Nuclear Instruments and Methods in Physics Research B 207 (2003) 275–282
www.elsevier.com/locate/nimb
Doppler broadened c-lines from exotic nuclei
H.O.U. Fynbo
Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark
Received 11 March 2002; received in revised form 9 January 2003
Abstract
A method for calculating Doppler broadened peak shapes relevant for a number of different physical phenomena is
presented. Peak shapes for different lifetimes and using different evaluations of the stopping power are given. A new
example of the occurrence of Doppler broadened c-lines in the decay spectra of exotic atomic nuclei is presented and it
is discussed how information may be obtained from an analysis of the line-shape.
Ó 2003 Elsevier Science B.V. All rights reserved.
PACS: 21.10.Tg; 23.40.Bw; 29.30.Kv
Keywords: Doppler broadening; Peak shapes; Stopping power; Lifetimes
1. Introduction
The Doppler effect is employed in several techniques for determining lifetimes of excited nuclear
states. In most of these techniques the excited state
of the nucleus is created in a nuclear reaction
employing a unidirectional beam of at least several
MeV and consequently the Doppler effect will
cause a shift of the emitted c-ray. This is the case
for the recoil distance method (RDM), the
Doppler shift attenuation method (DSAM) and
the Doppler broadened line shape analysis method
(DBLA), see [1] for a review of these techniques.
We have developed a method for using the
Doppler effect for c-rays emitted after b-delayed
neutron emission in the b-decay of 11 Li. In this
E-mail addresses: [email protected], [email protected]
(H.O.U. Fynbo).
URL: http://www.phys.au.dk/~fynbo.
case the recoil direction after neutron emission is
isotropic and hence the Doppler effect induces a
broadening of the c-peak in contradistinction to
the usual shift. The size of this Doppler broadening is directly related to the energy of the emitted
neutron and consequently by performing a detailed fit of the peak shape, information may be
extracted which would otherwise require a b–c–n
triple coincidence experiment [2–4].
We have realized that the formulas derived for
this method may be of interest for a larger class of
phenomena where similar Doppler broadening
occurs. In the inverted Doppler shift attenuation
method (IDSA), which is used to determine stopping powers [5–8], a thermal neutron induced reaction 10 B(n; a)7 Li is used to produce a c-decaying
state in 7 Li and the analysis of the Doppler
broadened line shape is then used to extract information about the stopping power. These authors give very good illustrations of the physics
0168-583X/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0168-583X(03)00570-6
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H.O.U. Fynbo / Nucl. Instr. and Meth. in Phys. Res. B 207 (2003) 275–282
behind the shape of Doppler broadened peaks but
give no expression for the shape. A completely
different application was suggested by Grenacs
et al. [9] in connection with l-capture. c-rays
emitted after l-capture are Doppler broadened
due to the emission of a neutrino with typically 50
MeV energy. If the lifetime of the c-emitting state
is sufficiently short to neglect the slowing down of
the recoiling nucleus, the shape of the Doppler
broadened peak is directly related to the neutrinogamma angular correlation, which is of fundamental interest. Later Pratt [10] gave some
approximations for how to take into account the
energy loss of the recoiling ions. There has recently
been some very interesting applications of this
method [11–14], but none of these authors describes how the energy loss is accounted for. Finally, in the Gamma Ray Induced Doppler Effect
method (GRID) the tiny recoil from the emission
of the first c-ray in a c–c cascade induces a
Doppler broadening of the peak shape of the
second c-ray [15]. Although the energy scale is very
different for this method some of the relevant
formulas are similar.
Here we give our results for the shape of such
Doppler broadened c-peaks to remedy the fact
that, to our knowledge, no consistent derivation is
available in the literature. In Section 2 we present
our derivation of the peak shape of Doppler
broadened c-peaks and in Section 3 we discuss
different treatments of the stopping power and give
some useful analytical results. In Section 4 we
discuss the qualitative features of the peak shape
and its dependence on the main determining parameters and finally in Section 5 we discuss the
occurrence and application of Doppler broadened
c-lines in the decay spectra of exotic nuclei.
2. Derivation of Doppler broadened c-peak shapes
We consider the general case of a nucleus generated in some medium in a c-emitting excited
state and with a velocity (recoil) distribution which
is isotropic in the laboratory system.
