Units

In SAMSON, physical quantities are strongly typed. All physical quantities have associated units types, and the use of regular C++ types (e.g. double, float, etc.) is discouraged for the development of new SAMSON Elements (although it is natural when integrating existing source code).

In order to use units types, the SBQuantity.hpp header should be included:

#include "SBQuantity.hpp"

but this file is often indirectly included via other files (e.g. "SBAtom.hpp", etc.).

Thanks to metaprogramming techniques, conversions and checks are performed at compile time, and have no runtime impact on performance.

Definitions

The following definitions are used in SAMSON:

• A base unit is used to measure one type of physical quantities (e.g. meter is a base unit for measuring length).
• A system of units is a set of base units used together. For example, there are seven base units in the International System of Units (SI): meter (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount of substance) and candela (luminous intensity).
• A scaled base unit is a multiple of a base unit (a power of ten). For example, nanometer is a scaled version of meter.
• The scale of a scaled base unit is the corresponding power of ten, e.g. the scale of nanometer is -9, relative to the meter.
• An exponentiated unit is a power of a unit, such as nanometer^2.
• The exponent of an exponentiated unit is the corresponding exponent, e.g. the exponent of nanometer^2 is 2.
• Derived units are products of exponentiated units. For instance, newton is defined as kg.m.s^-2, while nanonewton is defined in SAMSON as yg.pm.fs^-2 (yoctogram.picometer.femtosecond^-2), for reasons detailed below.

SAMSON uses scales to avoid loss of precision when dealing with typical quantities encountered in nanosciences. For example, internally, nanometer is an instance of a template type parameterized by the scale -9, which stores a value (as a double) in nanometers.

Internally, SAMSON uses the international system of units (SI), but is able to perform conversion to and from other units (e.g. convert kilocalories per mole to zeptojoules).

Basic usage

Declaration and definition

Quantities are declared in typical C++ fashion, for example using types defined in the SAMSON SDK:

SBQuantity::angstrom l; // l is a length expressed in angstroms
SBQuantity::zeptojoule e; // e is an energy expressed in zeptojoules
SBQuantity::dimensionless d; // d is a dimensionless quantity

SAMSON has default type definitions (via typedefs) for common dimensions. For example, the following two statements are equivalent:

SBQuantity::zeptojoule e1(10); // e1 is equal to 10 zeptojoules
SBQuantity::energy e2(10); // e2 is equal to 10 zeptojoules

Please refer to this section for the list of default types.

There is no implicit conversion from C++ types (e.g. double) to SAMSON quantities, in order to help developers avoid mistakes:

SBQuantity::angstrom d1 = 10; // error, does not compile
SBQuantity::angstrom d2(10); // OK, d2 is equal to 10 angstroms
SBQuantity::angstrom d3 = SBQuantity::nm(1); // OK, d3 is equal to 10 angstroms

Conversions

SAMSON automatically converts between compatible units in the same system, or between different systems when conversions have been defined:

SBQuantity::angstrom d = SBQuantity::nm(1); // d = 10 angstroms
SBQuantity::zJ e = SBQuantity::kcalPerMol(1); // e = 6.94769(...) zeptojoules

When units are incompatible, an error is detected at compilation time. This automatic dimensional analysis helps developers manipulate physical quantities correctly:

SBQuantity::angstrom d = SBQuantity::nm(1); // d = 10 angstroms
SBQuantity::force f = SBQuantity::nanonewton(1); // f = 1 nanonewton
SBQuantity::energy w1 = f/d; // error, does not compile
SBQuantity::energy w2 = f*d; // OK, w2 = 1000 zeptojoules

Operations

Typical operations are possible through overloaded operators:

SBQuantity::mass m=SBQuantity::yg(1); // m = 1 yoctogram
SBQuantity::velocity v=SBQuantity::pmPerFs(1); // v = 1 pm per femtosecond
SBQuantity::energy k = 0.5*m*v*v; // k = 0.5 zeptojoules

Note that operations are only defined for units that belong to the same system (SI, atomic units, etc.), so that conversions have to be performed before applying operations between quantities expressed in different systems:

SBQuantity::kcalPerMol e(1); // e = 1 kilocalorie per mole
SBQuantity::pm l(1); // l = 1 picometer
SBQuantity::nanonewton f1 = e/l; // error, incompatible systems
SBQuantity::nanonewton f2 = (SBQuantity::J)e/l; // OK, e is converted to SI

