Neutron Optics

Chapter

5 Neutron Optics Jay Theodore Cremer, Jr.

Contents

1. Neutron Phase and Group Velocity 2. Derivation of the Schr¨odinger Wave Equation for Neutron Wave 3. Derivation of the Schr¨odinger Wave Equation by the Electromagnetic Wave Equation Analogy 4. Derivation of the Schr¨odinger Wave Equation by Assumption of the Neutron Plane Wave Solution 5. Operator form of the Schr¨odinger Wave Equation for a Neutron Wave 6. Reflection and Transmission of Neutrons at Media Interfaces 7. Neutron Reflectometry 8. Measurement of the Complex Refractive Index via Refraction 9. X-ray and Neutron Interferometry 10. Interferometry and the Fizeau Effect 11. Pendell¨osung Oscillations and Anomalous Absorption in Perfect Crystals 12. Measurement of Refractive Index via Interferometry 13. The Maxwell–Boltzmann Distribution for the Reactor Source of Thermal Neutrons References

561 566 568 572 574 576 582 588 590 592 596 601 603 606

1. NEUTRON PHASE AND GROUP VELOCITY In the particle view, the neutron energy E is related to its rest mass m0 and momentum p by the Einstein relation E = γ m0 c 2 .

(1)

Advances in Imaging and Electron Physics, Volume 172, ISSN 1076-5670, DOI: 10.1016/B978-0-12-394422-1.00005-0. c 2012 Elsevier Inc. All rights reserved. Copyright

561

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Jay Theodore Cremer, Jr

The velocity-dependent relativistic gamma factor γ is related to the neutron velocity v and the speed of light c by v2 γ = 1− 2 c

!−1/2 .

(2)

The energy E of the neutron is related to the neutron momentum p by  2 E2 = m0 c2 + p2 .

(3)

Also, the non-relativistic thermal neutron kinetic energy K is related to its velocity v, where the neutron mass m at velocity v is approximated by its rest mass m ∼ = m0 , where K=

mv2 . 2

(4)

The neutron momentum p is then p = mv.

(5)

The non-relativistic kinetic energy K and momentum p are related by the neutron mass m ∼ = m0 : K=

p2 . 2m

(6)

A photon has rest mass m0 because at the speed of light c, the photon would have infinite total energy E with a finite rest mass. Hence, the total energy E of a photon is related to its momentum p by E = pc.

(7)

In a vacuum, the photon frequency f is related to its speed of light velocity c by the photon wavelength λ with the following equation: f =

c . λ

(8)

The photon energy E is related to its wavelength λ by E=

hc . λ

(9)

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The Einstein photon energy cp is equated to the Planck photon energy expression hf , and one applies the wave relation c = f λ, where E = cp

and E = hf .

(10)

Equating the two expressions of Eq. 10 yields the de Broglie relation p=

h . λ

(11)

Based on the work of Planck and Einstein, de Broglie was able to produce a wave representation of matter and energy. In the wave representation of matter, the de Broglie hypothesis relates to the momentum p of the neutron to the center wavelength λ of a localized wave packet. The neutron momentum p is related to the wave number k = 2π/λ of the center wavelength of the representative wave packet by p=

h = ~k. λ

(12)

The neutron momentum is the product of the neutron rest mass m0 and velocity v, where p = m0 v.

(13)

The neutron wavelength λ is then inversely proportional to its group or particle velocity v as follows: λ=

h . m0 v

(14)

Alternatively, the neutron particle velocity v is related to the neutron wave number k by v=

~k . m0

(15)


Given wavelength λ [A], the neutron particle velocity in v km sec is
3.956 v km/s = . λ [A]

(16)

In reciprocal space (K-space), the neutron energy E is E=

p2 ~2 k 2 = . 2m0 2m0

(17)

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In direct space (R-space), the neutron energy E is E=

h2 . 2m0 λ2

(18)

The neutron wave phase velocity, in terms of angular frequency w/E = ~ and wave number k, is vp =

wE 1 1 ~2 k2 ~k w = = = . k Ek ~ k 2m0 2m0

(19)

At non-relativistic speeds, the neutron wave phase velocity vp of an unconfined neutron is half the neutron particle velocity v in a vacuum: vp =

~k v = . 2m0 2

(20)

The neutron is confined to the region of the lens or detector, and thus the neutron is not a single sinusoid wave, but rather a superposition of waves. The amplitude modulation of the wave packet has group velocity vg =

dw . dk

(21)

Application of Eqs. 19 and 21 shows the neutron wave group velocity is equal to the neutron particle velocity: ~k dw dE = = v. (22) dE dk m0
The frequency distribution w k of the wave packet is written in terms of expansion  k about the packet center wave number k = k0 . The group velocity dw dk is the linear term of the expansion, and it is evaluated at packet center peak k = k0 . Hence, rapid variation of w as a function of k invalidates the use of the group velocity vg to represent the velocity of energy or information flow, because other terms, such as the quadratic term d2 w dk2 , become important and the packet velocity can no longer be characterized by its center velocity, vg . For more details, see Eisberg and Resnick (1974). In a vacuum, which has unity refractive index n = 1, a thermal neutron with phase velocity vp = v0 (and group velocity vg = 2v0 ) travels length L0 in time τ0 , where vg =

τ0 =

L . v0

(23)

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However, in a slab with refractive index n with the same length L0 as the vacuum, the thermal neutron wave propagates at phase velocity vp over distance L0 in time τ , where v0 v0 = n 1−δ

(24)

L0 nL0 nτ0 = (1 − δ)τ0 . = vp v0

(25)

vp = and τ=

For time interval τ0 , the neutron wave phase front propagates distance L0 in a vacuum and distance L in the slab, where L = vp τ0 =

v0 L0 L0 L0 = = . n v0 n 1−δ

(26)

A neutron wave packet with group velocity vg is identical to the corresponding neutron particle velocity v, and they are twice the neutron wave phase velocity vp , where v = vg = 2vp .

(27)

For the identical time interval τ0 , a neutron wave packet with group velocity vg or corresponding neutron particle velocity v, then travels distance Lg , where Lg = vτ0 = vg τ0 = 2vp

v0 L0 L0 L0 L0 =2 =2 = 2L = 2 . v0 n v0 n 1−δ

(28)

In a vacuum or material medium for the same time interval τ0 , the neutron wave packet (modulated sine wave), which is represented by total neutron wave function 9(r, t) and the corresponding neutron particle, propagates at twice the velocity v = vg = 2vp as phase velocity vp of the individual sine P wave, neutron wave function, and components ψi (r, t), where 9(r, t) = i ψi (r, t). Hence, in time interval τ0 , the neutron particle or equivalent wave packet 9(r, t) then travels a distance Lg = 2L, and the individual sine wave, neutron wave function, and components ψi (r, t) travel half the distance L.

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Jay Theodore Cremer, Jr

¨ 2. DERIVATION OF THE SCHRODINGER WAVE EQUATION FOR NEUTRON WAVE ¨ For the wave view of matter, Schrodinger derived the wave equation. ¨ In hindsight, the Schrodinger wave equation can be derived using the following five assumptions: 1. The wave equation should be consistent with the Planck equation. The Planck equation relates wave angular frequency w or period T = 2π/w to wave energy E via Planck’s proportionality constant h, where E ~ h T= E

w=

(frequency)

(29)

(temporal).

(30)

2. The wave equation should be consistent with the de Broglie relation. The de Broglie relation between a particle’s momentum p and the wavelength λ of the particle’s wave representation is given by Planck’s constant via h p p k= ~

λ=

(direct space)

(31)

(reciprocal space).

(32)

3. The wave equation should be consistent with the Hamiltonian equation. The Hamiltonian equation relates total particle energy E to its kinetic energy K and potential energy V(r), where E = K + V.

(33)

The kinetic energy K of the particle is related to the particle rest mass m0 and momentum p by K=

p2 . 2m0

(34)

4. The solution of the wave equation 9(r, t) as a function of position r x, y, z and time t should be linear.




If 91 (r, t) and 92 (r, t) are solutions of the wave equation and c1 and c2 are constants, then their linear sum is also a solution: 9(r, t) = c1 91 (r, t) + c2 92 (r, t) .

(35)

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This linearity allows the addition of wave functions of one or more particles together and produce constructive and destructive interference. This wave interference causes the diffraction and refraction observed in experiments with particles, such as the Davisson–Germer experiments on electron and neutron diffraction. 5. Assume that the total energy E of a particle is constant. The total energy E is per the following equation: E = K + V.

(36)

Taking the differential shows that the increase of kinetic energy is equal to the decrease in potential energy: 1K = −1V.

(37)

Given the work-energy theorem, relating force F and the differential distance 1x over which it acts give differential kinetic energy 1K: 1K = F1x.

(38)

Substitution into Eq. 38 for the change of kinetic energy 1K in terms of change in potential energy 1V of Eq. 37 gives F=−

1V . 1x

(39)

The spatial gradient (or grad) of the potential is ∇V(r, t) =

1V 1V 1V xˆ + yˆ + zˆ . 1x 1y 1z

(40)

The force F (r, t) should be equal, but the force should act in the opposite direction of the gradient, where F (r, t) = −∇V (r, t) .

