11. basic neutron physics for logging applications

8/12/2019 11. Basic Neutron Physics for Logging Applications

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1 1BASIC NEUTRON PHYSICSFORLOGGING APPLICATIONS

INTRODUCTIONThe use of neutrons to probe formations has had a long history in welllogging. The first neutron device appeared shortly after World War 11. Theinitial application was to determine formation porosity. Currently, in additionto logging tools that detect neutrons of several energy ranges in order todetermine formation porosity, there are tools which use pulsed neutrons toanalyze the absorption rate of the emitted neutrons, and gamma rayspectroscopy tools which detect neutron-induced gamma rays to produce alimited chemical analysis of the formation. The key to understanding theresponses of these tools is the interactions that are exploited. The purpose ofthis chapter is to describe these interactions in order to provide a basis forsucceeding chapters.As in the case of gamma rays, neutrons can interact with materials in anumber of different ways, each with an appropriate cross section to describeits probability of occurrence. The interactions of neutrons with matter aremuch more varied and complex than those of the gamma rays. Forsimplicity, we will confine ourselves to two groups of those interactions:scattering and absorption. Four types of cross sections to describe theseinteractions will be taken up, after a review of some useful terminology andthe kinematics of neutron elastic scattering.

Unlike gamma ray sources, which come from naturally occurring oreasily produced isotopes, neutron sources used in logging are the result ofdeliberate nuclear reactions. Several of these reactions will be discussed,

227


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228 Well Logging for Earth Scientistsalong with the techniques for the detection of neutrons.

FUNDAMENTALS OF NEUTRON INTERACTIONSThe reaction rate of neutrons with matter depends on four parameters. Thefirst two are the density (numberlvolume) of neutrons, n, and their velocity, v.The product of these two quantities is called the f lux (identical with Y usedearlier to describe gamma ray intensities), and the units are number ofneutrons per cm2-sec. The reaction rate also depends on the nuclear density,Ni of the particles with which they will interact, and finally upon the crosssection ci or the particular reaction. Thus an expression for R, the reactionrate (number of neutron reactions of type i I cm3), is given by:

R = n v oiNi .The density of particles of type i in a material of molecular weight M andbulk density p is:

6 . 0 2~10~~M PYi =

if there is a single nucleus of type i per molecule.N e u t r o n S p e e d(c rn iksec)

2200

2 2

0 22001 0 1 1 10 102 lo3 lo4 lo5 106 i 0 7 Energy (eV )

1keV 1M e VFigure 11-1. The classification of neutrons according to broad energy ranges andtheir corresponding velocities. From Ellis.'


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Neutron Physics for Lagging Applications 229Fig. 11-1 defines, in broad terms, the energy range of interest for

neutrons. For logging applications, this range is over about nine decades:from source neutrons of 5 to 15 MeV, in the broad fast neutron range above10 eV, to epithermal neutrons in the range of 0.2-10 eV, and thermalneutrons which are distributed around 0.025 eV at room temperature.

For later discussions of the time scale associated with the slowing-downprocess, it is useful to note the relationship between neutron energy and itsassociated velocity. To evaluate the velocity of a neutron, we can use, at lowenergies, the classical relationship between kinetic energy E, velocity v, andmass m:

(1)= - m v , 22so that the velocity v is given by:

If this expression for velocity is evaluated for thermal energies (0.025 eV),the result is 2200 mlsec or 0.22 cmlysec. Thus the velocity at any energy E(in eV) is given by:

v = 0.22 z (3)where v is in cmlysec. Therefore the speed of an epithermal neutron of2.5 eV is 2.2 cmlpsec, and for a near-source-energy neutron of 2.5 MeV, itsvelocity is 2200 cdysec. These velocities are also noted on Fig. 11-1.

