folding and unfolding nai spectra magne guttormsen department of physics and safe university of oslo
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Folding and unfoldingNaI spectra
Magne GuttormsenDepartment of Physics and SAFE
University of Oslo
Nuclear Physics Group seminars, Oslo 15th March 2010
Not perfect detectors
We measure electron energies, not ’s!
● Photo-electric effect
● Compton scattering
● Pair production
● Radiation from environment (Pb)
● Threshold
€
γ
Nuclear Physics Group seminars, Oslo 15th March 2010
The problem (I)
incomming
measured
N
€
γ
E
€
γN
E
e
e
Unfolded u
Raw r
u1 u2
Nuclear Physics Group seminars, Oslo 15th March 2010
The problem (II)
●Easy to fold
●Difficult to unfold
€
r(Eγ ) ≈ f (Eγ ) = u1 ⋅R1(Eγ ) + u2 ⋅R2(Eγ )
u(Eγ ) = unfold r(Eγ ) with all R(Eγ )
f (Eγ ) = R j (Eγ )u j
j
∑ = R ⋅u ⋅dE∫
known = r known unknown
Nuclear Physics Group seminars, Oslo 15th March 2010
The response matrix
Nuclear Physics Group seminars, Oslo 15th March 2010
The solution?
●Channel by channel
€
f1
f2
M
fN
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
=
R11R12L R1N
R21R22L R2N
L
RN1RN 2L RNN
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
u1
u2
M
uN
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
●Inverting response matrix R
€
f =Ru⇒
R−1f =R−1Ru = Iu⇒
u =R−1f
●But large fluctuations in u
Nuclear Physics Group seminars, Oslo 15th March 2010
Alternative solution
●Exploit that we know how to fold:
●Control by iteration:
€
f =Ru
€
(i) First trial function :u0 = r
(ii) First folded spectrum: f 0 =Ru0
(iii) Correct for how much we fail : u1 = u0 + (r − f 0)
(iv) Second folded spectrum: f1 =Ru1
(v) The third trial function : u2 = u1 + (r − f1 )
and so on until f i ≈ r.
Nuclear Physics Group seminars, Oslo 15th March 2010
Iterations
Nuclear Physics Group seminars, Oslo 15th March 2010
Compton subtraction method
We define the Compton spectrum c = r - v,
where photo-peaks (=u) + s.e. + d.e. + 511 keVare subtracted from the raw spectrum. The u-spectrum is obtained from about 30 iterations and has large fluctuations.Next step is to smooth c, and subtract it from the raw spectrum giving an unfolded spectrum with fluctuations comparable with the raw spectrum. For further details, see NIM-paper.
Nuclear Physics Group seminars, Oslo 15th March 2010
Examples of unfolding
Nuclear Physics Group seminars, Oslo 15th March 2010
Comparison
Raw
Old
New
Oldsmoothed
Summary
● Consistent unfolding and folding
● Reduced fluctuations with Compton subtraction method
● See M. Guttormsen et al., NIM A 374 (1996) 371-376