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Density Measurements:
Rutherford Backscattering Spectroscopy
1 Introduction
As part of the AXAF Reflectivity Study, it is of interest to perform density measurements on
the films used to coat the mirrors being investigated. The interest in density, ae, derives from
the fact that the critical angle for reflectivity, ` c , scales as ae 1=2 . The critical angle is inversely
related to energy (through the optical constants) so that, for a chosen angle of operation
(i.e. for a fixed grazing angle associated with a particular mirror design) the critical angle
determines the high energy cutoff of operation. To maximize the upper energy range, then,
it is important to achieve coating densities that are as high as possible.
Experimental measurements of reflectivity generally yield results which are lower than
those predicted by theoretical calculations in which densities equal to the bulk density of the
coating material are assumed. Thus, it is of interest to measure densities for coatings in an
effort to correlate density with x­ray reflectivity.
2 Theoretical Aspects
The technique for determining coating density using Rutherford Backscattering Spectroscopy
(RBS) involves bombardment of the coated sample with ff­particles. The fraction of incident
particles backscattered into a given annular region is directly proportional to the ``areal
density'' j defined by
j =
Z t
0
ae(x) dx = ¯
aet (1)
where ¯
ae is the mean density and t is the coating thickness (see below for a more thorough dis­
cussion of this relationship). Thus, determination of ¯
ae requires an independent measurement
of the coating thickness. For purposes of the AXAF Reflectivity Study, where possible, thick­
ness measurements will be made with a Scanning Transmission Electron Microscope (STEM)
by viewing cross­sectional samples of the coating prepared from silicon wafer chips coated
with the mirrors. Where STEM measurements can not be performed, thickness measure­
ments will be performed by surface profilometry. The thickness measurements are accurate
to ¸20 š A with the STEM and to ¸50­75 š A by profilometry. The uncertainty in the areal
density measurement by RBS is about 1%; hence, for thin coatings, the accuracy to which
the thickness can be measured determines the uncertainty in the final density determination.
It should be noted that this technique yields an average density for a given coating. X­ray
reflectivity measurements are sensitive to densities averaged over the associated skin depth
ffi which is a function of the x­ray energy and grazing angle (see Figure 1):
ffi = ¯ hc
fl
¸
E
(2)
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Figure 1: X­ray skin depth as a function of energy for gold coatings at a grazing angle of 30
arcmin.
where
¸ 2 = 1
2
(
sin 2 ` \Gamma ff +
Ÿ i
sin 2 ` \Gamma ff
j 2
+ fl 2
– 1=2
)
: (3)
Here ff and fl are derived from the complex dielectric constant Ÿ = 1 \Gamma ff \Gamma ifl. Thus, the
densities inferred from the RBS measurements (in conjunction with thickness measurements,
which also represent an averaging over a somewhat ``bumpy'' surface) may not provide the full
information required to predict x­ray reflectivity behavior. Still, the mean density is clearly
a quantity that should be maximized for good overall reflectivity, and thus the measurements
have merit. In addition, the measurements provide an indication of how successfully one can
obtain bulk material densities by thin film techniques.
A typical set­up for RBS measurements is shown in Figure 2. A beam of ff­particles is
accelerated to several MeV, collimated, and directed toward the sample film. A solid state
detector is used to detect the backscattered beam at some angle `. In practice, a sample
holder containing up to 20 samples is placed at the target position; individual samples are
chosen for investigation by properly adjusting the position of the sample holder. Typical
beam sizes, after collimation, are ¸ 1 mm.
The number of particles scattered into a cone bounded by ` and ` + d` is given by
dN(`)
I
=
/
úD 2 N 0
8A
! 2
4
sin ` d`
sin 4
i
`
2
j
3
5 j (4)
2

Figure 2: Schematic diagram of set­up for Rutherford Backscattering Spectroscopy measure­
ments.
where
D = zZe 2
4ú ffl 0 E
: (5)
Here I is the incident intensity of ff­particles (of charge z and energy E), Z and A are the
atomic number and atomic mass of the scatterer, and N 0
is Avogadro's number. Thus, as
noted above, for a fixed target material the fraction of particles scattered into a particular
annulus is proportional to the areal density j.
