Correction of X- and γ-ray spectra acquired by CZT-based detectors
Abstract
Room Temperature Semiconductor Detectors based on CdZnTe have been widely used in X- and γ-ray spectroscopy in the energy range 10–10000 keV owing to their compactness, high energy resolution and high stopping power. These detectors do not require any cooling which enables their applicability in various fields such as medical imaging, environmental monitoring, astrophysics, decommissioning, nuclear power plants monitoring and homeland security. The related advancements in material science (crystal growth and processing, contact deposition) and technology (low-noise pre-amplifier, signal processing) are undoubtedly consolidated. Nonetheless, the measurements obtained with such devices are still affected by spectral distortions, which are mainly related to partial transfer of the photon energy, incomplete charge collection and excessive noise. Therefore, a straightforward interpretation of experimental data is not always possible. The development of powerful data analysis methods has always accompanied experimental science, and hence also this sector. Nowadays, the extensive knowledge of the physical mechanisms underlying the functioning of CdZnTe- based detectors can be combined with the tremendous progresses of the last decade in data science to push further the performances of these devices. This thesis is focused on the development of algorithms and techniques to correct spectral distortions affecting CdZnTe-based devices and to extract reliable information from measurements.
Full text
Uni e si à degli S udi di Pa ma
Do o a o di ice ca in
Scienza e Tecnologia dei Ma e iali
Ciclo XXXIII
Co ec ion o X- and - ay spec a
acqui ed by CZT-based de ec o s
Coo dina o e:
Chia .mo P o . En ico Dalcanale
Tu o e:
Chia .mo Do . And ea Zappe ini
Do o ando: Nicola Sa zi Amadè
Anni Accademici 2017/2018 2019/2020
To my amily
The addi i e p ocess is me ely a cul i a ion o
memo y, which becomes mechanical.
Lea ning is ne e cumula i e; i is a mo emen
o knowing ha has no beginning and no end.
B uce lee,B uce Lee: A is o Li e
Acknowledgemen s
Fi s o all, I would like o hank Xnex s. .l. o he inancial suppo and, especially,
o gi ing me he oppo uni y o ge an insigh in o a eal indus ial con ex .
Iwouldlike oexp essmyg a e ul hanks omy esea chg oup: mysupe iso
And ea Zappe ini, who has always shown g ea ai h in me, and Manuele Be elli,
whose pa ience and e sa ili y s ill imp esses me.
A special hanks goes o my Ph.D. "buddies" Filippo Vu o and An onino Bu -
aca oli. I am uly happy ha I me such like-minded people wi h whom I sha ed
expe iences, hough s and doub s du ing hese yea s.
I would also like o hank Nicola Zambelli, Sil ia Zane ini and Giacomo Benassi
o hei willingness and suppo . Su ely, hey had a majo ole in my educa ional
pa h.
Thanks o he es o he old "SIGNAL" g oup and o all he people o IMEM and
o he ins i u es who helped me in my esea ch ac i i y, he lis is indeed long.
A special men ion goes o my lunch companions (B onz and Dedè especially) wi h
whom I had he mos in e es ing and s imula ing deba es on wha e e opic o he
human knowledge, I will eally miss hose momen s.
Finally, hanks o my amily and all he special people in my li e o hei whole-
hea ed suppo .
Pa ma,
Decembe 31, 2020
Nicola Sa zi Amadè
Con en s
Con en s I
Lis o Tables III
Lis o Figu es V
1In oduc ion 1
IFundamen als o CZT-based X-and- ay de ec-
o s 3
2 In e ac ion o adia ion wi h ma e 4
2.1 Radioac i edecay.............................. 5
2.2 X-and- adia ion ............................. 7
2.2.1 Pho oelec ic abso p ion . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Comp on sca e ing . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Pai p oduc ion........................... 12
2.2.4 To al a enua ion coefficien .................... 13
2.3 Elec on ea angemen . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Fluo escence............................. 14
2.3.2 Auge effec ............................. 15
2.4 Fas elec ons................................ 17
2.4.1 Elas ic sca e ing . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 Inelas ic sca e ing . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.3 B emss ahlung........................... 18
2.4.4 F ee ca ie gene a ion in a semiconduc o . . . . . . . . . . . . 19
3 Spec oscopic oom empe a u e semiconduc o de ec o s 21
3.1 Wo kingp inciples ............................. 22
3.1.1 Cha ge anspo and collec ion . . . . . . . . . . . . . . . . . . 23
3.1.2 Signalinduc ion........................... 24
3.1.3 Ene gy esolu ion . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Cadmium Zinc Tellu ide . . . . . . . . . . . . . . . . . . . . . . . . . . 29
I
3.2.1 Ma e ial p ope ies . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 CZT as adia ion de ec o . . . . . . . . . . . . . . . . . . . . . 30
II Resul s 32
4Un oldingo -spec a wi h gene ic algo i hm 33
4.1 Concep s o spec al un olding . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Calcula ion o he de ec o esponse unc ion . . . . . . . . . . . . . . . 36
4.2.1 Radia ion-ma e in e ac ion . . . . . . . . . . . . . . . . . . . . 37
4.2.2 Elec ic and weigh ing ields . . . . . . . . . . . . . . . . . . . . 37
4.2.3 Cha ge anspo and signal induc ion . . . . . . . . . . . . . . 38
4.3 Gene icalgo i hm.............................. 39
4.4 Expe imen al alida ion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.1 CZTde ec o s............................ 42
4.4.2 Response ma ices . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4.3 Un olding .............................. 45
4.5 Conclusions ................................. 52
5Radioiso ope ecogni ionusingcon olu ionalneu alne wo ks 53
5.1 P oblems and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Role o a i icial neu al ne wo ks . . . . . . . . . . . . . . . . . . . . . . 56
5.3 P oposedme hod.............................. 60
5.3.1 Final a chi ec u e . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.2 T aining............................... 63
5.4 Resul s.................................... 66
5.4.1 Tes se ............................... 66
5.4.2 Addi ional da ase s . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5 Conclusions ................................. 74
6P ojec sando he ac i i ies 76
6.1 Synch o onsession............................. 77
6.1.1 Ma e ials and me hods . . . . . . . . . . . . . . . . . . . . . . . 78
6.1.2 Expe imen al esul s . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Simula ion o peak pileup dis o ion . . . . . . . . . . . . . . . . . . . . 84
6.2.1 Pileup modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.2 Measu emen s and simula ions . . . . . . . . . . . . . . . . . . . 88
7Conclusions 92
Bibliog aphy 96
II
Lis o Tables
2.1 Lis o he main disin eg a ion modes which lead o he emission o
adia ion. The "⇤"supe sc ip deno esanexci eds a e.. .. . . . .. . 5
2.2 Lis o he p edominan in e ac ion mechanisms o - ays in ma e in
he ene gy ange 101–104keV........................ 7
2.3 Lis o he p edominan in e ac ion mechanisms o as elec ons in ma -
e in he ene gy ange 101–104keV..................... 17
3.1 P ope ies o semiconduc o s used o adia ion de ec o s a 25 C( e -
e encein he ex ). ............................. 31
4.1 Desc ip ion and alues o he pa ame e s used in he GA. . . . . . . . . 41
4.2 Fi ness alues o each decon olu ed spec a a e 104gene a ions. . . . 48
4.3 Pe o mances a diffe en i e a ions (500 uns each) o s anda d and
seedingGA.................................. 50
5.1 Raw ou pu (CL. and REG.) and p ocessed ou pu (PROC.) o he
ained ne in he case o a 137Cs sou ce wi h 103coun s. . . . . . . . . 65
5.2 Con usion ma ix o he spec a o he ain se wi h one iso ope. . . . 66
5.3 P edic ions (in pe cen age) on he spec a o he es se wi h wo iso-
opes in a 1:1 a io. The esul s a e a e aged o e all spec a belonging
o he same class, ega dless o he s a is ics (2⇥103,2⇥104and 2⇥105).
S anda d de ia ion is epo ed in b acke s. . . . . . . . . . . . . . . . . 67
5.4 P edic ions (in pe cen age) on he spec a o he es se wi h wo iso-
opes in a 3:1 a io. The esul s a e a e aged o e all spec a belonging
o he same class, ega dless o he s a is ics (4⇥103,4⇥104and 4⇥105).
S anda d de ia ion is epo ed in b acke s. . . . . . . . . . . . . . . . . 68
5.5 P edic ions (in pe cen age) on he spec a o he es se wi h wo iso-
opes in a 1:3 a io. The esul s a e a e aged o e all spec a belonging
o he same class, ega dless o he s a is ics (4⇥103,4⇥104and 4⇥105).
S anda d de ia ion is epo ed in b acke s. . . . . . . . . . . . . . . . . 69
5.6 Con usion ma ix o he da ase con aining spec a wi h 100 coun s
(100 spec a o each iso ope). Columns do no sum o 100 because o
alse mul iple iden i ica ion. . . . . . . . . . . . . . . . . . . . . . . . . 70
III
5.7 Raw ou pu (CL. and REG.) and p ocessed ou pu (PROC.) o he
ained ne in he case o a spec um o a 133Ba sou ce a enua ed by 5
mmo s eel.................................. 71
5.8 Con usion ma ix o he shielded da ase (5 mm o s eel) con aining
spec a wi h 104coun s. .......................... 71
5.9 P edic ions (in pe cen age) on he da ase con aining spec a wi h h ee
iso opes in a 1:1:1 a io wi h 3⇥103,3⇥104and 3⇥105coun s. The e-
sul s a e a e aged o e all spec a belonging o he same class. S anda d
de ia ion is 1.5%inallcases. ...................... 72
6.1 Binding ene gies o he K-shells and XK- ay emissions in keV o Cd
and Te. The alues o Zn a e no epo ed because i s con ibu ion is
negligible. .................................. 79
IV
Lis o Figu es
2.1 Schema ic ep esen a ion o an isome ic ansi ion. . . . . . . . . . . . . 6
2.2 Schema ic ep esen a ion o he pho oelec ic p ocess. The inciden pho-
on is comple ely abso bed and a pho oelec on is ejec ed wi h an angle
'in espec o he pho on di ec ion. . . . . . . . . . . . . . . . . . . . 8
2.3 Schema ic ep esen a ion o Comp on effec in he sca e ing plane. The
inciden pho on is de lec ed wi h an angle #whe eas he elec on is
ejec ed wi h an angle 'in espec o he pho on di ec ion. . . . . . . . 9
2.4 Diffe en ial sca e ing c oss sec ion pe elec on as a unc ion o he
sca e ing angle #a diffe en ene gies (Equa ion 2.15). . . . . . . . . . 10
2.5 Diffe en ial sca e ing c oss sec ion pe elec on ecoil a diffe en ene -
gies(Equa ion2.17). ............................ 11
2.6 Schema ic ep esen a ion o he pai p oduc ion p ocess. The inciden
pho on is comple ely abso bed and a posi on-elec on pai is c ea ed. . 12
2.7 a) Pho on abso p ion c oss sec ions o CZT o all in e ac ion mecha-
nisms in he 100–5⇥104keV ene gy ange. b) Abso p ion efficiency o
CZT as a unc ion o c ys al hickness a a ious pho on ene gies (co e-
sponding o he Cd and Te K- luo escences and he cha ac e is ic - ays
o he mos commonly used adionuclides). . . . . . . . . . . . . . . . . 13
2.8 Schema ic ep esen a ion o luo escence. . . . . . . . . . . . . . . . . . 14
2.9 Schema ic ep esen a ion o he Auge effec ................ 15
2.10 K-shell and a e age L-shell luo escence and Auge yields as a unc ion
o he a omic numbe . The dashed e ical line ep esen s he a e age
a omicnumbe o CZT............................ 16
2.11 Simula ed spec um (GEANT4) o a X- ay ube o an accele a ing
ol age o 100 keV and a ungs en a ge . The peaks ep esen he
cha ac e is ic K-andL- luo escence lines o he me al a ge . Inse :
zoomed po ion o he L- luo escence lines. . . . . . . . . . . . . . . . . 19
2.12 Ene gy dependence o he ini ial dimension o he cha ge cloud as a
unc ion o Ein he 101-103keV ene gy ange ob ained wi h Gean 4
calcula ions.................................. 20
V
Pa I
Fundamen als o CZT-based X-
and - ay de ec o s
3
Chap e 2
In e ac ion o adia ion wi h ma e
This chap e is de o ed o he desc ip ion o he mechanisms o in e ac ion o X-and
- adia ion wi h ma e . The de ec ion o high ene gy pho ons in RTSDs is indeed
possible h ough hei pa ial o comple e abso p ion wi hin he semiconduc o olume
whe e he ene gy ca ied by he incoming pho on is ans e ed o he c ys al h ough
mul iple sequen ial collisions. The pu pose is gi ing he eade he necessa y elemen s
o unde s and he o igin o ea u es which appea in ac ual measu emen s o adioac i e
sou ces.
In he i s pa , he p inciples o adioac i e decays and adioac i i y a e b ie ly
men ioned. Then, he ypes o in e ac ion o high ene gy pho ons wi h he abso bing
medium and he ela ed seconda y p ocesses a e desc ibed. Fo a ine and de ailed
explana ion o he physics on his opics, he in e es ed eade is e e ed o speci ic
ex books [1, 2, 3]
4
2.1 Radioac i e decay
A adioac i ep ocessconsis sinaspon aneous ans o ma iono anuns ablea omic
nucleus P( e e ed o as pa en ) o one o mo e nuclei D( e e ed o as daugh e s)
wi h a diffe en numbe o nucleons (mass numbe A)and/o p o ons(a omicnumbe
Z), depending on he disin eg a ion mode. In he p ocess, he nuclide eleases ene gy
in he o m o adia ion. The main decay modes a e epo ed in Table 2.1.
Table 2.1: Lis o he main disin eg a ion modes which lead o he emission o adia ion.
The "⇤" supe sc ip deno es an exci ed s a e.
Mode Decay P oduc s
↵-decay A
ZP![A4
Z2D]2+↵•↵pa icle
-decay:
•-decay
+-decay
A
ZP![A
Z+1D]+++¯⌫e
A
ZP![A
Z1D]+++⌫e
•elec on
•an i-neu ino
posi on
neu ino
Elec on cap u e A
ZP!A
Z1D⇤+⌫e•neu ino
P o on emission A
ZP![A1
Z1D⇤]+p •p o on
Spon aneous ission P!D1+D
2+d1+d2+... • agmen nuclei
•nuclea pa icles
delayed p ocesses:
•neu on emission (n)
p o on emission (+p)
⇧↵emission (↵)
A
ZP⇤!A1
ZD+n
A
ZP⇤!A1
Z1D+p
A
ZP⇤!A4
Z2D+↵
•neu on
p o on
⇧↵pa icle
The daugh e nucleus can o m in an exci ed nuclea s a e which elaxes h ough he
emission o one o mo e -pho ons. This decay is de ined as isome ic ansi ion since
he mass and a omic numbe s emain unchanged (Figu e 2.1):
[A
ZD]⇤!A
ZD+(2.1)
Depending on he ene gy o he exci ed s a e, he a omic numbe and he mul ipola i y
(i.e., he classi ica ion o he ansi ion based on he o al angula momen a o he ini ial
and inal s a es), he a e age li e ime may a y om ⇠1014 s o se e al yea s. Abo e
109s, he nuclide is conside ed me as able (con en ionally deno ed by he supe sc ip
m). The ene gy o he - ay co esponds o he diffe ence be ween he ini ial and inal
ene gy le els minus he ecoil ene gy o he nucleus which is usually negligible:
E=(EiE )E =E⇤E (2.2)
The de-exci a ion does no always lead o he emission o a -pho on. E en i wi h a
low p obabili y (/Z3/E⇤), he ene gy may be ans e ed o an elec on o he X-shell
5
Figu e 2.1: Schema ic ep esen a ion o an isome ic ansi ion.
(whe e X=K, L, M, . . . ) which is hen ejec ed (in e nal con e sion):
A
ZD![A
ZD]⇤+e(2.3)
The con e sion elec on ca ies an ene gy de ined as:
Ece =EEbwi h E>E
b(2.4)
whe e Ebis he binding ene gy o he o bi al elec on. In e nal con e sion occu s
mainly in he decay o low-lying exci ed s a es o hea y nuclei and wi h elec ons o
he inne shells since hey ha e he highes p obabili y o being wi hin he nucleus
(e.g., 241Am). Ano he possible p ocess, which ge s signi ican a high ene gies, is he
c ea ion o an elec on-posi on pai (in e nal pai c ea ion) whose kine ic ene gy Tis
T(e±)=E2mec2wi h E>2mec2'1022 keV (2.5)
whe e meis he elec on es mass and cis he speed o ligh in acuum. The subsequen
annihila ion o he posi on leads o a u he emission o wo 511 keV pho ons e e ed
o as "annihila ion adia ion" (e.g., 22Na, 40K).