It is instructive to consider first the case where
energy loss to the medium is neglected. In this case
the nuclei move through the medium with constant
energy Ei and the line shape dn=dEc is determined
as follows:
dn
dn
dX
ðEc Þ
ðEc Þ ¼
;
dEc
dX
dEc
ð1Þ
where Ec is the energy of the c-ray. The dependence of the angular distribution on the energy of
the photon arises from the form of the Doppler
shift
cos h ¼
Ec Ec0
;
Ec0 bi
ð2Þ
where bi ¼ vi =c is the velocity corresponding to Ei .
Using this we get
dX ¼ 2p sin h dh ¼ 2p dx;
x ¼ cos h
Ec ¼ Ec0 ð1 þ cos hbi Þ ¼ Ec0 ð1 þ xbi Þ;
thus
dEc Ec0 bi
¼
dX
2p
ð3Þ
and hence
dn
2p dn
ðEc Þ:
¼
dEc Ec0 bi dX
ð4Þ
The angular distribution should be normalized
as
Z
dX
4p
dn
¼ Nc ;
dX
ð5Þ
where Nc is the total number of emitted c-rays.
Thus, neglecting energy loss, the line shape simply
reflects the angular correlation between the c-ray
and the recoil via the transformation between cray energy and angle Eq. (2). The line shape extends between Ec0 ð1 bi Þ where bi is obtained
from the kinematics of the fundamental process
behind the Doppler broadening, either l-capture,
particle-emission or something else. In the case of
an isotropic angular correlation the line shape is
simply a rectangle. Results similar to Eq. (4) were
reached in [5,9] (Fig. 2) and [15] (Eq. (8)).
We now take the effect of energy loss to the
medium into account. In this case the recoils will
slow down and eventually be stopped, where the
energy loss can be described by the function dE=
dx. c-rays emitted at time t will give a distribution between Ecmin ¼ Ec0 ð1 bðtÞÞ and Ecmax ¼
H.O.U. Fynbo / Nucl. Instr. and Meth. in Phys. Res. B 207 (2003) 275–282
Ec0 ð1 þ bðtÞÞ described by Eq. (4). The contribution to the full shape at a given energy, Ec , is then
given by c-rays emitted between t ¼ 0 and t ¼ t0 ,
where t0 is determined by Ec ¼ Ec0 ð1 bðt0 ÞÞ. By
analogy with Eq. (4) this amounts to the integral
Z t0
dn
2p dn
ðEc Þ;
ðEc Þ ¼
dt
ð6Þ
dEc
E
c0 bðtÞ dX
0
where the normalization of the angular distribution is now
Z
dn
¼ AðtÞc dt:
dX
ð7Þ
dX
4p
The velocity dependence on time is readily obtained from the stopping power by integration
Z b
Z bi
db0
db0
tðbÞ ¼ mc
¼ mc
;
ð8Þ
ðdE=dxÞ
b
bi dE=dx
where ðdE=dxÞ is positive. Thus dt=db and tðbÞ
are in principle known functions. The discussion of
how to treat the required ðdE=dxÞ and how to
calculate the integral Eq. (8) from it, is deferred to
Section 3.
If the recoils come to rest at time tr , c-rays
emitted after this time will not be Doppler broadened and must be included as a d-function distribution normalized to represent the total remaining
activity
Z 1
dn
¼ dðEc ¼ Ec0 Þ
AðtÞ dt:
ð9Þ
dEcd
tr
For this we need the time tr determined by the
condition b ¼ 0. This is easily obtained from the
integral Eq. (8).
The activity AðtÞ can readily be determined given
the decay scheme of the system under consideration. Alternatively, parts of unknown decay
schemes may be determined from an analysis of the
peak shape, as for the case of the decay of 11 Li [3].
In order to compare the derived line shapes with
measured spectra, it is necessary to perform a
convolution with the response function of the detector.
In summary, the line shape depends on the recoil energy, the time dependence of the c-ray
emission, the angular correlation between the recoil and the c-ray and on the stopping power of
the medium. In the calculation above we have
277
neglected the change of direction of the recoils
from the nuclear stopping. In some DBLA applications this has been included by applying the
Blaugrund correction [16]. The effect of recoil
scattering for isotropic Doppler broadening is to
wash out angular correlations. For uniform angular correlations the line shape is unaffected,
whereas for nonuniform correlations the line shape
is changed towards the uniform case. The effect is
largest for low recoil energies (b < 1%) where nuclear stopping dominates. Hence, we expect to be
most sensitive to this effect when the timescale for
c-emission is long compared to the stopping time
and most c-rays are emitted at low recoil energy.
3. Treatment of stopping power
There is a long and ongoing interest in the calculation of energy loss of ions in matter which has
resulted in various parameterizations for the
stopping power which are amenable to practical
applications.
