Folding and unfolding NaI spectra Magne Guttormsen Department of Physics and SAFE

Folding and unfoldingNaI spectra

Magne GuttormsenDepartment of Physics and SAFE

University of Oslo

Nuclear Physics Group seminars, Oslo 15th March 2010

Not perfect detectorsWe measure electron energies, not ’s!● Photo-electric effect● Compton scattering● Pair production● Radiation from environment (Pb)● Threshold

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γ

Nuclear Physics Group seminars, Oslo 15th March 2010

The problem (I)incomming

measured

N

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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

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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 jj

∑ = 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

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f1

f2

MfN

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟=

R11R12L R1N

R21R22L R2N

LRN1RN 2L RNN

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

u1

u2

MuN

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

●Inverting response matrix R

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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:

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f =Ru

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(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 methodWe 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