Supporting information
Thermal radiation and fragmentation pathways of
photo-excited silicon clusters
Piero Ferrari, Ewald Janssens, Peter Lievens
Laboratory of Solid State Physics and Magnetism
KU Leuven, 3001 Leuven, Belgium
Klavs Hansen
Department of Physics, University of Gothenburg,
41296 Gothenburg, Sweden
and
Department of Physics and Astronomy,
Aarhus University, 8000 Aarhus C, Denmark
Content
A) Equations to calculate the flight time of delayed fragments
B) Assignments of the fragmentations pathways of Sin+ (n = 5–13, 15–19, 21)
1
A)
Equations to calculate the flight time of delayed fragments
The fragmentation pathways are assigned by comparison of the α dependence of the measured
flight times for different parent and fragments sizes, with  the scaling factor of the voltages on
the reflectron electrodes of the time-of-flight mass spectrometer and  = 1 corresponding to
optimal mass resolution. Modeled flight times originate from solving the equation of motion of
the clusters. A schematic representation of the system is given in Fig. S1 (note that this scheme
contains more details than that in Fig. 1).
Figure S1: Detailed schematic representation of the experimental procedure used to
assign the decay paths of cationic silicon clusters. Velocities along the flight path are
represented in red (note that the length of the arrows do not represent the speed).
Most important lengths (shown by black arrows) are L1 = 21 mm, L2 = 10 mm, L3 =
10 mm, L4 = 100 mm, L5 = 790 mm and LF = 1230 mm. The distance x, measured
from the first extraction electrode to the point of laser excitation is x = (12 ± 3) mm.
The applied voltages on the extraction and reflectron electrodes (for α = 1) are: V1 =
3550 V, V2 = 2700 V, R1 = 3593 V and R2 = 2336 V. Flight times through different parts
of the setup are given in green.
Symbol definitions:
𝑀
mass of the parent cluster
𝑚
mass of the fragment cluster
𝑣0
initial cluster velocity before excitation
𝑥
position from where charged clusters are accelerated
2
𝑣1
𝑡1
𝑣2
𝑡2
𝑡𝐹
𝑣3
𝑡3
𝑥′
𝑡4
𝑡5
𝑇
velocity at the second extraction plate on which V2 is applied
time elapsed from laser excitation to the second extraction plate
velocity at the end of the extraction zone
time elapsed between second and third extraction plates
time elapsed in the field free zone
velocity at the second reflectron plate on which αR2 is applied
time elapsed between the first and second reflectron plate
stopping point within the reflectron
time elapsed from the second reflectron plate until the stopping point
time elapsed from the exit of the reflectron to detection
total time-of-flight from laser excitation to detection
After production, the clusters are assumed to fly at a velocity 𝑣0 = 1200 𝑚/𝑠. Although this
value is an estimation, small changes do not affect flight times considering the much larger
kinetic energy of the clusters after ion extraction. After laser excitation, positively charged
clusters are accelerated by a potential difference V2-V1 resulting in a velocity v1 at the second
extraction plate (Eq. S1). The time elapsed from excitation to the entrance of the second
extraction stage (t1) is given by Eq. S2,
𝑣1 = √𝑣0 2 +
𝑡1 =
2𝑞(𝑉2 −𝑉1 )
(𝐿1
𝑀𝐿1
− 𝑥)
𝑀𝐿1
(𝑣 − 𝑣0 )
𝑞(𝑉2 − 𝑉1 ) 1
(S1)
(S2)
Then, clusters are accelerated by a potential difference V2. The terminal velocity (v2) and the
corresponding flight time in this stage (t2) are given by Eqs. S3 and S4, respectively.
2𝑞𝑉2
𝑀
(S3)
𝑀𝐿2
(𝑣 − 𝑣1 )
𝑞𝑉2 2
(S4)
𝑣2 = √𝑣1 2 +
𝑡2 =
During flight in the field free zone the velocity is constant. Therefore, the elapsed time is simply,
𝑡𝐹 =
𝐿𝐹
.