Pratt [10] gives some analytical results for
Doppler broadened peak shapes for two parameterizations of the stopping power dE=dx ¼ Mb=a1
and dE=dx ¼ ðkn b0 =bÞ þ ðke b=b0 Þ with b ¼ v=c the
velocity of the recoiling ion and b0 the Bohrvelocity. These expressions are only valid when
the electronic stopping power dominates and it is
therefore necessary to extend this work.
We consider first a direct extension of the parameterizations used by Pratt,
dE
¼ qf ðvÞ;
dx
3
b
b0
b
f ðvÞ ¼ kn þ ke k3
;
b0
b0
b
ð10Þ
with q the density of the stopping material [17,18].
In the region of validity of Eq. (10), we can determine an analytical expression for the time tðbÞ,
/ðbi =b0 Þ
;
/ðb=b0 Þ
ke 2k3 x2 K
/ðxÞ ¼
;
ke 2k3 x2 þ K
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mb0 c
K ¼ ke2 þ 4k3 kn ; T ¼
;
2qK
tðbÞ ¼ T log
ð11Þ
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H.O.U. Fynbo / Nucl. Instr. and Meth. in Phys. Res. B 207 (2003) 275–282
Fig. 1. The left part shows the stopping power for the stopping of 28 Al in 28 Si estimated with the LSS formalism (dashed), the SRIM
code (full curve), binary theory (dot-dashed curve) and from Eq. (10) (dotted curve). The electronic and nuclear contributions are
shown individually together with their sum (binary theory only gives the electronic stopping power). The right part shows the relation
between recoil energy and time for an initial recoil energy near 200 keV relevant for l-capture studies calculated from Eq. (8). Using the
LSS formalism and Eq. (10) this time is calculated analytically, whereas using SRIM it must be calculated numerically. In the binary
theory calculation the nuclear stopping power is calculated from the LSS formalism.
with T giving the time scale for the slowing down
process, typically of the order 100–1000 fs.
Alternatively one can describe the energy loss in
the Lindhard, Scharff and Shiøtt (LSS) formalism,
which is better adapted to the low energy region
near the maximum of the nuclear stopping power
[11,19], 1
dE
Zi Zm Ai
d
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 5:102 dx
dq
2=3
2=3
Am ðAi þ Am Þ Zi þ Zm
d
d
d
¼
þ
dq
dq n
dq e
pffiffi
fn
¼ fe k þ
;
/n 0:67 þ 2:07 þ 0:032
ð12Þ
where Ai , Am , Zi , Zm are the atomic number and
nuclear charge of ions and media, respectively, fe ,
1
Note, there are two misprints in the appendix of [13]. In the
expression for in Eq. (A.2) Ai should be Am and in Eq. (A.7) Ai
and Am should be interchanged. For l-capture applications
these errors have no effect as Ai ¼ Am .
fn , /n are correction factors which can be adjusted
in a fit to experimental values and is a dimensionless quantity proportional to the kinetic energy of the ion. The integral in Eq. (8) with the
LSS stopping power can also be solved analytically
for tðbÞ giving a somewhat more complicated
formula which will not be reproduced here.
Various codes exist which provide stopping
powers as numerical tables, the most widely used
being SRIM [20]. If a numerical table for the
stopping power is used, the integral Eq. (8) must
be performed numerically as well.
A new treatment of the electronic stopping
power was recently presented in the so called binary theory [21] where binary scattering theory is
applied for the interaction between the incoming
ion and electrons of the stopping material. The
advantage in this approach is that it applies to
collisions with all impact parameters and that the
calculation is nonperturbative. This means that
the calculation should be more accurate than the
SRIM and LSS evaluations.
For illustration we show in the left part of Fig. 1
the stopping power for the case of stopping of 28 Al
H.O.U. Fynbo / Nucl. Instr. and Meth. in Phys. Res. B 207 (2003) 275–282
ions in 28 Si with the stopping power estimated from
LSS, SRIM, binary theory and Eq. (10), where for
the latter the parameters were determined in a fit to
the SRIM values. This case is relevant for l-capture on a Si target where 28 Al is produced with an
initial recoil energy of 183 keV. In recent studies of
this case, both LSS [11,13] and SRIM [12] have
been used for the stopping power. There is a significant difference in the magnitude of the stopping
power between the SRIM and LSS estimates, while
Eq. (10) completely fails below 100 keV. The supposedly more accurate binary theory calculation
favours the LSS estimate of the electronic stopping
power. The right part of Fig. 1 shows the time as a
function of energy Eq. (8) beginning with the full
recoil energy at time zero and ending with the ions
at rest 500–700 fs later, depending on the assumed
stopping power. The difference between the different estimates is small, but it can have an important
effect on the peak shape, see Section 4. The time
calculated with Eq. (11) again differs markedly
below 50 keV. In the binary theory based calculation the nuclear stopping power was estimated
from the LSS formalism.