6
2.2 X-and- adia ion
Since pho ons pene a ing ma e a e abso bed o sca e ed in a single e en , i we
conside a collima ed beam o - ays ha incides on a de ec o , he emo ed numbe
o pho on (N)isp opo ional o he hicknesso he a e sedabso bingmedium
(x)and o henumbe o inciden pho ons(N):
N=µNx! N(x)=N0eµx (2.6)
The in eg a ion leads o an exponen ial beha iou whe e he p opo ionali y cons an
µ ep esen s he linea a enua ion coefficien exp essed as cm1. The mechanisms o
in e ac ion in he ene gy egion o in e es o RTSD, which spans om ew keV up o
ew MeV, a e basically h ee∗: pho oelec ic abso p ion, Comp on sca e ing and pai
p oduc ion (see Table 2.2).
Table 2.2: Lis o he p edominan in e ac ion mechanisms o - ays in ma e in he ene gy
ange 101–104keV.
P ocess In e ac ion wi h Effec s P oduc s
Pho oelec ic A omic elec ons Comple e
abso p ion •Pho oelec on
•Ionized a om
Comp on A omic elec ons Incohe en
sca e ing
•De lec ed pho on
•Ejec ed elec on
•Ionized a om
Rayleigh A omic elec ons Cohe en
sca e ing •De lec ed pho on
Pai p oduc ion Coulomb ield o
cha ged pa icles
Comple e
abso p ion •Elec on
•Posi on
Fo comple eness, Rayleigh sca e ing is also epo ed, al hough his p ocess is sig-
ni ican only a low ene gy in case o hea y elemen s bu i can be neglec ed in he
e alua ion o µsince he a e age de lec ion angle is small. Since hese mechanisms a e
independen , µconsis s in he sum o he a enua ion coefficien s o each p ocess:
µ=(⌧++)Ncm1(2.7)
whe e ⌧,and a e he c oss sec ions o he pho oelec ic, Comp on and pai p o-
duc ion effec s in cm2/a om, espec i ely, N=⇢mNA/M is he a om densi y pe cm3,
NAis he A ogad o cons an in a om/mol, Mis he mola mass in g/mol and ⇢mis
he mass densi y in g/cm3. The mass a enua ion coefficien , de ined as µ/⇢m,isalso
la gely used since i is independen om he ma e ial densi y and physical s a e. In
he ollowing sec ions, I will desc ibe he physical concep s which a e necessa y o he
e alua ion o ⌧,and .
∗Fo a comple e lis he in e es ed eade is e e ed o [1], Ch. 23, pg. 672
7
2.2.1 Pho oelec ic abso p ion
Figu e 2.2: Schema ic ep esen a ion o he pho oelec ic p ocess. The inciden pho on is
comple ely abso bed and a pho oelec on is ejec ed wi h an angle 'in espec o he pho on
di ec ion.
Pho oelec ic effec occu s only wi h a omic (i.e., bound) elec ons because a hi d
pa icle is necessa y o conse e momen um and he in e ac ion may occu wi h elec-
ons o any a omic shells. Howe e , he pho oelec on is mos p obably ejec ed om
he K-andL-shells (p o ided ha he pho on exceeds he binding ene gy) since he
p obabili y inc eases apidly wi h Eb. The ene gy o he pho oelec on is gi en by
T=EEbwi h E>E
b(2.8)
Depending on Eand Eb, he pho oelec ic c oss sec ion assumes diffe en analy ic
exp essions. I EEb, heBo napp oxima ionholdsand heabso p ioncoefficien
o he K-shell o an a om wi h Zelec ons can be calcula ed using non- ela i is ic
quan um mechanics and hyd ogen-like wa e unc ions o he a omic elec ons:
⌧K=0Z5↵425
2✓mec2
E◆7
2cm2/a om (2.9)
whe e 0=8⇡ 2
0/3is he c oss sec ion o Thomson sca e ing, 0=}↵/mecis he
classical elec on adius, ↵=e2/4⇡✏0}cis he ine-s uc u e cons an , }is he e-
duced Planck cons an and ✏0is he acuum pe mi i i y. I E⇡Eb,⌧Kcan s ill be
calcula ed unde he dipole ansi ion app oxima ion:
⌧K=0
(Z0.3)8
Z2↵523⇡✓mec2
E◆4exp(4⇠co 1⇠)
1exp(2⇡⇠)cm2/a om (2.10)
whe e in he nume a o Zis educed by he 1ssc eening cons an which is equal o all
Zand ⇠=(Eb/T)1
2. The abso p ion coefficien o he ou e shells can be calcula ed
using he same assump ions and in hese cases he dependence om he Zand E
is diffe en . The o al c oss sec ion ⌧is ob ained by adding he con ibu ion o each
shell (⌧L,M,...). Based on expe imen al da a, a high ene gies ⌧is usually ob ained by
mul iplying ⌧Kby a cons an :
8
⌧=⌧K+⌧L+⌧M'5
4⌧K(2.11)
On he o he hand, a low ene gies he c oss sec ion is discon inuous (abso p ion edges)
in co espondence o he elec onic ansi ions when E=EXwhe e EXis he elec on
binding ene gy and X=K, L, M, . . . . Empi ically, ⌧is oughly app oxima ed o
⌧'cons an ·Zn1
En2
whe e (4<n
1<5
n2⇠7/2(2.12)
whe e bo h exponen s n1and n2depend on E.O e all,i ise iden ha highZ
ma e ials a e p e e ed o enhance pho oelec ic abso p ion.
The pho oelec ic abso p ion is always accompanied by he elec on ea angemen
o he ionized a om (Sec ion 2.3) and he ejec ion o a pho oelec on (Sec ion 2.4).
2.2.2 Comp on sca e ing
Figu e 2.3: Schema ic ep esen a ion o Comp on effec in he sca e ing plane. The inciden
pho on is de lec ed wi h an angle #whe eas he elec on is ejec ed wi h an angle 'in espec
o he pho on di ec ion.
The heo y o he sca e ing o - ays by elec ons can be de i ed by sol ing he
equa ions exp essing he conse a ion o ene gy and momen um. Usually, he s uck
elec on can be conside ed o be unbound since EEb. Mo eo e , he ela i is ic
exp ession o mass and ene gy mus be adop ed because Eis no negligible in espec
o mec2(i is use ul o in oduce he adimensional pa ame e ⇣=E/mec2in he
ollowing o mulas). Thus, by equa ing he ene gies and momen a be o e and a e he
collision be ween he incoming pho on and he elec on a es we ob ain
8
>
>
>
>
>
<
>
>
>
>
>
:
E=E0
+T=E0
+mec2h121
21i
E
c=mec121
2cos '+E0
ccos #
0=mec121
2sin 'E0
csin #
(2.13)
9
whe e = /c is he elec on eloci y exp essed in uni s o he speed o ligh , Tis he
kine ic ene gy o he s uck elec on, E0
is he ene gy o he sca e ed pho on and #
and ' ep esen he angles o he sca e ed pho on and he ecoil elec on, espec i ely,
as shown in Figu e 2.3. E0
and Ta e he quan i ies o in e es since hey ha e a
undamen al ole in de ining he measu ed spec al shape, as explained in Chap e 4.2.
They can be ob ained by sol ing he sys em o conse a ion equa ions (Equa ion 2.13):
E0
=E
1+⇣(1 cos #)T=EE0
=E11
1+⇣(1 cos #)(2.14)
The diffe en ial sca e ing c oss sec ion pe elec on, also called Klein-Nishina o -
mula, can be ob ained by sol ing he Di ac equa ion o he elec on which, o unpo-
la ized inciden adia ion, gi es:
d
d⌦= 2
0
2✓E0
E◆2✓E
E0
+E0
Esin2#◆=
= 2
0
2(1
[1 + ⇣(1 cos #)]2"1+cos
2#+⇣2(1 cos #)2
1+⇣(1 cos #)#) (2.15)
Equa ion 2.15 educes o he classical Thomson c oss sec ion a low ene gies (i.e.,
⇣⌧1). The angula dependence o d/d⌦a diffe en ene gies is epo ed in Figu e
2.4: by inc easing E, hepho onismainlyde lec edalong hep ima ydi ec ion.
Figu e 2.4: Diffe en ial sca e ing c oss sec ion pe elec on as a unc ion o he sca e ing
angle #a diffe en ene gies (Equa ion 2.15).
10
By in eg a ing Equa ion 2.15 o e he solid angle and by mul iplying o he numbe
o elec ons in an a om, we inally ob ain he o al Comp on c oss sec ion:
=2⇡ 2
0Z⇢1+⇣
⇣22(1 + ⇣)
1+2⇣1
⇣ln(1 + 2⇣)+
+1
2⇣ln(1 + 2⇣)1+3⇣
(1 + 2⇣)cm2/a om
(2.16)
A low ene gies (E⇡Eb)s ongapp oxima ionsmus beadop ed odesc ibe hemo-
ions, dis ibu ions and binding ene gies o a omic elec ons. Consequen ly, heo e ical
calcula ions do no yield accu a e alues o o all Zand T.
Analogously o pho oelec ic effec , an ionized a om and an ejec ed elec on esul
om Comp on sca e ing bu , in addi ion, he incoming pho on is de lec ed wi h a
lowe ene gy. As al eady men ioned, we a e in e es ed in he ecoil ene gy o he
elec on because, i he de lec ed pho on escapes om he c ys al olume, i ep esen s
he only ene gy ac ually deposi ed in he RTSD, o ming he so-called Comp on edge.
Hence, i is use ul de ining he diffe en ial c oss sec ion o gi ing an elec on a ecoil
ene gy in he in e al be ween Tand T+dT (Figu e 2.5):
d
dT =2⇡mec2
E02
d
d⌦=
=⇡ 2
0mec2
(ET)2(✓mec2T
E2
◆2
+2✓ET
E◆2
+T(ET)
E3
T2mec2)(2.17)
Figu e 2.5: Diffe en ial sca e ing c oss sec ion pe elec on ecoil a diffe en ene gies (Equa-
ion 2.17).
11
2.2.3 Pai p oduc ion
Figu e 2.6: Schema ic ep esen a ion o he pai p oduc ion p ocess. The inciden pho on
is comple ely abso bed and a posi on-elec on pai is c ea ed.
Pai p oduc ion p ocess consis s in he comple e abso p ion o a pho on which is e-
placed by an elec on-posi on pai whose o al ene gy is equal o he - ay ene gy.
This in e ac ion occu s only in he Coulomb ield o cha ged pa icles and i Eis abo e
a ce ain h eshold. In he egion o in e es o RTSDs, his p ocess akes place mainly
in he nuclea ield when Eexceeds wice he es -mass ene gy o an elec on.
E=T+T++2mec2wi h E2mec2=1.022 MeV (2.18)
The solu ion o his physical p oblem can be ob ained assuming a negligible in e ac ion
be ween he elec on/posi on and he nucleus, ha is when Z↵/⌧1 o bo h
pa icles (Bo n app oxima ion). As a consequence, he ene gy dis ibu ion o elec ons
and posi ons a e symme ic since hey a e no subjec ed o he Coulomb epulsion
and a ac ion. Mo eo e , o E.20 MeV, he p ocess is mo e likely o occu a a
dis ance om he nucleus ha is smalle han he adius o K-shell, hus he sc eening
due o a omic elec ons can be neglec ed. The diffe en ial c oss sec ion o he c ea ion
o a posi on o kine ic ene gy be ween T+and T++dT+in he ield o a nucleus o
cha ge Ze is ound o be
d
dT+
=↵ 2
oZ2P(E,Z)
E2mec2(2.19)
whe e Pis a complica ed unc ion which a ies be ween 0 ( o E2mec2)and
⇡20 ( o E=1). The analy ical in eg a ion o Pis possible only in he ex emely
ela i is ic case. Mo e gene ally, we can w i e
=↵ 2
0Z2Z1
0
Pd T+
E2mec2=↵ 2
0Z2¯
Pcm2/nucleus (2.20)
whe e ¯
Pinc eases app oxima ely loga i hmically wi h E.Fo lowene gy(sligh ly
abo e he 2mec2 h eshold), Eq. 2.20 unde es ima es he c oss sec ion, especially o
high Zelemen s, because he Bo n app oxima ion does no hold. In his case, has a
s onge dependence han Z2.Insumma y,pai p oduc ionisp edominan inhea y
elemen s and a high pho on ene gy (E⇠se e al MeV).
12
✓dE
dx ◆B
=N 2
0Z2↵(T+mec2)4ln2(T+mec2)
mec24
3(2.29)
The app oxima e a io be ween ioniza ion/exci a ion and adia i e ene gy losses (Equa-
ions 2.28 and 2.29, espec i ely) shows ha B emss ahlung is signi ican only in hea y
elemen s a high ene gy:
(dE/dx)B
(dE/dx)c'ZT
1400mec2(2.30)
Figu e 2.11: Simula ed spec um (GEANT4) o a X- ay ube o an accele a ing ol age o
100 keV and a ungs en a ge . The peaks ep esen he cha ac e is ic K-andL- luo escence
lines o he me al a ge . Inse : zoomed po ion o he L- luo escence lines.
This physical p ocess unde lies he ope a ion o X- ay ubes whe e elec ons a e
accele a ed in acuum by an elec ic ield and subsequen ly collide wi h a me al a ge .
The X- ays esul ing om bo h B emmss ahlung and he elaxa ion o ionized a oms
o he a ge (Sec ion 2.3) cons i u e he spec um emi ed by he ube. Diffe en
ene gy dis ibu ions can be ob ained by a ying he accele a ing ol age, he a ge
and he il e s (an example is epo ed in Figu e 2.11).
2.4.4 F ee ca ie gene a ion in a semiconduc o
The mean ee pa h o as elec ons in ma e is small compa ed o he ypical c ys al
dimensions o RTSDs, hence he o al kine ic ene gy is e en ually eleased wi hin he
c ys al olume. Impac ioniza ions and exci a ions induce ansi ions om he alence
band o he conduc ion band and c ea e a non-equilib ium dis ibu ion o "ho " ca i-
e s. This p ocess is coupled o he emission o op ical phonons and he esidual kine ic
ene gy o he c ea ed pai is con e ed in o la ice ib a ions h ough he maliza ion
19
losses. This leads o an a e age elec on-hole pai c ea ion ene gy (Ee-h) which is la ge
han he semiconduc o ene gy gap (EG)andisgi enby hesemi-empi icallaw[6,7]
Ee-h=EG+hERi+hEKi'14
5EG+ (}!R)(2.31)
hERiis assumed o be p opo ional o he a e age numbe o emi ed Raman quan a
(}!R)pe gene a edpai and allsin he0.55 < (}!R)<1eV ene gy ange. hEKi
ep esen s he he maliza ion losses and, in he case o a simple wo band con igu a ion
in di ec band gap ma e ials wi h equal elec on and hole effec i e masses, is equal o
9/5EG. The numbe o pho ogene a ed elec on-hole pai s can be inally calcula ed as
Ne-h=E
Ee-h
(2.32)
Es ima ing he ini ial shape and dimension o he cha ge cloud as a unc ion o Ein
agi enma e ialisa duous,bo h heo e icallyandexpe imen ally,becauseo all he
b anching p ocesses in ol ed in he abso p ion o he p ima y adia ion. Ne e heless,
his can be achie ed by Mon e Ca lo simula ions whe e i is possible o ack all he
seconda y pa icles and he ene gy eleased in each poin o he c ys al (Figu e 2.12)
[8]. The cha ge densi y pe uni olume can be oughly app oxima ed o an iso opic
3D-gaussian:
⇢ch =Q0
3
0(2⇡)3/2exp ✓||~x~xb||2
22
0◆(2.33)
whe e Q0=eNe-h,~xbis he ba ycen e o he cha ge cloud and 0is he ini ial a iance.
Figu e 2.12: Ene gy dependence o he ini ial dimension o he cha ge cloud as a unc ion
o Ein he 101-103keV ene gy ange ob ained wi h Gean 4 calcula ions.
20
Chap e 3
Spec oscopic oom empe a u e
semiconduc o de ec o s
Since hei in oduc ion in he 1960’s, semiconduc o s de ec o s ha e ep esen ed a
b eak h ough in nuclea spec oscopy. The efficien and di ec con e sion o ionizing
adia ion o elec ic signal allowed us o ealize sensi i e and compac de ices wi h
incompa able pe o mances in espec o p e ious echnologies. Ini ially, g oup IV
elemen s (silicon and ge manium) we e he only ma e ials used o his pu pose bu
he ope abili y a c yogenic empe a u es ep esen ed a s ong limi which p e en ed
hei diffusion in ields whe e cos s and con enience we e o conce n.