(41)

That is, the force acts in a direction opposite to the spatial change of potential, similar to an induced current in a coil producing a magnetic field to counter the temporal change of the magnetic field linking the coil. This is an act of conservation of energy; without the minus sign, an increase in potential energy over a spatial interval would cause the kinetic energy to increase with increasing potential energy. In the case of a free particle, which has constant momentum and thus constant energy, one requires a

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Jay Theodore Cremer, Jr

zero gradient ∇V(r, t) = 0, in which the potential is constant V(r, t) = c throughout space, where constant c can also be zero. The combination of assumptions 1–3 produces the following form of the conservation of energy expression or Hamiltonian expression H, where total energy E is H=E=

~2 k 2 + V (r, t) = ~w. 2m0

(42)

Rearrangement yields the required outcome of the matter wave equation that satisfies the conservation of energy or Hamiltonian expression: ~ 2 1 k + V = w. 2m0 ~

(43)

The neutron of rest mass m0 = 938 keV is represented by a position r 
dependent wave function ψ of energy, E = h 2π w, and momentum, (r)  ¨ p = h λ, and follows the time-independent Schrodinger wave equation. ¨ The Schrodinger wave equation with an average nuclear interaction potential for a neutron is ! 2 m (E − V) 8π 0 ψ(r) = 0. ∇2 + h2

(44)

The average potential hVi inside the scattering medium of volume U consists of N isotopes, with each jth isotope possessing scattering length bj , and is

hVi =

N h2 X bj . 2π m0 U

(45)

j=1

¨ 3. DERIVATION OF THE SCHRODINGER WAVE EQUATION BY THE ELECTROMAGNETIC WAVE EQUATION ANALOGY One can look to the analogous wave equations to aid in the derivation of ¨ the Schrodinger wave equation. Consider the wave equation for an electromagnetic field propagating in the z-direction through a material zero charge, but constant conductivity σ , where ∇ · E⊥ = 0.

(46)

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In this case, a current density J⊥ is induced in the material by the incident electric field amplitude E⊥ , where J⊥ = σ E⊥

(47)


 E⊥ = E0 exp i k0 z − wt .

(48)

and

The electric and magnetic fields in CGS (centimeter-gram-second) units of an electromagnetic wave are equal, so the initial magnetic field is as follows: B⊥ = E⊥ .

(49)

The vacuum space neutron wave number k0 is k0 =

w . c

(50)

In CGS units, the resultant electric E⊥ (z, t) and magnetic B⊥ (z, t) fields in the material are identical, and so are their wave equations: ∇ 2 E⊥ −

1 ∂ 2 E⊥ 4π ∂E⊥ = kσ 2 2 c ∂t c ∂t

(51)

∇ 2 B⊥ −

1 ∂ 2 B⊥ 4π ∂B⊥ = kσ . 2 2 c ∂t c ∂t

(52)

where

In the material, a periodic solution is assumed for both the electric field E⊥ and magnetic field B⊥ . In the material, the wave number is no longer the free space wave number k0 , but rather is a wave number k, which arises as a result of the interference of the incident electromagnetic wave and the electromagnetic wave induced in the material, so that 
 E⊥ = B⊥ = E0 exp i kz − wt .

(53)

Substitution of the resultant electric and magnetic field amplitudes of Eq. 53 into the two wave equation expressions of Eqs. 51 and 52 yields the solution of the wave number for the electromagnetic wave in the material: c2 k 2 w2 − = iw. 4π σ 4π σ

(54)

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Jay Theodore Cremer, Jr

Solving for the neutron wave number k in the material gives r k = k0

r 4π 4π w 1 + ik0 σ 1 + iwσ 2 . = c c c

(55)

This verifies the validity of the assumed exponential solution in a vacuum and the material, where 
 E⊥ = E0 exp i kz − wt .

(56)

Only the wave number changes when the electromagnetic wave moves from a vacuum into a material. The electromagnetic wave moves at phase velocity c through a vacuum and phase velocity vp through the material. The index of refraction of the material is n=

c . vp

(57)

In the material, the neutron wave number k is the ratio of the neutron wave angular frequency w divided by the neutron phase velocity vp : k=

w . vp

(58)

The refractive index n is defined as the ratio of the neutron wave number in the material k divided by the neutron wave number k0 in a vacuum, where n=

k k0

(59)

Substitution in Eq. 59 for k from Eq. 55 yields s n=

1+i

4π σ . k0 c

(60)

The conductivity σ of the material can be complex, where σ = σr + iσi .

(61)

In this case, the real part σr contributes to wave attenuation, and the imaginary part σi contributes to the phase shift of the resultant wave in the material with respect to the incident wave from the vacuum.

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The solution of the wave equation now provides a basis for ¨ determining the Schrodinger wave equation by analogy. The solution of ¨ the Schrodinger wave equation is assumed to be an exponential wave expression similar to that of the wave expression solution for the above electromagnetic plane wave, as well as the wave equations for scalar and vector potentials. Thus, the wave amplitude solution for the matter wave 9(z, t) is assumed to have the same form as that of the above electromagnetic wave E⊥ (z, t) solution, where 
 9 (z, t) = exp i kz − wt .

(62)

Multiplication on both sides of this equation by matter wave amplitude 9 yields 1 ~ 2 k 9 + V9 = w9. 2m0 ~

(63)

The imaginary argument of the exponential expression for the proposed matter wave amplitude 9 (z, t) is linear in time t (via iw) and position z (via ikx). The first derivative of 9 (z, t) taken with respect to time t or position z, yields the exponential term multiplied iw or ik. The second derivative of 9 (z, t) with respect to time t or position z multiplies the first derivative with respect to time t or position z by another factor of iw or ik. This ¨ suggests the following proposed form of the Schrodinger wave equation: V ∂9 ~ 2 ∇ 9+ 9= . 2m0 ~ ∂t

(64)

Plugging the proposed wave amplitude solution 9(z, t) into the proposed ¨ Schrodinger wave equation yields −

~ 2 V k + = −iw. 2m0 ~

(65)

¨ Comparison of the required outcome of the Schrodinger wave equation, ¨ the outcome of the proposed Schrodinger wave equation, and its proposed solution suggests a modification of the proposed equation. The form of the solution 9 (z, t) is unchanged, but the form of the proposed equation is slightly modified. The k2 term of the proposed√equation is multiplied by –1, and the d9 dt term is multiplied by i = −1, so that the proposed, modified wave equation is −

~ 2 V ∂9 ∇ 9+ 9=i . 2m0 ~ ∂t

(66)

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Substitution of the unchanged, proposed wave amplitude solution 9 (z, t) of Eq. 62 thus yields the required outcome of the wave equation of Eq. 66 and its solution (namely, the conservation of energy E), where ~ 2 k 9 + V9 = ~w = E 2m0

(67)


 9(z, t) = exp i kz − wt .

(68)

where

¨ The expression of the conservation of energy by this Schrodinger wave equation suggests that it can be represented by the Hamiltonian operator H, which was developed in classical mechanics to express total energy. In this case, the Hamiltonian operator H extracts its eigenvalue; namely, the total system energy E from its eigenfunction, the system state eigenfunction 9 (z, t), where H9 = E9

(69)

and the Hamiltonian operator H is then H=

~2 2 k + V = E. 2m0

(70)

¨ 4. DERIVATION OF THE SCHRODINGER WAVE EQUATION BY ASSUMPTION OF THE NEUTRON PLANE WAVE SOLUTION ¨ The simplest method to obtain the Schrodinger wave equation for a neutron is to consider a neutron plane wave propagating in the positive z-direction, and assume a neutron wave solution 9 (z, t), where
9 = sin wt − kz . (71) Differentiate 9 (z, t) twice with respect to distance z to obtain ∂ 29 = −k2 9. ∂z2

(72)

Recall that for a neutron particle of rest mass m0 and particle and wave group velocity v, the neutron wave number k is k=

m0 v . h

(73)

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Substitution of Eq. 73 into Eq. 72 yields ∂ 29 9 2m0 9 m0 v2 . = − 2 m20 v2 = − 2 2 2 ∂z ~ ~

(74)

The non-relativistic neutron kinetic energy K is K=

m0 v2 . 2

(75)

The total energy E of the neutron is related to its kinetic energy K and potential energy V by K = E − V.

(76)

¨ One then obtains the Schrodinger wave equation, which describes the propagation of the neutron wave in a medium, whose refractive index varies with distance z, where 2m0 9 ∂ 29 + (E − V) = 0. 2 ∂z ~2

(77)

Similarly, we can derive the wave equation for an electromagnetic wave propagating in the positive z-direction, and assume a transverse electric field amplitude wave solution E⊥ (z, t), where E⊥ = sin (wt − kz).

(78)

As before, differentiate E⊥ (z, t) twice with respect to distance z to obtain ∂ 2 E⊥ = −k2 E⊥ . ∂z2

(79)

The phase velocity v of an electromagnetic wave, which propagates through a material with index of refraction n = c/v (with n = 1 for a vacuum), is related to its wave number k = 2π/λ, the vacuum speed of light c, and the refractive index n by k=

nw . c

(80)

Substitution of Eq. 80 into Eq. 79 yields n2 2 ∂ 2 E⊥ = − w E⊥ . ∂z2 c2

(81)

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Differentiation of E⊥ (z, t) twice with respect to time t yields ∂ 2 E⊥ = −w2 E⊥ . ∂t2

(82)

Comparing Eqs. 81 and 82 yields the electromagnetic wave equation propagating through a medium of refractive index n, where ∂ 2 E⊥ n2 ∂ 2 E⊥ − = 0. ∂z2 c2 ∂t2

(83)

¨ 5. OPERATOR FORM OF THE SCHRODINGER WAVE EQUATION FOR A NEUTRON WAVE ¨ The Schrodinger wave equation can be obtained by replacing the momentum p, potential energy V, and total energy E in the Hamiltonian expression H by operators p, V, and E, where p → i~∇

(84)

V→V

(85)

E → i~

∂ . ∂t

¨ The resulting time-dependent Schrodinger wave equation is ! ~2 2 ∂9 − ∇ + V 9 = i~ . 2m0 ∂t

(86)

(87)

¨ The Schrodinger wave equation describes the evolution in time and space of a particle of rest mass m0 , in which the particle and its trajectory is described by state function 9 (z, t), which has total energy E, which is the sum of the particle kinetic energy K and potential energy V, where K=

~2 k2 . 2m0

(88)

In contrast, X-ray wave amplitudes are described by their electromagnetic field amplitudes via Maxwell’s equations and the Lorentz force equation, which are both invariant under the Lorentz transformation. Maxwell’s equations describe the generation and propagation of electromagnetic fields from accelerated charges via the scalar and vector Li´enard–Wiechert

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potentials that are solutions of Maxwell’s equations. However, in the Lorentz force equation, the electron rest mass m0 must be replaced by the electron relativistic mass γ m0 to be invariant under the Lorentz transformation. The electron relativistic mass m is obtained by multiplying the relativistic gamma factor γ with the electron rest mass m0 , where m = γ · m0 = p

1 1 − (v/c)2

· m0 .

