Abstract. This paper presents measurements of the refractive index of a hygroscopic silica aerogel block at several wavelengths. The measurements, performed with a monochromator, have been compared with different parameterisations for n(λ), in order to determine the best chromaticity law for the aerogel. This is an important input for design and operation of RICH detectors with silica aerogel radiator

B Storaci

C Matteuzzi

D L Perego

M Calvi

M Musy

T Bellunato

CiteSeerX

Eur. Phys. J. C 52, 759–764 (2007) THE EUROPEAN
PHYSICAL JOURNAL C
DOI 10.1140/epjc/s10052-007-0431-3
Special Article – Tools for Experiment and Theory
Refractive index dispersion law of silica aerogel
T. Bellunatoa, M. Calvi, C. Matteuzzi, M. Musy, D.L. Perego, B. Storaci
Università degli Studi di Milano Bicocca and INFN, Piazza della Scienza 3, 20126 Milano, Italy
Received: 31 July 2007 / Revised version: 7 September 2007 /
Published online: 26 September 2007 − © Springer-Verlag / Società Italiana di Fisica 2007
Abstract. This paper presents measurements of the refractive index of a hygroscopic silica aerogel block at
several wavelengths. The measurements, performed with a monochromator, have been compared with dif-
ferent parameterisations for n(λ), in order to determine the best chromaticity law for the aerogel. This is an
important input for design and operation of RICH detectors with silica aerogel radiator.
PACS. 29.40.Ka; 78.20.Ci
1 Introduction
Silica aerogel has gained in the last decade a prominent
role in high energy physics as a radiator for Ring Imaging
Cherenkov (RICH) detectors. Its refractive index, tunable
at the production stage in the range 1.01–1.1, makes it
a suitable tool for the identification of charged particles
from 1GeV/c to a few GeV/c in momentum.
The refractive index of a dielectric material is wave-
length dependent. The dependence of the refractive index
of a dielectric medium on the photon wavelength, the so-
called dispersion law or chromaticity n= n(λ), is related to
the reconstructed angle θC by the Cherenkov relation [1]:
θC = arccos
[
1
n(λ)β
]
. (1)
The resolution of a RICH detector is intrinsically limited
by the chromatic dispersion of its radiator [2]. Therefore it
is important to know the dispersion law both for simulation
and for reconstruction purposes.
Silica aerogel, produced by the Boreskov Institute of
Catalysis (Novosibirsk) with a refractive index n ∼ 1.03
has been produced for the LHCb RICH detector [3–6].
This aerogel is hygroscopic and several tests have been
already performed during the R&D phase on its optical
properties, ageing and performances [7, 8].
Since most photon detectors work in the wavelength
range 200–700 nm, the behaviour of the index of refraction
within this range is the subject of this paper.
2 The refractive index
A common way to measure the refractive index n of
a dielectric solid material like silica aerogel is the prism
a e-mail: tito.bellunato@cern.ch
Fig. 1. Set–up of the prism method measurement of the silica
aerogel refractive index n
method which is shown in Fig. 1. Assuming the index of re-
fraction of air nair = 1, simple calculations in the classical
geometrical optics framework give the following formula:
θ = Φ+arcsin
{
n sin
[
α−arcsin
(
sinΦ
n
)]}
−α .
(2)
Here, θ is the deflection angle, Φ is the incident angle on
the aerogel surface, n is the refractive index of the solid
and α is the angle between the two adjacent sides of the
block. The measurements described in the following sec-
tions have been performed on an aerogel sample precisely
cut and polished after the production, with α= 90◦±0.1◦.
In the minimum deviation condition, (2) becomes:
n=
sin
(
α+θ
2
)
sin
(
α
2
) . (3)
760 T. Bellunato et al.: Refractive index dispersion law of silica aerogel
The dispersion law of aerogel can be parameterised
in several ways. The Lorentz–Lorenz dispersion law [1] is
given by:
n2−1
n2+2
= cf(Eγ) , with c=
4πa30ρNA
3M
, (4)
being a0 the Bohr radius, ρ the density of the medium,M
its molecular weight andNA is the Avogadro constant. The
