Thermal Analysis

Dr. Jyoti Prakash Dhal

 General thermodynamic relationships
Thermal analyses are usually run under conditions of constant pressure, the
underlying thermodynamic equation is the Gibbs-Helmholtz expression:
G0=H0-TS0
where G=free energy of the system, H=enthalpy of the system, S=entropy of the
system, T=temperature in kelvin.

The general chemical reaction

aA+bBcC+dD
Is spontaneous as written if G<0, is at equilibrium if G=0, and does not
proceed if G >0.

Thermal analysis involves the monitoring of spontaneous reaction.

Differentiating the Gibbs-Helmholtz equation with respect to
temperature (assuming S and H not vary with temperature):

d (G )
  S
dT

Show how to move from a stable situation (G>>0) to one where

reaction will occur.
S>0, an increase in temperature cause G<0,
S<0, decreasing the temperature will achieve the desired
spontaneous reaction.

Once the reaction is made to occur, thermal analysis may be used to

detect the process, yielding different and complementary information.
 Thermogravimetry (TG)

o Thermogravimetric (TG) is a branch of thermal analysis examining the

weight changes of a sample as a function off temperature (in the scanning
mode) or as a function of time (in the isothermal mode).

o The technique is useful strictly for transformations involving the absorption

or evolution of gases from a specimen consisting of a condensed phase.

o Changes in the mass of a sample due to various thermal events (desorption,

absorption, sublimation, vaporization, oxidation, reduction and decomposition)
are studied while the sample is subjected to a program of change in
temperature. Therefore, it is used in the analysis of volatile products, gaseous
products lost during the reaction in thermoplastics, thermosets, elastomers,
composites, films, fibers, coatings, paints, etc.
There are different types of TGA available:

i. Isothermal or Static TGA: In this case, sample is maintained at a constant

temperature for a period of time during which change in weight is
recorded.

ii. Quasi-static TGA: In this technique, the sample is heated to a constant

weight at each of a series of increasing temperature.

iii. Dynamic TGA: In this type of analysis, the sample is subjected to

condition of a continuous increase in temperature at a constant heating
rate, i.e., usually linear with time.
Measurements of changes in sample mass with temperature are made using
thermobalance. The balance should be in a suitably enclosed system so that
the atmosphere can be controlled. Thermobalance consists of an electronic
microbalance (important component), a furnace, a temperature programmer
and a recorder.

Block Diagram of a Thermobalance

A plot of mass versus temperature (thermogravimetric curves or TG curves)
permits evaluation of thermal stabilities, rate of reaction, reaction processes,
and sample composition.
The plot of mass change in percentage versus temperature or time (known as
TGA curves) is the typical result of TGA analysis as shown in Figure
General considerations
Suitable samples for TG are solids that undergo one of the two general
types of reaction:

Reactant(s)  Product(s)+ Gas (a mass loss)

Gas + Reactant(s)  Product(s) (a mass gain)

Processes occurring without change in mass (e.g., the melting of a

sample) obviously cannot be studied by TG.
 Instrumentation

LINSEIS L81
Thermogravimetric instrumentation should include several basic
components to provide the flexibility necessary for the production of
useful analytical data:
a) A balance,
b) A heating device,
c) A unit for temperature measurement and control,
d) A means for automatically recording the mass and
temperature changes,
e) A system to control the atmosphere around the sample.
 The Thermobalance
Two typical designs of the thermobalance are :

Specimen powder is placed in

refractory pan ( often porcelain or
platinum). The pan, in the hot zone
of the furnace, is suspended from a
high precision balance. A
thermocouple is in close proximity
to the specimen but not in contact, so
as not to interfere with the free float
of the balance. The balances are
electronically compensated so that
the specimen pan does not move
when the specimen gains or loses
weight.
 The Cahn balance design is shown
in figure. The balance arm is
connected at the fulcrum to a
platinum taut band. Which is held in
place by roller pins extending
orthogonally from the balance arm.
This design permits the motion of
the balance arm to be essentially
frictionless. The taut band is
reflected (twisted) by the current
through the coil surrounding it.

