1

Induced Radioactivity at Accelerators

Lecture 1: Basic Principles & Activation of Accelerator Components

J. Donald CossairtPh.D., C.H.P.

Applied Scientist IIIFermi National Accelerator Laboratory

2

Introduction• Goal: Describe radioactivation at accelerators• Will Discuss in Lecture 1

– Basic physical principles– Production of radioactivity in accelerator components

• Will Discuss in Lecture 2– Production of radioactivity in environmental media (e.g., air, soil.

rock)– Propagation of radioactivity in environmental media– Use of induced radioactivity in radiation measurements

• Beware: Will use cgs (centimeter, gram, second) units extensively, consistent with general practice!

3

General principles• Cross section is an important concept

– Represents the effective size of the atom or nucleus for a specified process

• Consider a beam ofFluence Φ (particles cm-2)Incident on a slab of material of thickness dx (cm)Material has number density of N (atom cm-3)

N=ρNA/A with density ρ (g cm-3)Avogadro’s number NA (6.02 x 1023 g-mole-1)A = atomic mass number of the material

4

General principlesChange (loss) dΦ in fluence due to

interactions in dx is-dΦ =σ NdΦ dx (1)

σ (cm2) is the cross sectionCross sections are

– Used to describe many kinds of processes– In this context activation– Often tabulated in barns or submultiples;

mb, μb, pb, nb, fb, etc.– 1 barn = 10-24 cm2

5

General principles

• Back to -dΦ =σ NdΦ dx (1)• Assuming

– Only one process present (e.g. activation)– One starts with initial fluence Φ0

• After finite material thickness x, integrate to get

Here σ is the activation cross section.

(( ) 2)N xox e σ−Φ = Φ

6

The activation equation

• Activation can occur at all accelerators• Usually have a reaction threshold; an

energy that must be exceeded to allow the process to proceed

• Characterized by cross sections– Nearly always energy-dependent, incident

particle-dependent, etc.• Aside: Nuclear reactions can produce

non-radioactive nuclides as well!

7

The activation equation

• Activity A refers decay rate:The # of atoms decaying per unit time– Customary unit: Curie (Ci) = 3.7x1010 s-1

– SI unit is Bequerel (Bq) = 1 s-1

– 1.0 Bq is small, use multiples; e.g. Gbq (109 Bq)

• Specific activity;– Activity per unit mass (e.g., Bq kg-1)– Activity per unit volume (e.g., Bq m-3)

8

The activation equation• Radioactive decay- a random process

– Characterized by a mean-life, τ (units of time)

– Decay constant λ = 1/τ[Unfortunately, symbol λ is used for lots of other stuff

in health physics!]

• If Ntot(t) atoms are present at time t, since decay is random,

( ) 1( ) ( ) ( ) (3)tottot tot tot

dN tA t N t N tdt

λτ

= − = =

9

The activation equationIf at time t=0, Ntot(0) atoms are present,

then at later time t=T (a simple integration) :

History leads to tabulation of half-lives t1/2rather than mean-lives t:

(5)

National Nuclear Data Center at BNL reference gives latest accepted values of t1/2.

( ) (0)ex (p( ) (0)exp( ) 4)tot tot totA T N T A Tλ λ λ= − = −

τ = = =12

10 693

14421 2 1 2 1 2ln ../ / /t t t

10

The activation equationSteady irradiation, most simple case

– Start at t=0– Constant flux density, energy, geometry, etc. – Material is uniform and uniformly irradiated– Neglects self-absorption– Irradiation period ti ends at t=ti– Followed by decay period, called cooling time

denoted tc that begins at t=ti and ends at t=ti+tc

– Use a single cross section σ (averaged over components (e.g., energy, particle types, as needed)

11

The activation equationNumber of atoms/unit volume at time n(t) of the

nuclide produced will be given by:(6)

– n(t) is, e.g., atoms cm-3, of produced atoms

– N is number density of target nuclei in consistent units.φ is the flux density of incident particles (e.g., cm-2s-1)

Eq. (6) balances the loss by decay (1st term) against the gain from production (2nd term).

dn tdt

n t N( )

( )= − +λ σφ

12

The activation equationEq. (6):has the solution for 0<t<ti:

(7)

Fromspecific activity is easy to get, just multiply by λ:

If you really want Ci cm-3, divide by 3.7x1010 Bq Ci-1.