In he ea ly 1970’s, wide band gap semiconduc o compounds (CdTe and HgI2)
s a ed appea ing in his con ex since hey equi ed minimal o no cooling. Ne e he-
less, a he ime he ac ual c ys als p ope ies we e diffe en om he p omising ones
p edic ed by heo y. Only a e wo decades o o e all imp o emen s he majo p ob-
lems ela ed o c ys al g ow h, su ace p ocessing and con ac deposi ion we e pa ially
sol ed, hus opening he possibili y o la ge-scale p oduc ion sho ly a e [9].
Nowadays, CdTe and CZT domina e he pano ama o RTSD owing o hei po a-
bili y, high ene gy esolu ion (e e y yea close and close o Si-based sys ems), la ge
a omic numbe and high densi y [10, 11]. In his chap e , I will b ie ly desc ibe he
mechanisms unde lying he unc ioning o semiconduc o adia ion de ec o s and which
ole CZT has in his con ex .
21
3.1 Wo king p inciples
Figu e 3.1: Schema ic ep esen a ion o he signal gene a ion in a semiconduc o -based adi-
a ion de ec o : a) in e ac ion o he - ay wi h he c ys al; b) elec on-hole pai s p oduc ion;
c) Cha ge d i ing and collec ion.
Single-pho on coun ing semiconduc o de ec o s ac as solid s a e ioniza ion chambe s
whe e he abso bed adia ion p oduces a cloud o nega i e (elec ons) and posi i e
(holes) ee cha ges acco ding o he in e ac ion mechanisms desc ibed in Chap e 2
(Figu e 3.1a). The biased me al con ac s deposi ed on he su aces o he c ys al
gene a e an elec ic ield wi hin i s olume which sepa a es and d i he ca ie s (Figu e
3.1b). The mo ion and he e en ual collec ion o hese cha ges induce a ansien
cu en on he ead-ou elec ode which is hen ampli ied and analyzed by p ope
elec onics (Figu e 3.1c). This echnology p esen se e al s eng hs:
•Ex emely compac and po able de ices can be ealized hanks o he highe
s opping powe in espec o gaseous o liquid de ec o s.
•The elec ic p ope ies o he semiconduc o can be inely uned and op imized
o ob ain he desi ed a ibu es.
•The ene gy deposi ed by he pho on is di ec ly con e ed in o elec ic cha ge,
p o iding highe signals in espec o indi ec con e sion p ocesses like scin illa-
o s.
•Spec oscopic in o ma ion can be ob ained by p ocessing he signal p oduced by
each pho on, in con as o de ices which in eg a e he pho ogene a ed cu en
o e all he exposu e ime.
•The con ibu ion o da k cu en (i.e., he cu en which lows in he de ice in
absence o adia ion) can be neglec ed by se ing a p ope h eshold and only
he signals which exceed i a e p ocessed, hus p o iding a highe signal- o-noise
a io in espec o de ice which ope a es in in eg a ing mode.
•Loca ing he in e ac ion posi ion is possible wi h a p ope elec ode pa e ning
(e.g., pixela ed de ec o s o X- ay imaging, 3D d i s ips spec ome e s).
22
3.1.1 Cha ge anspo and collec ion
The pho ogene a ed elec ons and holes d i owa ds he anode and he ca hode,
espec i ely, wi h a eloci y gi en by
~ =µ~
Ewi h µ=e
m⇤¯⌧and 1
m⇤=1
}2
@2E
@k2(3.1)
whe e µis he ca ie mobili y, ¯⌧is he a e age sca e ing ime wi h he la ice, im-
pu i ies o de ec s and m⇤is he effec i e mass which is gi en by he cu a u e o he
elec onic band in k-space. The linea ela ion be ween eloci y and elec ic ield holds
below sa u a ion which occu s a ields highe han he ypical ones used in ac ual de-
ices (⇡15000 V·cm1a oom empe a u e (RT) o CdTe [12]); hence, he mobili y
is assumed o be cons an . The elec ic ield wi hin he c ys al olume depends om
se e al ac o s such as applied bias ol age, geome ical con igu a ion o he elec odes,
c ys al dimensions, p esence o spa ial cha ge and de ec s and, inally, he me al con-
ac s. Theo e ically, calcula ing he ield p o ile in each poin o he c ys al is a duous
and usually expe imen al echniques a e employed (Pockel effec , Lase Induced T an-
sien Cu en Technique). In he case o iso opic band dispe sion, he ajec o y o a
poin cha ge can be simply calcula ed as
~x( )=~x0+Z
0
~ ( 0)d 0=~x0+µZ
0
~
E(~x( 0))d 0(3.2)
whe e ~xand ~ a e he ins an aneous posi ion and eloci y and ~x0is he ini ial posi-
ion. Since he pho ogene a ed cha ge cloud is no poin -like, he equa ion o mo ion
should be sol ed o he whole cha ge dis ibu ion by conside ing also he diffusion and
Coulomb epulsion o he ca ie s. Howe e , i we neglec sel -shielding effec s and we
conside a nea ly cons an elec ic ield in he olume egion o he cloud, he b oaden-
ing o he cloud can be decoupled om i s ajec o y. I we conside a gaussian ini ial
cha ge dis ibu ion (Equa ion 2.33), he ba ycen e ~xb ollows Equa ion 3.2 whe eas
he ime e olu ion o he a iance is gi en by [8]
2( )=2
0+2D +µNe
2p5✏Z
0
d 0
p2( 0)(3.3)
whe e 0is he ini ial a iance, D=µkBT/e is he diffusion coefficien , kBis he
bol zmann cons an , Tis he empe a u e, Nis he numbe o cha ges and ✏is he
elec ic pe mi i i y o he semiconduc o . In mul i-elec ode de ices hese assump ions
do no hold in he p oximi y o con ac s since he cloud can spli and be collec ed by
mul iple elec odes, gi ing ise o he “cha ge sha ing” effec [13, 14, 15]. Usually, his
effec is ele an in pixela ed de ec o s when he cloud dimension is no negligible in
espec o he pixel size. Du ing my PhD, I pa icipa ed in a synch o on session wi h
he pu pose o in es iga ing his kind o dis o ion as desc ibed in Chap e 6. Figu e 3.2
epo s he ime e olu ion o he cloud dimension o diffe en ene gies o he p ima y
pho on [8].
23
Figu e 3.2: Time e olu ion o he cha ge cloud dimension due o diffusion and Coulomb
epulsion in he 101–103keV ene gy ange.
The p esence o de ec s in eal semiconduc o s (e.g., acancies, disloca ions, g ain
bounda ies) may p oduce ene gy le els inside he o bidden ene gy gap o which ac as
aps o ecombina ion cen es o elec ons and holes. Thus, ee ca ie s a e cha ac-
e ised by a ini e li e ime ⌧and an ini ial cha ge Q0decays in ime acco ding o:
Q( )= ⌧
⌧+Q0+
⌧+Q0e
⌧
(3.4)
whe e he effec o de- apping (de ined by he cha ac e is ic ime )hasalsobeen
indica ed. Usually, is longe han he ansi ime and i s effec can be neglec ed,
hence esul ing in an exponen ial decay. The o e all quali y o he semiconduc o
is ep esen ed by he µ⌧p oduc om which, assuming a cons an elec ic ield, he
mean d i leng h can be calcula ed as =µ⌧|~
E|.I ⌧is compa able o he ansi
ime o , analogously, i is compa able o he de ec o hickness, incomple e cha ge
collec ion occu s. I he e a e la ge diffe ences in he anspo p ope ies o elec ons
and holes, i is possible o neglec he con ibu ion o he wo se ca ie by choosing a
p ope elec ode con igu a ion and o ealize single pola i y cha ge sensing de ices (see
Sec ion 3.1.2).
3.1.2 Signal induc ion
The induc ion on he ead-ou elec ode by he pho ogene a ed ca ie s (i.e., he ac ual
ou pu signal o he RTSD) is desc ibed by he Ramo-Shockley heo em [16, 17]. I
s a es ha he induced cha ge and he co esponding induced cu en by a mo ing
poin cha ge Qa he ime a e gi en by
24
Qind( )=QW(~x( ))
Iind( )=dQind( )
d ·d~x
d~x=Qd~x
d ·dW(~x( ))
d~x=Q~ ( )·~
EW(~x( )) (3.5)
The weigh ing po en ial Wand he weigh ing ield ~
EW= Wuniquely depend on
he geome ical con igu a ion o he elec odes and hey can be calcula ed by sol ing
he Laplace equa ion wi h Di ichle bounda y condi ions in which all elec odes o
he de ice a e g ounded excep he conside ed one which is a uni po en ial. The
comple e desc ip ion o he signal induc ion p ocess can be achie ed by subs i u ing in
Equa ion 3.5 he cha ge densi y and he ela ion be ween he eloci y and he elec ic
ield (Equa ions 2.33 and 3.1, espec i ely) and by conside ing he ime e olu ion o
he cha ge cloud (Equa ions 3.2, 3.3 and 3.4):
Iind( )=µZZZV
⇢ch (Q( ),~x
b( ),( )) ~
E(~x( )) ·~
EW(~x( ))dV (3.6)
I should be no ed ha elec ons and holes possess diffe en p ope ies (m⇤,µ,e c.)
and he equa ions epo ed in Sec ions 3.1.1 and 3.1.2 mus be sol ed o bo h ca ie s.
Finally, by in eg a ing Iind( )o e he ca ie s ansi ime ( eand h o elec ons
and holes, espec i ely), he o al collec ed cha ge Qcoll can be ob ained as well as he
Cha ge Collec ion Efficiency (CCE):
Qcoll =Z e
0
Ie( )d +Z h
0
Ih( )d ! CCE =Qcoll
Q0
(3.7)
CCE is he mos impo an pa ame e which de e mines he o e all pe o mance o he
de ice. Expe imen ally, he in eg a ion is pe o med by a Cha ge Sensi i e P eampli-
ie (CSP) which p oduces a ol age ou pu signal ha is p opo ional o he collec ed
cha ge. The p eampli ied pulses a e hen p ocessed by he subsequen s ages o he ana-
log o digi al ead-ou chain and a e inally con e ed o an ene gy- esol ed spec um.
The ime equi ed o p ocess each e en is limi ed by he coun ing a e and is usually
sho compa ed o he cha ge collec ion imes. Depending on he in e ac ion posi ion,
cha ge pulses may no each he ull heigh be o e being shaped, esul ing in a ballis ic
de ici which deg ades he spec a. The p o ile o Wcan be op imized in o de o
sense only he cha ges ha a el in he p oximi y o he ead-ou con ac . The e o e,
only he ype o ca ie which eaches he ead-ou elec ode induces a signal. Single
pola i y cha ge sensing can be achie ed by educing one o bo h con ac s dimensions
(na ow s ips o small pixels, om which he name "small pixel" effec ) bu o he con-
igu a ions exis ( i ual F isch g id [18, 19], coplana g id [20, 21], d i s ip [22, 23]
quasi-hemisphe ical [24]). The quali a i e end o he elec ic and weigh ing ield o
hese geome ies is epo ed in [8].
25
3.1.3 Ene gy esolu ion
The ene gy esolu ion o he de ec o o a mono-ene ge ic inciden adia ion is ex-
p essed as he Full Wid h a Hal Maximum (FWHM)o hemeasu ed ullene gy
peak (o pho opeak) di ided by i s cen oid, usually indica ed in pe cen age:
R(E)=100·FWHM
E(%) (3.8)
The pho opeak is he measu ed coun s dis ibu ion in he case o a ull ene gy de-
posi ion, ei he ia a single pho oelec ic in e ac ion o any combina ion o all he
mechanisms (see Figu e 3.4). Se e al noise sou ces con ibu e o b oaden he pho o-
peak and, in gene al, FWHM is gi en by he sum in quad a u e o all he luc ua ions
in ol ed in he gene a ion and p ocessing o he signal [3, 25]:
FWHM2=FWHM2
noise +FWHM2
Fano +FWHM2
ap +FWHM2
da k +... (3.9)
FWHMnoise is ela ed o he noise o he ead-ou sys em and is usually he dominan
e m. FWHMFano is ela ed o he s a is ical luc ua ions in he gene a ion o he e-h
pai s (Fano noise) and i ep esen s he lowe bound o FWHM since i depends only
om he semiconduc o p ope ies (elec on-hole pai c ea ion ene gy Ee-h). The e m
FWHM ap is ela ed o cha ge anspo and collec ion: depending on he in e ac-
ion posi ion, ca ie s incu in diffe en apping p ocesses along hei pa hs and poo
anspo p ope ies lead o incomple e cha ge collec ion and asymme ic pho opeaks
(" ailing effec "). FWHMda k is ela ed o he luc ua ions o he da k cu en whose
ex en depends om he semiconduc o p ope ies, he quali y o su aces, he me al
con ac s and he applied bias ol age. In ac , in in insic semiconduc o s in he mal
equilib ium, he ee ca ie densi y pe uni olume in he conduc ion and alence
bands is calcula ed acco ding o he o mula [26]:
n=p=ni/T3/2exp (EG/2kBT)(3.10)
whe e niis he in insic ca ie concen a ion, nand pa e he elec ons and holes
densi y, espec i ely, Tis he empe a u e and kBis he Bol zmann cons an . nand p
in u n de e mine he conduc i i y ()and esis i i y(⇢=1/)and,hence, hed i
cu en densi y ~
Jd i i an elec ic ield is p esen wi hin he semiconduc o :
=e(nµe+pµh)!~
Jd i =~
E=~
E
⇢(3.11)
whe e is a scala in iso opic semiconduc o s. This is he c ux o Room Tempe a u e
(RT) ope abili y and he eason why he esea ch mo ed o wide band gap compen-
sa ed semiconduc o s. Co espondingly, low Ee-has well as low he mal exci a ion o
ca ie s (desc ibed by Equa ions 2.31 and 3.10, espec i ely) a e he key ea u es be-
hind he excellen signal- o-noise a io o cooled Si- and Ge-based de ec o s. In mos
semiconduc o s, he p esence o de ec s on he su aces o he c ys al c ea e elec onic
26
s a es in he ene gy gap, esul ing in local doping and leading o highe conduc i i y
wi h espec o he bulk. The su aces con ibu e can be educed by passi a ion (oxida-
ion o he supe icial s a es) o e en neglec ed by su ounding he ead-ou elec odes
wi h a g ounded gua d- ing con ac which collec s he su ace cu en . Mo eo e , an
impo an ole is played by he me al-semiconduc o junc ion: depending on he me al,
diffe en ypes o band alignmen may occu , esul ing in blocking (Scho ky) o ohmic
con ac s acco ding o he wo k unc ions o he wo ma e ials and he na u e o he
supe icial s a es. The o me a e cha ac e ised by an ene gy ba ie which p e en s he
injec ion o ca ie s om he me al o he semiconduc o s, hus educing he leakage
cu en (diode-like beha iou ). On he o he hand, hey may p oduce cha ge accu-
mula ion a he in e ace (usually a high adia ion lux) which dis o s he elec ic
ield and deg ades he cha ge collec ion. In his case, he la e ype o con ac is
usually p e e ed. The beha iou o Scho ky ba ie diodes can be modeled acco d-
ing o he in e acial laye - he moionic-diffusion heo y [27]. The measu emen o he
cu en / ol age (I/V) cha ac e is ics is always manda o y o de e mine he op imal
applied ol age in ope a ing condi ions which is always a ma e o comp omise since
highe biases ensu e be e cha ge collec ion bu inc ease he da k cu en as well.
The measu ed spec um is he esul o all he effec s desc ibed in Chap e s 2 and
3. An example o 137Cs spec um measu ed wi h a CZT de ec o is epo ed in Figu e
3.3. The physical explana ion o he ea u es p esen in he measu emen and o o he
possible dis o ions is epo ed in Figu e 3.4 whe e he effec o he su ounding is also
highligh ed (image cou esy o [28]).
Figu e 3.3: Expe imen al spec um o a 137Cs adioac i e sou ce measu ed wi h a CZT-
based de ec o . 137Cs decays by -emission o he exci ed s a e o 137Ba; he subsequen
isome ic ansi ion and elec on ea angemen p oduce he adia ion indica ed in he igu e.
27
Figu e 3.4: Schema ic ep esen a ion o pho oelec ic and Comp on in e ac ion and sub-
sequen mechanisms in a semiconduc o de ec o and objec s in i s su oundings ( op) and
co esponding ene gy spec um (bo om). 1) Fluo escence peak; 2) backsca e peak; 3)
Comp on edge; 4) double Comp on e en s; 5) escape peak; 6-7) pho opeak ( ull ene gy depo-
si ion); 8) ailing effec .
28
adia ion o ene gy be ween E0
j1and E0
j.Asal eadyexplainedin hep e iouschap e s
and exempli ied in Figu e 3.4, Rcan ha e a complex s uc u e and is ypically quasi-
singula , especially a high ene gies whe e pho oelec ic in e ac ion is less p obable.
