Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

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6.7 Neutron transport in matter 303

The calculation will result in an estimation of the smallest sphere of

239

that will support a nuclear chain reaction. The calculated “critical” radius of

this sphere turns out to be

=0.056 m , (6.30)

corresponding to a calculated critical mass

=14.7kg . (6.31)

The parameters of the calculation and its result are shown in Table 6.4.

In spheres of radius less than the critical radius, neutrons produced in

ﬁssion of

239

Pu generally leave the sphere before producing a further ﬁssion,

resulting in k<1. For spheres of greater radius, the neutrons generally pro-

duce further ﬁssions (and further neutrons). The number of neutrons increases

exponentially.

The size of the critical radius is just somewhat larger than the mean free

path of neutrons in

239

Pu. This is as expected since it is the mean free path

that determines whether neutron escape freely or remain to create further

ﬁssions.

Table 6.4. Characteristics of

239

Pu needed in the calculation of the critical radius

and mass. All cross-sections are given for 2 MeV neutrons.

Elastic scattering cross-section σ

=3.45 b

Neutron-induced ﬁssion cross-section σ

ﬁs

=1.96 b

Radiative capture cross-section σ

n,γ

=0.080 b

Total absorption cross-section σ

abs

= σ

n,γ

+ σ

ﬁs

=2.04 b

Total cross-section σ

tot

= σ

+ σ

abs

=5.87 b

Neutrons produced per ﬁssion

ν =2.88

Neutrons not radiatively absorbed k =



=2.74

Density ρ

239

=19.74 × 10

kg m

−3

neutron mean free path l =(σ

tot

239

)

−1

=0.0343 m

Critical radius R

=0.056 m

Critical mass M

=(4/3)πR

239

=14.7kg

We consider neutron transport under the following assumptions.

• The medium is static (i.e. we neglect any small thermal motions); it has

a constant density, and it is spherically symmetric, centered at the origin

r =0.

• Neutron–neutron scattering is negligible (since the density of neutrons is

much smaller than the density of the medium) .

• Neutron decay is negligible, i.e. the neutron lifetime is very large compared

to the typical time diﬀerences between two interactions.

304 6. Fission

• The nuclei of the medium are of only one species, (

239

Pu to be speciﬁc) they

have a much larger mass than the neutron mass. Therefore, in neutron–

nucleus collisions, the neutron kinetic energy is unchanged. In such a col-

lision, the direction on the neutron velocity can change, but not its mag-

nitude.

The neutron distribution is characterized by their density in phase space

= f (r, p,t) , (6.32)

where dN is the number of neutrons in the phase space element d

The space density of neutrons and the current describing the spatial ﬂow of

neutrons are the integrals over the momentum

n(r,t)=



f(r, p,t)d

p , (6.33)

J(r,t)=



vf(r, p,t)d

p . (6.34)

In the absence of collisions, neutron momenta are time-independent and

the ﬂow of particles in phase space is generated by the motion of particles at

velocities v = p/m. In that case, the density ρ satisﬁes the Liouville equation

∂ρ

∂t

+ v · ∇ρ =0 .

In the presence of collision processes, the Liouville equation (6.7.1) be-

comes

∂ρ

∂t

+ v · ∇f = C(f) , (6.35)

where C(f) is the term arising from collision processes, for which we will ﬁnd

an explicit form shortly.

The elastic scattering and absorption rates λ

and λ

abs

are products of

the elementary cross-sections, the density of scattering centers n

239

,andthe

mean velocity v

= vn

239

abs

= vn

239

abs

. (6.36)

The absorption is due to both (n, γ) reactions and to ﬁssion

abs

= σ

(n,γ)

+ σ

ﬁs

. (6.37)

The collision term is therefore

C(f (p)) = n

239





v(p



) f(r, p



,t)

dσ



→ p) (6.38)

−[λ

+ λ

abs

]f(r, p,t)+S(r, p) .

The ﬁrst term accounts for neutrons coming from the elements of phase

space d



which enter the element of phase space d

p by elastic

6.7 Neutron transport in matter 305

scattering. The second term represents the neutrons which leave the element

p either by elastic scattering or by absorption. The last term S(r, p)

is a source term, representing the production of neutrons by ﬁssion. We will

write its explicit form shortly.

6.7.2 The Lorentz equation

We assume that all neutrons have the same velocity, v, i.e. that the function

f(r, p) is strongly peaked near values of momentum satisfying |p| = m

This is the case in breeders and in ﬁssion explosive devices. The homogeneity

assumption is also reasonable in ﬁrst approximation in these examples.