Of the four principal types of interaction, the first two are generallyreferred to as moderating interactions, or interactions in which the energy (orspeed) of the neutron is reduced. One of these is known as elastic scattering,and the other as inelastic scattering. Let us consider elastic scattering first.Classical mechanics (elastic billiard ball analysis) can be used to describe themoderating power of the struck nucleus. The energy of the neutron isreduced more efficiently in collisions with nuclei of mass not too differentfrom the mass of the neutron. Thus hydrogen and other low atomic masselements are quite effective in reducing fast neutron energy.

DetectorScat tered

IncidentNeu t ron

I. .NucleusFigure 11 2. The idealized scattering of a neutron with a target nucleus. FromR y d h 3


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230 Well Logging for Earth ScientistsThe physical variables for describing elastic scattering can be obtained bya consideration of the con cep t of center of mass. Fig. 11-2 shows thelaboratory view of a collision between a stationary nucleus and a neutron

moving with velocity v. After collision, the neutron has deviated from itsinitial direction by an angle 8 nd has some reduced velocity, v'.

ICM Im+eutron

Figure 11 3. The scattering reaction drawn to suggest the center of mass (CM)system. From Rydin?Another approach is to define the center of mass as shown in Fig 11-3.This new coordinate system is fined by:

Mxo = m(x-x,),where M is the mass of the target, and m is the mass of the neutron. Thecoordinate x , is given by:

mx Xx,=-=-m + M 1 + Aafter substituting I for the mass of the neutron and A for the mass of thenucleus. The velocity of the center of mass v,,, as seen in the laboratorysystem, can befound rom:

V .dr 1 d n 1v,, =-t 1 + A dt l + AIncidentNeut ron

.iiTargetN u ; e u b , Neutronncident.

Neut ronReco-Nucleus

ScatteredNeut ron

0c TargetNucleus

Nucleusc

VbFigure 11 4. The scattering reaction drawn from the perspective of the laboratoryand center of mass systems. From Rydin?

Two views of the reaction are shown in Fig 11 4: the laboratory view onthe lej2 and the center of mars system on the right. In the center of mass


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Neutron Physics for Logging Applications 231system, the two particles are seen to be approaching each other withvelocities v , and V . These two velocities are given by:

1 Al+A l+Av , = v v,, = 1 ) v = - ) v ,

andV .1l+Av - - v , , = - -c -

The total momentum in the center o mass system is given by:A 1A+ l+Amv, + MV, = .v -Av ,

which is seen to be zero . This unique result, fo r an elastic collision viewed inthe center of mass system, means that the neutron and nucleus enter andleave the reaction with the same veloc ities and are oppositely dire cted .An analysis of conservation of energy shows that the neutron energy E,after scattering through an angle 6 in the center of mass system, can berelated to the energy Eo efore the collision by the following :3

EE O ( A + 1)2

A2 2AcosO + 1From this expression, it is seen that the minimum energy after collision is afraction a of the initial energy, where a is related to the mass A of thescattering nucleus by:

2a =[s]

Fig. 11-5 illustrates, for most of the elements of interest, the permittedranges of reduction in neutron energy on a single collision. It is seen that forthe most common earth formation elements the maximum energy reductionper collision for the heavy elements is about 10-25%. How ever, for the caseof hydrogen, the entire neutron energy can be lost in a single collision. Thissensitivity of elastic scattering energy loss to hydrogen is exploited in neutronporosity devices.In the case of inelastic scattering, a portion of the energy of the incidentneutron goes into exciting the target nucleus. This reduces the energy of theincident neutron. The target nucleus will usually produce one or morecharacteristic gamma rays upon de-excitation. This type of reaction alwayshas a threshold energy (below w hich it will not happen). The de-excitationgamma ray is exploited in the measurement of the carbon-to-oxygen ratio inearth formations.Another general category of neutron interaction is known as absorption.It also is divided into two types: radiative capture and reactions whichproduce nuclear particles. In radiative capture, unlike the moderating


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232 Well Logging for Earth Scientists

