Decryption of Igneous Rock textures:
Crystal Size Distribution Tools
P. Armienti
Dipartimento di Scienze della Terra, via S. Maria 53
- Pisa, Italy-
American Mineralogist – RiMG 69 Short Course .
San Francisco Dec 2008
8,9
ETNA
Luglio -Agosto 2001
6,7
3
4,5
0
1.5
3 km
1
2
Modal data can be represented as
Crystal Size Distributions
n(L)
(the number of crystals per unit size per unit volume):
The statistic reliability of data has always to be checked before attempting any interpretation.
We must be sure to have analyzed an area that is sufficient to guarantee that the largest
particle detected has the "true" number density. On the contrary the CSD plots will be
"sparsely" populated by points at larger diameters (right hand truncation).
In this case n(L) does not represent measurement of the number density but is related to
"single" casual findings of particles of that size in the explored section
17.5
The minimum area to be analyzed
depends on the number of particles
the resolution (width of the interval
between the classes - DL) according
to the formula:
Ln n(L)
15
Area: 3 cm2
12.5
10
Area
7.5
5
0.0
0.2
0.5
0.8
1.0

1
2 n ( L ) * L * L * L
(Armienti et al 1994)
D(cm)
in the figure, for particles with:
L= 0.15 cm ; L= 0.0036 cm; n(L) ≈ e8 cm-4
The minimum area must not be less then 2 cm2 to have a statistical significance.
5
Ln N(L) av 3
Thin section 7 cm
0
2
Brick 122 cm 2
Photo 808 cm 2
-5
Vesicles in lava
flow from dike
-10
-15
0
5
10
15
20
25
30
D (mm)
Right Hand Truncation may occurr at different length scales
30
30
A
Ln n(L)
Plagioclase of 1974 lavas
25
Ln n(L)
20
20
0.0
0.0005
0.0010
D (Cm)
10
0.0015
0.0020
1974 - olivine
1974 - clinopyroxene
1992 - clinopyroxene
0
0.0
0.02
0.04
0.06
D (cm)
0.08
0.10
areas of
crystals mm^2
7,85E-05
7,85E-05
7,85E-05
7,85E-05
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
Equiv.
Diameter
(mm)
READ ME
0,009997459 0,009996961
0,009996961 2,529460531
0,009999668
0,009997386
0,129971177
0,129970721
0,129966216
0,129966365
0,129965091
0,12996902
0,129965045
0,129963646
0,129967873
0,129966428
1,33E-02
1,33E-02
1,33E-02
1,33E-02
1,33E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
4,91E-02
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
1,07E-01
0,12996565 # of classes
0,129964469
158
0,129970716
159
0,129961747
160
0,129972371
161
0,249934792
162
0,249943004
163
0,249930765
164
0,249945026
165
0,24993526
166
0,249935178
0,249933694
0,249945372
0,24993946
0,249930278
0,249930312
0,249936357
0,249936807
0,249933609
0,249932772
166
0,249941554
0,249930117
0,249943677
0,24994244
0,249946557
0,249933199
1
0,249932365
1
0,369892448
1
0,36990469
1
0,369903067
1
0,369922202
1
0,369904309
1
0,369897966
1
0,369921807
1
0,369905512
1
0,36991479
1
0,369897503
1
0,369904498
1
Trials
166
total area of the sample
3500
# of size classes
19
40
40
40
40
40
40
40
40
40
6,25E+02
classi
f cum/A Na i
0,12 0,016946
14
14 1
decimali delta 0,139939
29
15 2
4 0,265926
51
22 3
0,394907
77
26 4
0,12 0,526882
104
27 5
0,66185
122
18 6
0,799812
139
17 7
2
0,940767
160
21 8
1,084716
185
25 9
1,231659
258
73 10
1,381595
3121
54 11
1,534525
386
74 12
1,690449
435
49 13
SE(E6<>"";E6;INT(10^E4*(C3-C2)/(D2))/10^E4)
1,849366
4480
13 14
0
(Cryst from CSD) measured Cryst
40
delta
 areas
1,303825639
1,303825639
1,303825639
1,303825639
1,303825639
1,303825639