In that case, the diﬀerential elastic scattering cross-section is

dσ



(p → p



)=p

−2

δ(p − p



)

dσ

dΩ

. (6.39)

We assume, for simplicity, that the scattering cross section is isotropic

dσ

dΩ

4π

. (6.40)

Using (6.40) we ﬁnd that the Boltzmann equation (6.35) and (6.38) re-

duces to the Lorentz equation

∂f

∂t

+ v · ∇f = λ

(

f − f ) − λ

abs

f + S(r, p) , (6.41)

where

f(r,p,t)=

4π



f(r, p,t)dΩ

, (6.42)

is the phase-space density averaged over momentum directions.

It is useful to integrate the Lorentz equation over the direction of the

momentum (or the velocity) d

Ω

(we do not integrate on p itself since the

modulus of p or v is ﬁxed by assumption). This leads to

∂n

∂t

+ ∇ · J = −λ

abs

n +4πS(r) , (6.43)

where 4πS(r) is the momentum integral of S(p, r).

In the or local quasi-equilibrium regime which is of interest here, the

mean free path between two collisions is small compared to the size of the

medium.

We make the usual approximation of Fick’s law, where the current

is proportional to the density gradient:

J = −Dv∇n. (6.44)

The diﬀusion coeﬃcient D is related to the elastic-scattering rate or to the

mean free path by

A comparison of the mean free path and the critical radius in Table 6.4 indicates

that this is, admittedly, not an excellent approximation.

306 6. Fission

D =

3λ

. (6.45)

The value of this coeﬃcient will be justiﬁed in Appendix D. In general, the

diﬀusion coeﬃcient depends on the velocity and on the position r,butthis

diﬃculty is by-passed within our assumptions.

Inserting (6.45) into (6.44) and (6.43) leads to

∂n

∂t

− Dv∇

n = −λ

abs

n +4πS(r) . (6.46)

In the absence of absorption and sources, we recover a Fourier diﬀusion equa-

tion.

6.7.3 Divergence, critical mass

Consider a very simplistic fast-neutron reactor in which we assume that the

neutrons have all the same velocity v, and a kinetic energy of ∼ 2MeV.

These neutrons evolve in a homogeneous ﬁssile medium which contains n

239

Pu nuclei per unit volume. The motion of the neutrons is a random walk.

At each collision they can be absorbed by the nuclei of the medium, with

a cross-section σ

abs

, or they can be scattered elastically with a cross-section

The total cross-section is σ

tot

= σ

abs

+ σ

, and the total reaction rate is

tot

= n

239

tot

= v/l , (6.47)

where l isthemeanfreepathoftheneutronsinthemedium.

The source term in (6.43) corresponds to the rate of neutron production

by ﬁssion. If σ

ﬁs

is the ﬁssion cross-section (σ

ﬁs

<σ

abs

), and ν is the average

number of neutrons produced in a ﬁssion, the rate of increase of the density

n(r) due to ﬁssions is

4πS(r)=

νn

239

n(r) vσ

ﬁs

. (6.48)

If we insert the expression (6.44) for J and this source term into (6.43),

we obtain the evolution equation for n

∇

n +

(

νλ

ﬁs

− λ

abs

)

n =

∂n

∂t

. (6.49)

If we set k = νσ

ﬁs

/σ

abs

,asdonepreviously,weobtain

∇

n + B

n =

∂n

∂t

, (6.50)

wherewehavedeﬁned

=(k − 1)

abs

. (6.51)

Traditionally, in neutron transport equations one works with the so-called neu-

tron scalar ﬂux φ = vn. In order to avoid introducing too many symbols, we stick

to the variable n since this makes no diﬀerence in the simple case considered here.

6.7 Neutron transport in matter 307

(We assume that k ≥ 1.)

Sinceweassumethatthemediumisﬁnite,spherical,ofradiusR, vn

depends only on the distance r from the center. The conditions we must

impose on vn are the following : vn ≥ 0forr ≤ R,andvn(0,t) is ﬁnite.

However, equation (6.50) is only valid inside the medium. There are no

incoming neutrons from the outside. In diﬀusion theory, a simple but accurate

empirical way to simulate this condition is to impose that n vanishes at an

“extrapolated distance ” R

n(R

,t) = 0 with R

= R +0.71l, (6.52)

where l is the mean free path 1/nσ

tot

Of particular interest is the stationary solution (critical regime) of (6.50),

i.e. a solution for which (∂n/∂t = 0). We then have to solve ∇

n + B

n =0

or, in spherical coordinates,

rn + B

n =0 . (6.53)

Setting u(r)=rn(r), this equation is readily solved:

u(r)=α sin Br + β cos Br , (6.54)

and, since vn must be regular at the origin,

n(r)=α

sin Br

. (6.55)