A flag beneath the balance arm interferes with infrared light propagating from
a source to a photo-cell detector. A servo mechanism feedback control loop
adjusts the current in the coil and hence position of the flag in order to maintain
constant illumination level at the detector. The current sent to the coil in order
to maintain the flag position is proportional to weight loss or gain by the
specimen. A DC voltage proportional to current is provided for external data
acquisition.
If reactive (or corrosive ) gases are passed through the specimen chamber or
gases are released by the specimen the chamber containing the balance is often
maintained at a slightly more positive pressure via compressed air or inert
gases; this is in order to protect the balance chamber and its associated
electronic components from exposure to corrosive gases.

The balance chamber is not completely protected since gases released from the
specimen can still diffuse into the balance chamber.

Further maintaining the specimen in a pure inert gas or other special gases is
limited by back-diffusion of air through the exit port. To protect against this
the exit gas is bubbled through a fluid. The fluid will permit exiting gas to
bubbled out, but will not permit back-diffusion og gases.
Balances must remain precise and accurate continuously
under extreme temperature and atmosphere conditions, and
should deliver a signal suitable for continuous recording.

*   Null-deflection weighing mechanisms are favoured in TG as

they ensure that the sample remains in the same zone of the
furnace irrespective of changes in mass.
 
*   Sensitivity of balance  1g for a 1g maximum load balance.
 
*   The output weight signal may be differentiated electronically
to give a derivative thermogravimetric curve (DTG)
 The Heating Chamber
• The furnace is normally an electrical resistive heater;
• Some basic requirements of the heating chamber are :
be non-inductively wound
be capable of reaching 100 to 200°C above the maximum desired working
temperature
have a uniform hot-zone of reasonable length
reach the required starting temperature as quickly as possible
not affect the balance mechanism through radiation or convection
• In order to overcome the problem of possible temperature gradient, infrared
or microwave radiation have been used in some equipment.
infrared heating : use halogen lamp, temperature up to 1400°C, heating rate
can be as high as 1000°C/min, accuracy is about ±0.5°C.
Microwave heating : large sample can be used because uniform heating
generated within sample but temperature measurement and power control are
difficult.
Constant heating rate
 The atmosphere
Sort, pressure and flow rate of the gas in the sample chamber influence the
following parameters:
• Sample reaction
Sample reactions with the gas (oxidation in the presence of oxygen).

• Heat transitions
Different heat conductivity of the gases used in an experiment.

• Buoyancy and current effects

Different density and flow rate of the gases used in an experiment.

For all thermoanalytical investigations it is very important to report the

sort, the pressure and the flow rate of the gases used in the experiment.
Thermal decomposition temperatures for CaCO3 in different gas
atmospheres
*  Thermbalance are normally housed in
glass or metal system to allow for
operation at pressures ranging from high
vacuum (< 10-4 Pa) to high pressure
(>3000 kPa) of inert, oxidizing,
reducing or corrosive gases.
*  Care must be taken to correct for
buoyancy arising from the lack of
symmetry in the weighing system
*  Thermal convection is responsible for
noise in the signal of TG. The use of
dense carrier gases at high pressures in
hot zones with large temperature
gradients give the most noise. Fitting of
convoluted baffles was found to be most
successful in reducing thermal
convection.
 The sample
 Sample form, defect content, porosity and surface properties has
influence to the behaviour on heating, e.g. single crystal sample give
different response from powdered sample.
 Large sample size cause problems like heat transfer, and gas
exchange with the surrounding is reduced; in general, the use of
small (~ 20 mg) specimen is preferable if sensitivity of balance
permits
 Sample should be powdered and spread thinly and uniformly in
the container
Crucibles
Decomposition temperatures of CaCO3 as function of crucibles
 Temperature measurement and calibration
 Platinum resistance thermometers or
thermocouples are used for
temperature measurement.
 Large difference between sample
temperature (Ts) and furnace
temperature (Tf) can exist, sometime as
high as 30°C. Calibration is thus
needed.
 The difference or lag is more
marked when operating in vacuum or
in fast flowing atmosphere and with
high heating rate.
Temperature calibration for small furnace can be done by making use of the
melting point or Curie points of a range of metals and alloys.