1,303825639
1,303825639
1,303825639
2,011277
2,176181
2,344079
2,514971
2,688857
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1,30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
458
-1
475
482
488
-2
492
0
0
-3
0
0
0
-4
0
0
0
-5
0
0
0
-6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(Na/^1,5 ln Nv
measured derived from CSD
0,114685 -2,2 6,908870025 17,85014
24,76
0,00536 -5,2 47321,64534 8782,268
56103,91352
0,003634 -5,6
1774
492
2266,00
0,00258
-6
1064,4
0,001772 -6,3 4,145322015
0,000685 -7,3
0,000473 -7,7
0,000509 -7,6
0,000534 -7,5
0,002204 -6,1
0,00118 -6,7
0,001617 -6,4
0,000754 -7,2
9E-05 -9,3
0,5
10
17
7
6
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
15
16
17
18
19
5,35E-05
0,000105
2,49E-05
1,78E-05
8,76E-06
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
-9,8
-9,2
-11
-11
-12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1,5
2
2,5
0
0,405280657
0,535492606
0,510865365
0,151003382
0,037456346
0,007235877
0,004937735
0
Serie1
0,025788893
0,008688268
0,022530101
0,01669102
0,002425863
3
0,000774724
0,004578798
0,001373406
0,001398649
0,001201431
Balance equations
for the number of crystals
number of crystals
in unit volume of size =
between L and L+  L
{
(grow th input - growth output)
+
(flux in - flux out)
}
(V2n2-V1n1)L=(G1V1n1-G2V2n2)t+(Qini-Qono)Lt
Or else:
(Vn) (G Vn)
+
= Qi ni - Qo no
t
L
n(L,t) crystal density
G Crystal growth rate
V Volume
L Crystal size (e.g. diameter)
t Time
Qi ni - Qo no Flux in - Flux out
Balance equations
for the number of crystals
Case 1:
Continuosly taped and refilled magma chamber:
In flux of crystals = 0 (crystal free magma)
Out flux of crystals = Q =V/ ( = time of recharge)
(Gn) n
=

L
For constant G :
n= no exp (-L/G)
or
Ln(n) = Ln (no) -L/G
Balance equations
for the number of crystals
Ln(n) = Ln (no) -L/G
Case 1:
Continuosly taped and refilled magma chamber:
6
unfolded 09-96 k
Ln n(L)
4
Ln N(R) av 3
valori usati per la regressione
2
Ln n(L) = -7.971 L + 5.886
0
-2
0
0.5
1
1.5
2
L = Equivalent Radius (mm)
STROMBOLI - eruzione 1996
7.5
6
Ln n(L)
unfolded str 90
Ln n (L)
0
-2.5
Ln n (L) av 3
2
Ln n(L)
Ln n(L)
unfolded str 151
4
Ln n(L) av 3
2.5
0
-2
-5
0
0.25
0.5
0.75
1
-4
1.25
0
0.5
1
Equiv alent Radius (mm)
2
6
unfolded153
unfolded 152
4
2.5
Ln n(L)
Ln n(L)
2
Ln n(L)
Ln n(L)
1.5
Equiv alent Radius (mm)
5
0
0
-2.5
-2
-5
0
0.25
0.5
0.75
1
-4
1.25
0
0.5
Equiv alent Radius (mm)
1
1.5
Equiv alent Radius (mm)
6
5
unfolded str 45
unfolded 09-96
LnN(R) av 3
4
Ln n(L)
2.5
Ln n(l) av 3
Ln n(L)
Ln n(L)
Ln N(R)
2
0
0
-2.5
-2
0
0.5
1
1.5
-5
2
0
0.5
Equiv alent Radius (mm)
1
1.5
Equiv alent Radius (mm)
5
5
unfolded 04-96 a(2)
Ln N(R)
04-96 a1
4
Ln N(R) av 3
Ln n(L)
2.5
Ln n(L)
STROMBOLI
5
0
Ln N(R) av 3
3
2
-2.5
1
-5
0
0.5
1
Equiv alent Radius (mm)
1.5
0
0
0.25
0.5
0.75
Equiv alent Radius (mm)
1
1.25
8
Stromboli
lava flow Jan 15 2003
Plagioclase CSD
18745 crystals
6
4
2
Ln (N(L)) = -6,6242 D + 4,6224
0
-2
-4
-6
-8
Ln (N(D)) ( mm-4)
D (mm)
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
 of Stromboli has been estimated to be 19
years on the basis of Sr isotopic data
. This would imply a plagioclase growth
rate on the order of 2x10-11cm s-1 much
lower the known on basaltic systems, on
average in the range 10-9-10-10 cm s-1
(Kirkpatrick et al., 1976; Lofgren, 1980;
Armienti et al., 1994).