𝑣2
(S5)
Inside the reflectron an opposite potential is applied, decreasing the clusters velocities. Note
that assuming a delayed fragmentation from mother cluster (mass M) to fragment clusters
(mass m), this mass should now be used to calculate the flight times for the subsequent zones.
3
In the first stage of the reflectron, a potential difference of αR2 is applied, leading to a velocity
v3 at the second plate of the reflectron (Eq. S6). The corresponding flight time to travel the
distance L3 is given by Eq. S7,
𝑣3 = √𝑣2 2 −
𝑡3 =
2𝑞𝛼𝑅2
𝑚
𝑚𝐿3
(𝑣 − 𝑣2 )
𝑞𝛼𝑅2 3
(S6)
(S7)
In the second stage of the reflectron, clusters are stopped at a distance x’ measured from the
second reflectron plate. This distance is given by Eq. S8 and the time elapsed from the second
reflectron plate to the stopping point x’ by Eq. S9.
𝑥′ =
𝑚𝐿4
𝑣 2
2𝑞𝛼(𝑅1 − 𝑅2 ) 3
(S8)
𝑚𝐿4
𝑣
𝑞𝛼(𝑅1 − 𝑅2 ) 3
(S9)
𝑡4 =
Finally, clusters exit the reflectron entering the field free zone again until they reach the
detector, with a corresponding flight time given by,
𝑡5 =
𝐿5
𝑣2
(S10)
The total time of flight, from laser excitation to detection is thus:
𝑇 = 𝑡1 + 𝑡2 + 𝑡𝐹 + 2(𝑡3 + 𝑡4 ) + 𝑡5 .
(S11)
Since t3 and t4 depend on α, the total time-of-flight of the clusters, T, also depends on α. The
experimental α dependence of the time-of-flights is compared with calculated values for
different parent and fragment clusters, resulting in the assignment of the pathways.
4
Assignments of the fragmentations pathways of Sin+ (n = 5–13, 15–19, 21)
B)
In Figs. S2 to S9 the experimental dependence of the flight time of the delayed fragments T on
α is compared with calculated values for fragmentation pathways of different parent sizes but
with the same fragment size. Flight times for different fragment sizes differ much more and
thus the fragment size can easily be determined as is illustrated in Fig. 3 of the main text.
45
50
Experiment
+
+
Si5  Si4 +Si1
44
+
+
+
+
Si6  Si4 +Si2
Flight time / s
Flight time / s
Experimental
+
+
Si6  Si5 +Si1
48
Si7  Si5 +Si2
+
+
+
+
Si8  Si5 +Si3
Si7  Si4 +Si3
43
49
42
41
47
46
45
44
40
43
39
0.66
0.68
0.70
0.72
0.74
0.76
42
0.66
0.78
0.68
Scale of reflectron voltage
0.70
0.72
0.74
0.76
0.78
Scale of reflectron voltage
Figure S2. Assigned fragmentation pathways Si5+ → Si4++Si1 and Si6+ → Si5++Si1.
53
57
Experimental
+
+
Si7  Si6 +Si1
52
+
+
+
+
56
Si8  Si6 +Si2
+
+
Si9  Si7 +Si2
+
+
Si10  Si7 +Si3
Si9  Si6 +Si3
51
Flight time / s
Flight time / s
Experimental
+
+
Si8  Si7 +Si1
50
55
54
49
53
48
0.68
0.70
0.72
0.74
0.76
52
0.68
0.78
Scale of reflectron voltage
0.70
0.72
0.74
0.76
Scale of reflectron voltage
52.4
52.0
Figure S3. Assigned fragmentation pathways Si7+ → Si6++Si1 and Si8+ → Si7++Si1.