 A series of high purity wires may

be suspend in the region where the
hanger of sample pan
specimen crucible would normally
be located. If the furnace furnace
temperature is slowly raised
through the melting point of a
particular wire, a significant weight
loss will be recorded when the wire different metal wires
melts.
 A series of fusible wire, such as :
indium (156.63°C), lead (327.5°C), thermocouple
zinc (419.58°C), aluminium
(660.37°C), silver (961.93°C), and
gold (1064.42°C) should give a
reasonable calibration curve.
 Calibration can also be done by placing a series of ferromagnetic
materials in the specimen basket and a magnet below or above it,
external to the furnace. When each material goes through its Curie
temperature (ferro- to paramagnetic transition), a sharp ‘weight’
change will be indicated.
Interpretation of TGA curves

TGA curves are typically classified into seven

types according to their shapes. Figure shows
schematic of various types of TGA curves.

Curve 1: No change: This curve depicts no mass

change over the entire range of temperature,
indicating that the decomposition temperature is
greater than the temperature range of the
instrument.
The sample undergoes no decomposition with
loss of volatile products over the temperature
range shown but solid phase transformation,
melting ,etc can not be detected by TG.

Schematic of various types

of TGA curves
Interpretation of TGA curves
Curve 2: Desorption / drying: This curve shows
that the mass loss is large followed by mass
plateau. This is formed when evaporation of
volatile product(s) during desorption, drying or
polymerization takes place. If a non-interacting
atmosphere is present in the chamber, then curve
2 becomes curve 1

Curve 3: Single stage decomposition: This curve

is typical of single-stage decomposition
temperatures having Ti and Tf.

Curve 4: Multistage decomposition: This curve

reveals the multi-stage decomposition processes
as a result various reactions. Multi-stage
decomposition with relatively stable
intermediates : provide information on the
temperature limit of stability of reactants and
Schematic of various types
intermediate products and also stoichiometry of TGA curves
Interpretation of TGA curves
Curve 5 Multi-stage decomposition with no
stable intermediate product. However heating-
rate effect must be considered. At low heating
rate, type (Curve 5) resemble type (Curve 4). At
high heating rate, type (Curve 4) and (Curve 5)
resemble type (Curve 3) and lose all the details.

Curve 6: Atmospheric reaction: This curve

shows the increase in mass. This may be due to
the reactions such as surface oxidation reactions
in the presence of an interacting atmosphere.

Curve 7: Similar to curve 6, but product

decomposes at high temperatures. For example,
the reaction of surface oxidation followed by
decomposition of reaction product(s).
Schematic of various types
of TGA curves
Processes that leads to Weight Gain and loss in TGA experiment:
Differential thermogravimetry (DTG)
In most of the cases subsequent decomposition processes give overlapping
decomposition stages. In certain cases , a particular decomposition reaction is
not yet finished when an other (higher temperature one) commences. In this
case a reliable qualitative and quantitative evaluation of the TG curve is not
possible without having its first derivative (i.e. the DTG curve).

The area of peak is directly

proportional to the mass loss
over the same temperature
range. The DTG peak at any
temperature gives the rate of
mass loss (dm/dT in mg/min).
With the DTG curve Ti, Tf and
Tmax (the temperature of the
maximum mass loss rate ) can
be precisely determined in
overlapping reaction.
TG & DTG curve of CuSO4.5H2O
Quantitative evolution of thermogravimetric curves
For single transformations and consecutive reactions which do not overlap the
height of the TG curve can be determined accurately. For overlapping reactions
quantitative evolution requires the use of DTG curve. The local minimum in the
DTG curve can serve as a basis for the separation of two individual mass
changes associated with two overlapping reaction.

Quantitative evolution of the TG

curve in the case of two overlapping
reactions ( A & B) resulting TG and
DTG curves. Ma and Mb are the
mass changes in the reactions A &
B, ma and mb are the error due to
overlap.
Proximate analysis of coal
On heating in nitrogen atmosphere to 950 oC, the amount of moisture and
volatiles can be determined. Changing the purge gas to oxygen and holding the
temperature at 700 oC for around 15 minutes the fixed carbon and ash content
can be obtained.

Proximate analysis of coal with thermogravimetry

Calcium, Strontium and barium can be determined in the presence of
one another after precipitation in the form of oxalates.

On heating moisture (H), crystallization

water (E), carbon monoxide from the
decomposition of calcium oxalate (F),
and carbon dioxide from calcium
carbonate (G) and strontium carbonate
(L) are liberated. The amount of
Calcium, Strontium and Barium can be
determined if the dry weight (D) is
known.

