The Journal of
Physical Chemistry ~~~~
0 Copyright, 1991, by the American Chemical Society
VOLUME 95, NUMBER 23 NOVEMBER 14, 1991
LETTERS New Analytical Formula for Neutron Reflection from Surface Films Xiao-Lin Zhou,t.*Sow-Hsin Chen,*It and Gian P. Felcherf Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Argonne National Laboratory, Argonne, Illinois 60439 (Received: June 3, 1991)
The closed-form expressions for the neutron reflectance and transmittance of a thin film deposited on a bulk substrate are derived from the Green’sfunction solution of the Schroedinger wave equation in the material medium with an optical potential under the approximation that the curvature of the scattering length density profile in the film is small. This closed-form solution reduces to all the known approximations in various limiting cases and is shown to be more accurate than the existing approximations.
1, Introduction The specular reflection of a plane wave from a one-dimensional system illustrated in Figure 1 is a very useful phenomenon. By measuring the amplitude of the reflected wave, one can learn a great deal about the structure of the system. The wave can be electromagnetic (X-rays, light, or radiowave), sound, or that associated with slow neutrons, but the equation governing the wave motion turns out to be of the same form, namely, a one-dimensional Helmholtz equation (the theory presented in terms of neutrons in this paper applies to other types of waves as well):
U”(z) U”(z)
+ ko2U(z)= 0
of incidence, X the DeBroglie wavelength of neutrons in vacuum, and ps the scattering length density of the substrate. The z component of the neutron wave vector in the film is k(z), a function of the scattering length density profile p ( z ) of the film defined by k2 = ko2 - 2mV/h2 = ko2- 4 r p ( z ) Through the use of the Green’s function’ corresponding to eqs l.la-c, the neutron reflectance r and transmittance t can be derived as2 0
(for z 5 -d) in air
+ ko2U(z)= -(k2 - ko2)U(z) (for -d
(1.la)
r
5 z I0) in film
(for 0 Iz) in substrate
(1.lc)
where U(z) denotes the one-dimensional wave function, ko = 2 7 sin O/X, and k, = (ko2+f,)’12 withf, = -4rps. ko and k, are the z component of the neutron wave vector in free space and in the substrate medium, respectively. O here denotes the grazing angle
’MlT. f
Argonne.
0022-3654/91/2095-9025$02.50/0
(1.2a)
0
t
(1.lb)
U”(z) + k,2U(z) = 0
- R = i / 2 k 0 J -d dz’f(z9 U(z’)[e“w+ - T = ( i / 2 k 0 ) T ] dz’f(z’) U(z’)e-‘k@’ 4
(1.2b)
where flz) = k2 - ko2 = -2mV(z)/h2 = - 4 ~ p ( z ) ( 1 . 2 ~ ) (1) Tai. C.-T. Dyadic Green’s Functions in Electromagnetic Theory; Intext Educational Publishers: 1971; pp 35-36. (2) Zhou; X.-L.; Chen. S.-H.; Felcher, G.P. Inverse Problem in Neutron Reflection; invited lecture presented at The NATO Advanced Research Workshop on Inverse Problems in Scattering and Imaging, April 15-22, 1991, Cape Cod, MA.
0 1991 American Chemical Society
9026
Letters
The Journal of Physical Chemistry, Vol. 95, No. 23, I991 ib2
- 2ikib - ik‘ = 0
(2.5b)
The proper solutions of (2.5a,b) are
d1-
(2.6a)
= ikll - j / l - ik’/k2)
(2.6b)
i, = -ik{l i b
-
Clearly (2.6a,b) vanish for k’= 0, meaning that the variation of A and B is completely due to the variation of the scattering length
density profile. As a result, we get
sample Figure I . Reflection of a plane wave by a structured layer superimposed on the bulk substrate.