(3)( ) ( ) tot totA t N tλ=

dn tdt

n t N( )

( )= − +λ σφ

( )n tN t( ) = −

σφλ

λ1 e-

( ) -3( ) 1 (Bq cm ) (8)ta t N e λσφ −= −

13

The activation equationAt the end of the irradiation (t=ti), the specific

activity is (9)

Characterized by a buildup from zero to a saturation concentration = Nσφ, the production rate. Warning: A lot of people don’t understand this, think radioactivity continues to build up forever to infinite values!!!!

{ }a t N ti i( ) exp( )= − −σφ λ1

14

The activation equationAfter irradiation ceases, specific activity will decay

according to the mean-life as a function of tc.The result is the activation equation:

Can get total activity A– For homogenous target multiply by the volume– For complex case with non-uniform flux densities,

target materials etc., do by sub-volumes.– May need to integrate over energy spectra, particle

types, etc.

{ }{ } -3( ) 1 exp( ) ex (Bq cmp( )( )) 10c i ca t N t tσφ λ λ= − − −

15

Activation at electron accelerators-general phenomena• The electrons produce photons copiously• Through secondary reactions, photons produce

charged particles and neutrons (hadrons) subject to strong (i.e., nuclear) force

• These secondary particles produce activation• Important fact: Most dose received by workers

is due to activated components, not direct beams unless shielding against prompt radiation is deficient!

• Topics in this chapter can help with ALARA!

16

Activation at electron accelerators-general phenomena• Bremsstrahlung mechanism of photon production is

generally dominant.– Electron cross sections are generally 2 orders of magnitude

less than photon cross sections.– Photons produce neutrons that activate.

• Reference by Barbier (1969) still useful for global results

• In general the larger cross sections are for reactions with the fewest nucleons removed– Especially true at low energies.

• Examples from Barbier are given in Fig. 1.

17

Fig. 1

18

Fig. 1

19

Results for electrons at low energies

• Swanson: Approximation B of electromagnetic shower theory (Rossi-Griesen, done for basic physics purposes)

• Ee < 35 MeV– Rapidly varying cross sections, energy-dependent

• Reactions considered:(γ,n), (γ,p), (γ,np), & (γ,2n)

• Other reactions ignored due to– small cross sections– higher energy thresholds

20

Results for electrons at low energiesInduced activity as function of energy follows neutron yields. See Fig.

2

21

Results for electrons at low energies

Swanson• Assumed target absorbs ALL of the beam power• Irradiated for an infinite time

with no cooldown; tc = 0• Thus calculated saturation activities• Normalized to incident beam power (kW-1)• Likely accurate to about + 30 %• In this regime, point source approximation is good• Also get specific gamma ray constants Γ

– Connects absorbed dose rate at 1 m to activation

– Includes all photons even internal bremsstrahlung and β +annihilation

it = ∞

22

Results for electrons at low energiesTable 1

23

Results for electrons at low energiesTable 1

24

Results for electrons at higher energies• With increased energy, more reaction channels open• Swanson calculated results, again using shower theory

– Provided Table 2, similar to Table 1– Good within about X 2 if well above reaction threshold

– Gives specific gamma constants Γ– Assumes electrons totally absorbed in material– No self-shielding– As before, for saturation conditions; infinite irradiation, zero

cooldown; tc = 0, – Works well because of lack of energy dependence in reactions

it = ∞

25

Results for electrons at higher energiesTable 2 (excerpt)

26

Results for electrons at higher energiesTable 2 (excerpt)

27

Results for electrons at higher energies

• Barbier has calculated cooling curves– For infinite irradiations at 1 electron s-1

– Assumed point source conditions– Gives absorbed dose rates (mGy h-1)– Given per MeV of incident electron energy

(works because of limited energy dependence, amounts to scaling with the beam power.)

it = ∞

28

Results for electrons at higher energiesFig. 3 shows an example of these cooling curves

29

Activation and proton and ion accelerators

• Activation will happen for protons > few MeV• For other ions, similar general behavior at specific

energies above a few MeV/amu (or MeV/nucleon)• There are exothermic reactions where activation occurs

at much lower energies.• General observation: If prompt radiation is properly

shielded, dose due to work with activated components will dominate personnel exposures.

• At low energies, one has reactions like (p,γ), few and multi-nucleon transfer reactions.

30

Activation at proton and ion accelerators

Nuclear reaction Q-value Qv is importantFor reaction m1+m2->m3+m4

equivalently denoted m2(m1,m3)m4

Qv=[(m1+m2)-(m3+m4)]c2 (11)[Get mass values from tables of nuclear masses, or mass differences, e.g. from

Tuli (2005).]