energy dependence of the molar refractivity f(Eγ) can be
parameterised in terms of a multi-pole Sellmeier function
as:
f(E) =
N∑
j=1
Fj
E2j −E
2
γ
, (5)
where Fj and Ej are the Sellmeier coefficients and N is
the number of poles to be used to fit data. Equation (5)
can be written as a function of wavelengths. One- and two-
pole approximations for the dependence as a function of λ
are:
n2(λ)−1 =
a0λ
2
λ2−λ20
, (6)
n2(λ)−1 =
a1λ
2
λ2−λ21
+
a2λ
2
λ2−λ22
. (7)
A two-pole Sellmeier heuristic formula that param-
eterises the refractive index as a function of the energy Eγ
of the incident photon is often found in literature.
3 The experimental set-up
Refractive indices at several wavelengths have been meas-
ured using the experimental arrangement shown in Fig. 1.
A monochromator1 coupled to a Xe-UV lamp has been
used as light source. It selects wavelengths from 200 nm to
900 nm, with a resolution of 1 nm.
The deflection angle was determined by measuring the
displacement x with a CCD camera2 located at a dis-
tance L = 47.5 cm from the aerogel. The camera is fixed
to a support sliding on a rail transversely with respect
to the light beam to allow the positioning of the spot in-
side the CCD sensor. The fine measurement of the de-
flection was performed calculating the centre of gravity of
the beam spot image on the sensor with a typical reso-
lution of the order of 20 µm. Typically, the light devia-
tion x was about 3 cm in the plane containing the CCD
sensor.
To obtain a better defined spot on the CCD display,
measurements were performed at a generic fixed incident
angle Φ and not at the minimum deviation position. With
this configuration the refractive index n has been evaluated
by solving numerically equation (2).
1 Jobin Yvon Horiba H10, 1200 g/mm grating.
2 SONY® XC-ST70CE with 768×512 pixels and a sensitive
area of 1.025×0.850 cm2.
4 Results
A precise measurement of the incident angle Φwas not pos-
sible; its value has been estimated by (2) as the value giving
n= 1.0283 at λ= 632.8 nm, as previously determined with
the minimum deviation method using a He-Ne laser as
light source.
For each selected wavelength λ, the value of the refrac-
tive index n (λ) has been obtained from (2).
The evaluation of the uncertainty in the measurement
of the refractive index, σn, has been performed using
a Monte Carlo simulation in which all variables in (2) are
normally distributed around their measured values. The
standard deviation of each distribution is assumed as the
experimental uncertainty on each variable. Local small
flatness variations mimic deviations of α from right angle,
therefore they are implicitly taken into account in the
simulation.
The Φ uncertainty is the same for all the experimen-
tal data so that it is considered as a correlated error and
its contribution to the refractive index uncertainty σn has
been considered separately as explained at the end of this
section.
Data have been fitted with (6). The data and the fit
function are shown in Fig. 2. The results for the two pa-
rameters a0 and λ0 are shown in Table 1. An attempt to
fit experimental data with a two-pole Sellmeier formula in
the range 350–700 nm gives two superimposed poles show-
ing that there is no sensitivity to discriminate the second
pole.
Aerogel is a mixture of air, silica particles and some
residual impurities such as water and ethanol. As a first
approximation it can be considered as a less dense form
of fused silica. In order to verify this hypothesis, the con-
stants a0 and λ0 from aerogel have been compared to those
from quartz [9]. Data are listed in Table 1. The pole found
for the aerogel is close to the pole of the pure fused sil-
ica. However, since in our case the pole is shifted towards
shorter wavelengths, in the wavelength range investigated
the dispersion curve is less steep than the one resulting
from a pedestrian application of the Lorentz–Lorenz law to
fused silica data. This, in turn, results in a smaller chro-
matic dispersion.
In the pure fused silica model, the a0 parameter is re-