(89)

The Lorentz invariant force equation then describes the electric and magnetic forces acting on charge particles that lead them to accelerate and emit electromagnetic fields. There is tight coupling between Maxwell’s equations and the Lorentz force equation, which are constrained by special relativity and boundary conditions of potential gradients. This tight coupling is evident in the undulators, wigglers, and bending magnets that change electron speed and/or electron direction—change in direction, change in speed, and a change in speed and direction are three ways to get acceleration and X-rays. Traditional X-ray sources in hindsight are relatively crude compared to the narrow-band undulator sources, in which traditional X-ray sources accelerate a beam of electrons, let them run into a tungsten anode and emit wide-bandwidth, bremsstrahlung X-rays. Reflection, refraction, and diffraction occur at boundaries between two media in which the total energy E remains unchanged, but the potential energy V can vary in type (electric, magnetic, nuclear, gravity, etc.); in addition, the magnitude and spatial gradient of the potential energy can vary between media, and even within the media. As a result, the kinetic energy K and potential energy V change from one medium to the next, and even inside media, but this is provided that the magnitude V (r) and gradient ∇V for the potential energy changed appropriately as well. From ¨ one medium to the next, the Schrodinger wave equation (in its differential equation form shown above or in its operator shown below) describes the time and space variation of the wave function amplitude 9 (z, t) with this very convenient constraint of total energy E, which makes simple solutions possible. ¨ The above Schrodinger wave equation is for non-relativistic appli¨ cations, and the above form of the Schrodinger wave equation is not ¨ invariant under the Lorentz transformation. The Schrodinger wave equation for relativistic particles must have the total energy E replaced by the special relativity identity between total energy E and momentum p, rest mass m0 , and speed of light c, where E→

q

c2 p2 + m20 c2 .

(90)

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6. REFLECTION AND TRANSMISSION OF NEUTRONS AT MEDIA INTERFACES The neutron plane wave is described by its ray, a line in the direction of travel perpendicular to the wave front. The neutron ray also represents the particle trajectory of the neutron. Consider a neutron particle that propagates in a vacuum, where the potential energy is zero V = 0, such that its velocity is s v0 =

2E . m0

(91)

If the neutron particle is incidental at a material surface at angle φ0 with respect to the surface normal, then the particle refraction angle φ1 is governed by the speed of the neutron particle and the continuity of its tangential velocity component across the interface. The potential energy V that the neutron particle encounters in the material yields the neutron particle velocity s v1 =

2 (E − V) . m0

(92)

The neutron particle tangential velocity remains unchanged across the boundary so that v0 sin φ0 = v1 sin φ1 .

(93)

Using Eq. 91 and substitution of Eq. 92 into Eq. 93 yields the description of the corpuscular or neutron particle refraction at an interface r sin φ1 v0 E = = . (94) sin φ0 v1 E−V In comparison, in the description of the same neutron refraction, but using the neutron wave approach, one applies Snell’s law: n0 sin φ0 = n1 sin φ1 .

(95)

The index of refraction of the vacuum n0 = 1 and the refractive index of the material are the ratio of the wave speed in the transmitting medium divided by the wave speed in the incident medium: n1 =

v1 . v0

(96)

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Hence, the wave description produces the same refraction result as this particle description. Neutron reflection and transmission is exploited by neutron waveguides, mirrors, and curved surfaces. The Fresnel equations for neutron reflection and transmission at the surface interface between two media are derived as follows. As a function of vector position r, time t, frequency w, and wave vector k, a neutron plane wave with amplitude ψi (r) is at incident angle θi measured relative to the surface tangent plane at the point of incidence. In general, a portion of the neutron wave is reflected and a portion transmitted. The transmitted neutron plane wave is refracted by the interface to refracted or transmitted angle θt with respect to the surface tangent plane. The transmitted wave has the same frequency w incident wave; however, the transmitted wave amplitude Et and wavelength λt (and associated wave vector kt and wave number magnitude kt ) differ from the incident wave. The remainder of the incident neutron wave is reflected from the surface at angle θr with respect to the surface tangent plane. The reflected wave has the same frequency w and wavelength λr (and associated wave vector kr with wave number magnitude kr ) as the incident wave. However, the reflected wave amplitude differs from the incident wave. The incident wave and reflected wave each has the same wave number, wavelength, and thus phase velocity, as they are traveling in the same vacuum medium (where hVi = 0). Hence, plugging in the incident and reflected wave amplitudes yields ki2 = kr2 =

8π 2 m0 E h2

(97)

where hVi = 0.

(98)

The incident neutron total energy is also the neutron kinetic energy, where E=

m0 v2 . 2

(99)

The neutron energy speed, v in terms of its wave number k, is v=

hk . 2π · m0

(100)

The index of refraction n is defined as the ratio of the wave speed vt in the refracting medium, divided by the wave speed vi in the incident medium.

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The neutron velocity is proportional to its wave number (v ∝ k), and thus, the index of refraction n is also the ratio of the respective wave number in the refracting medium kt divided by the wave number in the incident medium ki , where n=

kt vt = . vi ki

(101)

The incident, transmitted, and reflected neutron waves at the reflective surface are, respectively, ψi = Ei exp (wt − ki · r)

(102)

ψr = Er exp (iwt − ikr · r)

(103)

ψt = Et exp (iwt − ikt · r)

(104)

where ki =

2π λi

kr =

2π λr

kt =

2π . λt

(105)

The incident wave number ki and reflected wave number kr are equal: ki = kr .

(106)

However, the incident number ki and transmitted wave number kr are not equal: ki 6= kt .

(107)

The mirror or lens surface cannot change the tangential velocity component of the neutron wave. Hence, the tangential component of the incident wave ki should be equal to the tangential component of the refracted (or transmitted) wave kt , where ki cos θi = kt cos θt .

(108)

Snell’s law for neutron refraction between two media is now expressed in terms of the grazing angles, which are measured relative to the tangent surface, is cos θi = n cos θt .

(109)

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In comparison, the prior expression of Snell’s law that used angles measured relative to the surface normal—the 90◦ rotation of reference converts the sine to cosine. Incidence angles θi less than or equal to the grazing incidence critical angle θc lead to the total reflection of incident neutrons, where θt = 0 and cos θc = n = 1 − δ

δ  1.

(110)

The critical angle θc is a small except for ultracold neutrons so that θ2 cos θc ∼ =1− c =1−δ 2

θc  1

&

δ  1.

(111)

The critical angle of reflection for neutrons (and X-rays) is θc =

√ 2δ.

(112)

To calculate the neutron reflectivity of the surface, one can (for mathematical convenience) locate the surface at z = 0 in which the z-axis is perpendicular to the surface acting as the surface normal. Then, one invokes two continuity relationships. The first relationship is the continuity of the tangential components of the incident, reflected, and transmitted neutron waves at the surface, located at z = 0, where ψi + ψr = ψt .

(113)

ψr ψt = . ψi ψi

(114)

Rearranging gives 1+

The second relationship is the continuity of their normal derivatives of the neutron waves with respect to the surface normal z-direction, where dψi dψr dψt + = . dz dz dz

(115)

The normal derivative boundary condition evaluated at the surface at z = 0 yields (−ki sin θi )ψi + (kr sin θr )ψr = (−kt sin θt )ψt .

(116)

Only the components of the normal components of the wave vectors along the z-axis are involved in the differentiation of the arguments

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|ki · r|, |kr · r|, and |kt · r| of the exponentials. Note the sign convention of the z-components: The incident and transmitted wave z-components propagate in the same negative z-direction; whereas, the reflected wave z-components propagate in the opposite, positive z-direction. Since ki = kr

and θi = θr ,

(117)

The normal derivative boundary condition yields 

 kt sin θt ψt ψr −1 =− . ψi ki sin θi ψi

(118)

Substitution for ψt /ψi from Eq. 114 and solving for ψr /ψi gives the surface reflection fraction or reflectivity: R=

ki sin θi − kt sin θt ψr = . ψi ki sin θi + kt sin θt

(119)

The surface transmission fraction, or transmissivity T, is found by substitution of Eq. 119 into Eq. 118 or using, R + T = 1, where: T=

ψt 2ki sin θi = . ψi ki sin θi + kt sin θt

(120)

The probability that an incident neutron is reflected from a reflective surface PR is the ratio of the square of the amplitude ψr2 of the reflected neutron wave function divided by the square of the amplitude ψi2 of the incident neutron wave function, or 2 ki sin θi − kt sin θt 2 ψr |ψr |2 PR = |R|2 = = = k sin θ + k sin θ . ψi |ψi |2 t t i i

(121)

The probability of neutron transmission into the surface material PT is the ratio of the square of the amplitude ψt2 of the transmitted neutron wave function, divided by the square of the amplitude ψi2 of the incident neutron wave function or PT = T2 , where 2 2 ψt |ψt |2 2ki sin θi . PT = |T| = = = 2 ψi ki sin θi + kt sin θt |ψi | 2

(122)

With these results, the probabilities of reflection and transmission sum to unity, which is required for conservation of the number of neutrons at the mirror interface, where PR + PT = 1.