Mathematical functions

For convenience, many usual mathematical functions have been overloaded for quantities, e.g.:

SBQuantity::energy a(-1.0); // a = -1 zeptojoule
SBQuantity::energy b = abs(a); // b = 1 zeptojoule
SBQuantity::dimensionless x(-2.0); // x = -2
SBQuantity::dimensionless y = exp(x); // OK, x is dimensionless
SBQuantity::area d(9); // d = 9 picometer^2
SBQuantity::length e = sqrt(d); // e = 3 picometers

Note that some functions only compile for dimensionless quantities (e.g. exp, log, cos, sin, etc.).

All overloaded functions have the same syntax as their regular C++ counterpart, except for pow and root, which use a template syntax:

SBQuantity::length a(2.0); // a = 2 picometers
SBQuantity::squareLength b = pow<2>(a); // b = 4 picometer^2
SBQuantity::length c = root<2>(b); // c = 2 picometers

Making new units

Numerous units have been predefined in SAMSON (see the documentation of SBDQuantity). However, when a new type is required, templates may be used to form it through inversions, products, divisions, powers and roots:

// a is expressed in inverse moles
SBQuantityInverse<SBQuantity::mole>::Type a;
// b is expressed in moles.seconds
SBQuantityProduct2<SBQuantity::mole, SBQuantity::second>::Type b;
// c is expressed in grams per mole
SBQuantityInverse<SBQuantity::mole>::Type> >::Type c;
// d is expressed in grams per mole
SBQuantityDivision<SBQuantity::gram, SBQuantity::mole>::Type d;
// e is expressed in picometer^6
SBQuantityPower<6, SBQuantity::picometer>::Type e;
// f is expressed in angstroms
SBQuantityRoot<3, SBQuantity::cubicAngstrom>::Type f;

For convenience, templates have been defined for products of up to seven units (SBQuantityProduct2, ..., SBQuantityProduct7), e.g.:

// a is expressed in moles.seconds.grams
SBQuantityProduct3<SBQuantity::mole, SBQuantity::second,SBQuantity::gram>::Type a;

Of course, typedefs may be used to help declare variables:

// gramPerMole is a new unit
SBQuantityInverse<SBQuantity::mole>::Type> >::Type gramPerMole;
gramPerMole a; // a is expressed in grams per mole

Note that products and divisions have the following limitation: they must involve units within the same systems and, if two unit types have non-zero exponents for a given base unit, then the associated scales must be the same. It is thus acceptable to declare:

// f is in force units
SBQuantityDivision<SBQuantity::energy, SBQuantity::length>::Type f;

since both SBQuantity::energy and SBQuantity::length involve picometers (scale -12), but the following is incorrect:

// this is incorrect
SBQuantityDivision<SBQuantity::energy, SBQuantity::angstrom>::Type f;

since the calculation involves two different scales: picometers in the energy (scale -12), and angstroms (scale -10).

Defined systems

Non-SI unit systems

• Atomic units (AU)
• Dalton
• Electronvolt
• KilocaloriePerMole

It will often be useful to make new units based on existing units. For example, assume you want to define

Defining new systems

When integrating existing code as SAMSON Elements you can define your own unit systems and units.

List of default types

SAMSON has a list of default types, for common dimensions. For example, instead of declaring a length as follows:

SBQuantity::nanometer l1; // l1 is a length

it might be preferable to write:

SBQuantity::length l2; // l2 is a length

These dimensions are typedefs for units that are natural in nanoscience, when expressed in the international system of units. For example, length is a typedef for picometer, so that, in the example above, l1 and l2 have different types (but conversions are possible). As a result, when defining a variable (i.e. when assigning a value to it), it is strongly encouraged to use units instead of dimensions:

// unclear and risky (the definition of length could change)
// OK (even if the definition of length would change)

Default types

Here is the list of default types defined in SAMSON.

Base types

• SBQuantity::length == SBQuantity::picometer
• SBQuantity::mass == SBQuantity::yoctogram
• SBQuantity::time == SBQuantity::femtosecond
• SBQuantity::intensity == SBQuantity::nanoampere
• SBQuantity::temperature == SBQuantity::kelvin
• SBQuantity::amountOfSubstance == SBQuantity::mole
• SBQuantity::luminousIntensity == SBQuantity::candela

There are also inverse, square, inverse square base units defined, e.g.:

• SBQuantity::inverseLength
• SBQuantity::squareLength
• SBQuantity::inverseSquareLength
• SBQuantity::cubicLength
• SBQuantity::inverseCubicLength
• SBQuantity::inverseMass
• SBQuantity::inverseTime
• SBQuantity::squareTime
• etc.