51.6
51.2
50.8
50.4
50.0
0.69
0.70
0.71
0.72
0.73
0.74
5
0.78
59.5
59
Experimental
+
+
Si9  Si8 +Si1
59.0
+
+
+
+
58
Si10  Si8 +Si2
58.0
57.5
+
+
+
+
Si10  Si6 +Si4
57
Si11  Si8 +Si3
Flight time / s
Flight time / s
58.5
Experimental
+
+
Si9  Si6 +Si3
Si11  Si6 +Si5
56
55
54
53
57.0
52
56.5
51
0.72
0.74
0.76
0.78
0.54
0.56
0.58
Scale of reflectron voltage
0.60
0.62
0.64
0.66
Scale of reflectron voltage
Figure S4. Assigned fragmentation pathways Si9+ → Si8++Si1 and Si10+ → Si6++Si4.
59
Experimental
+
+
Si11  Si6 +Si5
64
58
Flight time / s
Flight time / s
56
55
54
53
0.64
Experimental
+
+
Si10  Si7 +Si3
+
+
+
+
+
+
+
+
Si12  Si6 +Si6
62
57
Si13  Si6 +Si7
60
58
Si11  Si7 +Si4
Si12  Si7 +Si5
0.66
0.68
56
0.70
0.72
0.74
0.46
Scale of reflectron voltage
0.48
0.50
0.52
0.54
Scale of reflectron voltage
Figure S5. Assigned fragmentation pathways Si11+ → Si7++Si4 and Si12+ → Si6++Si6.
6
0.56
0.58
66
Experimental
+
+
Si12  Si7 +Si5
+
+
+
+
Si13  Si7 +Si6
64
+
+
+
+
Si15  Si10 +Si5
68
Flight time / s
Si14  Si7 +Si7
Flight time / s
Experimental
+
+
Si14  Si10 +Si4
70
62
60
58
Si16  Si10 +Si6
66
64
62
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
Scale of reflectron voltage
0.68
0.70
0.72
0.74
0.76
0.78
0.80
Scale of reflectron voltage
Figure S6. Assigned fragmentation pathways Si13+ → Si7++Si6 and Si15+ → Si10++Si5.
75
Experimental
+
+
Si15  Si10 +Si5
70
+
+
+
+
Experimental
+
+
Si16  Si11 +Si5
Si16  Si10 +Si6
Si17  Si10 +Si7
Flight time / s
Flight time / s
68
66
64
+
+
+
+
Si17  Si11 +Si6
72
Si18  Si11 +Si7
69
66
62
0.66
0.68
0.70
0.72
0.74
0.76
0.64
0.78
0.66
0.68
0.70
0.72
0.74
0.76
Scale of reflectron voltage
Scale of reflectron voltage
Figure S7. Assigned fragmentation pathways Si16+ → Si10++Si6 and Si17+ → Si11++Si6.
7
0.78
0.80
74
Experimental
+
+
Si17  Si11 +Si6
+
+
+
+
76
Si18  Si11 +Si7
72
Experimental
+
+
Si18  Si12 +Si6
Si19  Si11 +Si8
+
+
+
Flight time / s
Si20  Si12 +Si8
70
68
72
66
68
0.66
0.68
0.70
0.72
0.74
0.76
0.66
0.68
Scale of reflectron voltage
0.70
0.72
0.74
Scale of reflectron voltage
Figure S8. Assigned fragmentation pathways Si18+ → Si11++Si7 and Si19+ → Si12++Si7.
85
Flight time / s
Flight time / s
+
Si19  Si12 +Si7
84
Experimental
+
+
Si20  Si15 +Si5
83
Si21  Si15 +Si6
+
+
+
+
Si22  Si15 +Si7
82
81
80
79
78
77
0.66
0.68
0.70
0.72
0.74
0.76
0.78
Scale of reflectron voltage
Figure S9. Assigned fragmentation pathways Si21+ → Si15++Si6.
8
0.76