4. Peak shapes
To get a qualitative understanding of Eq. (6) we
show in Fig. 2 the peak shape calculated for lifetimes of 50, 500 and 5000 fs for the case of the
2170 keV line from the deexcitation of the 2201.5
keV level in 28 Al fed in l-capture on 28 Si. For
simplicity we assume the angular correlations to be
isotropic, as is often the case. The detector response function has been included through a
Gaussian convolution with a resolution of 0.5 keV
(sigma). For each lifetime the peak shape is calculated using stopping powers from SRIM, LSS,
Eq. (10) and binary theory, where for the latter we
use LSS to estimate the nuclear stopping power.
The difference between the SRIM and LSS calculations is small, but it would give significant differences if this was used to determine lifetimes in a
fit, e.g. by changing the lifetimes in the LSS calculated peak shapes to 40 fs and 440 fs perfect
agreement is found with the SRIM calculated
shapes at 50 and 500 fs, respectively. This indicates
279
Fig. 2. Line shapes calculated for lifetimes of 50, 500 and 5000
fs for the case of the 2170 keV line from the deexcitation of the
2201.5 keV level in 28 Al fed in l-capture on 28 Si. We show results using stopping powers from the LSS formalism (dashed
curve), SRIM code (full curve), binary theory (dot-dashed) and
Eq. (10) (dotted curve). For the binary theory calculation the
nuclear stopping power is estimated from the LSS formalism.
the size of the systematic error introduced by the
uncertainty in the magnitude of the stopping
power which is slightly larger than that estimated
in [13]. The peak shapes for 50 fs lifetime calculated with SRIM and LSS stopping powers are in
good agreement with those given in [11–13].
As abscissa we use both the c-ray energy and the
Doppler velocity b ¼ ððEc Ec0 Þ=Ec0 Þ, the latter
allowing a direct reading of the magnitude of the
initial recoil energy in cases where this is unknown,
as demonstrated in Section 5. In all cases the line
shape extends roughly as expected between bi
and þbi . The effect of the detector resolution is to
expand the range slightly beyond these points.
When the lifetime is much smaller than the stopping time, the line shape is close to the no-stopping
case of a rectangle (in the isotropic angular correlation case), whereas when the lifetime is much
larger than the stopping time scale, the line shape
approaches the expected Gaussian shape (assumed
response function of the detector).
5. Application to the b-decay of
11
Li
Doppler broadened c-lines have been found in
the decay spectra of exotic nuclei near the neutron
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H.O.U. Fynbo / Nucl. Instr. and Meth. in Phys. Res. B 207 (2003) 275–282
drip line [22]. In these cases the b-decay feeds neutron unbound states in the daughter which, after
neutron emission, can feed excited states in the bndaughter. The c-lines from the deexcitation of these
states are Doppler-broadened by the recoil from
the neutron emission and the magnitude of the effect is directly related to the energy of the neutron.
Hence, by analysing the Doppler broadened peak
shapes, information may be obtained about the
energy of the neutron and thereby indirectly which
state in the b-daughter was involved [2,3].
In the b-decay of 11 Li, 30% of all decays feed
excited states in the bn-daughter 10 Be and it is
therefore essential for correctly reconstructing the
decay to understand which of the neutrons in the
observed neutron spectrum leads to the corresponding states in 10 Be. Two recent attempts on
disentangling this decay reached completely different assignments for this significant part of the
decay [23,24]. In [23] the 5958.3 keV 2þ state in
10
Be is assumed to be fed by neutron emission
from two known states at 6.51 and 6.7 MeV in
11
Be, whereas [24] assumes feeding from a new
state at 8 MeV.