Qv > 0 is an exothermic reaction (releases energy, rare)

Qv < 0 is an endothermic reaction (most are endothermic)Endothermic reactions have a threshold below which they will not

occur:1 2

2

(12) th vm mE Q

m+

=

31

Activation at proton and ion accelerators

Barbier has addressed proton and ion activations• Global calculations, many types of particles• Used approximations to address data “gaps” as of 1969• Still very helpful for rough estimates• Inclusive processes;

– Low energies, few and multi-nucleon transfers– Higher energies includes spallation processes– At still higher energies include pion (π+ ) or kaon ( K+) production– In general reaction thresholds for more complicated processes or

those involving more nucleons or “exotic” particles, will be higher.Results: cross sections given in Figs. 4 and 5 (examples)

Results for light elements are useful for environmental considerationsResults for iron and copper are useful for accelerator components.

32

Activation at proton and ion acceleratorsFig. 4

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Activation at proton and ion acceleratorsFig. 4

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Activation at proton and ion acceleratorsFig. 5

35

Activation at proton and ion acceleratorsFig. 5

36

Activation at proton and ion acceleratorsCohen has studied thick target yields “systematically” for Thick targets are those which range-out the incident particle/ionFig. 6 is a representative plot showing general features.

it = ∞

37

Activation at proton and ion acceleratorsTable 3 gives values for specific processes.An excerpt from Table 3 is given below:

38

Methods of systematizing activation due to high energy hadronsAs the energy (or specific energy) rises

• Neglect of secondary reactions becomes important• Cannot limit to few and multi-nucleon transfer reactions unless

less than about 40 MeV (or 40 MeV/nucleon for ions)• Thresholds for producing many more nucleons are exceeded as

energy rises• Finally, at high energies (i.e., > than about 1 GeV or 1

GeV/nucleon), all radionuclides with mass numbers < that of the target can be produced.

• Many variations in the details.

39

Systematizing activation …

As the energy (or specific energy) risesTable 4 lists typical radionuclides seen at proton/ion accelerators

– Only includes those with 10 min < t1/2< 5 y– Pure β- (electron) emitters are not included (not of much external

exposure impact with volume activation at accelerators)– Pure β- (positron) emitters are included (two 0.511 MeV γ-rays with

each decay)

40

Systematizing activation …Table 4(excerpt)

41

Systematizing activation …Table 4 (excerpt)

42

Systematizing activation-Sullivan-Overton approximationA clean simple, method of addressing the production of many radionuclides

would sure be useful!Sullivan and Overton have shown us how.Consider absorbed dose rate δ(ti,tc) due to activation [ref. Eq. (10) ]

Here:

φ is the flux density of particles incident on the material

• G depends onbeam energybeam particle typesecondary particle productiongeometry (point source, line source, etc.)energy of γ-rays emittedγ-ray attenuation coefficients

( , ) 1 exp( ) exp( ) ( 13)i c i ct t G t tδ φ λ λ⎡ ⎤= − − −⎣ ⎦

43

Sullivan-Overton approximation

Will perform an integration over multiple radionuclidesTake dm as the # of radionuclides with decay constants λ between

λ and λ +d λIncrement in absorbed dose due to them is d δ(ti,tc). Then have

Commonly, G can be assumed to be independent of λ. Then integrate

where λο is the smallest decay constant (longest mean-life) considered.

Nature is simpler than it appears to be! See Fig. 7.

( , ) 1 exp( ) exp( ) ( 13)i c i ct t G t tδ φ λ λ⎡ ⎤= − − −⎣ ⎦

( , ) 1 exp( (15) exp( )) i c i cd t t dmG t tδ φ λ λ⎡ ⎤= − − −⎣ ⎦

( , ) 1 exp( ) exp( ) (16)i c i co

dmt t G d t tdλ

δ φ λ λ λλ

∞⎡ ⎤= − − −⎣ ⎦∫

44

Sullivan-Overton approximationFig. 7

45

Sullivan-Overton approximationFig. 7

46

Sullivan-Overton approximation

For t1/2 between about 10-3 and 103 days a good fit to these values is:

Here N (t1/2) is # of radionuclides < t1/2

a and b are fitting parametersSince decay constant, half-life, and mean-life have one-to-

one correspondance:

This is the keystone of the Sullivan-Overton approximationwhere now m(λ) is the # of radionuclides > t1/2.