H 10 20 30 40A , Atomic Mass

Figure 11-5. The allowed distribution of neutron energy after a single elasticscattering with nuclei ranging in mass from H t Ca. The energyscale is normalized t the incident neutron energy. From Ellis.

interactions considered above, the neutron (usually near thermal energies) isabsorbed by the target nucleus, producing a compound nucleus. This nucleusde-excites instantly with the emission of characteristic gamma rays. Thistype of reaction is exploited in pulsed neutron logging tools or in gamma rayspectroscopy of the induced gamma rays for chemical analysis.The category of particle reactions is quite broad; it is sufficient to say thatthe interaction of neutrons with some nuclei can provoke the emission ofparticles such as alphas, protons, ps, or even additional neutrons. Thesereactions, although common, have a very small probability for occurringrelative to the other interactions of interest in logging described above.Usually they are possible only above a relatively high neutron energy.The complexity of the cross sections for neutron interactions is illustratedin Fig. 11-6, which schem atically indicates the variations with energy. Thetop figure refers to the total cross section as a function of neutron energy E,and the four following figures indicate how this can be decom posed. Thefirst, (n,n), refers to elastic scattering, which is shown to be rather constantwith energy except for some resonances at low energies. The next sketchshow s inelastic interactions, (n,n), show ing some characteristic thresholdbelow which this reaction is not possible; the third sketch is one of the manyparticle reactions possible, (n,a); and the final (although there could beothers) is the radiative capture, (n,y), which is seen to increase in probabilityat low energies.


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Neutron Physics for Logging Applications 233

TotalCross sect ion

ElasticScat te r ing(n, n )

InelasticScat te r ingn . n )FastReactionn , a

ThermalCapturen , Y)

I-eutron Energy, E

Figure 11 6. A schematic illustration of the energy variation of the total neutroncross section and four of its components. From Ellis.'NUCLEAR REACTIONS AND NEUTRON SOURCESSince neutron sources are almost never found in nature, it is appropriate tobriefly discuss the techniques for creating them. There are two types in usein logging: so-called chemical, or encapsulated, sources, and acceleratorsources.The classic reaction, which resulted in the discovery of the neutron, wasthe bombardm ent of beryllium by alpha particles. It can be written as:

4Be + 2He + 6C+ n 5.76 MeV .This forms the basis for the cheapest, easiest, and most reliable method forneutron production. The physical explanation of this reaction is beyond thescope of interest of the present work and may be found in References 2 and4. The practical construction of this kind of chemical neutron source consistsof mixing a naturally occurring a-emitter with an appropriate light elementhaving a large (a,n)* cross section. Som e a-em itters which have been usedfor this purpose are Pu, Ra, Am, and Po. Three common target elements areBe, B, and Li. The actual spectrum (energy distribution) of emitted neutronsis quite complicated. It depends somewha t on the geometric details of thea-emitter and target, but the peak of the neutron distribution is generallyaround 4 MeV.

This shorthand, (a,n), indicates a reaction of an a particle with an unspecified nucleus,resulting in the production of a neutron and ano ther unspecified nucleus.


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234 Well Logging for Earth ScientistsAnother method of exploiting particle-induced reactions in the production

of neutrons is by the use of charged particle accelerators. In one realization,currently in use in well logging, deuterium and tritium ions are acceleratedtoward a target impregnated with the hydrogen isotopes deuterium (D) andtritium (T). The reaction is written as:

2D 3T 4He n + 17.6 MeV .The cross section for this reaction has a maximum at about 100 keV of 2Dprojectile energy. This dictates the required accelerating voltages in such adevice.