Mo eo e , he measu emen can be con amina ed by pho ons whose ene gy alls ou side
he uppe and lowe h esholds o he obse ed spec um. Finally, he p oblem is
u he complica ed by he p esence o expe imen al noise, de ined as he de ia ion
om he expec ed alue, which modi ies Equa ion 4.4 in o
~
S=R·~
D~
+~✏ o Si=
M
X
j=1
RijDjj+✏iwi h i=1,...,N (4.5)
I solu ions ~ssa is ying R·~s=0o R·~s=~
(wi h i⌧✏i)exis , heycanbeadded o
he ue solu ion ~
wi hou in alida ing Equa ion 4.5 bu , since he p oblem is uns able
o small luc ua ions, hese addi ional e ms may domina e in espec o ~
, husmaking
he numbe o po en ial solu ions in ini e wi hin e o bounds. Consequen ly, i we jus
apply R1 o an obse ed spec um, small a ia ions can p oduce unphysical ea u es
in he un olded spec um such as eno mous oscilla ions and nega i e alues (Figu e
4.1). The e o e, he p oblem is ill-condi ioned and ill-posed. This is he eason why
leas -squa es me hods usually ail and decon olu ion me hods, which a e mo e s able
wi h espec o unce ain ies, a e employed. Se e al app oaches ha e been p oposed in
a ious con ex (leas -squa es [46, 47], Mon e Ca lo [48, 49], i e a i e [50, 51, 52, 53, 54],
Bayesian [55, 56, 57], neu al ne wo ks [58, 59, 60], gene ic algo i hm [61, 62]).
Figu e 4.1: a) Ideal syn he ic spec um (i.e., a column o R) o a monoene ge ic adia ion
o 500 keV (blue cu e) and i s decon olu ion a e applying R1(o ange cu e). b) Same
spec um a e adding a negligible gaussian noise (blue cu e) and i s decon olu ion a e
applying R1(o ange cu e).
35
4.2 Calcula ion o he de ec o esponse unc ion
Al hough he obus ness and effec i eness o he un olding me hod a e o p ima y
impo ance, an accu a e esponse ma ix is he key o ob ain ine esul s. O he wise,
he decon olu ion will ail o succeed, ega dless o he chosen algo i hm. Rcan be
ob ained bo h wi h expe imen s and calcula ions. The o me me hod is de ini ely he
mos accu a e since i allows o po ay he exac beha iou o he de ec o : i consis s
in measu ing spec a o monoch oma ic sou ces a diffe en ene gies and using hem
o map Rin he whole desi ed ene gy space by in e pola ion. Howe e , i is also he
less iable one: i s ly, i is ex emely esou ce demanding and ime consuming since i
equi es acili ies like synch o ons; secondly, Rwould be cha ac e is ic o ha speci ic
de ec o and adap ing i o ano he one may no be i ial, e en i hey ha e he same
cha ac e is ics. Fo hese easons, syn hesized esponse ma ices a e usually employed.
Simula ions o he spec al esponse o a de ec o is a common p ocedu e since
when he i s li hium-d i ed Si and Ge semiconduc o de ec o s appea ed on he
scene [63, 64, 65, 66]. A ha ime, he simula ions conside ed only he effec s o
he in e ac ion o adia ion wi h he c ys al. The in oduc ion o high Zcompounds
like CdTe, which p esen ed wo se anspo p ope ies in espec o Si o high pu i y
Ge, imposed o include he deg ada ion effec s inhe en o he collec ion efficiency
o pho ogene a ed cha ges and o signal p ocessing [67, 68, 69, 70, 71]. Finally, as
he pho oli hog aphic echniques p og essed, he simple plana -plana geome y was
abandoned and mo e complex elec ode pa e ns we e ealised in o de o neglec he
con ibu ion o slow ca ie s (single pola i y cha ge sensing). Consequen ly, signal
induc ion was necessa ily included in he modelling [70, 72].
The pu pose o he ans e unc ion o a adia ion de ec o goes beyond spec al
decon olu ion, e en i i is i s p ima y unc ion. Since he p ocessing o CZT c ys als
s ill p esen s se e al c i icali ies (c ys al cu ing and polishing, me al con ac deposi ion
and pa e ning, as explained in Chap e 3.2), he ab ica ion o de ice wi h unusual o
complex dimensions and/o elec ode geome ies can be isky, especially in an indus ial
con ex whe e o limi he ailu e a e is a p io i y. The e o e, he design phase and
enginee ing o each aspec which can in alida e he unc ioning o he de ice is e en
mo e c ucial in sa ing ime and cos s. Beyond his, imp o ing he ene gy and spa ial
esolu ion and cha ge collec ion efficiency is de ini ely possible h ough simula ions
only. In his con ex , my esea ch g oup de eloped in collabo a ion wi h Xnex s. .l.
asimula ion oolki able o ep oduce hespec al esponseo hesede icesby i s
p inciples me hods [8] which includes each s ep o he signal gene a ion p ocess:
•in e ac ion o he pho on wi h he semiconduc o
• anspo and collec ion o ca ie s
•signal induc ion and p ocessing
Since hese p ocesses can be ea ed independen ly, he simula o is composed o mul-
iple blocks, each ha ing diffe en asks.
36
4.2.1 Radia ion-ma e in e ac ion
The i s block is de o ed o he simula ion o he anspo o pa icles h ough ma e
ia Mon e Ca lo calcula ions, ideal o ep oduce he s ochas ic na u e o he physical in-
e ac ion mechanisms desc ibed in Chap e 2, and is based on Gean 4 [73], la gely used
in he ield o X-and- ay spec oscopy [72, 74, 75, 76, 77]. G4EmS anda dPhysics and
G4EmLi e mo ePhysics packages conside all he equi ed elec omagne ic in e ac ions
in he ene gy ange om ew keV o ew MeV.: pai p oduc ion, Comp on sca e ing,
Rayleigh sca e ing and pho oelec ic effec o pho ons; elas ic and inelas ic sca e -
ing, B emmss ahlung and annihila ion o elec ons and posi ons. The c oss sec ions
o each mechanism a e calcula ed ia o mulas, pa ame isa ions o in e pola ion o
da abases. Gean 4 allows o de ine he geome ical con igu a ion o he whole sys em:
c ys al dimensions, di ec ion and shape o he pho on sou ce, p esence o collima o s
o il e and o he possible elemen s. Gean 4 acks each p ima y o seconda y pa icle
and eco ds he in o ma ion on each collision: he in e ac ion posi ion and ene gy e-
leased wi hin he sensi i e olume a e sa ed in o a ile. The equi ed numbe o e en s
gene a ed o simula e each column o Ris ypically in he o de o 106 o ob ain high
s a is ical signi icance.
Figu e 4.2: Example o Gean 4 simula ion: he ed pa allelepiped ep esen s a CZT c ys al
and g een lines ep esen inciden o sca e ed pho ons.
4.2.2 Elec ic and weigh ing ields
The ask o he second block is calcula ing he elec ic and weigh ing ields wi hin he
c ys al olume by sol ing he Poisson’s equa ion
2'=⇢
"(4.6)
whe e 'is he elec ic po en ial, ⇢is he cha ge densi y and "is he elec ic pe mi i i y
o CZT. Gi en he geome y o he sys em (c ys al and elec ode dimensions and
pa e ning) and he bounda y condi ions (applied bias ol age a each elec ode), he
37
elec ic po en ial can be calcula ed by ini e elemen s me hod. This app oach is alid i
he pho ogene a ed cha ges do no signi ican ly pe u b he elec ic ield (quasi s eady-
s a e condi ion). Consequen ly, he Poisson’s equa ion can be decoupled om he
con inui y equa ions o elec ons and holes. This condi ion is gua an eed since CZT is
less likely o suffe om sel -pola isa ion, which would lead o a cha ge accumula ion,
and he consequen o ma ion o a dead laye below one o he con ac s.
Simila ly, he weigh ing po en ial can be calcula ed by sol ing he Laplace’s equa-
ion wi h bounda y condi ions in which all elec odes a e g ounded excep he consid-
e ed one which is se a uni a y po en ial [16, 17]). These ope a ions a e pe o med by
he so wa e COMSOL Mul iPhysics (elec os a ics package) which allows he design
o he desi ed geome y o he sys em and au oma ically c ea es an adap i e mesh in
he olume ( ypical elemen dimensions span om 50 nm o 100 µm). A e he calcu-
la ion, he po en ials a e ex apola ed and expo ed on a uni o m 3D cubic g id wi h
⇡5µms eps,dependingon hede ec o geome yanddimensions.Twoexamplesa e
epo ed in Figu e 4.6.
4.2.3 Cha ge anspo and signal induc ion
Basically, his block has he pu pose o sol ing Equa ions 3.5, 3.6 and 3.7 o calcula e
he cha ge anspo and signal induc ion. Fi s ly, he ajec o ies o elec ons and
holes a e calcula ed by means o O dina y Diffe en ial Equa ion (ODE) sol e which
exploi s he ini ial condi ions p o ided by he GEANT4 block and he ield p o ile
calcula ed wi h COMSOL. Al hough cha ge clouds ha e ini e dimensions, in mos
cases he mo ion equa ions o he ba ycen e can be decoupled om he b oadening o
he cloud. This simpli ies he calcula ions e en i i implies ha clouds canno spli
and be collec ed om mul iple elec odes (cha ge sha ing effec ). The knowledge o
ajec o ies allows us o calcula e he induced cu en ia Ramo–Shockley’s heo em.
The ope a ions ca ied ou by he elec onic ead-ou chain a e applied o he cu en
ansien o shape he pulse and calcula e i s heigh . Finally, he spec um can be
econs uc ed by posi ioning each coun in he co esponding ene gy channel. The
effec o noise b oadening can be pe o med in wo ways: (a) con ol ing he pulse
heigh spec um wi h a gaussian ke nel (o any o he sui able ke nel); (b) adding he
noise di ec ly o he simula ed cu en pulses. The o me me hod is usually p e e ed
because he ke nel wid h is cha ac e is ic o he ead-ou sys em and i can be easily
measu ed. In he la e case, he noise spec al densi y mus be known. This block
has been implemen ed in MATLAB.
38
4.3 Gene ic algo i hm
Gene ic Algo i hm (GA) is inspi ed by he da winian heo y o na u al selec ion and is
ab ancho hee olu iona ycompu a ion amily[78,79,80].I emula es hep ocess
o selec ion pe o med by a hos ile en i onmen on a popula ion o indi iduals: he
ones wi h he bes cha ac e is ics a e mo e likely o pass down hei gene pool o he
offsp ing whe eas un i indi iduals die wi hou ep oducing ("su i al o he i es ").
As a esul , he o e all i ness o he popula ion will inc ease h ough gene a ions. In
compu e science, he indi iduals (o solu ions) a e a ays o bi s/in ege s/ eal numbe s
which unde go ou basic ope a ions:
•ini ializa ion o he i s gene a ion
•selec ion o indi iduals o ma ing
•gene a ion o new indi iduals
•mu a ion o he genome in new indi iduals
Se e al me hods o pe o m hese ope a ions ha e been de eloped and he choice de-
pends on he speci ic p oblem o sol e. A p ope objec i e unc ion ep esen s he
i ness: wi h he passing o gene a ions he popula ion e ol es owa ds be e solu ions
in o de o maximize (o minimize) .GAsa eusuallyemployedinop imiza ionand
sea ch p oblems and hey ha e al eady been applied o un old neu on spec a [61, 62].
Since he p inciples a e he same, I w o e my own code o deal wi h -spec a. In his
con ex , indi iduals ep esen po en ial inciden ene gy PDF. They consis o ec o s
o non-nega i e alues wi h uni a y a ea and hei genes sj, ha is heno malized
numbe o coun s in each channel, ul ima ely de e mine how he measu ed spec um
would appea h ough he con olu ion wi h he esponse unc ion R. Thus, he i ness
is de ined as
=1
N
N
X
i=1 M
X
j=1
Rij ·Djs0
jSi!2
(4.7)
whe e s0is a po en ial solu ion. The e o e, he bes indi idual is he one ha minimizes
he no malised esidual sum o squa es be ween i s con olu ion wi h Rand he obse ed
spec um S.Ino de omaximize heexplo a iono hesolu ionspace, heini ial
popula ion is andomly gene a ed and i is composed o Nsindi iduals. The selec ion
is pe o med by unning " ou namen s" in which he i ness alues o he con es an s,
andomly chosen among he popula ion, a e compa ed. The winne is he one wi h he
lowes and ob ain he igh o p oc ea ing:
winne =a g min( (s0), (s00)) (4.8)
39
whe e s0and s00 a e wo andomly d awn indi iduals. The numbe o ou namen s is
gi en by he Ns·Pcp oduc whe e Pcis he c osso e p obabili y (see Table 4.1). Then,
he Ns·Pcselec ed indi iduals a e combined in pai s o p oduce an equal numbe o
offsp ing: gi en wo pa en s ~
P1and ~
P2(i.e., he winne s o wo diffe en ou namen s),
he offsp ing ~
O1and ~
O2a e de ined as he weigh ed a e age o he pa en s:
~
O1=w~
P1+(1w)~
P2
~
O2=(1w)~
P1+w~
P2
(4.9)
whe e 0<w<1is a cons an (see Table 4.1). Each new gene a ed solu ion is mu a ed
by eplacing he alues o andomly selec ed channels wi h a andom alue in he
in e al [0,m·sj]whe e sjis he alue o he j- h channel and mis a uned pa ame e
(see Table 4.1). The numbe o mu a ed genes is gi en by he Pm·Np oduc whe e Pm
is he mu a ion p obabili y (see Table 4.1). Usually, a smoo hing s ep is equi ed since
highly oscilla ing solu ions can s ill gi e good esul s because o he quasi-singula i y
o R[81], al hough his ope a ion would also b oaden ue peaks. Howe e , in his
case smoo hing is no equi ed hanks o he choice o he c osso e me hod which
allows o a enua e g ea diffe ences among indi iduals wi hou smea ing ou he eal
cha ac e is ic lines. A e no malising he a ea o he mu a ed offsp ing, he i ness is
e alua ed o all he new indi iduals and he en i e popula ion is esized by disca ding
he wo s ones (eli is selec ion). These ope a ions a e epea ed un il a maximum
numbe o i e a ions is eached o when alls below a ce ain h eshold. The alues
o he pa ame e s used in each s ep a e epo ed in Table 4.1 and he low cha o he
code is shown in Figu e 4.3. A calib a ion p ocedu e has been pe o med o une hem
o ob ain he bes o e all pe o mances in e ms o unning ime ( ime equi ed pe each
gene a ion) and con e gence speed (numbe o gene a ions equi ed o each he goal).
As a ma e o ac , inc easing he size o he popula ion and he numbe o offsp ing
would indeed educe he numbe o gene a ions equi ed since he solu ion space is
be e explo ed bu a he expense o mo e compu a ions. Analogously, excessi e
mu a ions would p oduce wo se indi iduals han he co esponding pa en s, hus no
p oducing ac ual imp o emen s. On he o he hand, sligh mu a ions would educe
he con e gence speed. The e o e, he pai s [Ns,P
c]and [Pm,m]ha e been join ly
op imised by p og essi ely sampling he alues o bo h pa ame e s in a selec ed ange
whe eas whas been uned sepa a ely. The alues which p o ided he bes esul s on a
e e ence spec um ha e been selec ed. In eali y, all pa ame e s a e in e -co ela ed,
o a g ea e o lesse ex en , bu i would be a duous o simul aneously une all o
hem. The algo i hm has been implemen ed in MATLAB.
I is wo h no ing ha Ris always used di ec ly o old po en ial solu ions wi hou
he need o in e ing i . As a ma e o ac , he ask o GA is nei he in e ing R
no inding he ma hema ical solu ion o his p oblem: i sea ches o an app oxima e
and physically easonable solu ion which bes ma ches he measu ed spec um. The
physical alidi y o he solu ion is gua an eed by he ac ha he mu a ion p ocess
can only p oduce non-nega i e alues.
40
GA pa ame e s
NsSize o he popula ion 50
PcC osso e p obabili y 0.8
PmMu a ion p obabili y 0.01
wC osso e weigh 0.3
mMu a ion uppe bound 3
Table 4.1: Desc ip ion and alues o he pa ame e s used in he GA.
Figu e 4.3: Flowcha o he GA used in his wo k. The p elimina y and inal s eps a e
highligh ed in yellow whe eas he ope a ions in ol ed in he loop o e he gene a ions a e
highligh ed in o ange.
41
4.4 Expe imen al alida ion
The algo i hm has been alida ed on expe imen al spec a measu ed wi h wo diffe en
CZT-based de ec o : 137Cs and 133Ba measu ed wi h he d i s ip de ec o s; 241Am
and 57Co measu ed wi h he single pixel de ec o ;.