(123)

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Neutrons can be propagated from the reactor or spallation source to the user station over moderate distances (10–100 meters) via a neutron wave guide tube. The neutron wave guide tube, like a light wave guide, transmits neutrons of whose trajectory paths along the guide undergo 100% reflection for a grazing incident angle θ that is less than the neutron critical angle θc (λ) for neutrons of wavelength λ. The critical angle of neutron incidence θc (λ), at which or below which occur 100% reflection of neutrons with wavelength λ, is proportional to the neutron scatter length density ρb(λ) of the guide material, where ρ is the effective number density of the guide material scattering units (atoms or molecules) and b(λ) is the effective scatter length (femtometer units) of the scatter unit, where r θc = λ

ρb . π

(124)

The scatter length density ρb(λ) can be expressed in units of (#/cm2 ) or ˚ 2 ). There are some absorptive losses that attenuate the neutron beam (#/A as it undergoes grazing incidence reflection from one point to the next in the guide from the source to the user station. Free neutrons have an average lifetime of 885.7 sec, and free ultracold neutrons, which have energies below 300 (neV corresponding to a velocity of 7.6 m/s and a wavelength of 52 nm, are reflected at all angles from most materials, including normal (perpendicular) incidence. The low density of the ultracold neutrons causes them to behave as an ideal gas with a temperature of 3.5◦ milli-Kelvin (mK). All neutrons with rest mass mn = 1.675 × 10−27 kg are subject to gravity (acceleration g = 9.8 m/s2 ), and 300-neV neutrons gain a potential energy Ug (neV) with an increase in height h[m], where Ug (neV) = mn gh = 102 h.

(125)

However, the downward fall of ultracold neutrons due to gravity is very significant. Beryllium and beryllium oxide are good materials for neutron waveguides due to their relatively large refractive index decrement δ and relatively low cross sections for absorption and incoherent scatter. A. I. Frank, a pioneer in ultracold neutron optics, measured refractive indices n = 1 − δ for 17 amino acids and their deuterated derivatives for 5 m/s, 20 m/s, and 50 m/s ultracold neutrons. The refractive n index was as low as 0.570 (for tryptophan) to 0.375 (for deuterated tryptophan); see Frank (1987). The neutron spin produced a magnetic dipole moment µ (direction of µ points south S to north N), which interacts with the magnetic field B (B points north to south) to align the longitudinal component of the magnetic dipole moment point in the same direction of the applied magnetic field (lower-energy state) or in the opposite direction of the applied

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Jay Theodore Cremer, Jr

magnetic field (higher-energy state). The difference 1E in the higher Ehi and lower Elo energy states increases with applied magnetic field B[T] as 1E = Ehi − Elo = 60 neV/T.

(126)

7. NEUTRON REFLECTOMETRY Reflection of neutrons is a simple, effective method of measuring the neutron scatter length of a material, and thus the material refractive index for neutrons. A collimated beam of neutrons is totally reflected at gazing angles relative to a planar mirrorlike surface of the material of interest until the critical angle is reached. Measurement of the critical angle θc with a narrow, finely collimated, monochromatic neutron beam with wavelength λ that is at grazing incidence to a very flat, planar surface composed of a single isotope with atom number density ρ then yields the neutron scatter length b via r θc = λ

ρb . π

(127)

With measured critical angle θc , one can determine the atom density ρ of a single isotope material of known scatter length b, given a monochromatic neutron beam of wavelength λ. In the case of compounds, the critical angle θc yields the effective scatter length density ρb of the material. If the material is composed of N isotopes in which each nth isotope has scatter length bn and atom number density ρn , then the scatter length density ρb of the material is ρb =

N X

ρn bn .

(128)

n=1

In neutron scatter, the Christiansen filter is used to measure the neutron scatter length bp of a powdered material with a known atom, molecule, or particle density ρp that is dissolved (if soluble) or mixed in suspended state (insoluble) in a liquid of known scatter length bl and atom or molecule number density ρl . The boundary between the liquid medium and the particle medium creates a refractive interface with a relative index of refraction n that is the ratio of the refractive index of the particle np divided by that of the liquid nl , where n=

np . nl

(129)

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A neutron beam wavelength λ, the refractive index of the particle np , and the liquid nl are incident on the liquid with the suspended or dissolved powder in which the small angle scatter is measured as a function of the particle number density ρp , where λ2 λ2 ρp bp ρp bp  1 np ∼ =1− 2π 2π λ2 λ2 nl ∼ ρl bl ρl bl  1. =1− 2π 2π

(130) (131)

The relative refractive index n between the liquid and particle media is then defined by n∼ =

λ2 2π ρp bp . λ2 1 − 2π ρl bl

1−

(132)

The binomial approximation is applied as follows: λ2 n∼ ρp bp = 1− 2π

!

λ2 1+ ρl bl 2π

!


λ2 ∼ ρl bl − ρp bp . =1− 2π

(133)

When the scatter length density of the liquid and particle media are equal in which n = 1, then the minimum small-angle scatter angle is attained and measured, where ρl bl = ρp0 bp .

(134)

The particle number density ρp0 , which corresponds to the minimum measured scatter angle, then allows the determination of the particle neutron scatter length bp from the known neutron scatter length bl and number density ρl of liquid atoms or molecules, where bp =

ρl bl . ρp0

(135)

The accuracy of this method is about 0.1%. A large variety of liquids can be used, such as water, a combination of heavy water (D2 O) and water (H2 O), or organic liquids of known atom/molecule number density ρl and scatter length bl . This allows measurement of the neutron scatter length bp for a large variety of materials that can be dissolved, mixed, or suspended to create a particle-liquid refractive interface for each particle in the liquid.

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Jay Theodore Cremer, Jr

Thermal neutron scatter is dominated by the hydrogen in water (H2 O, in which hydrogen has a scatter cross section of 82 barns. In D2 O, by contrast, the thermal neutron cross section is 7.6 barns. By varying the ratio H O D O of the number density ρl 2 /ρl 2 of the D2 O and H2 O molecules in the liquid, and varying the density of the dissolved or mixed particle number density ρp , one can achieve equality between the liquid and particle scatter density in the neutron-interrogated sample, where ρp = ρp0 and the scatter angle is minimized that allows determination of the particle scatter length bp . For samples placed in water or on the water surface, contrast variation in small X-ray scatter can be accomplished by the replacement of H2 O by heavy water D2 O to alter the chemical composition of the sample. H2 O and D2 O have the same chemical properties so that the chemistry of the sample is not altered. In comparison, in X-ray diffraction and scatter contrast, variation must be done by altering the chemical composition of the sample by adding a heavy metal or altering the chemical composition of the liquid by adding a solute such as a salt or a sugar compound. Gravity refractometry presents a 100-fold increase in accuracy of the measurement of the neutron scatter length b compared to the ordinary mirror method described previously. The gravity refractometry method is based on a horizontal beam of neutrons with initial horizontal velocity vh that falls a vertical distance H at gravitational acceleration g onto a mirror surface of the material of interest (e.g., liquid mercury or molten lead). Conservation of energy states, the total neutron energy E, which is the sum of the initial horizontal, non-relativistic neutron kinetic energy Kh and neutron potential energy Uh , remains constant as the neutron falls from initial height H to the ground (H = 0). The neutron loses potential energy but gains vertical kinetic energy as it falls. However, the neutron horizontal kinetic energy remains constant (Kh1 = Kh2 ) during its fall, where E = Kh1 + Kv1 + Uh1 = Kh2 + Kv2 + Uh2 .

(136)

The initial neutron vertical kinetic energy is zero with Kh1 = 0. However, as the neutron initial potential energy Uh1 = mn gH decreases to zero potential energy Uh2 = 0 at the ground, the neutron of rest mass mn acquires vertical kinetic energy Kv2 . The neutron strikes the mirror surface at terminal vertical velocity vv . The energy conservation equation is, thus,

E=

mn v2h 2

+ mn gH =

mn v2h 2

+

mn v2v + 0. 2

(137)

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Solving for vertical terminal velocity vv at the mirror surface (H = 0) gives p vv = 2gH. (138) The horizontal distance L that the neutron travels before hitting the mirror material is the horizontal velocity vh multiplied by the time tv required for the neutron to fall vertical distance H with gravitational acceleration g, where s 2H 1 2 gt = H ⇒ tv = (139) 2 v g and s L = vh tv = vh

2H . g

(140)

The grazing angle of incidence θi of the neutron, which is measured relative to the mirror surface plane, is the inverse tangent of the final neutron vertical velocity vv divided by its assumed constant horizontal velocity vh . The experiment is set up for a small grazing incident angle θi so that the ratio of the velocities suffices, where p 2gH v v −1 vv θi = tan ⇒ θi ∼ = . (141) = vh vh vh We now express the prior derived reflectivity magnitude |R| in terms of the grazing incident angle θi and critical angle of incidence θc for total reflection at the mirror surface of the material of interest, where ki sin θi − kt sin θt 2 . |R| = (142) ki sin θi + kt sin θt Equivalently, 2 q ki sin θi − kt2 − kt2 cos2 θt |R| = q . ki sin θi + kt2 − kt2 cos2 θt

(143)

One recalls the boundary condition at the mirror (or lens surface), in which the tangential velocity component of the neutron wave cannot change upon crossing the interface, where ki cos θi = kt cos θt .