Given that, in the bombardment of a thin film coating with energetic ff­particles, consid­
erable penetration into deeper layers (such as the undercoating or substrate material) may
occur, it is necessary to identify which scattered particles are associated with the material
under investigation. This may be accomplished by considering the kinematics of the elastic
collisions. The ratio of scattered energy to incident energy for the ff­ particles is dependent
upon the scattering angle ` and the mass m t of the target scatterer:
E sc
E 0
=
2
6
4
i
m 2
t \Gamma m 2
ff sin 2 `
j 1=2
+m ff cos `
m t +m ff
3
7
5
2
: (6)
Thus, for a monoenergetic input beam, the energy of the particles scattered from a given
target material is well defined; energy discrimination can be used to identify and count only
the scattered particles of interest. Of course, both the finite angular size and the finite
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energy resolution of the detector will result in a distribution of energies associated with a
particular scattering material. Furthermore, the finite thickness of the coating layer will
broaden the energy distribution due to dE
dx
losses associated with inelastic collisions between
the ff­particles and electrons; the thicker the coating, the larger the broadening of the energy
spectrum. Separation of adjacent energy peaks in the spectrum may become difficult for
certain combinations of scattering materials. In general, such resolution difficulties can be
circumvented by increasing the energy of the ff­particles.
3 Experimental Technique
The procedure for determining areal density by RBS involves a comparison of the coating
scattering results with those from a sample with a known areal density. Such samples are
produced by ion implant whereby the number of ions deposited on a sample are determined
by a charge sensitive monitor; the implant is confined to a small area and is uniform over
this area so that the areal density may be calculated. The standard sample used for com­
parisons in this study consists of an implant of gold into silicon with an areal density of
2 \Theta 10 15 cm \Gamma2 . A scattering run gated for a fixed number of incident ff­particles (actually,
a fixed integrated charge ­ typically, 50 ¯C) yields the calibration number of counts in a
fixed detector subtending some known fraction of an annular region. Energy spectroscopy of
the ff­particles scattered into this annulus allows discrimination of particles scattered from
different elements (e.g. scattered from the silicon substrate rather than the gold). For each
sample, then, a comparison run is made, gated for the same number of input ions. The ratio
of counts in the gold peak from the sample to that from the calibration source is equal to
the ratio of areal densities:
N
N cal
= j
j cal
: (7)
For typical measurements (e.g. gold/chromium on silicon), 2 MeV ff­particles are used for
the scattering. This allows clear separation of peaks from the silicon substrate, the chromium
undercoat, and the gold (or iridium) coating (see Figure 3). For some samples, this peak
separation is not possible at 2 MeV (e.g. AuPd) and operation at 3 MeV is necessary.
As mentioned above, coating thickness plays a role in the energy distribution of the
scattered ff­particles. Thus, while the high energy side of the distribution is well­defined,
the low energy side is determined by the dE
dx
losses associated with penetration into the
coating. This is illustrated in Figure 4 which shows comparison profiles for two coatings of
different thicknesses. In Figure 4(a), results from a sample coated with ¸ 300 š A of gold
over a thin layer of chromium are plotted. The broad region associated with the substrate is
more complicated than that shown in Figure 3 due to the presence of a silicon­oxide layer on
the substrate. In Figure 4(b), results from a ¸ 700 š A gold coating are shown. Note that the
upper­edge of the gold peak is coincident for the two plots, as expected, but that the lower­
edge is pushed to lower energies for the thicker coating due to penetration losses. Similarly,
the entire chromium peak is shifted toward lower energies for the thicker coating because
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Figure 3: RBS output showing peaks associated with scattering from gold coating, chromium
undercoating, and silicon substrate. Events in the gold peak are used for comparison with a
calibrated sample in order to determine the area density.
complete penetration of the gold layer is required for any scattering off of the chromium
layer. Of course, this effect can be seen in the profile associated with the substrate as well.