The low growth rate must be the result of
an average behaviour which accounts for
the peculiar textural and chemical zoning of
the plagioclase.
This zoning leads to assume that the
growth rate G derived from CSD data is a
mean value resulting from growth and
dissolution episodes:
G  G f   G f 
where G+ and G - are mean growth and dissolution rates, respectively, and f + and f –
are the volume fractions of the system in which crystals grow or dissolve respectively
+
we can assume f to be proportional to the volume fraction of magma supplied in a characteristic time:
f += (V-Q*t)/V
21)
where Q is the magma supply rate, which, in a steady system, is equal to the mass eruption rate and
t is a characteristic time during which resorption events may occur due to mixing of undegassed magma
with the resident magma.
From the obvious relations f + + f - =1 and V/Q = , it follows:
f + = 1-t/ and f - = t/
t
t
from which: G  G  1    G   
 

 
)
)
1,20
which states that the net growth rate is a mean
of positive and negative growth rates weighted
for the times in which the two conditions hold.
we can compute the amount of dissolution
allowed for crystals (G - *t).
Thus a plot t/ vs (-G -*t) for different
G+, (Fig. 17), will provide, an estimate of the
characteristic size of the crystal that will be
completely resorbed.
1,00
0,80
G+ 10^-10
G+ 10-9
g+ 10^-8
0,60
0,40
0,20
0,00
1,E-04
1,E-03
1,E-02
1,E-01
1,E+00
maximum dissolution (cm)
1,E+01
Balance equations
for the number of crystals
Case 2 Cooling of a magma batch
a batch of volume V
- with no migration of crystals (no fractionation)
- the only input of crystals to a given class is due to
nucleation and growth
e.g. : a rising batch of magma
(Vn) (GVn)
t + L = Q i n i - Q o n o
reduces to
ŽVn
ŽVn
+G
= JV(L)
Žt
ŽL
Balance equations
for the number of crystals
Case 2 Cooling of a magma batch
J(t L) V(tL)
n (t,L) =
G (t L) V(t)
The ratio V(tL)/V(t) is a function of the total crystallinity
and in many cases can be neglected
: e.g. for Etnean lavas P.I. ≈ 20 ÷ 30%,thus 1.25<V(tL)/V(t)<1.43
in the worst case Ln 1.43 = 0.36 implies a correction of 0.36 for J/G.
Balance equations
for the number of crystals
Case 2 Cooling of a magma batch
Equation
For interpretation of
CSDs
||
||
\/
n(t,L) =
J(tL)
G(tL)
All the crystals simultaneously undergo the same variations of growth rate and this implies that crystals
born in different times cannot have the same size. This trivial observation has many important consequences
and allows to use eq. 27 for petrologic interpretations. In fact, the condition that crystal size depends on
time spent to grow and that N(L) reflects the ratio J/G at the time in which crystals of size L appeared
thus:
different trends of N(L) vs L reflect changes in the ratio J/G during crystallization
and, due to dependence of J and G from undercooling, a plot of N(L) vs L corrresponds to a plot of
undercooling vs time.
Balance equations
for the number of crystals
Case 2 Cooling of a magma batch
t
- Crystals size depends on the time:
Lt   Gdt
to
.