(144)

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Jay Theodore Cremer, Jr

The reflectivity magnitude |R| becomes 2 q ki sin θi − kt2 − k2 cos2 θi i |R| = q . 2 2 2 ki sin θi + kt − ki cos θi

(145)

At the critical incident angle of reflection θi = θc , the refracted (transmitted) angle is zero, where θt = 0. This allows one to obtain an expression for transmitted wave number kt in terms of the incident wave number ki and critical incident angle θc , where ki cos θc = kt .

(146)

Substitution for kt of Eq. 146 in the reflectivity magnitude |R| of Eq. 145 gives sin θ − pcos2 θ − cos2 θ 2 c i i |R| = p . sin θi + cos2 θc − cos2 θi

(147)

The small-angle approximation applied to the expression for the reflectivity magnitude |R| then gives 2 q θi − θ 2 − θc2 i |R| = (148) q 2 2 θ + θ − θ i c i where sin θ ∼ =θ

and

  θ 2∼ cos2 θ = 1 − = 1 − θ 2. 2

(149)

Simplifying, one obtains the reflectivity magnitude |R| as a function of the ratio of the critical incidence angle for total reflection divided by the incident angle θc /θi , where p 1 − 1 − (θ θ )2 2 c/ i |R| = (150) p . 1 + 1 − (θc /θi )2 From these derivations (measured relative to the mirror surface), the critical incident angle for total reflection θc and the neutron incidence angle at the mirror resulting from the fall of an initial horizontal neutron beam are r p 2gH ρb mn λ p ∼ and θi = = 2gH. (151) θc = λ π vh 2π~

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The ratio of the critical angle for total reflection θc divided by the gravity fall incident angle is θc = θi

s

2π ~2 ρb . m2n gH

(152)

One can thus define a particular height Hc for total reflection where Hc =

2π ~2 ρb m2n g

(153)

and θc = θi

r

Hc . H

(154)

The critical height Hc for neutrons is independent of the neutron beam initial horizontal velocity vh and the wavelength λ. That is, Hc is dependent only on the mirror material scatter length density ρb, which is the product of the atom number density ρ and neutron scatter length b. The critical height Hc allows a horizontal beam of neutrons of any initial horizontal velocity vh to fall onto a mirror surface at the critical grazing incidence angle θc for total neutron reflection. The reflectivity magnitude |R| as a function of the ratio of the critical height for total reflection divided by the chosen, initial height Hc /H is, then, 2 q 1 − 1 − (Hc /H)2 |R| = q . 1 + 1 − (Hc /H)2

(155)

Finally, the gravitational potential energy of the neutron at the critical height Hc is equal to the strong force nuclear scatter potential energy U of the mirror material, where U = mn gHc .

(156)

The effective nuclear strong force nuclear potential for a material is thus measured by finding the critical height Hc at which 100% reflection occurs and in which an incident horizontal neutron beam falls onto the mirror surface by gravity. One measures the completely reflected neutron beam at the proper reflection angle θr , which is equal to the critical incidence angle of for total reflection θc , where θr = θc .

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Jay Theodore Cremer, Jr

Frank and Nosov (2004) developed a one-dimensional magnetic gravitational trap for ultracold polarized neutrons in the higher-energy spin state using a Vladimirski˘ı magnetic mirror (Vladimirski˘ı, 1961). The Vladimirski˘ı mirror is a vertical (z-directed), one-dimensional magnetic field that varies sinusoidally along horizontal direction x. The sum of the magnetic and gravitational potentials has a minimum, and thus forms a potential well for the higher-energy spin-state neutrons, but no minimum and potential well forms for the lower-energy spin-state neutrons. Vertical (z-directed) magnetic field amplitudes that vary sinusoidally from –1.0 to 1.0 tesla on a planar surface in the horizontal x-direction with submillimeter periods allow the ultracold neutrons in the higher-energy spin state to be quantized in the magnetic-gravity potential well.

8. MEASUREMENT OF THE COMPLEX REFRACTIVE INDEX VIA REFRACTION The complex refractive index for neutrons (or X-rays) in a material can be measured via the refraction of an incident neutron beam from a planar slab of known thickness. Alternatively, the slab thickness can be determined via the slab refractive index. One starts with a slab of thickness d with refractive index n(λ) for neutrons (or X-rays) of wavelength λ given by n = 1 − δ + iβ.

(157)

A beam of incident plane wave neutrons at incident angle θi is refracted to angle θt relative to their respective outward-pointing surface normals. By Snell’s law: n=

sin θi . sin θt

(158)

Absorption is not involved in the refraction angles, so one obtains the decrement δ (λ) for neutrons (X-rays) where δ =1−

sin θi . sin θt

(159)

On the downstream or back surface of the planar slab is a neutron (X-ray) position-sensitive detector (PSD) to measure the incident beam position without the slab and then with the slab. The change in the neutron (X-ray) beam position measured by the PSD allows the determination of the incident and refracted neutron (X-ray) angles relative to the reference z-axis

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in which the neutron beam incidence is perpendicular to the slab surfaces and the back detector plane. A neutron (X-ray) PSD is placed on the front or upstream surface of the slab (facing the source plane) can measure the beam intensity at the front surface. And then removal of the front-surface PSD allows measurement of the refracted intensity of the beam that arrives at the downstream slab surface. The length of flight L of the neutron beam refracted at θt through the slab of thickness d is, then, L=

d . cos θt

(160)

From Snell’s law, with n = 1 − δ:   sin2 θi2 1 − cos2 θt .

(161)

One obtains cos θt = 1 −

sin θi n

(162)

and sin θi L=d 1− 1−δ 

−1 .

(163)

The refracted neutron (X-ray) beam amplitude At at the downstream slab surface is related to the amplitude Ai at the upstream slab surface as follows:   2π (164) At = Ai exp i L (1 − δ + iβ) . λ The transmitted neutron (X-ray) intensity T is 2   Ai 4π T = = exp − Lβ = exp (−µL) . A λ

(165)

t

The linear attenuation µ(cm−1 ) coefficient for the neutrons (X-rays) is, then, µ=

4π β. λ

(166)

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Jay Theodore Cremer, Jr

The slab linear attenuation coefficient µ(λ) or dimensionless, imaginary part of the slab refractive index β(λ) for neutrons (X-rays) is then obtained from the transmission fraction T so that β=−

λ loge T . 2π L

(167)

9. X-RAY AND NEUTRON INTERFEROMETRY Visible light interferometry is widely used for metrology in visible light optics and increasingly with ultraviolet (UV) and soft X-rays. X-ray interferometry is an important potential tool in astronomy, X-ray diagnostics, and metrology in general. Neutron interferometry can be used to provide accurate measurement of neutron scatter lengths and is applied to research in quantum mechanics and gravitation. Two categories of interferometry devices are (1) division of wave fronts and (2) division of amplitude. In visible light optics, the division of wave fronts is accomplished via the Young double-slit interferometer, and the division of amplitudes is accomplished by the Michelson interferometer. Interferometry in optics in general, and X-rays and neutrons specifically, require splitting an incident ray at a single point into two rays that follow separate paths that then converge to a single point, in which the beams are then recombined with constructive and destructive summing of their amplitudes. The squared magnitude of the summed amplitudes produces a modulated intensity pattern that is characteristic of the phase shift differences incurred between the two ray paths from a single source point to a single image point. Consider a monochromatic, small disk source of wave amplitude A0 , in which two very small diameter beams of equal amplitudes A0 /2 (conservation of neutrons) emerge at angles θ and −θ about the center z-axis. The beams travel a long object distance r0 and intercept a neutron lens as plane waves. The lens refracts the two beams to converge to a locus of points in the image plane at image distance ri . The transverse coherence length of each beam exceeds the radius of the image disk that conjugate to the source disk. The focal length f of the lens is half the object r0 and ri image distances so that r0 = ri the magnification is unity M = 1. The beams have traveled different total path lengths L1 and L2 from the source to image planes. Each beam is transverse coherent, such that the beams undergo constructive and destructive interference in the image plane. The amplitude Ar of the recombined pair of beams at the image plane is, then,      A0 2π 2π Ar = exp i L1 + φ1 + exp i L2 + φ2 (168) 2(r0 + ri ) λ λ

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591

where r0 + ri ∼ = L1 , L2 .

(169)

One assumes that the beam path lengths L1 and L2 are approximately equal to the axial distance r0 + ri between the source and image planes in the amplitude magnitude term. The intensity Ir of the recombining pair of beams at the image plane is, then,    2π 1 Ir = |Ar | = I0 1 + cos (w1 − w2 ) · t + (L1 − L2 ) + (φ1 − φ2 ) 2 λ (170) 2

where  I0 =

A0 (r0 + ri )

2 .

(171)

At room temperature, the thermal neutron velocity v = 2.2 × 105 cm/s and the wavelength is λ = 1.8 × 10−8 cm/s, so that w = 7.7 × 1013 rad/sec .

(172)

For a 10% bandwidth with w1 = w and w2 = w (1 ± 0.05), the variation of the neutron wave frequency is 3.8 THz, where w1 − w2 = w − w (1 ± 0.1) = ± 0.1 w = 3.8 × 1012 rad/s.