1/ 2 1/ 2( ) ln (16( ) ) N t a b t= +

( ) ln ( 7) 1m a bλ λ= +

47

Sullivan-Overton approximation

Proceeding:

Substituting [into Eq. (15) ]

Changing variables; α=λtc and α’=λ (ti+tc); helps

( )( ) ) 18dm bd

λλ λ

=

( , ) 1 exp( (15) exp( )) i c i cd t t dmG t tδ φ λ λ⎡ ⎤= − − −⎣ ⎦

( ) (1

( , ) 1 e

9

xp( ) exp( )

exp( ) exp )

i c i c

c i c

o

o o

dt t Gb t t

d dGb t t t

λ

λ λ

λδ φ λ λλ

λ λφ λ λλ λ

∞

∞ ∞

⎡ ⎤= − − −⎣ ⎦

⎧ ⎫= − − − +⎡ ⎤⎨ ⎬⎣ ⎦⎩ ⎭

∫

∫ ∫

( )(( , ) 20)

o c o i ci c t t t

e et t Gb d dα α

λ λδ φ α α

α α

′− −∞ ∞

+

⎧ ⎫′= −⎨ ⎬′⎩ ⎭

∫ ∫

48

Sullivan-Overton approximation

The integrands are now the same, we can rearrange the limits of integration:

From tables of integrals, get the solution:

Substituting:

Evaluating:

( )(( , ) 20)

o c o i ci c t t t

e et t Gb d dα α

λ λδ φ α α

α α

′− −∞ ∞

+

⎧ ⎫′= −⎨ ⎬′⎩ ⎭

∫ ∫

( )(2, 1)( ) o i c

o ci c

t t

t

et t Gb dαλ

λδ φ α

α

−+= ∫

22 2 3 32

11

(22ln ... 1! 2 2! 3 3

)!

ax xx

xx

e dx ax a x a xxx

⎡ ⎤= + + + +⎢ ⎥× ×⎣ ⎦

∫2 3( )

( )

ln ... 4 1

(238

)o i c

o c

o ci

o c

t t

t

t t

t

e dαλ

λ

λ

λ

α α αα αα

−++

⎡ ⎤= − + − +⎢ ⎥

⎣ ⎦∫

( , ) ln ... (24) i ci c o i

c

t tt t Gb tt

δ φ λ⎡ ⎤⎛ ⎞+

= − +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

49

Sullivan-Overton approximation

For large mean-lives, λο -> 0.

where several constants have been merged into B.

Thus

( , ) ln ... (24) i ci c o i

c

t tt t Gb tt

δ φ λ⎡ ⎤⎛ ⎞+

= − +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(2( , ) ln 5)i ci c

c

t tt t Bt

δ φ⎛ ⎞+

≈ ⎜ ⎟⎝ ⎠

50

Gollon’s Rules of Thumb

Gollon collected 4 Rules of Thumb for high energy hadronsRule 1: For gamma-emitters encountered at accelerators

(i.e., ranging from 100 keV to 10 MeV):

dD/dt (Gy h-1) is absorbed dose rate at r (m) from point source

S (GBq) is the source strength

Eγi are individual photon energies present inclusive of branching fractions

[For dD/dt (rads h-1), 0.4 replaces 1.08 x 10-4. An integration must be performed for non-point sources. If distance r is in U. S. feet, S is in Curies, and dD/dt is in rads h-1, this is the familiar “6CE” rule.]

where

( )42 (1.08 1 26)0 i

i

dD S Edt r

γ−= × ∑

51

Gollon’s Rules of Thumb

Rule 2: In many materials• About 50 % of nuclear interactions produce a

radionuclide atom with a half-life > a few minutes• About 50 % of these have a half-life > about 1 day.

radionuclide atom exceeding about 1 day.

Terminology: Individual inelastic nuclear interactions are called stars.

One commonly speaks of total star density (stars cm-3) or star density rate (stars cm-3 s-1) or star density per incident particle (stars cm-3 incident particle-1)

Thus about 25 % of the nuclear interactions produce a

52

Gollon’s Rules of Thumb

Rule 3: Eq. (25), the result of the Sullivan-Overton approximation:

Rule 4: In a hadronic cascade, each proton or neutron produced about 4 inelastic interactions for each GeV of incident kinetic energy.