In the following we use the c-spectrum from the
b-decay of 11 Li from an experiment performed at
the ISOLDE facility at CERN in 1994, the second
of the two experiments referred to in [22], where
also experimental details on the setup can be found
(the 11 Li beam was stopped in an Al stopper). In
Fig. 3 we show the c-line from the deexcitation of
the 5958.3 keV state which clearly can be seen to
be Doppler broadened with bi ¼ 6 7 103 . The
background peak seen at 2615 keV most likely
comes from 212 Po produced in the decay chain of
224
Ra produced in the previous ISOLDE run. The
high background level under the Doppler broadened peak at 2590 keV results from higher energy
lines from the b-decay of 11 Li. Background analysis is of high importance in DBLA studies since
small background contributions may affect the
results of line shape fits, see [1] for a good discussion. A detailed background analysis was not
performed in the present case since the possible use
Fig. 3. Doppler broadened c-line from the deexcitation of the 5958.3 keV state in 10 Be fed in the b-decay of 11 Li. The analysis of the
line-shape allows to deduce that this state is fed by neutron emission of a state in the b-daughter 11 Be at 8–9 MeV. The line at 2615 keV
is a contamination. The inset shows the relevant states in 10 Be. Note the suppressed zero on the ordinate to emphasize the Doppler
broadened part of the data.
H.O.U. Fynbo / Nucl. Instr. and Meth. in Phys. Res. B 207 (2003) 275–282
of DBLA techniques was not realized until after
the experiment.
The 5958 keV state can be fed directly via neutron emission from a state in 11 Be and in cascade
from higher lying states in 10 Be. From the positions of the corresponding c-lines limits can be
placed on cascade contributions which justify them
to be neglected. For the 3368 keV c-line there are
both cascade and direct contributions [2,3], see
also [25,26] for examples of the difficulties associated with the feeding of Doppler broadened clines. Since for the present case the stopping
powers estimated with LSS and SRIM agree
within 10% we only show results using SRIM (note
the binary theory estimate is 25% larger than LSS
and SRIM [21]). Initially the free parameters in the
v2 fit were the lifetime of the 5958.3 keV state, the
position of the feeding level in 11 Be and the angular correlation parameter A2 . The latter was restricted to be between 7=20 as determined from
[27]. The fit revealed a strong correlation between
the lifetime and A2 with a near linear dependence
such that the lifetime was ’60 fs for A2 ¼ 7=20
and ’180 fs for A2 ¼ þ7=20. The corresponding range for the fitted position of the feeding level
in 11 Be was 8.6–9.1 MeV, with v2 between 32.6 and
34 for 44 degrees of freedom. The curve shown
in Fig. 3 is the result of a fit with the initial recoil energy of the excited 10 Be fixed at 215 keV
which corresponds to neutron emission from the
known 8.816 MeV state in 11 Be. The lifetime of
the 5958.3 keV state obtained from the fit was
61(18) fs and the angular correlation parameter was at its minimum value A2 ¼ 0:35. This
fitted value of the lifetime is consistent with the
known upper limit of 80 fs [28]. The angular correlation is responsible for giving the peak the more
rounded shape compared to the curves shown in
Fig. 2.
The fit is consistent with [24], whereas the decay
scheme suggested in [23] would give a too low recoil energy and can be excluded: the initial
Doppler velocity corresponding to the 8 MeV 11 Be
state suggested in [24] is 5:5 103 c and it is
9:8 104 c and 2:2 103 c for the 6.51 and 6.7
MeV states suggested in [23]. A detailed background treatment is not estimated to change this
conclusion significantly.
281
6. Summary and outlook
We have presented a general derivation of
Doppler broadened c-lines which takes into account energy loss to the stopping medium, angular
correlations and the statistical deexcitation process
of the c-ray emission. Essential for the calculation
of the peak shape is the availability of the relevant
stopping power. We discuss various estimates of
this quantity and illustrate how the uncertainty in
its magnitude affects the calculated peak shapes.
Our analytical results extend on previous work
presented in this journal [9,10] for isotropic
Doppler broadening. Finally we introduce a new
example of the occurrence of Doppler broadened
c-lines in the decay spectra of exotic nuclei and
illustrate how information may be obtained from
an analysis of the line shape.
For each of the He, Li, Be and B isotopic chains
examples can already now be found for Doppler
broadened c-lines [29] (8 He), [22] (11 Li), [30] (14 Be),
[31] (17 B). The number of isotopes produced with
sufficient yield for decay spectroscopy to be possible is currently increasing rapidly. It is therefore
likely that the method presented here could be
applicable for a large range of isotopes close to the
neutron drip-line where delayed neutron emission
is frequent. Eventually the limit will be reached
where the mass of the bn-daughter is so high that
the Doppler broadening is too small to be measured.
Acknowledgements
The author gratefully acknowledges K. Riisager
for suggesting this problem and for useful discussions and comments on the manuscript, P. Sigmund for kindly providing his predictions from
binary theory and S. Egorov for discussions on the
use of Doppler-broadening in l-capture.
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