A ” = i,2A
(2.7a)
B” = ib2B
(2.7b)
Substitution of (2.7a,b) into (2.3a,b) gives A(z) =
is a function proportional to the sample scattering length density profile p(z), and R and T given by
I[ u - tJz$A
dz]
fi
(2.8a)
(1.3a) (1.3b) are respectively the Fresnel reflectance and transmittance of the half-space substrate in the absence of the film. I n the following section, we shall develop an approximate solution for U(z) and substitute this approximation into ( I .2a,b) to obtain the closed-form expressions for r and t . In section 3, to establish the accuracy of the closed-form formulas thus obtained, we compare them with the exact numerical results as well as with the Born and distorted wave Born approximations. 2. Closed-Form Expressions for rand t We derive in this section an explicit formula for r a n d t from
(1.2a,b) by solving for U(z)approximately under the continuity conditions at the boundaries. In view of the fact that there are two independent waves: the left-going and right-going waves, each satisfying the Helmholtz equation, one can write in general U(z) = A(z) exp( I ’ i k ( z ) dz)
+ B(z) exp(-&’ik(z)
dz) (2.1)
Substituting (2.1) into ( 1 . I b) gives two independent equations for the amplitude functions A and B: A ” + 2ikA’+ ik’A = 0
(2.2a)
B”- 2ikB’- ik’B = 0
(2.2b)
Solving (2.2a,b) formally, we obtain A(zj =
L[a fi
;IzK fi
dz]
6
[ b + i l z K d z ]
The integral equations have the following solution: A =
a
fi exp(
b B = - exp(
fi
-1’2 1’2
dz)
(2.9a)
dz)
(2.9b)
If one substitutes (2.9a,b) into (2.2a,b), one obtains that (2.9a,b) are valid when the neutron momenta are constrained by the conditions i:=o (2.10a)
i;=o
However, the accuracy of the expressions for r and f to be- obtained can go beyond the limitation of (2.10a,b) since the coefficients a and b in (2.9a,b) are to be determined consistently to satisfy the boundary conditions. The next step is to determine the constants u and b using the continuity conditions of U ( z ) and U’(z) at the film-substrate boundary ( z = 0) and the air-film boundary (2 = -d): U(0) = t (2.1 la) U’(0) = ik,t
(2.11b)
u(-d)= e-ikod +
(2.1 IC)
U’(-d) = jk,,{e-iW - reik0d)
(2.3a)
(2.3b)
J:(
fig-ikdtf exp( l:(ik a =
I
+ rbrfexp(
- i,2/2ik) dz)
2ik -
i,2 + i,z
for B
(2.1 2b)
where
(2.4b)
Substituting (2.4a,b) into (2.2a,b) and demanding that A and B be nontrivial, we have
La2 + 2 i K i , + ik’ = 0
)
(2.12a)
7) dz
b = rbU
E i b
(2.1 Id)
It is easy to show that u and b take the explicit expressions
where a and b are integration constants. Now, we define the differential operators in (2.2a,b) in the following way, assuming that they are nonzero only for nonvanishing value of k’: d/dz i, for A (2.4a) d/dz
(2.10b)
(2.5a)
k , - k f d l - ik;/kt rf =
ko + kfd-
(2.1 3 b)
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9027
Letters tf =
2kO
ko +
Here k f , k;, and kb, k 6 are the values of k and k’of the film evaluated at the front interface at z = -d and at the back interface at z = 0, respectively, and rf and tr are the effective Fresnel reflectance and transmittance of the front interface and f b the effective Fresnel reflectance of the back interface. Combining (2.l), (2.9a,b), and (2.12a,b), we obtain after further simplification
U ( r ) = A,[ exp( L’ikdfb
dz)
+
expj-i’ikd-
dz)] (2.14a)
e-jk&tfexp( J o-di k d l + i k ” / k Z = 1
+ 41 -
+ r g r exp( J I i k [ d -
C= dz)
Substitution of (2.14a,b) into (1.2a,b) gives us closed-form expressions for the reflectance r and transmittance t :
- R = -lodz’f(z’) i
-
1
I-
o
t - T = z T J d d z ’ f(z?e-ik@’Ao[exp(c+(z’))+ fb
exp(c-(z?)] (2.15b)
with c+(z? = f x ’ ; k . \ l l x d z
(2.1%)
where R and T are the known air-substrate interface Fresnel reflectance and transmittance and are given by eqs 1.3a,b, f b is the film-substrate interface effective Fresnel reflectance given by eq 2.13a, and A. is a constant defined by eq 2.14b, which can be evaluated when the film scattering length density profile is specified. Even with the limitation defined by (2.10a,b), the expressions for r and t are fairly accurate because the wave function inside the film is constrained to obey the rigorous boundary conditions. 3. Comparison among Approximations Let us consider some special cases of (2.15a,b). They give the obviously correct result r = R and t = T for the case of a single-step substrate, because in this caseflz) is identically zero. For a uniform sample layer of scattering length density po and thickness d deposited on a substrate, (2.1 5a,b) reduce to the exact results for r and t T 3i.e.