(2( , ) ln 5)i ci c

c

t tt t Bt

δ φ⎛ ⎞+

≈ ⎜ ⎟⎝ ⎠

53

Gollon’s Rules of Thumb

Examples in the use of these rules may be helpful

1) In a short target, about 10 % of an incident beam of 1011

protons s-1 interacts for several months of stable operation.

(activity) is 2.5 x 109 Bq (68 mCi)If each decay emits a 1 MeV γ-ray, Rule 1 gives

dD/dt = 0.27 mGy h-1 at 1 m (if a point source).

Use Rule 2, after 1 day of shutdown, the decay rate

54

Gollon’s Rules of Thumb

More examples

2) Rule 3

can be useful for predicting dose rates later in a shutdown from a measurement at one point in time after the shutdown begins. Note Rule 3 does not invoke “point source” conditions!

Can do this by using one measurement at known values of ti and tc to evaluate Bφ , then do later calculations.Helpful in planning radiological work!

(2( , ) ln 5)i ci c

c

t tt t Bt

δ φ⎛ ⎞+

≈ ⎜ ⎟⎝ ⎠

55

Gollon’s Rules of Thumb

More examples3) Rule 4 and Rule 2 can be used together

A beam of 1012 400 GeV protons s-1 produces 4 X 400 x 1012 stars s-1 (interactions s-1) in a beam absorber (Rule 4)

25 % will produce an atom of radionuclide with half-life > 1 day )(Rule 2). Get:

Assume r = 10 m achieves point source conditions, Rule 1 leads to:

if each decay emits a 1 MeV photon.

1514 50.25 atoms 1.6 10 stars 4 10 Bq = 4 10 GBq

star se( 7)

c2×

× = × ×

54 -1

2

4 101.08 10 (1 MeV) 0.43 Gy h 1

(28)0

dDdt

− ⎛ ⎞×= × =⎜ ⎟

⎝ ⎠

56

Barbier danger parameter

Barbier invented this quantity D the absorbed dose rate at the surface of a thick block uniformly irradiated.

If such a source subtends a solid angle Ω at a point of concern;

For contact with a semi-infinite slab, the fractional solid angle factor (Ω/4π)=1/2

The physical interpretation of the danger parameter can be understood as the absorbed dose rate found inside a cavity in an infinite volume irradiated by unit flux density (1 particle s-1

cm-2)Figs. 8 and 9 are examples of danger parameters.

(29 ) 4

dDdt

φπ

Ω= D

57

Barbier danger parameterFig. 8

(29 ) 4

dDdt

φπ

Ω= D

58

Barbier danger parameterFig. 8

(29 ) 4

dDdt

φπ

Ω= D

59

Barbier danger parameterFig. 8

(29 ) 4

dDdt

φπ

Ω= D

60

Barbier danger parameterFig. 9

(29 ) 4

dDdt

φπ

Ω= D

61

Barbier danger parameterFig. 9

(29 ) 4

dDdt

φπ

Ω= D

62

Barbier danger parameters

Life is not always simple in using Danger parameter curves:

• Activation in a large object can occur from a multitude of directions (e.g., in a cascade)

• Numerous energies of particles can be present• Multiple particle types• We are lucky that activation cross sections are as flat

above threshold as they are!• Values of cross sections often do not vary over too large a

domain.

63

Experimental cooling curvesFig. 10

64

Use of Monte Carlo star densities for activation

• Leveling-off of cross sections is MOST useful• Can often estimate activity without integrating

over energy• If one has a good estimate of flux density

above the reaction threshold of interest• Can use an average effective cross section• Fortuitously, cross section occurs precisely in

the energy domain (10s to a few 100s of MeV) where Monte Carlo codes are most reliable.

65

Use of Monte Carlo star densities for activation

At some location r connect flux density above threshold φ(r) (cm-3s-1) to

star density production ratedS(r)/dt (stars cm-3s-1):

ρ (g cm-3) is the material densityand λ (g cm-2) is the interaction length, not the

decay constant.