4.4.1 CZT de ec o s
Figu e 4.4: a) 3D model o he d i s ip de ec o . b) Ac ual CZT de ec o bonded o he
in e media e elec onic boa d.
The i s de ec o consis s o a 20 ⇥4.5⇥6mm3c ys al ealized by ep ocessing
s anda d spec oscopic g ade CZT ma e ial pu chased om REDLEN Technologies.
The c ys al p esen s a s ip elec ode geome y: he anode is segmen ed in se en
250 µm s ips wi h an in e gap o 550 µm whe eas he ca hode is ull a ea (Figu e
4.4). Con ac s ha e been ealized wi h gold elec oless deposi ion in aqueous solu ion.
S a ing om he mos ex e nal o he cen al one, du ing he measu emen s he s ips
a e pola ized a 450 V, 300 V, 150 Vand0V, espec i ely,and heca hodea
450 V. This geome y allows he ca ie s o d i owa ds he collec ing cen al s ip
and gua an ees a high ene gy esolu ion abo e 100 keV ega dless o he i adia ion
di ec ion. In ac , his de ec o has been moun ed on an unmanned ae ial ehicle wi h
he pu pose o examining con amina ed a eas [82, 83]. The analog eadou sys em has
been de eloped by due2Lab s. .l. The eadou channel includes a C ema CR110 CSP
(decay ime =140µs), a shape ampli ie (shaping ime =2µs) and a peak de ec o .
The ene gy esolu ion is 3.9%FWHM @662 keV.
Figu e 4.5: a) 3D model o he single pixel de ec o . b) Ac ual CZT de ec o bonded o
he in e media e elec onic boa d.
42
The second de ec o consis s o a 4.1⇥4.1⇥2.8mm3de ec o ob ained by ep o-
cessing a s anda d CZT c ys al pu chased om REDLEN Technologies (Canada) wi h
a ull-a eaca hode,a2⇥2mm2pixel and a gua d ing on he anode wi h a gap o
50 µm(Figu e4.5). Senso e ab ica ionhasbeenca iedou bydue2labs. .l.;gold
con ac s we e ab ica ed using he elec oless deposi ion p ocess om alcoholic solu ion
as desc ibed in [84, 85]. The de ec o was biased a 850 Vandi wasi adia ed om
he ca hode side. The CZT senso uni (D2L001) and he single-channel digi al pulse
p ocesso uni (D2L009-1) a e pa o he Hype spec al X- ay Spec ome e (HXS)
de eloped by due2lab. The ene gy esolu ion is 4.0%FWHM @60 keV.
4.4.2 Response ma ices
Figu e 4.6: T ans e sal sec ions o he elec ic po en ial (le ) and weigh ing po en ial ( igh )
o he d i s ip ( op) and single pixel de ec o s (bo om). In he case o he d i s ips
de ec o , he "small pixel" effec is e iden .
The esponse unc ions o bo h de ec o s ha e been ob ained using he simula ion ool
desc ibed in Sec ion 4.2 in he 0–750 keV and 0–150 keV ene gy anges o he d i s ips
and single pixel de ec o s, espec i ely, each di ided in 256 channels. In he o me
case, a iso opic i adia ion om he whole solid angle has been conside ed because his
con igu a ion is he mos simila o he senso eal ope a ion en i onmen , whe eas in
he la e only he ca hode side has been exposed in which he ajec o ies o inciden
43
pho ons a e o hogonal in espec o he ace o he de ec o (sou ce a in ini y). This is
agoodapp oxima iono hegeome icalcon igu a iono he ealmeasu emen .Ascan
be seen in Figu e 4.7, Comp on sca e ing quickly becomes he dominan effec o he
d i s ips de ec o which la ens he in ensi y o he pho opeak. On he o he hand,
o he single pixel de ec o he main dis o ions a e he escape peaks (off-diagonal
lines) due o he small olume o he c ys al. Addi ionally, a high ene gies hole ailing
becomes signi ican because he p obabili y o in e ac ion nea he anode inc eases.
Fluo escence peaks (ho izon al lines) can also be no ed, due o he abso p ion o CZT
XK- ays in he olume below he pixel which a e p oduced by ionised a oms in he
olume below he gua d ing. Expe imen al spec a ha e been e-binned on he ene gy
ec o s o R.
Figu e 4.7: Response ma ices on a loga i hmic alse colo scale (le ) and some o hei
columns as example ( igh ) o he d i s ip ( op) and single pixel de ec o s (bo om).
44
Figu e 4.15: 137Cs un olded spec um (g ay cu e) compa ed o he measu ed spec um
( ed cu e), he olded spec um (black dashed cu e) and he abula ed in ensi ies o he
-emissions (blue ba s) and X-emissions (o ange ba s) o 137Cs on a loga i hmic scale.
Figu e 4.16: 133Ba un olded spec um (g ay cu e) compa ed o he measu ed spec um
( ed cu e), he olded spec um (black dashed cu e) and he abula ed in ensi ies o he
-emissions (blue ba s) and X-emissions (o ange ba s) o 133Ba on a loga i hmic scale.
51
4.5 Conclusions
In his chap e , a me hod o un old -spec a based on GA is p esen ed and expe imen-
ally alida ed on spec a o ou adionuclides measu ed wi h wo diffe en CZT-based
de ec o s. This app oach effec i ely akes ad an age o all he coun s p esen in he
obse ed spec a o econs uc and enhance e en he weakes pho opeaks. P ima ily,
his is possible hanks o he simula ion ool used o p oduce he spec al esponse
unc ion whose accu acy is con i med by hese esul s. The simplici y o he GA op-
e a ions ensu es low compu a ion and high speed and allows o quickly ind he bes
app oxima e solu ion wi hin he limi s o he accu acy o R. The p oblem o in e -
ing Ris bypassed because i is always used di ec ly o old new po en ial solu ions,
hence a oiding he need o egula isa ions. The ex en o oscilla ions is limi ed by
he a i hme ic ecombina ion pe o med in he c osso e p ocess which a e age noise
and excessi e mu a ions o he pa en s. The me aheu is ic p ocedu e pe o med by
GA allows us o speed up he solu ion sea ch compa ed o Mon e Ca lo app oaches.
Simul aneously, he andom componen pe mi s a be e explo a ion o he solu ion
space wi h espec o o he i e a i e and p ede e mined me hods whe e, gi en a ce -
ain inpu , he algo i hm always con e ges o he same ou pu . The s eng h o GA is
indeed combining hese wo aspec s. The s ochas ic op imiza ion is as e en wi hou
p io assump ions bu exploi ing he physical in o ma ion con ained in he measu e-
men s allows o u he accele a e he sea ch p ocess. Finally, he algo i hm has been
success ully applied on spec a wi h low s a is ics wi hou he need o a smoo hing
p ocess.
This me hod is no dependen on he adia ion sou ce and, po en ially, i can be
applied also on con inuous ene gy dis ibu ion (e.g., X- ay spec a). S udies a e on-
going o conside he cha ge sha ing effec in he simula ion o Rin o de o es he
algo i hm on spec a acqui ed by pixela ed de ec o s. Finally, his app oach can be
ex ended o diffe en classes o adia ion de ices like scin illa ion de ec o s which a e
widely used in se e al con ex s hanks o lowe p oduc ion cos s. Howe e , he ene gy
esolu ion is usually low in comme cial de ices (FWHM &7% @662keV)and he
use o a obus analysis algo i hm is e en mo e ele an .
52
Chap e 5
Radioiso ope ecogni ion using
con olu ional neu al ne wo ks
The iden i ica ion and, abo e all, he quan i ica ion o he adioiso opes p esen in a
-spec um is a ask s ill hea ily affec ed by e o s. The majo obs acle is ep esen ed
by he low ene gy esolu ion o de ices ypically used in ields o he han scien i ic e-
sea ch whe e de ec ion speed, cos -effec i eness and sys em po abili y a e o p ima y
conce n (indus y, en i onmen al moni o ing, secu i y and sa e y). Fu he mo e, in
many p ac ical cases he ea ly iden i ica ion when s a is ical luc ua ions a e dominan
is manda o y. This impedes he applica ion o me hods based on he spec al shape
(peak sea ching, empla e ma ching, egions o in e es , spec al un olding) and usually
human in e en ion is equi ed o ob ain eliable esul s. In a ce ain sense, humans
ou class machines in some asks: he expe ience and capaci y o abs ac ion o a ained
spec oscopis enable him/he o ecognize he ele an in o ma ion p esen in a mea-
su emen and o guide an algo i hm on he igh pa h, hence minimizing iden i ica ion
e o s and alse posi i es. This is he eason why algo i hms mimicking human men al
p ocesses ha e been in oduced in his ield, especially in he las decade. In addi ion,
in machine lea ning app oaches he in ensi e compu a ional wo kloads is shi ed o he
aining phase in which a p edic i e model is syn hesised, hence allowing as ou ine
analysis on new da a [88, 89].
Fi s ly, his chap e gi es an o e iew o he main app oaches in adioiso ope ecog-
ni ion and i desc ibes which ole a i icial neu al ne wo ks ha e in his con ex . Then,
ano elapp oachbasedonmul i-objec i edenselyconnec edcon olu ionalneu alne -
wo k is p esen ed. The po en iali ies o his algo i hm a e demons a ed on simula ed
spec a o a CZT-based de ec o .
53
5.1 P oblems and challenges
I s a by explaining he diffe ence be ween adioiso ope iden i ica ion and quan i ica-
ion. The o me consis s in iden i ying he iso opic inge p in s in he measu emen s
(e.g., posi ion o pho opeaks) and in de e mining which iso opes a e p esen and which
a e absen ; his p ocess is e e ed o as "classi ica ion" in da a science ja gon. The
la e consis s in es ima ing he iso opic composi ion o he measu ed spec um (e.g.,
no only posi ion bu also a ea o pho opeaks) and i is a mo e challenging ask, ob i-
ously; his p ocess is e e ed o as " eg ession" in da a science ja gon. I is impo an
o cla i y hese concep s be o e p oceeding u he because in one case we ob ain a
label and in he o he case we ob ain a nume ical alue.
Ma hema ically, a -spec um gene a ed by mul iple adioiso opes can be consid-
e ed as a linea combina ion o he spec a o each adia ion sou ce. Le Ndeno es he
numbe o channels he measu ed spec um is di ided in o and M he o al numbe o
possibly p esen iso opes, he measu ed spec um can be exp essed as
~c=R·~ao ci=
M
X
j=1
Rijajwi h i=1,...,N (5.1)
whe e ~c=[c1,...,c
i,...,c
N]Tis he ec o con aining he numbe o coun s in each
channel, ~a=[a1,...,a
j,...,a
M]Tis he iso opic composi ion ec o and Ris he
esponse ma ix which is cha ac e is ics o he conside ed de ec ion sys em. The j- h
column o R ep esen s he ideal measu ed spec um o he j- h iso ope in which he
sou ce-de ec o geome y is assumed o be always he same and non-linea effec s such
as pileup a e also no conside ed. Gene ally, hese assump ions hold o CZT-based
de ec o s whe e he i adia ion di ec ion is known and he in ol ed adia ion luxes a e
low in espec o he dead ime o he sys em. The numbe o channels in aw spec a
is app oxima ely o a ew housands. On he o he hand, he comple e iso ope lib a y
has app oxima ely 200 adio-iso opes bu his numbe can be dec eased: depending on
he speci ic applica ion (e.g., indus ial, medical, secu i y, nuclea ene gy), he lis o
possible iso ope is usually known which allows o ocus he sea ch on a gi en subspace
(M⇡20–30) [90]. The p oblem is complica ed u he due o se e al unknown ac o s
such as he dis ance and he shielding. As a ma e o ac , a enua ion is one o
he mos oublesome aspec s because he p esence o abso be s modi ies he spec al
shape. In his case, he esponse ma ix is ew i en as
R! R(µ)
Rij ! Rij ·exp (X
z
µz(i) z)(5.2)
whe e µz(i)is he a enua ion coefficien a he i- h channel o he z- h abso be
(e.g., s eel, lead). The iso opic weigh s can be calcula ed s a ing om he numbe o
coun s in each ene gy channel by in e ing Equa ion 5.1. Howe e , he in e sion o R
54
is a duous, analogously o he case o spec al un olding: dis o ions, luc ua ions and
unce ain ies may lead o e oneous and unphysical esul s as explained in he p e ious
chap e . Ano he possible way is no in e ing Rbu , ins ead, i ing i s in e se using
expe imen al spec a whose iso opic composi ion is known (leas -squa es me hod).
Va ious app oaches ha e been p oposed o sol e ei he he classi ica ion o eg es-
sion p oblem and mos o hem can be ga he ed in wo main ca ego ies: peak sea ch
and ma ch and empla e ma ching [91]. In he o me , a e a p ope smoo hing s ep
and he emo al o backg ound, he peaks a e iden i ied and analysed. This p ocess is
no i ial in case o spec a wi h a low numbe o e en s and wi h low/a e age ene gy
esolu ion since s a is ical luc ua ions and b oad peaks may p e en dis inguishing a
small signal om noise. Then, nume ical a ibu es/ ea u es a e calcula ed (e.g., peaks
a ea). The quali y and he numbe o ea u es as well as he dimensions o he lib a y
a e c ucial in o de o ca y ou he subsequen classi ica ion s ep since speed and ac-
cu acy a e compe ing ac o s. These ea u es a e used o selec he co ec solu ion
in a lib a y o nuclea da a h ough compa ison. Se e al classi ica ion algo i hms ex-
is (decision ees, neu al ne wo ks, Naï e Bayes, Nea es Neighbou , Suppo ec o
machines) and he choice o he bes one depends on he p e ious phases. The la -
e me hod consis s in building a lib a y o iso ope spec a in diffe en con igu a ions
which mus be cha ac e is ic o he de ec ion sys em used o he measu emen s. An
algo i hm sea ches o he bes linea combina ion o solu ions p esen in he lib a y
which bes ma ches he measu ed spec um. Since his combina o ial p oblem can be
complex, a p e ious s ep mus educe he dimensionali y o da a and emo e noise and
dis o ions which could mislead he compa ison. Also in his case, many algo i hms
can be used and hey can be di ided in o heu is ic and sys ema ic. Bo h me hods a e
analogous in he sense ha , i s ly, he aw measu emen is p e-p ocessed o educe
he ex en o noise and/o he dimensionali y o da a, al hough his passage is also
in insically accompanied by in o ma ion losses. Secondly, he ex ac ed ea u es a e
used o pe o m he classi ica ion o eg ession (Figu e 5.1). The e o e, de e mining he
iso opic composi ion in a -spec um is gene ally a mul i-s ep p ocess. The algo i hms
can be ully au oma ed o can equi e he in e en ion o a ained spec oscopis o
assis and guide hem in he mos delica e s eps.
Figu e 5.1: Flowcha o he "peak sea ch and ma ch" and " empla e ma ching" me hods
o adioiso ope ecogni ion in -spec a.
55
5.2 Role o a i icial neu al ne wo ks
Ideally, he algo i hm should sa is y he ollowing equi emen s:
•i should be ully au oma ed;
• he esponse should be as in espec o he ime equi ed o acqui e he spec um
•i should p ocess di ec ly he aw spec um wi hou in e media e s eps
•i should be obus in espec o noise and dis o ions
• he aining phase should be simple in e m o speed and calcula ions
•i should sea ch o bo h local and global pa e ns o efficien ly ex ac he ele-
an ea u es
An addi ional equi emen is pe o ming bo h iden i ica ion and quan i ica ion. The
eason why bo h ope a ions a e necessa y is he ins abili y o he in e se p oblem
(Equa ion 5.1): he weigh s should be es ima ed only o he adionuclides ac ually
p esen and no o he absen ones. In summa y, he ideal ou pu should consis s
o a no malised ec o o non-nega i e alues (one o each possible iso ope). Thus,
he ideal algo i hm should pe o m bo h a mul i-label classi ica ion (i.e., each iso ope
could be ei he p esen o absen ) and a eg ession o he iso opic ac ions. Recen ly,
a i icial neu al ne wo k-based algo i hms (ANN) appea ed in his con ex bo h in
scien i ic a icles [89, 92, 93, 94, 95] and in pa en s [96, 97, 98]. The pu pose o ANN is
eplacing one o all he s eps epo ed in Figu e 5.1 and sa is ying, pa ially o en i ely,
he a o emen ioned equi emen s hanks o hei peculia i ies. Fi s ly, he compa ison
wi h a lib a y is no pe o med o each new measu emen ; ins ead, he compu a ional
ime is mo ed o he aining phase. A e , he p ocess is comple ely au oma ed and he
esponse is p ac ically immedia e. Secondly, i is possible o design he ne o di ec ly
analyse he aw measu emen and o p o ide he desi ed esul s as ou pu . Tha being
said, he gene aliza ion abili y is de ini ely he p ima y s eng h o ANN. Howe e , aw
spec a a e usually composed o a ew housands o channels and i could be difficul o
ain a ne wo k wi h such a la ge numbe o inpu pa ame e s [95]. Fo example, he
a chi ec u e p oposed by Kamuda e al. p esen s ⇡106pa ame e s o spec a wi h
1024 channels [92, 94]. Fo his eason, a p elimina y s ep o dimensionali y educ ion
is usually adop ed [89, 93]. Mo eo e , ANN conside s he aw spec a as a whole: his
leads o a la ge numbe o lea nable pa ame e s because he ne wo k does no exploi
he spa ial ela ions among channels. In con as , Con olu ional Neu al Ne wo ks
(CNN), mainly used o image ecogni ion pu poses, a e cha ac e ised by wo aspec s
which allows o o e come hese p oblems: pa ame e sha ing and local connec i i y.