Another case is a free liquid surface such as the surface of water with a scattering length density profile varying continuously from zero in the air to the constant bulk value of the liquid. Since rf = 0 , rb = 0 , and tf = 1, (2.15a) reduces to the following form:
+
[eik@’
t=
Tf,beikd 1
e-ik,yf
+ RfRbe2jkd
(3.4d)
and it will be seen that (3.4d) gives an almost exact result for the case of free liquid surfaces. We now compare (2.15a) with the Born approximation. The Born approximation can be derived from (1.2a) in the following limits: we consider a commonly occurring case of reflection from a freely standing liquid surface situated at z = -d with the bulk liquid below it extending to positive infinity. For sufficiently large ko = 2n sin 6/A = Q/2, where Q is the magnitude of the wave vector transfer at scattering angle 0, such that k = d k o 2 - 4xp Z= ko, we can put kr Z= ko and A. = I . We have also R = f b = 0, from the geometry. Then (2.15a) reduces to (3.5a) Integration by parts gives (remembering that f = -47rp) (3.5b) If we demand that (3.5b) gave the correct result r = R for a single-step substrate for which dp(z’j/dz’ = Ap6(z?
(3.5c)
where Ap = ps - pairis the height of the step and 6 is the Dirac 6 function. Then, substituting ( 3 . 5 ~ )into (3.5b) gives r(step) = R = 47rAp/Q2
(3.1)
(3.4c)
RfRbeaikd
X
2ko -d [eik@’+ Re-ik@’]Ao[exp(c+(z?)+ f b exp(c-(z?)] (2.15a) i
The integration excludes the discontinuities at z = -d and z = 0. 7 and i are the uniform layer Keflectance and transmittance given by (3.1) and (3.2) with k = k and Rf = ( k , - k f ) / ( k ,+ k f ) , Rb = (kb- k,)/(kb + ks), Tf = 2ko/(ko + k f ) ,and Tb = 2kb/(kb + ks). The factors T = 2ko/(ko k s ) ,and C is given by T#i(k-ko)d
dz)
(2.14b)
r
where I; = (k: - 47rp)’/*is the effective wavenumber in the film with an average density p . Using the same limits used in deriving eqs 3.5a,b, we obtain from eqs 2.15a,b the following simple results:
+
with the coefficient A. given by
A0
(3.3)
(2.13~)
kd-
(3.5d)
Dividing (3.5d) into (3.5b) gives
(3.2)
where k = (ko2- 47r~,)]/~ is the constant wave number in the film and Rf, Tfand Rb, Tb are the known Fresnel reflectance and transmittance at the front and back interfaces, respectively. We now turn to the case of a very thin film deposited onto a uniform substrate, where the front and back interfaces of the film are strongly reflecting. Since the layer is very thin, we can approximate the exponential factor by (3) Born, M.; Wolf, E. Principle ofoprics; Pergamon Press: Oxford, 1975.
which is the well-known formula for liquid surface^.^ A characteristic feature of this approximation is that the reflectivity (Le., the squared modulus of r ) is inversely proportional to Q to the fourth power. Next we discuss the distorted wave Born approximation (DWBA).S It consists of replacing the excitation wave inside the film by the so-called distorted wave. This is a wave that would (4) Als-Nielsen, J . 2.Phys. B: Condensed Marfer 1985, 61, 41 1-414. ( 5 ) Vineyard, G. H . Phys. Reu. B 1982, 26, 4146.
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The Journal of Physical Chemistry, Vol. 95, No. 23, 1991
Letters
.......... exponential ea+-
e-
-m
e.
2
j
4104
Dlrtorted Wave
-
3 10.6
..
I
.................
210-5-
a
substrate
film 0’ -1Mx)
I
I
-1OW
500
I I
I
0
500
Depth cwrdinate 2 (A)
Figure 2. Three profiles used in comparison: linear, exponential, and error function.
exist in a simpler film model such as a uniform layer obtained by averaging the profile. This single-layer DWBA can also be obtained fro-m (2. ISa] by approximating the integral in the phase factors by ikz’with k obtained from smoothing the profile. The result is not very simple. Finally, we would like to point out that the solution (2.14a) contains the well-known WKB method as a very special case in which k’/k2