Thermogravimetric determination
of Calcium, Strontium and barium
after precipitation in the form of
oxalates
Applications of TGA:
a) Thermal stability of the related materials can be compared at elevated
temperatures under the required atmosphere. TGA curve helps to explicate
decomposition mechanisms. 

b) Materials Characterization: TGA curves can be used to fingerprint materials

for identification or quality control. 

c) Compositional analysis: By a careful choice of temperature programming

and gaseous environment, many complex materials/ mixtures can be analyzed
by decomposing or removing their components. For example: filler content in
polymers; carbon black in oils; ash and carbon in coals, and the moisture
content of many substances.

d) Kinetic studies: A variety of methods can be used to analyze the kinetic

features of weight loss or gain through controlling the chemistry or predictive
studies.

e) Corrosion studies: TGA provides a means of studying oxidation or some

reactions with other reactive gases or vapors.
Examples of TGA curves:
Figure shows the heat decomposition mass curve of Whewellite (calcium
oxalate monohydrate).

TGA curve of Whewellite

Examples of TGA curves:
Thermal decomposition of calcium oxalate monohydrate studied by TGA
(a) Ca(COO)2 H2O → [200°C] → Ca(COO)2 + H2O (g);
(b) Ca(COO)2 → [500°C] → CaCO3 + CO (g); and,
(c) CaCO3 → [800°C] → CaO + CO2 (g).

Figure depicts the mass change corresponding to each reactions of calcium

oxalate monohydrate.
Examples of TGA curves:
Figure displays three step weight loss of the synthesized hydrozincite.
Step 1: ~ 4% weight loss in the temperature range of 30 - 250°C due to
moisture, trapped carbon dioxide, and ammonia removal. 
Step 2: ~ 10% weight loss around 250 - 325 °C due to the loss of three water
molecules. 
Step 3: 16 % weight loss around 325 - 440°C, close to the loss of two
CO2 molecules.
Analytical calculations
Under controlled and reproducible conditions, quantitative data can be
extracted from the relevant TG curves. Most commonly, the mass change is
related to sample purity or composition.

Example: A pure compound may be either MgO, MgCO3, or MgC2O4.

A thermogram of the substance shows a loss of 91.0 mg from a total of
175.0 mg used for analysis. What is the formula of the compound? The
relevant possible reactions are
MgO  No reaction
MgCO3  MgO+CO2
MgC2O4  MgO+CO2+CO
Solution: % Mass loss Sample=(91.0/175.0)(100%)=52.0
% Mass loss if MgCO3=(44/84.3)(100%)=52.2
% Mass loss if MgC2O4=((44+28)/112.3)(100%)=64.1
If the preparation was pure, the compound present is MgCO3.
 Differential Thermal Analysis (DTA)

In most cases, heating a system (an element, a compound, a mixture) causes

physical and chemical changes. Possible transformations are summarized in the
following table:
DTA: Differential thermal analysis is defined by ICTAC (International
Conferation for Thermal Analysis) as a technique in which the temperature
difference between a substance and a thermally inert reference material is
measured as a function of the temperature , where the substance and reference
materials are subjected to a controlled temperature programme.

The thermally inert material used is usually α-Al2O3 preheated to 1500 ⁰C.

In principle DTA can be applied to any solid state system associated with
changes of state, since these changes arte accompanied with an enthalpy
change, as well.

Physical transformations (such as crystal phase transition, glass transitions,

liquid crystal transformations, solid state heat capacity transitions, fusion,
vaporization, adsorption and desorption) and chemical changes (dehydration,
decomposition, oxidation, reduction, combustion, polymerization) can be
investigated by DTA. Phase diagrams to determine eutectic points or purity of
metals can be constructed by this technique.
 Block diagram of DTA
The sample and the reference are placed symmetrically in the furnace. The
furnace is controlled under a temperature program and the temperature of the
sample and the reference are changed. During this process, a differential
thermocouple is set up to detect the temperature difference between the sample
and the reference. Also, the sample temperature is detected from the
thermocouple on the sample side.
 Measurement principles of DTA
Graph (a) shows the temperature change of the furnace, the reference and the
sample against time. Graph (b) shows the change in temperature difference
(ΔT) against time detected with the differential thermocouple.

ΔT signal is referred to as the DTA signal.

