Washington, D.C., 1972).
37 S. H. Simon, Y. Dobrosavljevic, and R. M. Stratt, J. Chern. Phys. 93, 2640
(1990).
38W. Kleeman and F. Fischer, Z. Phys. 197, 75 (1966).
39y. Lupei, L. Lou, G. Boulon, and A. Lupei, J. Phys. Condensed Matter 5,
L35 (1993).
40G. Davies, J. Phys. D 4, 1340 (1971).
41 Y. R. Shen, T. Gregorian, Th. Troster, O. Schulte, and W. B. Holzapfel,
High Pressure Res. 3, 144 (1990).
42 J. C. Gacon, I. C. Souillat, J. Seriot, and B. Di Bartolo, Phys. Status Solidi
A 39, 147 (1977).
43R. Baumert, J. Pelzl, and S. Hufner, Solid State Commun. 16,345 (1975).
44C. R. A. Catlow and A. M. Stoneham, J. Phys. C 16,4321 (1983).
45 Which should be considered because the surface tractions of a finite crystal should vanish (for more details, see Ref. 34). Formula (29) with formula (30) can be converted to the same form as given previously in Ref.
10 if one makes the following substitution for the strain sensitivity coefficient; 11 11/(11 V/3 V)p=B,,+ Byy + Bxx.
46p. W. Anderson, cited as private communication in W. J. C. Grant and M.
W. P. Strandberg, Phys. Rev. 135, A715 (1964).
47D. NicolJin and H. Bill, Solid State Commun. 20, 135 (1976).
48T. Gregorian, Y. R. Shen, T. Triister, and W. B. Holzapfel, Eur. J. Solid
State Inorg. Chern. 28, 117 (1991) .
49M. Fischer, M. Sieskind, A. Polian, and A. Lahmar, 1. Phys. Condensed
Matter 5, 2749 (1993).
50 R. M. Macfarlane, R. M. Shelby, and A. Winnacker, Phys. Rev. B 33, 4207
(1986).
51 J. R. Hardy, 1. Phys. Chern. Solids. 29, 2009 (1968).
52D. L. Orth and J. L. Skinner, J. Phys. Chern. 98, 7342 (1994).
36 Handbook
J. Chem. Phys., Vol. 101, No. 12, 15 December 1994
Downloaded 08 Dec 2010 to 128.95.64.60. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions