CALCULATION OF ULTRASOUND EXCITED BY A PULSED
THERMAL SOURCE DISTRIBUTED ALONG THE DEPTH
DIRECTION
J. He and X. R. Zhang
State Key Lab. of Modern Acoustics and Institute of Acoustics, Nanjing University,
Nanjing 210093
Y. Y. Huang and D. C. Xian
Synchrotron Radiation Lab., Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing, China
ABSTRACT. When an intensive pulsed laser or ion beam impinging a material surface gradually
loses its energy and heating the material along the passing path of the beams, a pulsed ultrasonic wave
can be excited due to a thermal source distributed along the depth direction. We deduce an expression
and calculate the epicenter displacement of ultrasound. Investigation results show that: the waveform
of displacement depends on the distribution depth, the type of thermal source, the properties of
samples, and the ratio between the thickness of the absorption layer and the duration of excitation
beam. The detail process, results and discussions are presented.
INTRODUCTION
It is well known that the ultrasonic pulses can be excited by thermal sources
respectively caused by particles beams, such as the beams of photons, electrons, protons,
and ions. Scientists have investigated the acoustic radiation induced by intensity
modulated beam of H + ions and Ar+ [1, 2], time-resolved elastic waves generated by
modulated ion beams [3]. Lyamshev [4] reviewed the works studied before 1992. Teichert
et al. [5] reported the evidence for acoustic waves induced by focused ion beams. Recently
people paid their attention on the ultrasonic pulse generated by fast intensity heavy ion
beam [6-9]. They used 3xl010 Uranium ion beam at energy of 50 MeV/u to 200 MeV/u,
2xlOn 40Ar18+ with duration of 50 ns about 50-100 kg/g specific energy and 1-2 Tw/g
specific power in solid peak, and Xe+ ion 20Mev/u etc., as excitation sources. Al, Cu, lead
or PMM A used as targets. Kambara et al. reported that the waveform of the pulse does not
depend on the material and thickness of the sample much, but depends more on the
position of the irradiation. Deemer reported the time-resolved acoustic waveforms during
implantation are similar to those generated by a laser in the thermoelastic generation
regime [10].
However, as we know that the interaction between laser beam with solid take place in
the surface of the solid, i.e., the surface electrons absorb the photon's energy when a laser
beam impinge a solid surface, except for the weak absorption material. The ion beam can
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
289
penetrate into solid with a penetrating depth and associated damage profiles. So, we must
consider the mechanism of the ultrasonic pulse generated by pulsed fast ion beam as a
thermoelastic source distributed along the depth direction. We regard the ion excitation
source as a thermal source distributed along the ion beam-impinging path. Thus the pulsed
ultrasonic waveform is excited by a thermal source distributed along the depth direction.
The depth distribution of the ion, associated damage profiles, determines the distribution
of the thermal source. We divided the depth into several parts, each parts as a sublayer
source, and deduced an expression to calculate the epicenter displacement of ultrasound by
weight addition of the contributions coming from the sublayers. The results show that the
waveform depends on the material properties. The detail process, results and discussions
are presented in this paper.
THE CALCULATION OF THE PROJECTED RANGES RP OF HEAVY IONS
The ratio of energy loss of ion (i.e., stopping range) is given by
^_^V N z B
dx
(1)
m0v
where E is the instantaneous energy of the particle, e is the electron charge, ze the particle
charge, x the distance along the track of the ions, m0 the mass of static ion, z the number of
atoms of absorption medium, N the atomic number absorbed by medium per unit
number/cm3,1 the potential energy of ion for absorbed atom, B= (In 2mov2 - Inl (l-ji2)-|32),
(5 the relative velocity of light. We calculate the energy E and energy loss ratio dE/dx of
ions as the function of the implanting depth. The dE/dx can result thermal source in target.
We calculate the stopping range (depth distribution of the energy loss) of 12C+, 40Ar+,
136
+ 238 +
Xe , U ions etc. implant into Al, Si, PMMA and water by using TRIM96
calculations respectively. Some calculation results are shown in Figure 1 and Figure 2 as
examples.
From Figur.l, we can see that the distribution of the energy loss along the
penetrating track of the ions depends on the kind of the ion and the properties of targets.
From Figure 2, we see that the penetrating depth for water is larger that that for metals.
THE CALCULATION OF THE WAVEFORMS
The Velocity of the Ion
Generally, the velocity of heavy ion depends on the energy of the ion and given by
Vion=cjl——————————5-2
(2)
(l + E/931.19)
290
20000-1
140012 +
C implant Al j
12001000-
(a)
i
800600- ^
400/
• 50 Mev /'
200- /
„.'