lated to the density of the material. Under this approxima-
tion, using (4) it is possible to calculate the density of the
aerogel tile from the relation:
ρ(aerogel) =
a0(aerogel)
a0(SiO2)
n2(SiO2)+2
n2(aerogel)+2
ρ(SiO2) . (8)
For the parameters listed in Table 1 one can determine
a density ρ(aerogel) = (0.158±0.001) g/cm3. This value is
in coarse agreement with ρ0 = (0.149±0.004) g/cm3 pro-
vided by the manufacturer. Therefore, it is possible to con-
sider, for our purposes, that aerogel is a binary mixture of
pure fused silica and air with a one-pole dispersion law in
the range 350–700 nm.
A comparison with a similar measurement performed
on a hydrophobic aerogel tile has also been done. In this
T. Bellunato et al.: Refractive index dispersion law of silica aerogel 761
Fig. 2. Chromatic disper-
sion n = n(λ) measured in
a hygroscopic silica aerogel
tile. A one-pole Sellmeier pa-
rameterisation has been fitted
to experimental data
Fig. 3. Experimental data
obtained with Φ, Φ+σΦ and
Φ−σΦ and fitted to the one-
pole Sellmeier formula
Table 1. Parameters of one-pole Sellmeier equation for aerogel
from fit and for pure fused silica from [9]
Aerogel Pure fused silica
a0 0.05639±0.00004 1.0963±0.00002
λ0 (nm) 83.22±1.25 93.76±0.01
case, a0 = (0.0536± 0.0003) and λ0 = (91.82± 3.82) nm
have been measured [10]. As before, the pole position is
directly comparable with the one from the hygroscopic
aerogel, while the a0 coefficient cannot because this pa-
rameter is density-dependent. This comparison indicates
to a 2σ level that the dispersion curve is flatter for hygro-
scopic aerogel in the wavelength range 350–700 nm.
762 T. Bellunato et al.: Refractive index dispersion law of silica aerogel
Table 2. Results of the fit with one-pole Sellmeier formula for
different input values of Φ
a0 λ0 (nm)
Φ−σΦ 0.05579±0.00003 83.15±0.95
Φ 0.05639±0.00004 83.22±1.25
Φ+σΦ 0.05698±0.00004 83.29±1.25
The uncertainty on the entrance angle, σΦ = 8mrad,
has been determined from the measurement of the refrac-
tive index at λ= 632.8 nm with the laser. To evaluate the
effect of σΦ, the aerogel refractive index has been deter-
mined using the reference value Φ= 536mrad and Φ±σΦ.
The three datasets (λ, n(λ)) have been then fitted
with (6). Results are shown in Fig. 3 and Table 2. The vari-
ation of Φ produces a shift, but the shape is not affected.
The effect on λ0 is masked by its uncertainty, but the a0
scaling parameter is slightly different in the three cases.
So the uncertainty on Φ mimics a density effect. Our de-
termination of the parameter is a0 = (0.05639±0.00004±
0.0006) where the statistical error is the first termwhile the
systematic part is given by the second term.
5 Particle identification and chromaticity
The effects of the dispersion law of the aerogel on the
particle identification performances in a RICH detector
have been investigated with a dedicated simulation. The
GEANT4 toolkit has been used [11]. An existing simula-
tion package used to study the refractive index inhomo-
Fig. 4. Dispersion laws used
in the simulation
geneity, described in detail elsewhere [8], has been adapted
to the current scope.
The inputs to the simulation are the optical properties
of the aerogel tile and its size, 200×200×50mm3. The pro-
gram then generates a bunch of charged particles which
traverse the aerogel perpendicular to the wider surfaces.
The Cherenkov photons produced in the aerogel are traced
along their path, can undergo Rayleigh scattering and are
focused by a tilted spherical mirror (R = 949mm) onto
a photon detector whose quantum efficiency has been set to
be the same as for the LHCb photon detectors [12].
Mixtures ofK− and π− beam (60 k particles in total for
each run) with a momentum of 10 GeV/c have been gener-
ated. Four different parameterisations of the dispersion law
of the aerogel have been used:
1. n1 = const, wavelength-independent;
2. n2 = n2(λ), the one-pole Sellmeier formula measured in
this paper;
3. n3 = n3(λ) derived from the the Clausius–Mossotti