Matters that do not change in the measurement temperature range (usually α-
alumina) are used as reference.
When the furnace heating begins, the reference and the sample begin heating
with a slight delay depending on their respective heat capacity, and eventually
heat up in according to the furnace temperature.

ΔT changes until a static state is reached after the heating begins, and after
achieving stability, reaches a set amount compliant with the difference in heat
capacity between the sample and the reference. The signal at the static state is
known as the baseline.
When the temperature rises and melting occurs in the sample, for example, the
temperature rise stops as shown in graph (a) and the ΔT increases. When the
melting ends, the temperature curve rapidly reverts to the baseline.

At this point, the ΔT signal reaches the peak, as shown in graph (b).
From this, we can detect the sample’s transition temperature and the reaction
temperature from the ΔT signal (DTA signal).

In graph (b), the temperature difference due to the sample’s endothermic change
is shown as a negative direction and the temperature difference due to the
sample’s exothermic change is shown as a positive direction.
The temperature difference between the sample and the reference material ΔT
(= TS – TR) measured by two identical thermocouples connected in series
opposition is recorded to obtained the DTA curve.

Example: DTA trace of Kaolinite mineral, Al2Si2O5(OH)4

According to the sign convention,
the DTA peak for an endothermic
reaction is negative , while an
exothermic process gives rise to a
peak in the opposite (positive)
direction. The endothermic peak for
kaolinite at approx. 600 ⁰C indicate
dehydroxylation, while exothermic
change at approx. 900 ⁰C shows a
structural change accompanied with
release of heat.

In practice, ΔT signal is plotted against the temperature of the sample, the inert
material or the furnace temperature while a constant rate of heating is used.
As suggested in Figure, DTA peaks result from both physical changes and
chemical reactions induced by temperature changes in the sample.

Physical processes that are endothermic include fusion, vaporization,

sublimation, absorption, and desorption. Adsorption and crystallization are
generally exothermic.

Chemical reactions may also be

exothermic or endothermic.
Endothermic reactions include
dehydration, reduction in a
gaseous atmosphere, and
decomposition.

Exothermic reactions include

oxidation in air or oxygen,
polymerization, and catalytic
reactions.
Instrumentation for DTA/Block Diagram
The DTA apparatus consists of the
following components
• Sample and reference holder with
thermocouple assembly.
• Furnace temperature controller (to
increase the furnace temperature steadily)
• Furnace atmospheric control system (To
maintain a suitable atmosphere in the
furnace and sample holder)
• Low level DC amplifier
• Recording device(Recorder)
Sample size: 1~20 mg
Reference: inert materials
(e.g., Al2O3 , SiC, glass, etc.)
• Differential temperature sensor (to measure the temperature difference
between the sample and reference material) the sample and reference holder are
kept inside the furnace and the temperature of the furnace and sample holder is
controlled by using furnace controller.
Figure shows a schematic setup of the DTA. The substance under investigation
and an inert reference are heated under the same conditions simultaneously in a
furnace. The increase of the temperature in the furnace should be linear. Both
sample and reference are connected to thermocouples. These thermocouples are
connected to each other. The difference of the voltages, which are correlated to
the differences of the temperatures, are measured.

Figure. Schematic setup for the DTA: 1. furnace, 2. sample substance, 3. inert
reference, 4. control unit for heating, 5. temperature difference gauge, 6.
temperature gauge
As shown in Figure, the control unit shows no thermo-voltage as long as there
is no heat consumed or released, because there is the same temperature Ts = Ti
in the sample and in the inert reference and hence ΔT = 0.
If there is an endothermic reaction in
the sample, the sample temperature
increases slower than the reference
temperature (Ts1 < Ti1) and the gauge
shows a thermo-voltage according to
a temperature difference ΔT1 = Ts1 -
Ti1, which is negative.
If there is an exothermic reaction in
the sample, the sample temperature
increases faster than the reference
temperature (Ts2 > Ti2) and the gauge
shows a thermo-voltage according to
a temperature difference ΔT2 = Ts2
-Ti2, which is positive. DTA curves: a) Ts: temperature-time-curve of the
sample, Ti: temperature-time curve of the
The DTA curve can be determined by reference, b) temperature difference-time-curve
subtraction of the curve Ts and Ti.
Factors which influence the results of the DTA experiments
Influence of the Furnace Atmosphere

The equilibrium temperature of reversible transitions is dependent on the

pressure in the furnace or the partial pressures of the components. A pressure
change results mainly in a temperature shift of the peak.
Secondly, the gases of the furnace atmosphere may react with the substance or
with volatile products from decomposition of the sample. With a change of the
composition of the furnace atmosphere, some characteristic peaks may
disappear or extra peaks may appear.