Such a thickness effect can, of course, be used to estimate the actual coating thickness; such
estimates require the coating density as input, however, and are thus not appropriate for
cases in which the density is not known. Note that the low energy end of the substrate
profile is coincident for the two samples; this hard cut­off represents an energy threshold for
the detector.
4 Preliminary Results and Conclusions
The scattering results for a number of samples measured in a test run are listed in Table I.
The first column lists the sample mirror associated with the measurement (recall that it is
not the mirror itself measured, but rather a witness sample coated with the mirror). The
second column indicates the coating type (gold, nickel, etc.) while the third column indicates
the nominal thickness of the coating. More accurate results for the thickness will become
available upon completion of the STEM investigation of the samples.
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Figure 4: Scattering profiles for a)¸ 300 š A and b)¸ 700 š A of gold over a thin chromium
layer, atop a silicon substrate (with a thin silicon­oxide layer). Effects of energy loss due to
finite coating thickness can be seen, as explained in text.
Table I. RBS Results for Deposition Effects Samples
Integration Net aet aet t ( š A) ae
Sample Region Counts (atoms cm \Gamma2 ) (gm cm \Gamma2 ) (ff­ step) (g cm \Gamma3 )
CT#1 770­811 8624 2 \Theta 10 15 6:54 \Theta 10 \Gamma7
DE1 754­820 818334 1:90 \Theta 10 17 6:22 \Theta 10 \Gamma5 525 11.85
DE2 750­822 878102 2:04 \Theta 10 17 6:68 \Theta 10 \Gamma5 490 13.63
DE3 746­822 782028 1:81 \Theta 10 17 5:92 \Theta 10 \Gamma5 547 10.82
DE4 759­824 559315 1:30 \Theta 10 17 4:25 \Theta 10 \Gamma5 590 7.20
DE5 759­822 627542 1:45 \Theta 10 17 4:74 \Theta 10 \Gamma5 517 9.17
DE6 759­822 609584 1:41 \Theta 10 17 4:61 \Theta 10 \Gamma5 370 12.46
There are several factors associated with the measurement process which may cause
systematic errors in the density determinations outlined above. Thus, the numbers presented
should be considered very preliminary; further analysis, and quite possibly remeasurement,
will be required to determine realistic values.
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ffl Improper beam alignment can cause only partial illumination of the sample. In this
case, a lower density would be inferred in that the incoming flux would still be counted,
but some fraction would miss the sample thus yielding a smaller number of scattered
particles than expected. Such an effect can generally be identified in that the particle
beam leaves a slight mark on the sample. Thus, it can be determined whether or not
the beam was striking the sample close to an edge.
ffl Normalization difficulties occur if the beam monitor is not used. This did occur for
some of the acquired data because the monitor was being repaired. Typically, one
merely times successive runs and normalizes the data based upon a comparison of
the counts in the substrate region of the profile. However, since different coating
thicknesses can push this distribution toward lower energies, and since the lower end
cut­off is instrument related (i.e. it is the same regardless of the coating thickness) the
actual number of counts in the two regions should not be identical for identical input
beam fluxes. Samples obtained while operating in this mode will be remeasured.
ffl An overall uncertainty exists which is associated with the uncertainty in the areal
density of the calibration sample. This uncertainty is not known for the current sample;
procurement of a calibration sample for the AXAF reflectivity is desirable so that
comparisons can be made with a sample for which documentation can be obtained.
ffl Measurements made on coatings other than gold are difficult to assess. Since only a
gold calibration sample exists, the areal density associated with samples other than
gold must be inferred based upon the ratio of densities of the various materials to that
of gold. However, since it is the densities that are being investigated, this process
contains an ambiguity. The disirable mode of operation would be to obtain calibration
standards for each of the coating materials under investigation.
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