(173)

The maximum Imax and minimum Imin intensities of the recombined waves are Imax = I0

and Imin = I.

(174)

The contrast C is defined as C=

Imax − Imin = 1. Imax + Imin

(175)

The interference pattern rapidly oscillates between Imax and Imin at 3.8 THz, in which the time-averaged interference pattern is then, Ir = |Ar |2 =

   1 2π I0 1 + cos (L1 − L2 ) + (ϕ1 − ϕ2 ) , 2 λ

(176)

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Jay Theodore Cremer, Jr

where 1 < cos [(w1 − w2 ) · t] >= T

ZT

1 cos [1w · t] = T

0

ZT



t cos 2π T

 = 0. (177)

0

The spacing between adjacent maxima Imax and minima Imin is one-half period P, and the spacing between maxima pairs or between minima pairs is then P = λ.

(178)

The interference pattern is shifted by phase 8, where 8 = φ1 − φ2 .

(179)

The phase shift 8 can arise if one path contains an object whose length and refractive index causes a phase shift. The optical length difference 1L in the two paths can be varied so as to generate the interference pattern as a function of 1L, where n1 and n2 are the refractive indexes of the mediums in paths 1 and 2 and 1L = n1 L1 − n2 L2 .

(180)

The intensity of the interference pattern of the combined waves is then    1 2π Ir = |Ar | = I0 1 + cos 1L + 8 . 2 P 2

(181)

10. INTERFEROMETRY AND THE FIZEAU EFFECT Interferometry by division of the wave front, which used a Young doubleslit interferometer, was done by Klein et al. (1981). Their purpose was to measure the Fizeau effect with neutrons. In the Fizeau effect, waves undergo a phase shift due to the movement of the medium in addition to the stationary medium. Given non-relativistic, thermal, or cold neutron vacuum velocities vn and wavelength λ, and for moving medium velocity v, thickness D, and refractive index n, the neutron phase shift 18 incurred in a moving medium relative to a stationary medium is 18 =

2π v n − 1 D . λ vn n

(182)

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The Fizeau interferometer, which is used to measure the phase shift effects of visible light in a moving medium, is similar to a Fabry–Perot interferometer. The Fabry–Perot interferometer uses two mirrors, one of which is totally reflective and the other is partially reflective and transmissive, thereby allowing the splitting and then recombining of the wave amplitudes. The Fizeau effect for neutron waves in a moving medium, which is measured by an interferometer, is described as follows. Assume that relative to the interferometer lab frame, the neutron velocity is vi , and the velocity of the phase-shifting medium velocity is ui . The velocity of the neutron vm relative to the medium is vm = vi − ui .

(183)

The neutron vi and medium vm velocities are assumed to be nonrelativistic, and thus the Galilean transformations are applied. The neutron particle has mass m and wave vector ki , and by the de Broglie relation, the particle and wave momentum pi , which is measured relative to the interferometer lab frame, is related via pi = mvi = ~ki .

(184)

The neutron momentum pm and wave vector km , measured relative to the moving medium, is pm = mvm = ~km .

(185)

Taking the difference of these two equations yields m (vi − vm ) = ~ (ki − km ).

(186)

Use of the medium velocity ui in Eq. 186 for the vi − vm term in Eq. 186 gives an equation for the wave vector km that is measured relative to the moving medium. The equation is expressed in terms of the incident neutron wave vector ki in a vacuum and the medium velocity ui that is measured relative to the interferometer lab frame, where km = ki −

m ui . ~

(187)

The phase shift χ is scalar and thus invariant in the Galilean transformation. As a result, the phase shift has the same value χ (ki ) in the medium and the interferometer frames when the medium is at rest, relative to the

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Jay Theodore Cremer, Jr

interferometer lab frame. And when the medium is moving relative to the interferometer lab frame, then the phase shift changes to χ (km ), in which χ(km ) is the same in both frames. Thus, the phase shift 1χ between the moving medium and the stationary medium is 1χ = χ (km ) − χ (ki ).

(188)

The moving medium is rectangular, with parallel input and output planar surfaces separated by medium thickness D. The neutron beam is incident at angle φ measured relative to the normal of the input surface of the moving medium. The path length L of the neutron beam through the moving or stationary medium is then L=

D . cos φ

(189)

The z-component kiz of the incident neutron wave vector ki is perpendicular to the parallel input output faces of the medium, where kiz = |ki | cos φ.

(190)

The medium has refractive index n, expressed in terms of the nuclear potential U and the neutron total energy E. Outside the medium in the field-free vacuum, the neutron total energy E is equal to its kinetic energy W, where n=1−

U U Um U =1− =1− =1− 2 2E 2W mvi ~2 ki2

(191)

and 1 2 mv (Outside medium) 2 i E = U + K = W (Inside medium). E=W=

(192) (193)

The neutron kinetic energy K inside the medium is decreased from its vacuum (and field-free) value W by the nuclear potential energy U that is acquired by the neutron inside the stationary or moving medium, where K + U = E = W.

(194)

The total neutron energy E is constant throughout its trajectory, and the nuclear potential energy U of the medium that is experienced by the neutron is assumed to be constant.

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The phase shift χ (ki ), acquired by the neutron in its trajectory length L through the stationary medium, is χ (ki ) = (n − 1) ki L =

UmD UmD (n − 1) ki D =− 2 =− 2 . cos φ ~ ki cos φ ~ kiz

(195)

The phase shift χ (km ), acquired by the neutron in its identical trajectory length L through the moving medium, is χ (km ) = (n − 1) km L =

UmD UmD (n − 1) km D =− 2 . =− 2 cos φ ~ km cos φ ~ kmz

(196)

The difference 1χ between the phase shift for the neutron passing through a moving medium relative to a stationary medium is, then, UmD 1χ = χ (ki ) − χ (km ) = ~2



1 kmz

1 − kiz

 .

(197)

Next, one assumes that the z-component of the neutron wave vector in the moving medium and stationary medium are approximately the same, kmz ∼ = kiz . The neutron refractive index n = 1 − δ of the medium is assumed to be almost that of the vacuum n = 1, where δ  1. Hence, 1 kmz

−

kiz − kmz 1 1kz = = 2 kiz kiz kmz kiz

δ  1.

(198)

Substitution of Eqs. 195 and 196 into the expression of Eq. 197 for the phase shift 1χ due to a moving medium, with use of Eq. 198, gives 1χ = χ (ki ) − χ (km ) =

UmD 1kz 1kz =− χ (km ). 2 kiz ~2 kiz

(199)

Transposition yields 1kz 1χ =− . χ (km ) kiz

(200)

The z-components of the difference between the neutron wave vectors measured relative to the moving medium kmz and the stationary interferometer kiz frames are 1kz = kmz − kiz .

(201)

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Jay Theodore Cremer, Jr

One then recalls the previously derived expression: kmz = kiz −

m uiz . ~

(202)

Substitution of Eq. 202 into Eq. 201, and substitution of the result into Eq. 200 gives 1χ muiz uiz = = . χ(km ) ~kiz viz

(203)

The phase shift χ(ki ), acquired by neutron trajectory length L through the stationary medium, differs by 1χ from that of a moving medium only when there is a nonzero component of the incident neutron velocity vi (or wave vector ki ) that is parallel to the direction of the moving medium velocity ui . To simplify matters, assume that the medium moves in the z-direction that is perpendicular to the two planar interfaces with the vacuum. The parallel, input, and output planar medium-vacuum interfaces are separated by the medium thickness D. And the neutron beam is incident on the input interface at angle φ that is measured relative to the surface normal. When the medium is moving, the neutron phase shift acquired by the neutron passage through the medium differs from that of a stationary medium by the fractional difference 1χ /χ , which is the ratio of the medium velocity ui divided by the neutron velocity component vi cos φ in the z-direction of the moving medium, where 1χ uiz ui = = . χ(km ) viz vi cos φ

(204)

¨ 11. PENDELLOSUNG OSCILLATIONS AND ANOMALOUS ABSORPTION IN PERFECT CRYSTALS In visible light optics, one commonly uses the division of wave amplitude via partially reflecting and transmitting mirrors. In this way, the incident light wave can be divided into two coherent parts that are directed to travel different paths and finally to recombine. The different path lengths and phase-shifting obstacles or media in the paths cause the recombining coherent waves to undergo constructive and destructive interference that produces an interference pattern. Mirrors for X-rays and neutrons work only for grazing incidence angles, and in the division of amplitude method, such as the Michelson interferometer, mirrors would produce very small beam separations, even for very long path lengths. Wide separations between the two beam paths are needed so that a phase-shifting material, magnetic field, or

