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```Optical properties of colloidal particles :
Silver colloidal particles evaporated in high vacuum on Formvar supporting film
(a)
(b)
(c)
- absorption spectrums,
(b) - sizes distribution histogramms,
(c) - electron microscopic photos of three typical layers
Optical properties of colloidal particles :
It is established that colloidal particles of silver with the size less than 3 nm in a
diameter lose metallic properties, silver becomes dielectric.
d , nm
Dependence of maximum
position in the distribution of
50
silver particles sizes upon
maximum in an attenuation
spectrum. (o) - for coated
40
particles, (x) - for ultra-thin
sections of particles, immersed
30
in gelatin. Dotted lines correspond
to boundary values for metallic
20
silver (derived from theoretical
calculations): 420 nm on the
10
wavelength axis, and 2.2 nm on
the diameter axis.
0
400
500
600
700
800
900
 , nm
Optical properties of colloidal particles :
Investigation of absorption spectra of colloidal particles in different media
d , nm
160
140
2
120
1
100
80
60
40
4
3
20
0
400
450
500
550
600
650
700
750
 , nm
Calculated dependencies of colloidal silver particles diameters upon the maximum of the absorption
spectra in different media: 1 - H2O; 2 - gelatin; 3 - AgCl; 4 - AgBr
Optical properties of colloidal particles :
For arguments ρ and mρ , called as z:
coefficients of в absorption ka ,
z f+1
z2
Ψf(z) = ---------------- .  -------- . (-----) k
scattering ks , reflection kr:
ka = kr - ks
1.3.….(2f+1)
k=1
z f+1 (f+1)
Theory G.Mie (1908)
2π
∞
∞
k!
2
(-1)k
z2
Ψf’(z) = ---------------- .  -------- . (-----) k
ka = -------- Im  f(f+1).(-1)f.(af – bf) ,
ka2
(-1)k
∞
1.3.....(2f+1)
f=1
k=1
∞
k!
1
.
----------------------(2f+3).….(2f+2k+1)
1
.
f+2k+1
------------------------- . -----------
2
(2f+3).….(2f+2k+1)
f+1
ρ = (2..a.na)/λ ,
i = (-1)1/2
(2f)!
ζf(z) = if+1 . eiz .  (-1)f . -------∞
f2.(f + 1)2
kr = -------- 
-------------.
2π
k=1
(|af|2
–
|bf|2)
,
ka2 f=1 2f + 1
where
2f + 1
ψf(ρ).ψf’(mρ) – m.ψf’(ρ).ψf(mρ)
af = i2f+1 . --------- . ------------------------------------------f(f+1)
ζf(ρ).ψf’(mρ) – m.ζf’(ρ).ψf(mρ)
2f + 1 ψf’(ρ).ψf(mρ) – m.ψf(ρ).ψf’(mρ)
bf = (-1)2f+1 . --------- . ------------------------------------------f(f+1)
ζf’(ρ).ψf(mρ) – m.ζf(ρ).ψf’(mρ)
where ψ, ζ – Bessel functions.
(2f)f.f!
f
ζf’(z) = ζf-1(z) - --- . ζf(z)
z
where ka = (2..na)/,
m = (n – i)/na,
For Ag particles with diameter 30 nm ka = kr
We should know the spectral dependence of reflection coefficient
n and absorption  of metal, reflection coefficient of medium na
and 2a –diameter of metal’s particles and wave length of light
in vacuum .
Optical properties of colloidal particles :
n5(469) = n5(422) = 0,063
n15(614) = n15(433) = 0,060
n25(729) = n25(443) = 0,059
n25(828) = n35(454) = 0,057
5(469) = 5(422) = 2,156.
15(614) = 15(433) = 2,257
25(729) = 25(443) = 2,318
35(828) = 35(454) = 2,448
d, nm
Correction of optical constants of silver
Dependencies of colloidal silver particles diameters
upon the maximum of the absorption spectra :
1 – theoretical calculated and 2 – experimental
produced dependences
Lines correspond to particles with diameters
5, 15, 25 and 35 nm.
,
nm
5
15
25
35
massive
420
440
460
480
500
520
540
560
580
600
620
700
750
800
850
2,138/0,063
2,146/0,063
2,153/0,064
2,159/0,064
2,165/0,066
2,169/0,068
2,173/0,072
2,176/0,075
2,178/0,080
2,179/0,085
2,180/0,091
-
2,138/0,063
2,154/0,060
2,169/0,057
2,183/0,055
2,197/0,054
2,201/0,053
2,222/0,054
2,232/0,054
2,242/0,056
2,251/0,058
2,269/0,061
2,285/0,080
-
2,138/0,063
2,156/0,058
2,176/0,053
2,196/0,049
2,212/0,046
2,228/0,044
2,244/0,042
2,259/0,041
2,272/0,040
2,285/0,041
2,300/0,042
2,338/0,052
2,355/0,065
2,368/0,081
-
2,138/0,063
2,161/0,056
2,184/0,050
2,206/0,044
2,226/0,039
2,247/0,035
2,266/0,032
2,284/0,029
2,302/0,027
2,318/0,025
2,334/0,024
2,399/0,028
2,416/0,036
2,438/0,048
2,454/0,065
2,138/0,063
2,321/0,059
2,503/0,056
2,685/0,053
2,865/0,051
3.045/0,050
3,224/0,049
3,401/0,049
3,579/0,050
3,756/0,052
3,931/0,054
4,624/0,069
5,051/0,085
5,472/0,104
5,887/0,128
Optical properties of colloidal particles :
Correction of the optical constants
of small silver particles, which are
strongly differ from properties of
massive silver
6
5
n, 
4
Dependence of optical constants for silver –
 (absorption coefficient, full line) and
n (refraction index, dotted line) upon the
wavelength .

3
2
The curves, corresponding to massive samples – 1
in black; silver particles with diameter
5 nm – in yellow, 15 nm - in blue,
0
25 nm – in green, 35 nm - in red.
n
300 400 500 600 700 800
 , nm
```
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