The limiting condition (6.52) imposes

= π. (6.56)

In other words, there is only one value R

of the radius R of the ﬁssile sphere

for which a critical regime exists (permanent or stationary regime) :

= π/B − 0.71 λ. (6.57)

For plutonium

D =1.14 10

−2

m ,

B =39m

−1

Therefore, there is a critical radius R

and a critical mass M

=5.63 10

−2

m , (6.58)

==ρ

239

(4π/3)R

=14.7kg . (6.59)

For R = R

, a stationary regime cannot occur. One can readily check this

by searching for solutions of the type :

n(r, t)=e

γt

f(r) , (6.60)

that:

308 6. Fission

–forR>R

, necessarily γ>0, this corresponds to a supercritical regime,

the system diverges and explodes;

–forR<R

, necessarily γ<0, this corresponds to a sub-critical regime;the

leaks (ﬁnite medium) are not compensated and the chain reaction cannot

take place. The neutron density decreases exponentially in time.

The calculation leading to the plutonium critical mass is oversimpliﬁed.

The actual values for critical masses of spheres of pure metals are M

=6kg

for

239

Pu and M

=50kgfor

235

U. These values can be reduced if the

material is surrounded by a non-ﬁssile medium consisting of heavy nuclei

so that neutrons have a high probability of scattering back into the ﬁssile

material.

6.8 Nuclear reactors

Fission was discovered in 1939 when Hahn and Strassman discovered the

presence of rare-earth elements in uranium after irradiation by neutrons.

L. Meitner and O. Frisch then interpreted this production as being due to

neutron-induced ﬁssion of uranium. This discovery was followed rapidly by

applications since, on December 2 1942, Enrico Fermi at the University of

Chicago produced a chain reaction in a system consisting in a periodic stack

of natural uranium spheres separated by graphite moderators. Fermi thus

demonstrated experimentally the notion of criticality of the size of the stack

in order to ensure a chain reaction. This was achieved with a very small

total power of the system, ∼ 1 W. Present power reactors attain powers of

∼ 3 GW. The increase in power does not present by any means the same

complication as in fusion, as we shall see in the next chapter. Indeed, in the

ﬁssion process all phenomena are more or less linear, in (great) contrast with

controlled fusion systems.

A ﬁssion reactor core consists of the following essential elements

• Fuel elements, generally consisting of bars containing natural uranium,

uranium enriched in

235

U, or

239

Pu. If there is to be a self-sustaining chain

reaction, the amount of fuel must be greater than the critical mass deﬁned

by geometric losses.

• A heat extraction system, generally a ﬂuid, e.g. water in thermal-neutron

reactors or sodium in fast-neutron reactors. Its role is to limit the tem-

perature of the core and, in power reactors, to transfer the core’s thermal

energy to electric generators.

• (Thermal-neutron reactors only) A moderating system to thermalize the

neutrons. This is most simply done by bathing the fuel bars in water. In

this case, the moderator also serves as the heat transporter.

In the following subsections, we will brieﬂy describe three basic types of

ﬁssion reactors, those based on thermal neutrons, fast neutrons, and proposed

schemes where reactors are driven by particle accelerators.

6.8 Nuclear reactors 309

6.8.1 Thermal reactors

Pressurized water reactors (PWRs). This is the most widely used cate-

gory. In PWRs the pressurized water is both the moderator and the coolant.

Typical characteristics are given in Table 6.5.

Table 6.5. Structure and operating characteristics of a typical pressurized water

reactor.

Volume V ∼ 27, m

Initial mass of U 3 × 24 ton = 72 ton

Initial mass of

235

U3× 0.84 ton = 2.52 ton

Initial number of

235

U nuclei N

235

(t =0) = 6.4 × 10

Mean neutron ﬂux φ =5×10

−2

−1

Mean neutron velocity v =



3kT/m

=2.5 × 10

msec

−1

Mean neutron density n = φ/v =2× 10

−3

Fission rate λ

ﬁs

= φσ

ﬁs

235

=9.4 × 10

−1

Thermal power P = λ

ﬁs

× 200 MeV = 3000 MW

Final mass of

235

U3× 220 kg = 660 kg

Final mass of

238

U3× 23 ton = 70 ton

Final mass of

239

Pu 3 × 145 kg = 435 kg

Final mass of ﬁssion products 3 × 400 kg = 1.2ton

The fuel rods are made with ceramic pellets of UO

, typically enriched to

about 3% in

235

U. The pellets are sealed inside tubes made of zirconium, a

material chosen for its low neutron absorption cross-section and its material

strength. The tubes are arranged in bundles in a steel structure inside which

the coolant circulates. The bundles are secured and are arranged vertically

in a rigid conﬁguration called the core structure, as shown on Fig. 6.9. The

core is inside a massive steel pressure container as shown on Fig. 6.10.