Rega ding he o me , CNN applies con olu ional ke nels o il e s, whose weigh s a e
adjus ed du ing aining, o he inpu image in each laye . The e o e, he numbe o
pa ame e s does no depend on he image size and i is usually a less han he numbe
56
o pixels o he inpu image. This allows o g ea ly educe he numbe o lea nable
pa ame e s since he same ke nels ope a e on he whole image. Mo eo e , wi h he same
numbe o pa ame e s, i is possible o ealize deepe ne wo ks, hence ob aining be e
abs ac ion easoning. Rega ding he la e , he ke nels a e connec ed only o a ac ion
o he pixels o he inpu image. Each po ion is p ocessed sepa a ely esul ing in an
effec i e ansla ion-in a ian ex ac ion o local pa e ns. The e o e, in con as o
s anda d ANN, CNN effec i ely exploi s spa ial ela ions among pixels and ex ac s he
so-called ea u es maps. These a e hen passed o ully connec ed laye s which ins ead
sea ch o global ela ions among all he inpu da a. As a ma e o ac , CNN belongs
o he deep lea ning amily in which he ea u es ex ac ion is au oma ically op imised
by he ne wo k i sel on he basis o he inal classi ica ion/ eg ession s ep. P ecisely
o hese easons, CNNs a e widely used o isual objec ecogni ion hanks o he
abili y o effec i ely ecognize pa e ns (shapes, con ou s, colou s, e c..) independen ly
om sligh dis o ions, ansla ions o ans o ma ions.
Figu e 5.2: Example o a simula ed -spec um o a 137Cs sou ce (a) and i s 1-D ep esen-
a ion on a colo scale (b).
In his amewo k, a -spec um can indeed be conside ed as a one-dimensional
image whe e each channel co esponds o a pixel (Figu e 5.2). Pho opeaks, Comp on
edges and Comp on con inua ac as ea u es which uni ocally de e mines he adioiso-
ope ha has gene a ed i and hey can be de ec ed by mo ing a p ope il e along
he -spec um, independen ly on whe e hey a e loca ed. In summa y, CNN allows
a obus ea u es ex ac ion di ec ly on he aw spec um wi hou in o ma ion losses
using ew pa ame e s. Admi edly, CNN mimics he ac ions pe o med by o he me h-
ods, as shown in he lowcha epo ed in Figu e 5.3. This app oach has al eady been
p oposed by Liang e al.,al houghin ha case hene wo k equi esap elimina yda a
p epa a ion (2D mapping) and i was designed o only pe o m he iden i ica ion [95].
57
Figu e 5.3: Flowcha o a gene ic con olu ional neu al ne wo k.
Las ly, he emaining unsol ed p oblem is ha neu al ne wo ks can sol e ei he
classi ica ion o eg ession p oblems: independen ly om he c i e ia used o ea u es
ex ac ion, he las s ep always consis s in wo o mo e ully connec ed laye s bo h
in ANN and CNN. A classic mul i-laye a chi ec u e can only pe o m classi ica ion
because non-linea ac i a ion unc ions p e en he ne o concei e he inpu as a
supe imposi ion o spec a o single iso opes [92, 94]. On he o he hand, pu e linea
neu al ne wo ks can be en a i ely used o eg ession: in his case he ne wo k would
consis in jus an inpu and an ou pu laye wi h Nand Mneu ons, espec i ely, and
no ac i a ion unc ions (Figu e 5.4). I we use spec a whose iso opic composi ion is
known, we can adjus he weigh s connec ing hese wo laye s in o de o p oduce he
desi ed ec o o weigh s gi en a ce ain inpu spec um acco ding o
aj=Wjiciwi h i=1,...,N and j=1,...,M (5.3)
The ne wo k linea ly combines he numbe o coun s in each channel o calcula e he
iso opic weigh s. Howe e , he lack o non-linea i ies impedes o sa is y he uni e -
sal app oxima ion heo em by which he ne wo k can app oxima e any unc ion ( he
numbe o lea nable pa ame e s is ixed o he N⇥Mp oduc and addi ional laye s
would be edundan in pu e linea NN). Addi ionally, cons ain s canno be applied o
he ou pu o a linea neu al ne wo k. As a ma e o ac , Equa ion 5.3 is exac ly he
in e se o Equa ion 5.1 and he weigh s Wji ep esen he elemen s o he in e se o R,
hence he app oach would be equi alen o a leas -squa es i ing [89].
58
Figu e 5.4: A chi ec u e o a linea neu al ne wo k wi h Nand Mneu ons in he inpu
and ou pu laye s, espec i ely. The ne wo k calcula es he iso opic composi ion by linea ly
combining he numbe o coun s in he measu ed spec um.
59
5.3 P oposed me hod
5.3.1 Final a chi ec u e
In he p e ious sec ion, he essen ial ing edien s o pe o m ea u es ex ac ion, iden-
i ica ion and quan i ica ion o adioiso opes wi h neu al ne wo ks a e desc ibed. As
al eady men ioned, he p oblem is wo old. Ne e heless, doing bo h classi ica ion and
eg ession is possible by using mul i-objec i e neu al ne wo ks whe e, du ing aining,
he same pa ame e s a e adjus ed o accomplish wo diffe en p edic ions. The ne wo k
o ks a e he ea u es ex ac ion and assigning a diffe en ask o each b anch. The
inal a chi ec u e, ob ained a e se e al ials and e o s, is he e desc ibed in de ail
o a lib a y o eigh adioiso opes ( he eason o his numbe is explained la e ) and
i is shown in Figu e 5.5.
Figu e 5.5: A chi ec u e o he mul i-objec i e densely-connec ed con olu ional neu al ne -
wo k used o adioiso ope iden i ica ion and quan i ica ion. The ask o each block is indi-
ca ed.
The inpu consis s in a ec o o 2048 elemen s which co esponds o he aw
spec um no malised by he o al numbe o coun s. The "Fea u es ex ac ion" pa
p esen s i e con olu ional blocks o which he i s ou a e Densely Connec ed (DC)
ha is he ou pu o each block is he inpu o all he subsequen ones [99]. The
main limi in deep ne wo ks is in o ma ion low h ough he a ious laye s because
he weigh s o he i s laye s a e he las ones o be upda ed du ing aining (back-
p opaga ion). Since he upda e is based on he pa ial de i a i e o he loss unc ion
(i.e., he quan i y ha we wan o minimise du ing aining) he p esence o se e al
laye s leads o he so-called " anishing g adien p oblem" which could slow o e en
s op he ne wo k om lea ning (i.e., he alues o weigh s do no change). In DC-CNN
60
Table 5.3: P edic ions (in pe cen age) on he spec a o he es se wi h wo iso opes in a
1:1 a io. The esul s a e a e aged o e all spec a belonging o he same class, ega dless o
he s a is ics (2⇥103,2⇥104and 2⇥105). S anda d de ia ion is epo ed in b acke s.
Iso opes (1:1) 57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
57Co+60Co 50.3(7) 49.7(7) 0 0 0 0 0 0
57Co+133Ba 50.2(7) 0 49.8(7) 0 0 0 0 0
57Co+137Cs 50.5(9) 0 0 49.5(9) 0 0 0 0
57Co+192I 50.7(16) 0 0 0 49.3(16) 0 0 0
57Co+204Tl 49.9(7) 0 0 0 0 50.1(7) 0 0
57Co+226Ra 50.1(8) 0 0 0 0 0 49.9(8) 0
57Co+241Am 51.5(9) 0 0 0 0 0 0 48.5(9)
60Co+133Ba 049.9(12)50.1 0 0 0 0 0
60Co+137Cs 050.2(9)049.8(9)0 0 0 0
60Co+192I 049.9(13)0 050.1(13)0 0 0
60Co+204Tl 050.3(6)0 0 049.7(6)0 0
60Co+226Ra 049.3(7)0 0 0 050.7(7)0
60Co+241Am 051.4(6)0 0 0 0 048.6(6)
133Ba+137Cs 0050.1(19)49.9(19)0000
133Ba+192I 0049.9(17)050.1(17)000
133Ba+204Tl 0053.9(9)0046.1(9)00
133Ba+226Ra 0050.0(12)00050.0(12)0
133Ba+241Am 0052.3(16)000047.7(16)
137Cs+192I 00049.5(13)50.5(13)000
137Cs+204Tl 00050.1(9)049.9(9)00
137Cs+226Ra 00049.1(9)0050.9(9)0
137Cs+241Am 00051.3(8)00048.7(8)
192I +204Tl 000049.9(10)50.1(10)00
192I +226Ra 000049.0(10)051.0(10)0
192I +241Am 000049.9(8)0050.1(8)
204Tl+226Ra 0000049.6(9)50.4(9)0
204Tl+241Am 0000052.4(8)047.6(8)
226Ra+241Am 00000052.4(9)47.6(9)
67
Table 5.4: P edic ions (in pe cen age) on he spec a o he es se wi h wo iso opes in a
3:1 a io. The esul s a e a e aged o e all spec a belonging o he same class, ega dless o
he s a is ics (4⇥103,4⇥104and 4⇥105). S anda d de ia ion is epo ed in b acke s.
Iso opes (3:1) 57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
57Co+60Co 74.4(4) 26.6(4) 0 0 0 0 0 0
57Co+133Ba 74.1(12) 0 25.9(12) 0 0 0 0 0
57Co+137Cs 74.6(5) 0 0 25.4(5) 0 0 0 0
57Co+192I 75.1(6) 0 0 0 24.9(6) 0 0 0
57Co+204Tl 71.6(6) 0 0 0 0 28.4(6) 0 0
57Co+226Ra 75.8(4) 0 0 0 0 0 24.2(4) 0
57Co+241Am 72.1(5) 0 0 0 0 0 0 27.9(5)
60Co+133Ba 073.9(5)26.1(5)0 0 0 0 0
60Co+137Cs 074.8(5)025.2(5)0 0 0 0
60Co+192I 074.1(10)0 025.9(10)0 0 0
60Co+204Tl 072.0(4)0 0 028.0(4)0 0
60Co+226Ra 074.4(5)0 0 0 025.6(5)0
60Co+241Am 073.5(6)0 0 0 0 027.4(6)
133Ba+137Cs 0075.0(6)25.0(6)0000
133Ba+192I 0074.5(9)025.5(9)000
133Ba+204Tl 0076.2(9)0023.8(9)00
133Ba+226Ra 0075.5(7)00024.5(7)0
133Ba+241Am 0073.5(7)000026.5(7)
137Cs+192I 00073.9(13)26.1(13)000
137Cs+204Tl 00072.1(6)027.9(6)00
137Cs+226Ra 00073.9(7)0026.1(7)0
137Cs+241Am 00072.7(7)00027.3(7)
192I +204Tl 000072.6(8)27.4(8)00
192I +226Ra 000074.9(9)025.1(9)0
192I +241Am 000070.7(10)0029.3(10)
204Tl+226Ra 0000074.9(6)25.1(6)0
204Tl+241Am 0000073.1(4)026.9(4)
226Ra+241Am 00000073.4(7)26.6(7)
68
Table 5.5: P edic ions (in pe cen age) on he spec a o he es se wi h wo iso opes in a
1:3 a io. The esul s a e a e aged o e all spec a belonging o he same class, ega dless o
he s a is ics (4⇥103,4⇥104and 4⇥105). S anda d de ia ion is epo ed in b acke s.
Iso opes (1:3) 57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
57Co+60
Co 24.8(6) 75.2(6) 0 0 0 0 0 0
57Co+133
Ba 24.9(7) 0 75.1(7) 0 0 0 0 0
57Co+137
Cs 25.9(7) 0 0 74.1(7) 0 0 0 0
57Co+192
I 26.0(8) 0 0 0 24.0(8) 0 0 0
57Co+204
Tl 25.1(5) 0 0 0 0 74.9(5) 0 0
57Co+226
Ra 24.5(5) 0 0 0 0 0 75.5(5) 0
57Co+241
Am 25.9(6) 0 0 0 0 0 0 74.1(6)
60Co+133
Ba 025.7(5)74.3(5) 0 0 0 0 0
60Co+137
Cs 025.5(6) 0 74.5(6) 0 0 0 0
60Co+192
I 025.4(4) 0 0 74.6(4) 0 0 0
60Co+204
Tl 026.5(4) 0 0 0 73.5(4)0 0
60Co+226
Ra 025.6(4) 0 0 0 074.4(4) 0
60Co+241
Am 027.1(3) 0 0 0 0 0 72.9(3)
133Ba+137
Cs 0026.0(13)74.0(13)0 00 0
133Ba+192
I 0025.2(15)074.8(15)00 0
133Ba+204
Tl 0026.5(8)0 073.5(8)0 0
133Ba+226
Ra 0025.8(9)0 0 074.2(9)0
133Ba+241
Am 0029.0(11)0 0 0071.0(11)
137Cs+192
I 00 025.0(5)75.0(5)00 0
137Cs+204
Tl 00 026.3(4)073.7(4)0 0
137Cs+226
Ra 00 025.2(6)0 074.8(6)0
137Cs+241
Am 00 027.1(6)0 0072.9(6)
192I +204
Tl 00 0 025.7(5)74.3(5)0 0
192I +226
Ra 00 0 025.0(8)075.0(8)0
192I +241
Am 00 0 026.2(7)0073.8(7)
204Tl+226
Ra 00 0 0 025.9(7)74.1(7)0
204Tl+241
Am 00 0 0 022.7(8)077.3(8)
226Ra+241
Am 00 0 0 0 026.7(4)73.8(4)
69
5.4.2 Addi ional da ase s
Figu e 5.10: Th ee spec a o he 102coun s da ase o 57Co, 137Cs and 60Co sou ces.
Fi s ly, I applied he algo i hm on a da ase con aining spec a wi h 102coun s. This
is an o de o magni ude less han he ones used o aining and he s a is ical noise
is dominan (Figu e 5.10). Ne e heless, he ne wo k is s ill able o iden i y he co -
ec adioiso opes, al hough in some cases i de ec s he p esence o mul iple sou ces
(Table 5.6). Mos o he alse posi i es occu in he case o 60Co because coun s a e
dis ibu ed in he whole ene gy ange: he s a is ics is so low ha , i some e en s a e
mo e condensed in ce ain zones, he ne wo k de ec s he p esence o ano he sou ce.
Table 5.6: Con usion ma ix o he da ase con aining spec a wi h 100 coun s (100 spec a
o each iso ope). Columns do no sum o 100 because o alse mul iple iden i ica ion.
ACTUAL CLASS
PREDICTED CLASS
Iso ope 57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
57Co 100 8 1 1 0 0 0 0
60Co 0100 0 0 0 0 0 0
133Ba 018 100 7 1 1 0 0
137Cs 029 18 100 4 0 0 0
192I 017 0 0 100 0 0 0
204Tl 0 0 0 0 0 100 1 0
226Ra 010 0 0 0 0 100 0
241Am 1 0 2 0 1 11 0100
Secondly, I es ed he ne wo k on spec a p oduced by sou ces shielded by 5 mm o
s eel (204Tl and 241Am a e no p esen since hei -emissions a e comple ely s opped
by he abso be ). A enua ion is one o he mos p oblema ic aspec s o adioiso ope
iden i ica ion because he spec al shape is modi ied (Figu e 5.11). The esul s a e less
70
s able because he ne wo k s uggles o calcula e he linea combina ion which bes
econs uc s he inpu spec um (Table 5.7). Ne e heless, i is s ill able o iden i y
he co ec adioiso opes wi h ew double iden i ica ions (Table 5.8).
Table 5.7: Raw ou pu (CL. and REG.) and p ocessed ou pu (PROC.) o he ained ne
in he case o a spec um o a 133Ba sou ce a enua ed by 5 mm o s eel.
57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
CL. 0.039 0.012 1.000 0.438 0.225 0.000 0.000 0.014
REG. 0.032 -0.002 1.300 0.053 0.100 -0.188 0.073 -0.222
PROC. 00 1 000 0 0
Table 5.8: Con usion ma ix o he shielded da ase (5 mm o s eel) con aining spec a wi h
104coun s.