equation assuming a simple binary mixture of air and
SiO2 [13];
4. n4 = n4(λ), a two-pole Sellmeier formula used in the
LHCb standard simulation [4].
The graph of these dispersion curves is shown in Fig. 4;
the density dependent factors have been tuned so that
the distributions match the value n= 1.0283 measured at
λ= 632.8 nm.
A hit by hit reconstruction algorithm [14] allows one
to build the distribution of the Cherenkov emission angle
θC from the hits on the detection plane. The information
used are the beam direction, the photon emission point, the
position in space of the centre of curvature of the mirror
T. Bellunato et al.: Refractive index dispersion law of silica aerogel 763
Fig. 5. Distributions of re-
constructed θC from mixed
K− and π− simulated beam
with n1 (black) and n2 (red)
runs
Fig. 6. Distributions of re-
constructed θC from pure
π− simulated beam with n1
(black) and n2 (red) runs
and its radius. All the photons are assumed to be produced
halfway inside the aerogel along the beam direction. In the
reconstruction algorithm the information from the disper-
sion law is not used.
The reconstructed Cherenkov angle distributions are
plotted in Fig. 5 for the first two cases n1 and n2. The dis-
tributions for n3 and n4 are similar to the n2 case. The
widths of the two peaks corresponding to pions and kaons
in the n1 = const run are due to spherical aberration from
the tilted mirror and the uncertainty on the emission point.
In the n2(λ) run the chromaticity effects, modulated by the
quantum efficiency of the photo-cathode, determine the
blurred shape of the distribution. In Fig. 6 only the pion
component is plotted.
764 T. Bellunato et al.: Refractive index dispersion law of silica aerogel
Table 3. Mean Cherenkov angle θC and single photon reso-
lution σθ (mrad) for the different dispersion laws and particle
species considered in the simulation
K− π−
θC (mrad) σθ (mrad) θC (mrad) σθ (mrad)
n1 229.8 0.6 234.3 0.5
n2 231.8 2.8 236.3 2.8
n3 231.6 2.1 236.1 2.0
n4 231.9 3.5 236.4 3.5
For each run and for each kind of particle, the average
value of the reconstructed angle θC and the standard devi-
ation of the single photon resolution have been determined
with a gaussian fit. Results are listed in Table 3. When n is
not constant in the simulation the widths of the two distri-
butions are larger and the peaks are shifted with respect to
the wavelength-independent case. The shift is of the order
of 2 mrad. The mean θC values do not differ much for the
different parameterisations, while the single photon reso-
lutions differ substantially.
6 Conclusions
The resolution of a RICH detector is limited by the chro-
matic dispersion of its radiator. The refractive index n(λ)
in the wavelength range 350–700 nm has been measured
using a monochromator and the prism method.
In this range the one-pole Sellmeier function is found to
be a good approximation for the dispersion law. The pa-
rameters obtained for the aerogel are compared with the
ones for pure fused silica. The pole for the aerogel is at
shorter wavelengths with respect to fused silica, while the
density-dependent parameter is found to be compatible
with the properly scaled fused silica one.
Simulation studies show that different assumptions on
the functional form of the dispersion law, although leading
to the same average Cherenkov emission angle, imply sub-
stantially different single photon resolutions. The particle
identification performance of a RICH detector strongly de-
pends upon this parameter. The determination of the cor-
rect law therefore is a powerful tool in the conceptual de-
sign of an aerogel-based Cherenkov detector.
Acknowledgements.We gratefully acknowledge the skilled tech-
nical support of R. Mazza and the fruitful discussions with
G. Lucchini about the experimental setup. We wish to thank
G. Carboni (Università di Roma Tor Vergata) who has kindly
provided part of the experimental equipment. Financial sup-
port from INTAS–5579 and HP–I3 contracts is acknowledged.
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