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other phase-altering obstacle can be placed in one of the beam paths. The resulting amplitude summing between the recombined beams leads to an interference pattern that characterizes the obstacle. An incident beam can be split into two widely separated beams by the combination of refraction and Bragg reflection in a large perfect crystal. With the development of pure, large, and almost perfect silicon crystals in the 1960s, the division of amplitude via Bragg reflection was developed for X-rays by Bonse and Hart (1965) and for neutrons by Rauch et al. (1974). With almost perfect silicon crystals, the two groups respectively developed an X-ray and neutron triple-Laue case (or LLL) interferometer. The LLL interferometer is fabricated from a single, large, perfect crystal. The rectangular crystal has two rectangular portions removed to form the letter “E.” Each horizontal line of the letter “E” is a planar slab with parallel surfaces and Bragg reflection planes that are perpendicular the slab surfaces. As a result, there is also a set of perpendicular crystal planes that are parallel to the slab surface. The z-directed gaps between the first slab, S, and the second slab, M, is equal to the z-directed gap between the second slab, M, and the third slab, A. In the kinematic theory of diffraction in crystals, the incident neutron (or X-ray) wave outside the crystal is assumed identical to the inside neutron (X-ray) waves incident on atomic planes in the crystal interior. Now consider an X-ray or neutron wave that is Bragg-reflected by a crystal plane in which the wave undergoes a second Bragg reflection by another parallel crystal plane. This double-Bragg-reflected X-ray or neutron wave now propagates in the same direction as the incident Xray or neutron wave. The kinematic theory does not account for the double-, triple-, or further-Bragg-reflected beam inside the crystal. The dynamic theory, however, does account for the multiple Bragg reflections and interference between the incident beam and multiple Bragg-reflected beams. In particular, we now consider the Borrmann effect or anomalous transmission for X-rays and the corresponding anomalous neutron transmission, which occurs only in perfect crystals. Consider the case of a perfect crystal in which the Bragg reflection planes are perpendicular to the planar crystal surface and thus parallel to the surface normals. This crystal configuration is termed the “symmetrical Laue case.” In the symmetrical Laue case, the Bragg forward reflection (the Laue case) can occur not only from the horizontal crystal planes that are perpendicular to the surface, but also from the vertical crystal planes that are parallel to the surface. The transmitted wave can be an unrefracted wave that passes without scatter through the crystal. Alternatively, the transmitted rays can be refracted at the entrance crystal surface and pass unscattered to the crystal slab exit surface, where the ray is again refracted; or the transmitted wave can be the result of a multiple-Bragg-reflected X-ray or neutron

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ray that propagates in the same direction as the incident ray through the crystal. As a result of the multiple Bragg reflections from the horizontal and vertical crystal planes, an incident X-ray or neutron wave is either transmitted directly through the crystal, or the wave is Bragg-reflected in the backward direction (the Bragg case) or Bragg-reflected in the forward direction (the Laue case). Each perfect crystal slab has a double-valued refractive index for refraction (i.e., transmission) and a double-valued refractive index for forward Bragg reflection (the Laue case) that are treated by the dynamical theory. The issue of perfect crystals for neutrons, dynamic theory of diffraction, and interferometers is covered by Sears (1989) and Squires (1996). X-ray interactions in perfect crystals are treated by Warren (1990). The perfect crystal of the symmetrical Laue case has horizontal Bragg reflection planes perpendicular to the planar crystal input and output surfaces, as well as vertical Bragg reflection planes parallel to those surfaces. Each incident neutron (X-ray) wave on the crystal slab produces two refracted (i.e., transmitted) waves and two Bragg-reflected waves that ¨ can interact, leading to the effects of extinction, Pendellosung interference, and anomalous absorption. The coherent, oscillating transfer of neutron flux occurs between the pair of refracted (transmitted) and Bragg reflected waves of wavelength λ. This oscillation is analogous to the energy transfer between a pair of coupled pendulums. Consider the case of the perfect crystal of thickness d, in which one ignores absorption because the crystal is optically thin for neutrons or X-rays. In this case, there is only the reflection and transmission of neutrons (X-rays). The neutron flux oscillation between the pair of refracted (transmitted) and Bragg-reflected waves of wavelength λ is observed by varying the crystal thickness d or the neutron wavelength λ. The fraction R of Bragg, forward-reflected neutrons (X-rays) and the fraction T of refracted (transmitted) neutrons (X-rays) are, respectively, R = sin2 y

and T = cos2 y

(205)

where y=

π d. 1hkl

(206)

¨ The Pendellosung period 1hkl is inversely proportional to the neutron (X-ray) wavelength λ and to the cosine of the Bragg reflection angle for the hkl reflection planes, where 1hkl =

π V0 cos θhkl . λ|Fhkl |

(207)

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¨ The Pendellosung period is also termed the extinction length 1hkl . Each crystal unit cell has volume V0 , and θhkl is the angle of incidence of the neutron wave, which is measured relative to the normal of the crystal slab surface, in which Bragg reflection (the Laue case) occurs from the hkl crystal planes. A Laue symmetrical crystal is cut so that the hkl crystal slab planes (e.g., 220) are perpendicular to the planar slab faces. The structure factor F (τ hkl ) for each of the N crystal unit cells is evaluated for the neutron scatter vector κ = τ hkl that leads to Bragg reflection from the set of hkl crystal planes, where F (τ hkl ) =

r X

bd exp (−Wnd ) exp (iτ hkl · dd ).

(208)

d=1

where bd is the neutron scatter length, which in the case of X-rays is replaced by the atomic form factor fd ; dd is the position vector relative to the unit cell reference corner; and Wnd is the Debye–Waller factor of the dth atom of the r atoms in each of the N unit cells that comprise the perfect crystal. The incident wave vector k0 at incidence angle θhkl to the slab surface normal yields diffracted wave vector k, in which scatter vector κ is equal to the reciprocal lattice vector τ hkl for the hkl set of Bragg reflection planes. The Debye–Waller factor Wnd is the thermally induced average displacement of the dth atom in the nth unit cell, whose reference corner is located at position Rn relative to the crystal center, where Wnd =

1 < (τ hkl · und )2 > . 2

(209)

The vibration amplitudes of the nth unit cell and the dth atom are expressed in terms of the set of 3Nr thermal vibration modes of the entire crystal in three dimensions. We now consider absorption in the perfect crystal, in which A is the fraction of incident neutrons (X-rays) absorbed by the crystal slab. The incident number of neutrons must equal the sum of the neutrons transmitted, reflected, or absorbed. Hence, neutron (X-ray) conservation requires R + T + A = 1.

(210)

For neutron path length D through the perfect crystal of thickness d and neutron linear attenuation coefficient µ for the crystal material, the absorption fraction A for anomalous absorption is given by A=

1 1 − exp (−2µD). 2 2

(211)

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That is, in a perfect crystal, the absorption independent of path length D and linear attenuation µ. The maximum fraction A of absorbed neutrons in even a very thick crystal is only 50% even as µD → ∞. In comparison, in normal absorption in an imperfect crystal, the absorption fraction A is given by Lamberts’s law, where A = 1 − exp (−2µD).

(212)

Thus, in a normal, imperfect crystal, the absorption fraction ranges from 0 to 1, and in a sufficiently thick or attenuating crystal, the transmission of neutrons goes to zero as µD increases. Anomalous absorption of X-rays or the Borrmann effect in perfect crystals was discovered by Borrmann (1941) and interpreted by Laue (1949). The anomalous absorption of neutrons was observed by Knowles (1956). For neutron or X-ray incident angles at the Bragg angle for the set of planes perpendicular to the planar crystal surfaces, the nodes of the interior coherent X-ray or neutron waves at the Bragg angle occur at positions of the absorbing nuclei. Given the linear attenuation µ coefficient for X-rays or neutrons in the perfect crystal slab and propagation path length L, the fraction of transmitted X-rays or neutrons is exp (−µL). Experimental observations showed that the X-rays or neutron rays were not extinguished in the crystal for path lengths of even µL = 20. The dynamic theory shows that at least 50% of the incident X-ray or neutron waves are transmitted by a perfect crystal, even for infinite thickness L → ∞ or linear attenuation µ → ∞. Until the advent of perfect crystals, the observation of anomalous absorption was not possible. That is, the pair of transmitted waves ψt+ and ψt− interfere by amplitude summation ψt+ + ψt− , and the pair of Bragg-reflected waves ψb+ and ψb− also interfere by amplitude summation ψb+ + ψb− . However, in a perfect crystal, the zero-amplitude nodes occur for one of the transmitted wave functions (ψt− = 0) at the absorbing crystal atoms when the other transmitted wave function ψt+ is at its maximum. And zero-amplitude nodes also occur for one of the Bragg-reflected wave functions (ψb− = 0) at the absorbing crystal atoms when the other Bragg-reflected wave function ψb+ is at its maximum. In mosaic crystals, primary extinction occurs as a result of multiple Bragg reflections in a single mosaic block, whereas secondary extinctions occur as a result of multiple Bragg reflections between different mosaic blocks. Each mosaic block is a perfect crystal in which the primary extinction is a dynamic scatter effect. The secondary extinction is a transport effect since the different mosaic blocks scatter incoherently. As a result, the mosaic crystal does not display the perfect crystal behavior of anomalous absorption, double refractive indexes for transmission and Bragg reflection, and the coupled interactions between the resulting four interior

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coherent waves produced by a single incident waves. Chapter 13 in the next volume will present the dynamic scatter of neutrons, which applies to X-rays as well, and provides derivations of neutron (X-ray) scatter in perfect and mosaic crystals.