( 0)) 3) (( dSdt

λφρ

=rr

66

Use of Monte Carlo star densities for activation

Want to connect to danger parameter DBut, M-C Codes may have a low-energy cutoff not equal

to the reaction thresholdGollon solved this:

The omega-factor ω (ti,tc) is connected to DHave:

ω (infinity,0)=9 x 10-2 (μGy h-1)/(stars cm-3s-1) (32a)

ω (30 d,1d )=2.5 x 10-2 (μGy h-1)/(stars cm-3s-1) (32a)

( 0)) 3) (( dSdt

λφρ

=rr

( ) ( ) (( , ) 4

31)i cdD dS t t

dt dtω

πΩ

=r r

67

Use of Monte Carlo star densities for activation

Other values of ω can be calculated (e.g Fig. 11)

( ) ( ) (( , ) 4

31)i cdD dS t t

dt dtω

πΩ

=r r

68

Use of Monte Carlo star densities for activation

Fig. 11

69

Special issues

Some special conditions limit the validity of this approach:

• Cracks in the shielding material “leak” radiation• Presence of abnormal amounts of thermal neutrons• Activation of enclosure walls

Gollon found a way to calculate other dose rates for different cooling times from values for infinite irradiations. Working from:

and rearranging: ( ), 1 exp( ) exp( ) (33)i c i ct t A t tμ μ μ

μ

δ λ λ⎡ ⎤= − − −⎣ ⎦∑

( ) ( ){ } ( ) ( ), exp( () exp , 3 ), 4 i c c i c c i ct t A t t t t t tμ μ μμ

δ λ λ δ δ⎡ ⎤= − − − + = ∞ − ∞ +⎣ ⎦∑

70

Connection to Monte Carlo “stars”

Use ratios of cross sectionsAt some point on space denoted by vector rRate of production of atoms cm-3 of nuclide ni(r) is

dS(r)/dt is the star density production rate at r.One is essentially scaling the production of the

radionuclide of interest to the total inelastic cross section, σin, or, with macroscopic cross sections; Σi/Σin. This is an approximation!

( ) ( ) ( ) (35)i i i

in in

dn dS dSdt dt dt

σσ

Σ≈ =

Σr r r

71

Uniform irradiation of enclosure walls

Topic arose about the time Fermilab was builtTypical concrete density is about 2.5 g cm-3.Of this about 0.04 g cm-3 is sodium and all

natural sodium is 23Na.23Na(n,γ)24Na reaction has large (535 mb) cross

section for thermal neutronsGet lots of 24Na

t1/2=15 hAlmost all decays emit 2 photons;

Eγ1=1.37 MeV, Eγ2=2.75 MeV

72

Uniform irradiation of enclosure walls

Eγ1=1.37 MeV, Eγ2=2.75 MeVRemember

Patterson estimated thermal neutron flux density φth in a concrete room as:

where Q is the # of fast neutrons emitted s-1, and

S is the interior surface area of the enclosure (cm2)

( )42 (1.08 1 26)0 i

i

dD S Edt r

γ−= × ∑

( )-2 -11.25 c )m 36s (thQ

Sφ =

73

Uniform irradiation of enclosure walls

Problem in accelerator enclosures:Dose rate falls with distance away from activated components – OK.Dose rate does not fall with distance away from activated walls.

Let’s see why:

74

Uniform irradiation of enclosure walls

See Fig. 12Assume wall is uniformly activated; emits φο flux densityin all directionsWant flux density at point P.

75

Uniform irradiation of enclosure walls

Consider flux at P due to

small element dASolid angle dΩ of dA atP is

Unit vector is

But, flux at P due to emission

from dA is

2

ˆ(37) dd •

Ω =′ −

A nr r

n̂ ˆ (38)′

=′

r - rnr - r

2 ) 3ˆ

4( 9dd οφφ

π•

=′A n

r - r

76

Uniform irradiation of enclosure walls

Since

and

Result: Flux density (and hence dose) is same at every point in the enclosure.

2 ) 3ˆ

4( 9dd οφφ

π•

=′A n

r - r

4od dφφπ

= Ω

44

4( 0 )o

odπ

φ φπ

Ω =∫

77

Uniform irradiation of enclosure walls

Result: Flux density (and hence dose) is same at every point in the enclosure.

• Most true early in a shutdown when 24Na still important• There are other thermal neutron activation products• Leads to disappointment:

Local shielding of beamline components may not helpRemoval of components may not helpThe distance part of “time, distance, & shielding” fails us.

44

4( 0 )o

odπ

φ φπ

Ω =∫

78

Acknowledgements

Adapted from lectures given at U. S. Particle Accelerator School on multiple occasions

Helpful comments received over the years fromKamran Vaziri (Fermilab)Alex Elwyn (Fermilab, retired)Sayed Rokni (SLAC)

Support was provided byMr. William Griffing, Fermilab ES&H Director, who encouraged my

participationMy Fermilab co-workers who have helped with many other tasksMy wife, Claudia, who allowed much work at home