0-^"
0
2000
600 Mev
15000-
\ /
Xe implant Al
........ ./600 Mev __-— - N
-\-""^""""*
~\
\.s 240 Mev
1000 Mev ^ 10000-
!
/
(b)
5000*1 \
6000
'-:
>j
i l l
0-
A000
2000Me>
* \S
0
8000 10000 12000
20
40
!
60
80 100 120 140 160
Distance from enter points (^irn)
Distance from enter points (jim)
120003000025000-
10000-
"^\t
-.,.. ^Mev
2000015000-
"^-,N (c)
X
\
8000-
2000 Mev
U implant PMMA
130 Mev
6000-
\f
A 220 Mev \
4000-
10000-
\\
\
5000-
13^Mevl |
0'
0
20
2000!
i
:
U implant Al '
40
60
80
Distance from enter points (jam)
Xe implant PMMA
70 Mev
0-
-10
100
-5
0
5
10
15
20
25
30
Distance from enter points (um)
FIGURE 1. The power losses of ion beam versus the distance from enter points of the beam, (a) for C+ ion
beam implanting into Al, (b) Xe+ ion beam implanting into Al, (c) for IT1" ion beam implanting into Al, and
(d) for U1" ion beam implanting into PMMA, respectively.
lm
200M
ZUUMev
~400Mev
400 Mev
600 Mev
1000- iooMev
' ^ 300 Mev /
500 Mev
800-
^
600-
/
400- 50 Mev
200-
•*^ •••**' i ——^
C implant water
-200 -|——,——,——,——,——,——j——,——,——,——,
-2.0xl03 0.0
2.0xl03 4.0xl03 6.0xl03 S.OxlO3
diatance from enter point (|im)
FIGURE 2. The energy losses of ion beam versus the distance from enter points of the beam, for C+ ion
beam implanting into water.
where c is the velocity of the photons (light velocity) in vacuum, c=2.998 108 m/s. E is the
energy of ion and the unit of E is MeV/amu (or Me/u). For example, as the 12C+ ion with
energy of 50 MeV/amu (Total energy is 600 MeV), the ion velocity is about 9.448xl07 m/s,
If the energy is IMeV (total energy 12 MeV), vion:r1.336xl07 m/s. It is faster than the
velocity of ultrasound in metals and water. It can be assume that the thermal source in
sublayer caused after ion arrived.
291
The Thermal Model
We assume that when an intensive ion beam impinging a material surface, gradually
loses its energy and heating the material along the passing path of the beams, a pulsed
ultrasonic wave can be excited by dividing the distribution layer into many sublayers (sub
sources), and considering the waves excited at different positions can only be reflected
when it arrived the surface of sample, as a first approximation. FigureS shows a transient
distribution thermal source consisted by several sub-sources, each have different sonic
transit time depending on the calculation results of the energy deposition points by using
strim96 software.
We deduce an epicenter displacement gn (hn+dn; dn; t) expression for nth sublayer
source, based on the laser ultrasonic pulse generated due to a surface modification by
transparent overlay model [11], as follows
d^2
_ dFf
dt
'
n
Fin(t) = — Ab2h~1H(t-t1)(a|39t)
dF^
n
dt
'
2 2__
1/2
'
n
dt
> n » n
(3a)
F2n(t) = —Ab 2 (h n +2d n )~ 1 H(t-t 2 )[ocy(y 2 +4 C 2 «P)^ 2 ] c=a(t 2 /t 2_ 1) i/2
(3b)
F 3 n (t)-4Ab 2 H(t-t 3 ){y(Ca(39t) 2 /[a(h n +d n ) + p]}c=f(t)
(3c)
where a=l/cL, b=l/cr,
K = ^- , A = — , cc = (a 2 +£ 2 ) 1 / 2 , (3 = (b2
y = (b2 + 2£ 2 ), and H is the step function.
5R = (y2 - 4£ 2 <xp)~ l , 11 = ahn, t2 = a(hn + 2dn),
C = f(t) is the inverse function of
t3 = adn+ b (hn +dn)
t = (a2 + £ 2 ) 1 / 2 d n + (b2 +£ 2 ) 1 / 2 (h n +d n )
Then we calculate the epicenter displacement of ultrasound by weight addition of the
contributions coming from the sublayers. The total displacement is given by
G = F(t)ic n g n ,
(4)
n=l
where C n = (—)n is the weight factor, F(t) is the laser profile in time domain,
ax
292
140012001000X 800jg 6004002000-
2n
7
—— c mplantAl
100 Mev
2n-l/
n
____—-—-•
^
7
1
0
0
50
100
150
A
Distance from enter points (jxm)
FIGURE 3. Schematic diagram of a distribution thermal source, where point 0 is the surface of target.