Despite the engineering difficulties of constructing such a device, theadvantages for logging are many. One is the relative high energy of theproduced neutrons. They are emitted at 14.1 MeV (not 17.6 MeV, becausesome of the energy of this reaction is given up to the alpha particle). Thesehigh energy neutrons are useful for producing other interesting nuclearreactions in the formation, as is discussed later. Another advantage is that asource of this type can be controlled, i.e., switched off and on at will. Thisprovides a degree of safety unparalleled for radioactive sources as well aspermitting measurements involving timing as a means of determining someinteresting nuclear properties of the formation, a topic covered in Chapter 12.USEFUL BULK PARAMETERSDespite the complexities of the cross sections shown in Fig. 11-6, whichgovern the details of the interactions, some gross properties can be specifiedfor neutron interactions with materials. The first is the macroscopic crosssection, which is defined as the product of the cross section oi)n questiontimes the number of atoms/cm3, N, i.e.:

where NAv s Avogadros number, P b is the bulk density, and A is the atomicweight. The dimensions of the macroscopic cross section C are inversecentimeters. Its reciprocal is the mean free path length between interactionsof type i. Frequently, in logging, special use is made of the macroscopicabsorption cross section evaluated at thermal energies. It is convenient todefine special units for it. These so-called capture units (cu) are 1000 timesthe C, as defined above. In this case, oiefers to the thermal absorption crosssection which dominates at thermal energies for most elements. Fig. 11-7shows the total mean free path in limestones of 0, 20, 40, and 100 PU(porosity units) as a function of energy for fast neutrons. At the energy ofchemical source emission (2-4 MeV), it is seen that there is very littleporosity dependence. It is only as the neutrons are slowed down that themean free path becomes strongly dependent on the hydrogen concentration ofthe formation.


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Neutron Physics for Logging Applications 235

5 .a2 -C ,2O

0.10.1 1 o 10

Neut ron Energy . MeVFigure 11 7. The mean free path of neutrons in limestones of various porosity andwater. They are given as a function of the neutron energy. FromEllis.

As mentioned earlier in the discussion of elastic scattering, low massnuclei are very effective in reducing the energy of the scattered neutron. Ascan be inferred from Fig. 11-5, the result of a collision can be considered, onaverage, as a percentage decrease of the neutron energy. This is usuallyexpressed as the average logarithmic energy decrement 6 which is definedby:5 )

where Ei is the initial energy and E is the energy of the neutron aftercollision. It can be shown from classical mechanics that the average logenergy decrement is simply related to the atomic mass, A, of the strucknucleus by:3

6 = ln(Ei)-ln(E) = n(E/Ei) ,

n

for large values of atomic mass A. The average log energy decrement allowsan estimation of the average number of collisions, n, to reduce the neutronfrom an initial energy Ei to some lower energy E, from the followingreasoning. If the sequence E1,E2, ,En represents the average energy aftereach collision, then we can write:ki ki k kn-1In -) = In(-- . . . - )En El E2 En= In(-) = n In(-)i EiEl El

(7)


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236 Well Logging for Earth Scientists

ModeratorH 1.0C 0.1580 0.12Ca 0.05H2O 0.9220 PU Limestone 0.5140 PU Limestone 0.115

Thus the average number of collisions is given by:

ii14.591.312130515.829.7132

1 EiEn

n = -ln(-) .The constant E can be computed for a mixture of elements by weighting thevalue of each ti for element i with the appropriate total scattering crosssection Nioi. Table 11-1 shows some typical values for the averagelogarithmic energy decrement and the number of collisions necessary toreduce source energy neutrons (4.2 MeV) to 1 eV.

*(avg. number of collisions from 4.2 MeV to 1 eV)Table 11 1. Average logarithmic energy decrement and average number ofcollisions for reducing neutron energy from 4.2 MeV to 1 eV forselected moderators.There are two more parameters which help to characterize neutroninteractions with bulk material. One parameter is known as the slowing-downlength, and the other as the thermal neutron diffusion length, Ld. Theslowing-down length is proportional to the root-mean-square distance fromthe point of emission of a high energy neutron to the point at which i t arrivesat the lower edge of the epithermal energy region. This distance can becalculated from a detailed knowledge of the cross sections of the constituentelements. Fig. 11-8 shows the slowing-down length as a function of water-filled porosity for limestone, sandstone, and dolomite.