- Number density n(L) reflects the ratio J/G when crystals of
size L nucleate
- Different trends of n(L) vs L reflect variations of J/G
during crystallization
 E  
 h Tm  To  



G  exp
1  exp 
R T 
 k To Tm 
 g m
  16 V 2 T 2 
  E*
c
o

 exp 
J  exp 
2
2
R T 
 3k T h T  T  
m
m
o
 g m


At small undercoolings
G=A(exp(BT -1)
Maaløe et al. (1989)
and
J=C(exp(DT -1)
J Cexp( DT )  1
N ( L)  
G Aexp( BT )  1
at constant undercooling (∆T= constant):
J= constant G=constant
n(L) = constant
25
Typical CSD of
Plagioclase crystals
Trend D
Trend C
Ln (N)
20
15
Trend B
Trend A
10
5
0.00
0.05
0.10
Equivalent diameter (cm)
0.15
0.20
THE SAME COOLING HYSTORY AFFECTS
ALL THE MINERALS SIMULANEOUSLY
20
Ln(N)
20
15
Sample 230592
Plagioclase CSD
Sampled at the vent
Ln (N)
25
Sample 230592
Clinopyroxene CSD
18
Sampled at the front
16
14
12
10
8
10
6
0.00
5
0.00
0.05
0.10
0.15
0.10
0.20
D(cm)
0.20
D(cm)
20
22
Sample 230592
Olivine CSD
Sample 230592
Oxyde CSD
20
Ln (N)
Ln (N)
25
15
18
16
14
12
10
10
5
0.00
0.05
D (Cm)
0.10
8
0.00
0.01
0.02
D (Cm)
0.03
0.04
J and G are functions of ∆T(t)
∆T = T - TLiquidus
TLiquidus
Varies with time as a response to the mass balance of the
exolution process (influence of volatiles on TLiquidus )
It can be modeled on the basis of solution models and
estimates of volatile contents during crystallization
Mt. Etna Lavas
Initial
(Estimated
2.3
Water Content
in fluid inclusions
wt%
)
Final Water Content
(LOI of lavas)
0.45
wt%
1225
1983
91 - 92
1200
19 74
1983 2Kb
1175
T
li quid us
°C
Male tto
1150
1125
1100
0
0.5
1
1.5
(H2O wt% )
2
2.5
Stromboli
Zieg and Lofgren experimental setting seems to imply a linear dependence of T from time :
ΔT (t) =  +  t (32)
Where   °C ) and  (°C s-1 ) are constants to be computed. The features of the CSDs may be
constrained with the equation 26:
N (t , L) 
J (t L ) V (t L )
G (t L ) V (t )
and adopting empirical relations derived from equations 28 and 29 to describe variations of J and G with
undercooling (Kirkpatrick 1977):
J = A exp(-B /Tm) (exp( -B’ T02/( Tm * ΔT2)))
(33)
G = C exp(-D / Tm) (1-exp( D’ ΔT/( T0 Tm)))
(34)
the unknown constants A, B,B’,C,D,D’, describe the kinetics of crystallization. We can write a set of
boundary conditions derived from the CSDs of Zieg and Lofgren (2006), namely:
t
(35)  Gdt  Lt (maximum length attained by crystals from in time t-t0)
to
t
(36)
 Jdt  N
tot
(total number of crystals)
to
t
(37)
J 4
 G 3 L dt  volume fraction (Marsh,. 1998)
3
to
(38) N ( L0 ) 
J0
G0
at t= to
J t max
Gt max
Equations 32 – 39 where numerically solved for the four Zieg and Lofgren (2006) CSD data set reported
in Fig. 23, providing the following solutions and fittings of relevant parameters for CSDs characterization.
(39) N ( L max) 
Zieg and Lofgren (2006)
T drop from 1550°C to 1495°C in 29 min
10
9
8
Ln (N(L)) mm^-4
7
Dwell time 60'
6
dwell time 30'
5
dwell time 0'
4
3
2
1
0
0
0,1
0,2
0,3
L (mm)
0,4
0,5
ETNA 2001
ETNA 2001
CSD
Plagioclasio