The Calculation Results and Discussions
The primary calculation results of waveforms excited by a pulsed thermal source
distributed along the depth direction is shown in Figure 4. In calculation, the function F (t)
is Gaussian. In the Figure 4 the symbol T denotes the duration of the pulsed ion beam.
From Figure 4, we can see that the waveform depends on the duration of the C+ ion beam.
Figure 5 and Figure 6(a) show the waveform depends on the duration T of the U+ and Ar+
ion beams respectively. By comparing the Figure 5 with Figure 4, it can be see that the
precursory pulse generated by U+beam is narrower than that generated by C+ ion beam,
when the target material and the duration of the ion beams are the same. It is due to the U+
ion is heavier than the C+ ion. We can say that the heavier the ions, the narrower the
precursory pulse is generated. Figure 6 (b) shows the waveform depends on the kind of the
ion beam, when the other conditions are the same. From Figure 6(b), it can clearly know
that the heavier the ion, the narrower the duration of the ionsound pulse is. For comparison,
we calculate the displacement waveform generated by the distribution thermal source and
—— i=100ns, o i- 150ns
----- T = 200ns,—— i =
= 1000ns
1000
2000
3000
4000
5000
1540 1560 1580 1600 1620 1640
Time(ns)
Time (ns)
FIGURE 4. The waveforms excited by pulsed C+ implanting with energy 50 MeV and six duration into
target Al with thickness of 10 mm, (a) shows waveform including L- and T-waves, (b) shows L- pulse only
for clearly seen.
293
that generated by single buried thermal source too. We assume that the single thermal
source is located at the position where the value of the dE/dx is maximum. The calculation
result is shown in Figure 7.
(a)
•a
0.0
_
1.5-1
——t=l 00ns
o T=150ns
——-x=200ns
........ T -500 ns
A <r=700ns
—
T=1000ns
(b)
1.0-
A
0.5-
J*L
0.0-
U implant Al
130 Mev, 10mm
-0.5-
U implant Al
130 Mev, 10 mm
"-1.0-
.1 0-
1000 2000 3000 4000 5000
0
— — - T =200 ns
........ T -500 ns
A i=700ns
---•• T=1000ns
1550
1580 1590
1560 1570
Time (ns)
1600
Time (ns)
FIGURE 5 The waveforms excited by pulsed U1" implanting with energy 130 MeV and six duration into
target Al with thickness of 10 mm, (a) shows waveform including L- and T-waves, (b) shows L- pulse only
for clearly seen.
1.51*1.0-
——— T=100ns
o T =150 ns
— —-1=200 ns
........ T =500 ns
* t=700ns
• • • - • • T =1000 ns
(a)
A
lo.5,
2.5-
2.01.51.00.500-0.5-1.0^
-1.5-
| 0.0§-0.5- Xe implant Al
70 Mev, 10mm
-1.0- ———•———.————•———i———•———,
1550
1560
1570
- - - Xe+ implanting PMMA
——— U+ implanting PMMA
—©— C+ implanting PMMA
'b'
oo°o0oo
o
O
1
° A°OO
°0oo0o°
-2.0-
1580
800
1000
1200 1400 1600 1800
Tims (ns)
Time (ns)
FIGURE 6 (a) The waveforms of L-pulse excited by pulsed Xe+with energy 130 MeV and six duration
implanting into target Al with thickness of 10 mm, (b) shows the comparison of the L- waveforms between it
is excited by pulsed Xe+, U4", and C+ beams implanting into PMMA.
——<f imlplant into Al
lOOMeV, 8ns, h=2 mm
§ 0.5^0.01-0.5———12C+ imlplant into Al
100MeV,8ns,h=2mm
one layer
-1.5-2.0
200
400
600
800
Time (ns)
FIGURE 7 The comparison of the waveform between it is excited by distribution source and by single
thermal source.
294
From Figure 7, it can be seen that the waveform generated by one thermal source is
different with that generated by distribution source. By comparing the Figure 4 to Figure 7
each other, it is obvious that the waveform also depends on the properties of the target
material.
CONCLUSIONS
As the above mentioned, some conclusion can be drawn: The expression for
calculation of epicenter displacement of ultrasound generated distribution thermal source
is deduced. The waveform is calculated by weight addition of the contributions coming
from the sublayer sources. Investigation results show that: The waveform of displacement
depends on the distribution depth and type of thermal source, the properties of samples,
and the ratio between the thickness of the absorption layer and the duration of excitation
beam. The shorter the thermal source, the duration of ultrasound is the narrower, when the
temperature increase and the sample thickness are fixed. The double polarity of the
waveform decreases with the increasing the duration of the thermal source even change
into a single polarity.
ACKNOWLEDGMENTS
The authors thank the National Science Foundation of China No. 19875062 and the
State Key Laboratory of Modern Acoustics Nanjing University, Nanjing, for financial
support.
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