In order to understand the variations of the slowing-down length seen inFig. 11-8, it is interesting to compare it with a random walk. The randomwalk in one dimension is shown in Fig, 11-9, which plots for three trials(three different neutrons) the distance from the starting point as a function ofthe number of equal-length steps taken. At each step the probabilities for aforward or backward displacement are equal. It is obvious that the averagedisplacement from the starting point for a large number of trials is zero.However, there is a distribution of terminal points around the origin. Ameasure of the width, or spread, of the distribution is the root-mean-squaredisplacement, which can be shown to be equal to times the length of thestep. Fig. 11-10 shows the probability distribution for three series of random


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Neutron Physics for Logging Applications 237walks, each containing a different number of steps. They all center aboutzero (no displacement from the origin), but the width increases as the numberof steps taken increases.

10 20 30 40 50 6 70 8 9 100Water-F i l led Poros i t y . p.u.

Figure 11 8. The slowing-down length of sandstone, limestone, and dolomite as afunction of porosity. From Ellis.

5iD

-10-I I I0 10 20 30-N (Steps taken)

Three trials of a random walk. Adapted from Feynman.6igure 11 9.


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238 Well Logging for Earth ScientistsAlthough the slowing-down of neutrons is a three-dimensional process,

and the free path between collisions varies somewhat, it can still be thoughtof as a random walk. One important feature which distinguishes the randomwalk in a zero porosity limestone from one in water is the number ofcollisions (the number of steps taken in the random walk). Fig. 11-11illustrates this idea, along with a few useful parameters. From Table 11-1,the number of co llisions in the slowing-down process in limestone is aboutnine times that in water. Consequently, if we associate the slowing-downlength with the root-mean-square displacement for a random walk, we expect

Probabi l i ty

N = 10,000 Steps

600 400 200 0 200 400 600D = Distance f rom star t

Figure 11 10. The probability distribution of terminal points for random walks withthree large step numbers. Adapted from Fey nm aa6

N = 132

0 p.u. LimeFigure 11 11. A schematic of the slowing-down trajectories of neubons i n water andlimestone, suggesting the connection with the random walk. Theratio of the slowing-down length in limestone to that in water followsthe expectation from the average number of collisions. From Ellis.


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240 Well Logging for Earth ScientistsMacroscopic ThermalAbsorption X-sections

BoronChlorineHydrogenManganeseIron 27.5

Table 11-2. Mass-normalized macroscopic thermal absorption cross section forselected elements.Another sometimes useful parameter, the migration length (Lm), has beenA & n d CX

It can be viewed as a distance which represents the combination of the pathtraveled during the slowing-down phase (L,) and the distance traveled in thethermal phase before being captured (Ld). This parameter provides aconvenient way of predicting the response of a thermal neutron porositydevice, which is discussed in more detail in the next chapter.

NEUTRON DETECTORSNeutrons are detected in a two-step process. First, he neutrons react with amaterial in which energetic charged particles are produced. Then the chargedparticles are detected through their ionizing ability. Thus most neutrondetectors consist of a target material for this conversion, coupled with aconventional detector, such as a proportional counter or scintillator, to achievethe measurement. Since the cross section for neutron interactions in mostmaterials is a strong function of neutron energy, different techniques havebeen developed for different energy regions. For well logging applications, atpresent, it is the detection of thermal and epithermal neutrons which is ofinterest. The detection schemes considered in this section are appropriate forthese low energy neutrons.

Nuclear reactions useful for neutron detectors must satisfy several criteria:the cross section for reaction must be very large, the target nuclide should beof high isotopic abundance, and the energy liberated in the reaction followingthe neutron absorption should be high enough for ease of detection byconventional means. Three target nuclei have been found to generally satisfythese conditions: I%, 6Li, and 3He. In the case of the first two targets, the(n,a) reaction is utilized, and for 3He, it is the (n,p) reaction.