12. MEASUREMENT OF REFRACTIVE INDEX VIA INTERFEROMETRY In this section, the perfect crystal interferometer is described, and its application for measurement of the neutron or X-ray refractive index of sample materials is presented. The perfect crystal interferometer requires samples with very parallel surfaces providing a constant sample thickness D. The sample cross-sectional area must be sufficiently large that the beam, which is incident at angle φ relative to the sample surface normal, allows the beam trajectory length d through uniform density sample material to remain in the material during the entire sample trajectory length d = D/ sin φ. Assume that a perfect crystal in the symmetrical Laue case has a particular set of hkl Bragg reflection planes that are perpendicular to the planar crystal surface. The set of crystal vertical planes perpendicular to the hkl horizontal planes leads to a pair of Bragg forward-reflected X-rays or neutrons and a pair of refracted (transmitted) rays. An incident neutron beam at incidence angle θ relative to the z-directed surface normal is refracted (transmitted) and Bragg-reflected as it enters the crystal slab at point A. As a result, two refracted waves emerge as a combined, single-refracted wave at angle θ relative to the exit surface normal pointing in the positive z-direction at point A0 . Entrance point A and exit point A0 are directly opposed to one another and separated by the planar crystal slab thickness D. The pair of transmitted and pair of Bragg forward-directed rays propagate in a narrow channel between the parallel planes that connect points A and A0 . The two collinear Bragg-reflected waves emerge as a combined, single wave at angle −θ relative to the same positive z-directed normal. The X-ray and neutron LLL interferometers are Mach-Zehnder interferometers. A single neutron or X-ray is incident at point A at angle θ relative to the backward-directed normal on the entrance surface of the first planar slab, S. On the direct opposite side of slab S at point A0 emerge two forward-directed rays at ±θ relative to the forward-directed exit surface normal. The two forward-directed Bragg scatter rays in the crystal sum and emerge as a single Bragg forward-scattered ray (the Laue case), and the two transmitted rays in the crystal sum and emerge from the crystal as a single transmitted ray. The transmitted and forward Bragg-reflected beams diverge at angle 2θ from the point A0 .

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The second planar slab, M, intercepts the two divergent rays from slab S (splitter). Then slab M (as in “mirror”) reflects the two incident beams in the forward direction (the Laue case) so the two beams converge to point X on the surface of a third planar slab A (analysis slab) with convergence angle 2θ . In this third (last) slab, A, the combined rays undergo transmission and Bragg forward reflection (the Laue case). The rays exit slab A at point X0 (opposite point X) as two diverging beams O and H with divergence angle 2θ . The two beams, O and H, that exit slab A are the result of interference between the recombined beams, and thus produce interference patterns. An object or rotating phase shifter is placed in one of the two beam paths between the second slab, M, and third slab, A. The rotating phase shifter can then vary the interference fringe pattern. And an object can be characterized by the interference patterns that it produces relative to the interference patterns without the object. A sample whose refractive index is to be determined is placed in one of the beam paths between the second slab, M, and this slab, A. The neutron (X-ray) beam is incident at angle φ relative to the sample surface normal and travels path length d through the sample thickness D and refractive index n, where d sin φ = D.

(213)

For accurate measurements, the sample surfaces must be extremely parallel to produce a constant sample thickness of D. For a particular neutron incidence angle φ, the phase shift 1χ produced by the sample with unknown refractive index n is χ = (n − 1)

2π D 2π d=δ . λ λ sin φ

(214)

Variation of the neutron (X-ray) beam incidence angle φ via rotation of the sample by 1φ results in a change in phase shift 1χ between the recombined beams at the A slab of the interferometer: 1χ = −δ

2π D1φ 2π D1φ cos φ = −δ cot φ. 2 λ sin φ λ sin φ

(215)

One varies the sample tilt angle 1φ until the interference fringes shift one full wavelength, which corresponds to phase shift 1χ = 2π . This yields

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603

the measurement of the refractive index decrement δ, where δ=−

λ sin φ tan φ . D1φ

(216)

˚ neutrons and sample thickness D = 0.5 cm with incident angle For 5-A ◦ 45 and 1φ = −0.05 rad, the decrement of the unknown material is then δ = 1.4 × 10−6 . A more practical setup is to have a phase shifter between the second and third interferometer slabs, M and A. One then places the sample of thickness D on the same beam leg but between the first and second interferometer slabs, S and M. The beam is incident on the sample at angle φ and on the phase shifter at angle −φ relative to their flat surface normals. Both the phase shifter and the sample are slabs with very parallel surfaces. To produce a fringe shift 1χ = 2π , the phase shifter is rotated angle 1φs with the sample in the beam, and then the phase shifter is rotated angle 1φ0 without the sample in the beam. The difference in the phase shifter angle with and without the sample yields the phase shift due to the sample alone, where 1φ = 1φs − 1φ0 .

(217)

The sample decrement δ for neutrons (X-rays) is, then, δ=−

λ sin φ tan φ . D (1φs − 1φ0 )

(218)

13. THE MAXWELL–BOLTZMANN DISTRIBUTION FOR THE REACTOR SOURCE OF THERMAL NEUTRONS Reactor sources produce a wavelength-dependent λ Maxwell–Boltzmann neutron flux distribution 8 (v), which is derived from the velocitydependent v flux distribution 8 (v) as follows. The Maxwell–Boltzmann neutron density n (v) distribution for neutrons of rest mass mn and velocities between v and v + dv is given by n (v) =

  4n0 v2 exp −v2r . 3√ vt π

(219)

The total, constant neutron density is n0 for all neutron velocities v in the distribution. The ratio vr is a particular neutron velocity v divided by the

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neutron peak velocity vt in the distribution: vr =

v . vt

(220)

The temperature of the Maxwell–Boltzmann distribution is T, and the peak neutron velocity vt of the distribution is given by s vt =

2kT . mn

(221)

Here, k is the Boltzmann constant, and the corresponding peak neutron de Broglie wavelength λt of the Maxwell–Boltzmann distribution is h h λt = √ = . m 2mn kT n vt

(222)

The Maxwell–Boltzmann distribution of beam neutrons of density n(v) emerges from the reactor, and the neutron beam flux 8(v) as a function of velocity v is the product of the neutron beam density n(v) and neutron velocity v, where 8(v) = nv =

  4n0 v3 2 exp −v . √ r v3t π

(223)

The increment of the velocity flux distribution 8(v)dv and the corresponding increment of the wavelength flux distribution 8(λ) dλ describe the same set of neutrons. As a result, the velocity 8(v) dv and wavelength 8(λ)dλ increments contain the same differential number dN of neutrons, where dN = 8(v)dv = 8(λ)dλ .

(224)

The de Broglie relationship between the neutron particle and the neutron wave representation is v=

h mn λ

(225)

h dλ. mn λ2

(226)

where dv = −

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Substitution for the expressions in Eqs. 225 and 226 for v and dv as a function of wavelength λ in 8(v)dv of Eq. 223 gives 

0 8(v)dv = 4n 3√ vt π

h mn

4

  dλ 2 exp −v . r λ5

(227)

Expression of vt in terms of λt by the de Broglie relation gives 4n0 2kT λ5t v2 8(v)dv = √ exp − 2 5 π h λ vt

! dλ.

(228)

Let λ0 be the ratio of the distribution peak wavelength λt divided by the neutron wavelength λ, where λ0 =

λt v = . λ vt

(229)

Using the expression for λ0 of Eq. 229 in the expression of Eq. 228 for 8(v)dv gives 8(v)dv =



   8kT √ n0 λ50 exp −λ20 dλ. h π

(230)

Having determined 8(v)dv , one can equate to 8(λ)dλ and then solve for the Maxwell–Boltzmann distribution neutron flux distribution 8 (λ) in terms of neutron wavelength λ. The neutron beam flux 8(λ) for neutrons with wavelengths between λ and λ + dλ is, then, 8(λ) =



   8kT √ n0 λ50 exp −λ20 . h π

(231)

The peak neutron flux 8p (λ) occurs at wavelength λ = λp , where r λp = λt

h 2 =√ . 5 5mn kT

(232)

Solving for λt gives r λt =

r 5 2 h h λp = λt =√ =√ . 2 5 5mn kT 2mn kT

(233)

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The flux distribution 8(λ) is normalized to unity at peak wavelength λp . With λ2t = 2.5 λ2p , the relative neutron flux distribution 8 (λ) becomes n0 kT 8(λ) = h

s

2 · 55 · π



λp λ

5

" exp 2.5 1 −

λ2p λ2

!# .

(234)

REFERENCES Bonse, U., & Hart, M. (1965). An x-ray interferometer. Applied Physics Letters, 6, 155–156. ¨ ¨ Borrmann, G. (1941). Uber Extinktionsdiagramme der Rontgenstrahlen (English translation: About the X-ray extinction diagram). Zeitschrift fur ¨ Physik, 42, 157–162. Eisberg, R., & Resnick, R. (1974). Chapter 3 De Broglie’s Postulates - Wavelike Properties of Particles of Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. John Wiley and Sons, New York. Frank, A. I. (1987). Optics of ultracold neutrons and the neutron microscope problem. Soviet Physics Uspekhi, 30(2), 110–132. Frank, A. I., & Nosov, V. G. (2004). Quantum effects in a one-dimensional magnetic gravitational trap for ultracold neutrons. JETP Letters, 79(7), 313–315. Klein, A. G., Kearney, P. D., Opat, G. I., Cimmino, A., & G¨ahler, R. (1981). Neutron interference by division of wavefront. Physical Review Letters, 46(14), 959–962. Knowles, J. W. (1956). Anamolous absorption of slow neutrons and X-rays in nearly perfect single crystals. Acta Crytallographica, 9, 61–69. ¨ Laue, M. V. (1949). Die Absorption der Rontgenstrahlen in Kristallen im Interferenzfall. Acta Crystallographica, 2, 106–113. Rauch, H., Treimer, W., & Bonse, U. (1974). Test of a single crystal neutron interferometer. Physics Letters, 47A, 369–371. Sears, V. F. (1989). Neutron Optics. Oxford University Press, Oxford, UK. Squires, G. L. (1996). Introduction to the Theory of Thermal Neutron Scattering. Dover Publications, New York. Vladimirski˘ı, V. V. (1961). Magnetic mirrors, channels, and bottles for cold neutrons. Soviet Physics Journal of Experimental and Theoretical Physics, 12, 740–746. Warren, B. F. (1990). X-ray Diffraction. Dover Publications, New York.