ACTUAL CLASS
PREDICTED CLASS
Iso ope 57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
57Co 100 0 0 0 0 0 0 0
60Co 0100 0 0 0 0 0 0
133Ba 0 0 100 0 0 0 0 0
137Cs 0 0 18 100 0 0 0 0
192I 0 0 0 0 100 0 0 0
204Tl 0 0 0 0 0 100 0 0
226Ra 010 0 0 0 0 100 0
241Am 0 0 0 0 0 0 0 100
Figu e 5.11: Example o he effec o shielding (5 mm o s eel) o a 133Ba sou ce.
Finally, I es ed he algo i hm also on a da ase con aining spec a wi h h ee
iso opes in a 1:1:1 a io (Table 5.9). Also in his case no classi ica ion e o s a e
de ec ed and he ne wo k co ec ly es ima es he ac ions.
71
Table 5.9: P edic ions (in pe cen age) on he da ase con aining spec a wi h h ee iso opes
in a 1:1:1 a io wi h 3⇥103,3⇥104and 3⇥105coun s. The esul s a e a e aged o e all
spec a belonging o he same class. S anda d de ia ion is 1.5% in all cases.
Iso opes (1:1:1) 57Co 60Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
57Co+60+Co133Ba 33.0 33.5 33.5 0 0 0 0 0
57Co+60
Co+137
Cs 33.7 33.5 0 32.8 0 0 0 0
57Co+60
Co+192
I 33.6 33.5 0 0 32.9 0 0 0
57Co+60
Co+204
Tl 32.0 33.2 0 0 0 34.8 0 0
57Co+60
Co+226
Ra 33.1 33.8 0 0 0 0 33.1 0
57Co+60
Co+241
Am 32.8 33.6 0 0 0 0 0 33.6
57Co+133
Ba+137
Cs 33.5 0 33.7 32.8 0 0 0 0
57Co+133
Ba+192
I 33.9 0 33.3 0 32.7 0 0 0
57Co+133
Ba+204
Tl 33.8 0 34.5 0 0 31.7 0 0
57Co+133
Ba+226
Ra 33.3 0 34.2 0 0 0 32.6 0
57Co+133
Ba+241
Am 33.4 0 34.6 0 0 0 0 32.0
57Co+137
Cs+192
I 34.1 0 0 32.8 33.0 0 0 0
57Co+137
Cs+204
Tl 32.4 0 0 32.6 0 35.0 0 0
57Co+137
C+226
Ra 33.4 0 0 33.1 0 0 33.5 0
57Co+137
Cs+241
Am 33.0 0 0 33.1 0 0 0 33.9
57Co+192
I+204
Tl 32.7 0 0 0 32.2 35.1 0 0
57Co+192
I +226
Ra 33.8 0 0 0 32.8 0 33.4 0
57Co+192
I +241
Am 32.7 0 0 0 32.0 0 0 35.3
57Co+204
Tl+226
Ra 32.1 0 0 0 0 35.8 32.0 0
57Co+204
Tl+241
Am 34.3 0 0 0 0 35.9 0 29.9
57Co+226
Ra+241
Am 34.3 0 0 0 0 35.9 0 29.9
60Co+133
Ba+137
Cs 0 33.5 33.8 32.6 0 0 0 0
60Co+133
Ba+192
I 0 33.9 33.1 0 33.0 0 0 0
60Co+133
Ba+204
Tl 0 34.6 34.6 0 0 30.9 0 0
60Co+133
Ba+226
Ra 0 33.7 33.5 0 0 0 32.8 0
60Co+133
Ba+241
Am 0 34.4 34.6 0 0 0 0 31.0
60Co+137
Cs+192
I 0 33.7 0 32.7 33.6 0 0 0
60Co+137
Cs+204
Tl 0 33.0 0 32.3 0 34.8 0 0
60Co+137
Cs+226
Ra 0 33.4 0 32.6 0 0 34.1 0
60Co+137
Cs+241
Am 0 33.3 0 32.8 0 0 0 33.9
60Co+192
I +204
Tl 0 33.2 0 0 32.6 34.2 0 0
60Co+192
I +226
Ra 0 33.7 0 0 32.8 0 33.5 0
60Co+192
I +241
Am 0 33.0 0 0 32.5 0 0 34.5
60Co+204
Tl+226
Ra 0 33.4 0 0 0 34.4 32.3 0
60Co+204
Tl+241
Am 0 35.4 0 0 0 35.6 0 29.0
60Co+226
Ra+241
Am 0 34.2 0 0 0 0 33.7 32.1
133Ba+137
Cs+192
I 0 0 33.5 33.2 33.3 0 0 0
133Ba+137
Cs+204
Tl 0 0 34.7 34.1 0 31.1 0 0
133Ba+137
Cs+226
Ra 0 0 33.6 33.1 0 0 33.3 0
133Ba+137
Cs+241
Am 0 0 34.6 33.7 0 0 0 31.7
133Ba+192
I +204
Tl 0 0 34.1 0 34.0 31.8 0 0
133Ba+192
I +226
Ra 0 0 33.5 0 33.0 0 33.5 0
133Ba+192
I +241
Am 0 0 33.6 0 33.2 0 0 33.2
133Ba+204
Tl+226
Ra 0 0 35.0 0 0 30.9 34.2 0
133Ba+204
Tl+241
Am 0 0 36.5 0 0 33.2 0 30.3
133Ba+226
Ra+241
Am 0 0 34.3 0 0 0 33.0 32.8
137Cs+192
I +204
Tl 0 0 0 32.6 32.8 34.6 0 0
137Cs+192
I +226
Ra 0 0 0 33.0 32.9 0 34.1 0
Con inued on nex page
72
Table 5.9 – con inued om p e ious page
Iso opes (1:1:1) 57Co 57Co 133Ba 137Cs 192I 204Tl 226Ra 241Am
137Cs+192
I +241
Am 0 0 0 32.4 32.5 0 0 35.1
137Cs+204
Tl+226
Ra 0 0 0 32.8 0 34.4 32.8 0
137Cs+204
Tl+241
Am 0 0 0 35.2 0 35.6 0 29.2
137Cs+226
Ra+241
Am 0 0 0 33.5 0 0 33.9 32.6
192I +204
Tl+226
Ra 0 0 0 0 32.3 35.1 32.6 0
192I +204
Tl+241
Am 0 0 0 0 35.0 36.5 0 28.5
192I +226
Ra+241
Am 0 0 0 0 32.7 0 33.5 33.8
204Tl+226
Ra+241
Am 0 0 0 0 0 35.6 34.9 29.5
73
5.5 Conclusions
This algo i hm allows o au oma ically iden i y he adioiso opes in -spec a in a sin-
gle s ep and o es ima e he ela i e ac ion o each de ec ed adionuclides. The main
s eng h o his app oach is exploi ing he iden i ica ion o he iso opic inge p in s
o egula ise he quan i ica ion s ep. This allows o o e come he a e age ene gy es-
olu ion pe o mances o CZT-based -de ec o s and p oduce accu a e p edic ions on
ex emely noisy spec a wi h low s a is ics, whe e s anda d algo i hms ail o succeed
o canno e en be applied. The algo i hm p esen s op imal pe o mances on all classes
o he es se (spec a wi h one o wo adioiso opes a diffe en s a is ics and a ios).
Addi ionally, i shows ema kable pe o mances on da ase s wi h much mo e com-
plex spec a no used o he aining (low s a is ics, h ee adioiso opes and shielded
sou ces). This p o es he imp essi e gene aliza ion abili y o he p oposed a chi ec u e
because he ne wo k makes p edic ions based on wha i lea ned om he aining
se . This is ex emely impo an in iew o u he es ing on diffe en lib a ies. My
esea ch g oup and me submi ed a pa en applica ion on his algo i hm.
This me hod combines some o he s a e-o - he-a echniques in da a science o
p oduce a eliable ool o he analysis o -spec a (densely connec ed blocks, all con-
olu ional neu al ne wo ks, ba ch no malisa ion and mul i-objec i e neu al ne wo k).
Fi s ly, he algo i hm exploi s he capabili y o CNN o inding and ex ac ing local
pa e ns in images. This in ui ion ep esen s he key o he effec i eness o he app oach
because he ela ion in he ene gy domain o he channels o a measu ed spec um is
analogous o he spa ial ela ion among he pixels in an image. Secondly, he quan i i-
ca ion o he ela i e ac ion o each iso ope is achie ed hanks o a double-objec i e
a chi ec u e. One b anch pe o ms he classi ica ion wi h which is possible o il e
he esul s o he eg ession b anch, occasionally unp edic able o e en unphysical.
Thi dly, he aining o he ne wo k is ein o ced by a ious expedien s, ecen ly p o-
posed in he li e a u e. The connec ion o each con olu ional block wi h all subsequen
ones (DC-CNN) allows a be e p opaga ion o he ex ac ed ea u es maps h ough
he laye s, which is usually a limi a ion o deep ne wo ks. Mo eo e , he numbe o
lea nable pa ame e s o he p oposed me hod is low compa ed o a classic ANN (mul i-
laye pe cep on). The e o e, he aining is ex emely as e en on a s anda d lap op
wi hou he need o cloud compu ing o g aphics p ocessing uni . A e ha , he only
inpu equi ed is he aw spec um, wi hou human in e en ion no in e media e da a
p ocessing. The me hod is ideal o po able o hand-held de ices in which he ene gy
consump ion and he compu a ional load mus be conside ed. In he case o CZT-
based de ices, he da ase can be ob ained om measu emen s, simula ions o bo h.
Syn he ic spec a a e usually p e e ed since he access o ce ain adioac i e sou ces
is limi ed, allowing g ea sa ings in e ms o ime and cos s. Howe e , in his case an
accu a e modelling o he esponse unc ion o he whole de ec ion sys em is equi ed.
CNN a e sensi i e o dis o ions o a limi ed ex en and a ce ain aul ole ance be-
ween simula ions and measu emen s is admissible. Ano he impo an consequence is
ha he me hod can be ex ended o any adia ion de ec o s whose esponse unc ion
74
can be simula ed.
Rega ding u u e wo ks, he me hod will be es ed on mo e adioiso opes wi h
he aim o eaching lib a ies o 20–30 sou ces. The only modi ica ion equi ed is he
numbe o neu ons in he wo ully connec ed ou pu laye s which mus ma ch he
numbe o iso opes (M). This will lead o a la ge numbe o pa ame e s, gi en by
he p oduc 4072 ·Mwhe e 4072 is he ou pu o he con olu ional pa (see Figu e
5.5). Ne e heless, his will no ep esen a p oblem since i would be compensa ed by
an equal inc ease o spec a in he da ase . The ac ha some adionuclides p esen
-emissions a nea ly he same ene gy is an appa en issue because he ne wo k does
no only exploi he posi ion o a ce ain ea u e bu also i s ela ion wi h he o he
ones. Fo example, he 2,0- ay o 137Cs and he 17,0- ay o ha e an ene gy o 661.66
and 662.28 keV, espec i ely: he co esponding pho opeaks canno be esol ed e en
conside ing he s a e-o - he-a s pe o mances o CZT-based RTSD (FWHM <1% ⇡
6keV). Ne e heless, he b anching a io is ex emely diffe en : in he case o 137Cs, i
is basically he only -emission (84.99 pho ons pe 100 nuclea disin eg a ions) whe eas
in he case o 239U hep obabili yislowe (0.17pho onspe 100nuclea disin eg a ions)
and, mo eo e , he spec um is popula ed by o he peaks wi h app oxima ely he same
in ensi y (21,0=819.26 and 22,0=844.1keV wi h 0.129 and 0.139 pho ons pe 100
nuclea disin eg a ions, espec i ely). Since he o e all spec al shape is diffe en , he
ne wo k should be able o dis inguish possible supe imposi ions. In some applica ions
(e.g., decommissioning) measu emen s a e long and he con ibu ion o he na u al
backg ound, p oduced by a mix u e o na u ally occu ing adioac i e ma e ials, is no
negligible. This al e s he spec al shape and could mislead he algo i hm. Howe e ,
he na u al mix u e is usually known and, consequen ly, he co esponding spec um.
The e o e, i is possible o add an addi ional class ela ed o his pseudo- adionuclide
wi h he pu pose o es ima ing i s ac i i y (p ocedu e al eady adop ed by o he wo ks
[92, 94]). Finally, he mos challenging goal is he es ima ion in he case o mix u e
o shielded spec a. The ac ha he same iso ope could appea in diffe en ways
is a limi o he eg ession b anch and he quan i ica ion is indeed p oblema ic. A
sui able aining wi h he same adioiso opes in a ious con igu a ions (diffe en ypes
o abso bing ma e ial and diffe en hickness) may be an op ion. Al e na i ely, he
a chi ec u e may be modi ied o de e mine i he measu ed spec um is a enua ed and
by which ma e ial: his would be undoub edly challenging bu s ill a iable op ion.
75
Chap e 6
P ojec s and o he ac i i ies
Du ing my Ph.D. I pa icipa ed in se e al p ojec s wi h o he esea ch g oups and com-
panies which sha ed he common pu pose o de eloping echniques o co ec spec al
dis o ions o CZT-based RTSD, making use o speci ic expe imen al measu emen s
and p ope models. In pa icula , in his chap e I will epo pa o he esul s o
he synch o on session in which I pa icipa ed, which was pe o med in collabo a ion
wi h he Depa men o Physics and Chemis y (DiFC) o he Uni e si y o Pale mo
(I aly) and he Ru he o d Apple on Labo a o y (Didco , UK). I will p esen a e -
wa ds he modelling o lux-dependen dis o ions which a e common in indus ial and
medical applica ions, pe o med in collabo a ion wi h Xnex s. .l., an I alian company
specialised in indus ial non-des uc i e inline inspec ions based on X- ays o quali y
con ols.
76
CSA a he cen e o he in e -pixel gap (R=0), and c1and c2a e calib a ed cons an s.
This allows o calcula e he cha ge de ici and eco e he co ec ene gy:
E=ECSA(R)+c2(1 R2)
1c1(1 R2)(6.2)
This p ocedu e does no depend on he ene gy o he pho on and, hus, i can be
applied o uncollima ed sou ces [111]. Cha ge Sha ing Co ec ion (CSC) is alid o
bo h he small and la ge a ay (Figu e 6.6c and 6.6d) e en in beam posi ion whe e
almos all e en s a e affec ed by sha ing. This is ex emely impo an o wo easons:
i s ly, coinciden e en s, which a e usually ejec ed, can be eco e ed; secondly, he
pa ame e Rcan be used o ex ac in o ma ion on he in e ac ion posi ion wi hin he
gap and, possibly, o achie e X- ay imaging wi h a sub-pixel spa ial esolu ion [112].
Figu e 6.6: (a) 2D sca e plo o he ene gy a e CSA o coinciden e en s (m=2) o
a collima ed beam posi ion a he cen e o he in e -pixel gap o he la ge a ay o he 3
mm de ec o plo ed e sus he cha ge sha ing a io. The ed line ep esen s he bes i ing
unc ion used o co ec cha ge losses. (b) Raw spec um o he cen al pixel (black line),
a e CSA (blue line) and a e CSC co ec ions ( ed line). (c) and (d) Raw (black lines) and
co ec ed spec a ( ed lines) o he cen al pixel o he pho on in e ac ion a he cen e o
he in e -pixel gap o he la ge ( op) and he small a ays (bo om) o he 3 mm de ec o .
83
6.2 Simula ion o peak pileup dis o ion
The abili y o p edic he beha iou o he de ice unde ce ain condi ions indeed un-
locks se e al possibili ies and in Chap e s 4 and 5 I showed wo me hods o exploi he
knowledge o he de ec o esponse unc ion. Howe e , he app oach desc ibes only
lux-independen dis o ions (Equa ion 4.5), which is no he case o oom empe a-
u e spec oscopic X- ay inspec ions (e.g., medical compu ed omog aphy, indus ial
inline analysis). In hese applica ion ields, educing acquisi ion imes is inc easingly
demanded and high pho on luxes allow highe s a is ics wi h he same in e al o
ime, al hough hey ine i ably in oduce spec al dis o ions. The limi s o he elec-
onic ead-ou chain in e ms o sampling equency, ini e pulse wid h and, especially,
dead ime esul in coinciden o pa ially o e lapped e en s whose pulse heigh does
no e lec anymo e he ene gy o he p ima y pho on (pileup effec ). Such spec al
deg ada ion impedes eliable quan i a i e analysis and a p elimina y co ec ion s ep
is necessa y. I DPP is employed, piled up e en s can be iden i ied by speci ic ea-
u es (e.g., anomalous ise ime o pulse wid h) and subsequen ly ejec ed [113, 110].