The boron reaction is widely exploited in the form of BF3 in aproportional counter. In this case the boron trifluoride serves as both the


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Neutron Physics for Logging Applications 241target and the ionization medium. For this application, the gas is enrichedin log, to attain a high detection efficiency. Another approach is to use aboron coating on the inner wall of a proportional counter, which may usesome other proportional gas more suitable than BF3 for applications involvingfast timing, for example.

Since a suitable lithium compound gas does not exist, the lithium reactionis not exploited in proportional counters. However LiI scintillators, similar tosodium iodide for gamma ray detection, are available. Due to the largeenergy released by the (n, a eaction, neutrons are registered at an energy ofabout 4.1 MeV, which provides a means of discriminating against the gammarays, which are also readily detected by the LiI crystal.

The most common neutron detector in well logging, however, is based onthe 3He (n,p) reaction. In this case, 3He is used as the target and proportionalgas in a counter. It is preferred since it has a higher cross section than theboron reaction and the gas pressure can be ma& much higher than for BF,without degradation of its proportional operation. The simplicity of aproportional counter is also preferred to the complications associated with ascintillator.

For the three reactions discussed above, the cross sections vary inverselywith the square root of the neutron energy, so that the detection efficiency forneutrons will vary in the same manner. The detectors employing thesereactions respond primarily to thermal neutrons. For some loggingapplications it is desirable to measure the epithermal neutron flux, whilebeing insensitive to thermal neutrons. This can be achieved by making aminor modification on any of the three types of detectors previouslymentioned. It consists of using a shield of thermal-neutron absorbing materialwith a large cross section, such as cadmium, around the detector. Thermalneutrons will be absorbed in the shield, but the reaction particles, whoserange is small (on the order of tenths of mm), do not reach the counter. Thehigher energy epithermal neutrons which manage to penetrate the shield aredetected with somewhat reduced efficiency.

1.2.3.4.

ReferencesEllis, D. V., Nuclear Logging Techniques in SPE PetroleumProduction Handbook, edited by H. Bradley, SPE, Dallas (in press).Weidner, R. T., and Sells, R. L., Elementary Modern Physics, Allynand Bacon, Boston, 1960, pp. 371-392.Rydin, R. A. Nuclear Reactor Theory and Design, UniversityPublications, Blacksburg, 1977.Evans, R. D., The Atomic Nucleus, McGraw-Hill, New York 1967,pp. 426-438.


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242 Well Logging for Earth Scientists5 . Kreft, A., Calculation of the Neutron Slowing Down Length inRocks and Soils, Nukleonika, Vol. 19 1974.6. Feynman, R. P., Leighton, R. B., and Sands, M. L., Feynmun

Lectures o n Physics, Vol. 1 Addison-Wesley, Reading, Mass., 1965.

Problems1. A neutron generator used in logging applications employs the D-Treaction illustrated in Fig. 11-13. The result of the reaction is twoparticles (a neutron and a 4He) which share 17.6 MeV of energy.

a.

b.Applying conservation of energy and momentum to the reactionproducts, calculate the neutron energy.If the reaction products were instead a neutron and a 3He, sharingthe same 17.6 MeV, what would the resultant neutron energy be?

I100 keVD T8 a

Figure 11 13. The D-T reaction used for producing 14 MeV neutrons.

2. Using the data of Figs. 11-8 and 11-12a.b.3 . From the data in Table 11-2 estimate the macroscopic thermalabsorption cross section of water. Assume that the oxygen can beneglected. Express the answer in capture units. How does it comparewith the standard value of 22 cu?

4. What is the mean free path of a thermal neutron in water? W hat is themean free path of a 4 MeV neutron in water?5 . Analysis of a shale core sample whose density is 2.60 g/cm3 indicates

that the concentration of boron is 400 ppm. What is its contribution, incu, to the total X of the sample?

Com pute the diffusion length in water.Com pute the diffusion length in 0-PU and 20-PU limestone.

11. basic neutron physics for logging applications

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