The maximum permissible power is determined by the requirement that

the temperature in the fuel rods be less than the melting point of UO

3100 K. Assuming the neutron ﬂux is constant inside the tube, the emitted

heat per unit volume, Q, is position independent. The integration of the heat

transfer equations gives

=4k(T

max

− T

) (6.61)

where a isthetuberadius,T

max

is the maximal temperature inside the

tube, T

is the temperature of surface of the tube and k is the conduc-

tion coeﬃcient of the uranium oxide. Using T

max

= 1800 C, T

= 450 C and

k =0.06 W cm

−1

,wehaveQa

= 300 W cm

−1

. For a diameter of 1 cm,

we obtain

Qρ =40Wg

−1

. (6.62)

310 6. Fission

Fig. 6.9. Fuel element holder for a PWR reactor. The ﬁgure shows the control rods

in place. When ﬁlled, the structure contains 264 zirconium tubes of length 4 m and

diameter 9.5 mm containing pellets of UO

6.8 Nuclear reactors 311

generator

turbine

cooling

tower

condensor

cooling

waterpump

containment

structure

generator

steam

pump

control rods

core

reactor

vessel

Fig. 6.10. Schematic of a pressurized water reactor.

(The density of the oxide is ∼ ρ =10gcm

−3

.) The size of the reactor is

therefore ﬁxed by the cooling conditions. For 3000 MW, we need a fuel mass

fuel

3000 MW

40 W g

−1

= 75 ton . (6.63)

This corresponds to 10

m of fuel tubes of 1 cm diameter.

The neutron transport considerations (eﬃcient thermalization, minimal

losses out the edges, etc.) imposes a moderation ratio (i.e. the ratio between

the volume of water and the total volume of the tubes) of roughly 3. The

reactor therefore has a diameter of at least three meters. Such large diameters

and heights are necessary in order to limit the neutron leakage.

Water is used both as a moderator and as a coolant ﬂuid. Its maximal

temperature in the vicinity of the zircalloy tubes is 300 C. In order to prevent

the water from boiling, it is permanently pressurized to 150 atmospheres. It

circulates and feeds a heat exchanger which leads to a secondary water circuit

which generates steam to drive turbines for the production of electricity.

The maximum eﬃciency with which the thermal energy of the steam can be

transformed to work on the turbines and then to electrical energy is limited

by Carnot’s principle

<1 −

, (6.64)

where T

and T

are the ﬁnal and initial water temperatures. T

is ﬁxed by

a cold source, generally a river or the ocean.

The local temperature increase of water has an obvious ecological impact. In par-

ticular it favors the development of various unwanted amebae and other micro-

organisms.

312 6. Fission

At the beginning of operation, ﬁssions are initiated with a small neutron

source and the ﬁssion rate is allowed to increase until the neutron ﬂux reaches

∼ 5 × 10

−2

−1

. Precisely how this is done is discussed below.

After ignition, the original fuel is gradually transformed into ﬁssion prod-

ucts and trans-uranium elements. The abundances of selected nuclei as a

function of time are shown in Fig. 6.11. The trans-uranium elements are pro-

duced by neutron capture. The path of neutron captures is shown in Fig.

6.12. By far the most common trans-uranium is

239

Pu which is produced by

radiative neutron capture on

238

U followed by two rapid β-decays (6.16).

110 330

550 770

990

Days in Reactor

Gram atomic weight per ton of fuel

Am x10

243

137

240

237

239

Fig. 6.11. The buildup of selected ﬁssion products and trans-uranium elements

inside a nuclear reactor [65]. Long-lived ﬁssion-products (e.g.

137

Cs) generally have

abundances that increase linearly with time. The trans-uranium nuclide

239

Pu is

produced from a single neutron capture on

238

U followed by two rapid β-decays and

therefore increases linearly at small time and then reaches an equilibrium when

its production rate is balanced by its destruction rate from induced ﬁssion and

neutron capture. The production of

237

Np requires two neutron captures (on

235

and therefore has an abundance that varies quadratically with time.

The fuel can last 3 years before it is too contaminated by ﬁssion products.

One third of the reactor fuel is renewed each year. Each recharge contains 24

tons of enriched uranium.

When it leaves the reactor, the fuel still contains 23 tons of uranium

enriched to 0.95%, i.e. 220 kg of

235

U and 145 kg of plutonium. 271 kg of the