The mechanisms of many chemical reactions are influenced by the atmosphere

and in particular by the gas pressure.
Example: DTA-curves of gypsum in a streaming nitrogen-steam-atmosphere
with different steam partial pressures. The dehydration reaction occurs in two
steps:
With increasing steam partial pressure, the
second step of the dehydration reaction
moves to higher temperatures. With higher
steam partial pressure, the DTA effects are
more pronounced and the peaks are
narrower.
The steam formed by the decomposition
reaction diffuses slower into the furnace
atmosphere than new steam is produced.
Above the sample surface the partial
pressure of steam increases. Consequently,
the equilibrium temperature of the
dehydration process increases. The local
steam pressure can only increase up to the
total pressure in the furnace. The lower the
given initial steam pressure is, the more the
local steam pressure may increase. Due to
sluggish diffusion, there are locally
different partial pressures of the gaseous DTA curve of gypsum with
decomposition product. different steam partial pressures
Influence of the Heating Rate

In addition, the DTA peaks as a

function of temperature T are
broadened, because the temperature
gradient in the sample increases at
higher heating rates. As a function of
time t, the peaks seem to get narrower.
At faster temperature rises, the
turnover needs less time, but the
temperature difference ΔT is
increasing. Therefore the area is
constant.

Influence of increasing temperature rise as a

function ΔT=f(t). The vertical marks
show the same sample temperatures.
At a higher heating rate, the DTA-peaks of chemical reactions move to higher
temperatures. In addition, there may be local changes in the furnace atmosphere
and on the sample surface as a result of the temperature dependence of the
reaction rate. The higher the heating rate, the less gas diffuses out of the sample
environment during the decomposition process. Consequently, the partial
pressure is locally higher and the decomposition temperature increases.

The broadening and the different shifting effects of the DTA at high heating
rates may result in overlapping of normal successively effects. For higher
resolution of the DTA curves, the peak temperatures are near the
thermodynamic equilibrium temperatures, thus, the heating rate should be as
low as possible. It is useful to notice that the peaks become flatter.
Influence of the Reference Substances

A reference substance for DTA measurements has to fulfil the following

conditions:

1. no transitions in the measured temperature range

2. similar heat conductivity and heat capacity as the measured substance

In general, it is impossible to fulfil both conditions 1 and 2 simultaneously,

because during the transition, heat capacity and heat conductivity are changing
in the sample substance.
Therefore, a reference is often chosen that remains as the product after heating
the sample.
Applications
Phase diagrams

An important use of DTA is for the generation of phase diagrams and the
study of phase transitions.

• An example is shown in Figure, which is a differential thermogram of

sulphur, in which the peak at 113 ⁰C corresponds to the solid-phase change
from the rhombic to the monoclinic form.

• The peak at 124 ⁰C corresponds to the melting point of the element.

• Liquid sulphur is known to exist in at least three forms, and the peak at
179 ⁰C apparently involves these transitions, whereas the peak at 446 ⁰C
corresponds to the boiling point of sulphur.
Differential thermograms (DTA) for sulphur.
• The DTA method also provides a simple and accurate way of determining the
melting, boiling, and decomposition points of organic compounds.
• Generally, the data appear to be more consistent and reproducible than those
obtained with a hot stage or a capillary tube.
• Figure shows thermograms for benzoic acid at atmospheric pressure (A)
and at 13.79 bar (B). The first peak corresponds to the melting point and the
second to the boiling point of the acid.

Differential thermogram (DTA)

for benzoic acid.
Curve-A taken at atmospheric
pressure.
Curve-B taken at a pressure
of 13.79 bar (200 lbs/in.2)
A DTA curve can be used only as a finger print for identification purposes but
usually the applications of this method are the determination of phase
diagrams, heat change measurements and decomposition in various
atmospheres.

-DTA is widely used in the pharmaceutical and food industries

-DTA may be used in cement chemistry, mineralogical research and in

environmental studies.

-DTA curves may also be used to date bone remains or to study archaeological
materials.