This allows o p ese e nice spec oscopic pe o mances e en a high luxes, al hough
aconside able ac iono e en sdoesno con ibu e o he inalspec a,hence educ-
ing he bene i o using high luxes o dec ease he acquisi ion ime. Implemen ing
co ec ions is no possible in DPP due o he in insic non-in e ibili y o he phe-
nomenon. On he o he hand, s a is ical app oaches exploi he in o ma ion con ained
in he whole spec um o edis ibu e coun s in he co ec channels. Th ough sui -
able app oxima ions, analy ic models ha e been p oposed in li e a u e and 1s o de
pileup dis o ions (i.e., wo o e lapped e en s) ha e al eady been success ully co ec ed
[114, 115, 116, 117, 118]. Ne e heless, ypical high- lux condi ions (up o 106–107pho-
ons/s/channel) a e affec ed by highe pileup o de s. A possible solu ion consis s in
p edic ing he esponse o he ead-ou sys em a gi en lux a e and exploi ing his
in o ma ion o en a i ely co ec he spec um. I success ul, spec al un olding can be
applied o emo e lux-independen dis o ions, ollowing a cascade model [119] (Figu e
6.7). The p esen wo k was de eloped in his amewo k in collabo a ion wi h Xnex
s. .l. (www.x-nex .com).
Figu e 6.7: Cascade model o simula e he esponse unc ion o a RTSD and o co ec
spec al dis o ions.
84
6.2.1 Pileup modelling
The ou pu signal o he CSP consis s o pulses wi h a as ise ime (⇠ ens o ns) and
long decay ime (⇠ ens o µs). A shaping s ep is manda o y o d as ically educe pulse
o e lapping while p ese ing he co ec ampli ude o each e en [3]. The ope a ion
can be pe o med ei he analogically by a linea ampli ie o digi ally bu he basic
p inciples a e he same. The empo al wid h o shaped pulses should be op imised o
be as sho as possible o minimize pulse o e lapping bu long enough o a oid ballis ic
de ici . The modelling o pileup s a s a he i s s age o he chain which does no
espond linea ly o he ou pu o he block ha p ecedes i . Assuming linea i y in
pho o-cu en gene a ion and pulse shaping, RTSD can be conside ed as ime in a ian
linea sys ems whose esponse S o simul aneous e en s is equal o he sum o he
esponse o he sepa a e e en s [114, 120]:
S(As( )+Bs( 0)) = S(As( )) + S(Bs( )) (6.3)
whe e sis he inpu signal o uni a y heigh , Aand Ba e ampli ude cons an s, Sis
he esponse o he sys em and 0is he empo al sepa a ion be ween he e en s.
This assump ion holds i he signal o each e en does no exceed he dynamic ange o
he sys em and in absence o c ys al pola isa ion which, ins ead, would p oduce ime
dependen esponse. In he case o CZT-based de ec o s hese assump ions a e usually
ue. A his s age, diffe en effec s a ise depending on how e en s pile up. Le E0and
E1be he heigh s o wo consecu i e pulses, 1 he ime in e al be ween hem and ⌧
he ins umen al dead ime. I 1<⌧, he woe en sa econside edasone.Depending
on he pulse shape and heigh s, he second e en may al e he eading o he i s
one and he elec onics eads a alue anging om E0and E0+E1(peak pileup).
By inc easing 1, he wopulses e u n obeconside edassepa a ee en sbu ,i he
signal o he i s one is no comple ely decayed, i al e s he eading o he signal o
he second one (" ail pileup"). The dead ime can be pa alyzable o non-pa alyzable:
in he o me case, each new e en which a i es wi hin he dead ime o he p e ious
one ex ends he ime in e al in which he elec onics is "dead" whe eas in he la e
case ⌧is cons an . He e, I will b ie ly epo he me hod I used o model he effec s o
pileup which is explained in de ail by Taguchi e al. [121], al hough he undamen al
concep s a e simila o o he wo ks [45, 120, 119]. The measu ed spec um wi h pileup
SP(E)can be exp essed as
SP(E)= ·P( ec| ⌧)
1
X
m=0
P(m| ec)·P(E|m)(6.4)
whe e is he inciden coun a e, is he acquisi ion ime, P( ec| ⌧)is he p oba-
bili y o e en s being eco ded, mis he pileup o de , P(m| ec)is he p obabili y ha
he eco ded e en s is due o a pileup e en o o de mand, inally, P(E|m)is he
p obabili y ha an e en o pileup o de mis measu ed wi h an ene gy E. The ac o
·P( ec| ⌧)simply ep esen s he numbe o coun s in he measu ed spec um.
85
Since he ime in e al be ween e en s ollows he Poisson dis ibu ion, P( ec| ⌧)can
be exp essed as ollowing:
P( ec| ⌧)=(1/(1 + ⌧)non-pa alyzable
exp( ⌧)pa alyzable (6.5)
whe e he ⌧p oduc de ines he a e age numbe o e en s in a ime in e al ⌧.Also
P(m| ec)assumes a diffe en exp ession depending on he ype o dead ime:
P(m| ec)=(( ⌧)mexp( ⌧)/m!non-pa alyzable
[1 exp( ⌧)]mexp( ⌧)pa alyzable (6.6)
In he case o non-pa alyzable de ec o , P(m| ec) ep esen s he p obabili y ha m+1
e en s occu s in a ime in e al ⌧whe eas, in he o he case, i is he p oduc o he
p obabili y o ha ing m+1 e en s sepa a ed by ime in e als sho e han ⌧and
he p obabili y o ha ing an addi ional e en sepa a ed by a ime in e al g ea e
han ⌧.I shouldbeno ed ha ealsys emsdono allexclusi elyin ooneo hese
wo ca ego ies, bu a he hey show a hyb id beha iou whe e one aspec p e ails.
None heless, Equa ion 6.6 can be used o de e mine a which lux pileup effec s a e
no negligible (Figu e 6.8).
Figu e 6.8: P obabili y o a pileup e en o o de ma diffe en alues o ⌧ o pa alyzable
and non-pa alyzable de ec o s. The ex endable dead ime o pa alyzable de ec o s quickly
leads o imp ac icable measu emen condi ions. A coun a e o 106e en s/s/channel and a
dead ime o 0.5 µsgi ea ⌧=0.5, o name bu one example.
The las e m, P(E|m),desc ibes heac ualspec alshape o eachpileupo de .
While peak pileup can be modelled s aigh o wa dly, he effec s o ail pileup canno be
de i ed easily. The solu ion p oposed by Taguchi e al. assumes s ong app oxima ions
and holds only o bipola pulses [121]. This model canno be applied he e because, as
explained la e , I adop ed a unipola pulse shaping echnique which gene ally p o ides
86
be e signal- o-noise- a io and less ail pileup bu also lowe pe o mances a high
luxes. The e o e, I used only he peak pileup model and I neglec ed he effec s o
ail pileup; any disc epancies in he inal esul s will be e alua ed in he ligh o his
app oxima ion. A p e ious s ep is necessa y o ob ain P(E|m)which is he calcula ion
o he measu ed ene gy Eas a unc ion o he piled up e en s o each pileup o de :
E(m)( 1,...,
m;E0,E
1,...,E
m).Basically, hisconsis sincalcula ing he esponseo
he elec onics o each possible combina ion o pulse ampli udes and ime in e als.
Fo app oxima ed pulse shapes (e.g., iangula , apezoidal), E(1) has an analy ical
o m bu i mus be de i ed nume ically o a bi a y pulse shape and highe o de s
[114, 120]. Fo m>1 his is compu a ionally un easible and Emis calcula ed by
combining wo e en s Em1and Em.Ano he equi edin o ma ionis hep obabili y
densi y unc ion o ime in e als o each pileup o de :
P( 1|m)= m
⌧m(⌧ 1)m(6.7)
Basically, P( 1|m)d 1desc ibes he p obabili y o ha e a second e en E1a e a ime
be ween 1and 1+d 1 om he i s e en E0a a ious m. Combining E(m)and
P( 1|m), hepileupma ixcanbecalcula edas
PU(m)(E,Em1,E
1)=ZP( 1|m)|E(m)( 1;Em1,E1)d 1(6.8)
Fo he sake o cla i y, PU(1)(E,E0,E
1) ep esen s he p obabili y ha wo pulses wi h
ene gy E0and E1, occu ing wi hin a ime in e al ⌧,a emeasu edasasinglee en
o ene gy E. The e o e, he in eg al conside s all possible 1which would lead o his
esul . I is impo an o no e ha PU(m)do no depend on and S(E0): heya e
cha ac e is ic o he conside ed elec onics and can be calcula ed only once. Finally,
he pileup spec a a e calcula ed ecu si ely as
P(E|m=1)=ZZ
E.S.
PU(1)(E,E0,E
1)S(E0)S(E1)dE0dE1
P(E|m>1) = ZZ
E.S.
PU(m)(E,Em1,E
0)S(E0)P(Em1|m1)dE0dEm1
(6.9)
whe e E.S. is he whole ene gy space ([0,1] o all a iables o in eg a ion) and Sis he
ene gy dis ibu ion o he e en s (i.e., he spec um a low lux). Fo m>1spec a a e
calcula ed by combining wo pulses, he i s o which is he esul s o all he p e ious
pileup o de s. This ecu si e app oach allows o g ea ly simpli y he calcula ions,
al hough gi ing an app oxima ed pic u e o he eali y. Finally, he weigh ed sum o
all P(E|m)gi es he pileup spec um (Equa ion 6.4).
87
6.2.2 Measu emen s and simula ions
The pileup model desc ibed p e iously was applied on measu emen s acqui ed du ing
he synch o on session (see he p e ious Sec ion 6.1). The cen e o he cen al pixel
o he la ge a ay (500 µm pi ch) o he 1 mm- hick de ec o was i adia ed wi h a
collima ed beam a 20 keV. This allowed o a oid dis o ions caused bo h by cha ge
sha ing and luo escence emissions and o ocus he a en ion on lux-dependen dis-
o ions. Pho on luxes a ied om 10 kcps up o ⇡500 kcps. The ou pu wa e o ms
o he "PIXIE" ASIC we e sa ed o iles o be p ocessed offline. Figu e 6.9 shows
examples o p e-ampli ied signals a diffe en luxes.
Figu e 6.9: Wa e o ms (signals p e-ampli ied by he CSP) a a ious pho on luxes (indica ed
in kilocoun s pe second o kcps) o a collima ed beam a 20 keV i adia ing he cen e o a
pixel o he 500 µm pi ch a ay o he 1 mm- hick PIXIE de ec o .
Figu e 6.10: Examples o he Single Delay Line shaping wi h a apezoidal il e pe o med
on sepa a ed and pa ially o e lapped pulses.
88
Pulse shaping was pe o med using he Single Delay Line (SDL) echnique which con-
sis s in sub ac ing he a enua ed and delayed signal o he o iginal one [3, 122]. The
a enua ion is equi ed in o de o a oid unde shoo and ob ain a pole ze o cancella-
ion. The signal is hen il e ed wi h a apezoidal ke nel o limi he ex en o noise
(Figu e 6.10). I is impo an o men ion ha he pulse wid h (⇡50 ns), gi en by
he shaping delay and he ising and dec easing edges o he apezoid, was no op i-
mised o high lux condi ions and i was chosen delibe a ely long o enhance spec al
dis o ions caused by pileup (Figu e 6.11).
Figu e 6.11: Measu ed spec a o mono-ene ge ic synch o on ligh a 20 keV a diffe en
pho on luxes on linea (le ) and loga i hmic scale ( igh ). By inc easing he coun a e,
a ious pileup o de s appea (up o 3 d o de ).
Gene ally, he dead ime can be ob ained by measu ing he numbe o obse ed
coun s e sus he numbe o expec ed coun s a ying he pho on lux. The esul ing
cu e can be i ed wi h a p ope model (Equa ion 6.5) and he co esponding alue
can be ex ac ed. SDL p ocessing has a pa alyzable dead ime o 279 ns in he p esen
case (Figu e 6.12). Howe e , his p ocedu e p o ides an a e age dead ime whe eas,
ac ually, i changes depending on he ampli udes, he supe imposi ion and he numbe
o in ol ed pulses [123]. As a ma e o ac , in he case o SDL he dead ime o m=1
should co espond o he wid h o shaped pulses: he elec onics is able o dis inguish
wo consecu i e e en s i he ime in e al is g ea e han hei wid h. The e o e, using
he alue ex ac ed wi h Equa ion 6.5 in Equa ion 6.7 would lead o a miscalcula ion
o he shape o E(1)( 1;E0,E
1).Fo his eason,unlikeo he wo ks[121,120,119],I
de ined wo diffe en dead imes, ⌧L=279ns and ⌧P=50ns, ela ed o coun losses
(Equa ion 6.5 and 6.6) and pulse disc imina ion (Equa ion 6.7), espec i ely. Since
highe pileup o de s a e ob ained ecu si ely, he same alue o ⌧Pwas used o e e y
m.
89
Figu e 6.12: Inpu and ou pu coun ing a e a a ious pho on luxes.
The ideal pulse shape wi h uni a y heigh o be used in he model was ob ained
by a e aging he signals o comple ely sepa a ed pulses om he measu emen s a 10
kcps. The simula ed pileup spec a we e ob ained by using he low lux spec um as
S(E0)(Figu e 6.13). Thanks o he use o he ue pulse shape, ob ained om p e-
ampli ied wa e o ms a low luxes, he o e all spec al shape is well ep oduced o all
pileup o de s. App oxima ed pulse shapes, al hough allowing analy ical exp essions
o E(m), do no p edic co ec ly he pileup spec um [117]. Mo eo e , he dis o ions
caused by ail pileup a e no no iceable, as expec ed, and he peak pileup model is
sufficien in he case o SDL shaping. Howe e , he weigh s P(m| ec)unde es ima e
pileup dis o ions a low luxes. The disc epancies change by a ying he coun a e and
pileup spec a a e unde es ima ed a high luxes. The e a e wo possible explana ions.
Fi s ly, he ecu si e model (Equa ion 6.9) assumes ha he supe imposi ion o wo
pulses p oduces a new pulse wi h he same shape and an ampli ude anging om
he ampli ude o he o me and he sum o he ampli udes o bo h pulses, which is
hen combined wi h ano he one. Since PU(m)(E,E0,...,E
m)can be calcula ed jus
once, i can be conside ed o igo ously ob ain hem combining he ue numbe o
pulses. Secondly, he alue o ⌧Pshould be adjus ed depending on he combina ion
o he in ol ed pulses. Tha being so, he model is p omising bu i s accu acy is no
good enough o a emp some co ec i e app oach. Analogously o spec al un olding
(Chap e 4), simila algo i hms can indeed be applied i he o wa d s ep (p edic ion
o lux-dependen dis o ions) is eliable.
90
Figu e 6.13: Measu ed and simula ed spec a a a ious pho on luxes.
91
Chap e 7
Conclusions
Du ing my Ph.D. I pa icipa ed in a ious ac i i ies hanks o he se e al p ojec s and
collabo a ions o my esea ch g oup. Being in ol ed in each phase o he ealiza ion o
a adia ionde ec o , omc ys alp ocessing obondingi o hededica edelec onics,
and in he expe imen al in es iga ion o i s p ope ies and pe o mances was inc edibly
o ma i e. I helped me in de eloping an o e all iew o he opic and he abili y o
co ec ly in e p e he measu emen s de i ing om his class o de ices. This enabled
he de elopmen o echniques o co ec spec al dis o ions and o ully exploi he
in o ma ion con ained in measu ed spec a.
Spec al un olding
The quali y and homogenei y o CZT c ys als is s ill imp o ing hanks o he p og ess
in ma e ial echnology. Despi e he imp essi e signal- o-noise a io achie ed by s a e-
o - he-a de ices, he in e ac ion o adia ion wi h he semiconduc o compound will
always limi he ull-ene gy abso p ion efficiency. The pa ial ene gy deposi ion, mainly
caused by Comp on sca e ing and pai p oduc ion, canno be o e come excep by using
absu dly la ge c ys als, which is s ill un easible. Spec al un olding can de ini ely help
in aking a u he s ep in he di ec ion o ealizing a de ec o de oid o ins umen al
a i ac s, especially in he case o s anda d comme cial de ices, whose pe o mances
a e poo e . In his hesis I p esen ed a ela i ely simple ye effec i e me hod based
on a gene ic algo i hm o pe o m his ask on spec a wi h a e age ene gy esolu ion.
The alida ion on expe imen al spec a ob ained wi h diffe en de ec o s is p omising
o u u e in es iga ions.
Radioiso ope iden i ica ion
A his his o ic momen machine lea ning is appea ing, o e whelmingly and elen -
lessly, in e e y scien i ic ield whe e da a analysis is s ill affec ed by high ailu e a es.
Radioiso ope iden i ica ion alls in o his ca ego y due o he ex eme complexi y and
a iabili y o ce ain eal scena ios. The e o e, he use o machine lea ning-based ap-
p oach is he na u al consequence, as wi nessed by he scien i ic wo ks published on
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