Derived types

• SBQuantity::distance == SBQuantity::length
• SBQuantity::inverseDistance == SBQuantity::inverseLength
• SBQuantity::squareDistance == SBQuantity::squareLength
• SBQuantity::inverseSquareDistance == SBQuantity::inverseSquareLength
• SBQuantity::cubicDistance == SBQuantity::cubicLength
• SBQuantity::inverseCubicDistance == SBQuantity::inverseCubicLength
• SBQuantity::electricCharge
• SBQuantity::voltage
• SBQuantity::electricField
• SBQuantity::magneticDensity
• SBQuantity::angularVelocity == SBQuantity::inverseTime
• SBQuantity::angularAcceleration == SBQuantity::inverseSquareTime
• SBQuantity::pressure
• SBQuantity::lengthMass
• SBQuantity::inverseLengthInverseMass
• SBQuantity::area == SBQuantity::squareLength
• SBQuantity::inverseArea == SBQuantity::inverseSquareLength
• SBQuantity::volume == SBQuantity::cubicLength
• SBQuantity::inverseVolume == SBQuantity::inverseCubicLength
• SBQuantity::position == SBQuantity::length
• SBQuantity::velocity
• SBQuantity::acceleration
• SBQuantity::force
• SBQuantity::energy
• SBQuantity::inverseForce
• SBQuantity::lengthPerForce
• SBQuantity::forcePerLength
• SBQuantity::energyPerSquareLength
• SBQuantity::momentum
• SBQuantity::inverseMomentum
• SBQuantity::momentOfInertia
• SBQuantity::inverseMomentOfInertia
• SBQuantity::electronDensity

Vectors

Here is the list of defined vector dimensions, and their corresponding types:

• SBVector3: SBPhysicalVector3<SBQuantity::dimensionless>
• SBPicometerPerSecond3: SBPhysicalVector3<SBQuantity::picometerPerSecond>
• SBRadian3: SBPhysicalVector3<SBQuantity::radian>
• SBRadianPerSecond3: SBPhysicalVector3<SBQuantity::radianPerSecond>
• SBLength3: SBPhysicalVector3<SBQuantity::length>
• SBInverseLength3: SBPhysicalVector3<SBQuantity::inverseLength>
• SBSquareLength3: SBPhysicalVector3<SBQuantity::squareLength>
• SBPosition3: SBPhysicalVector3<SBQuantity::position>
• SBVelocity3: SBPhysicalVector3<SBQuantity::velocity>
• SBAcceleration3: SBPhysicalVector3<SBQuantity::acceleration>
• SBEnergy3: SBPhysicalVector3<SBQuantity::energy>
• SBTorque3: SBPhysicalVector3<SBQuantity::energy>
• SBForce3: SBPhysicalVector3<SBQuantity::force>
• SBMomentum3: SBPhysicalVector3<SBQuantity::momentum>
• SBInverseMomentum3: SBPhysicalVector3<inverseMomentum>
SBDTypePhysicalVector3

Spatial vectors

Here is the list of defined spatial vector dimensions, and their corresponding types:

• SBVector6: SBPhysicalVector6<SBQuantity::dimensionless, SBQuantity::dimensionless>
• SBForce6: SBPhysicalVector6<SBQuantity::energy, SBQuantity::force>
• SBVelocity6: SBPhysicalVector6<SBQuantity::inverseTime, SBQuantity::velocity>
• SBAcceleration6: SBPhysicalVector6<SBQuantity::inverseSquareTime, SBQuantity::acceleration>
SBDTypePhysicalVector6

Intervals

Here is the list of defined interval dimensions, and their corresponding types:

• SBInterval: SBDTypePhysicalInterval<SBQuantity::dimensionless>
• SBLengthInterval: SBDTypePhysicalInterval<SBQuantity::length>
• SBSquareLengthInterval: SBDTypePhysicalInterval<SBQuantity::squareLength>
• SBPositionInterval: SBDTypePhysicalInterval<SBQuantity::position>
• SBVelocityInterval: SBDTypePhysicalInterval<SBQuantity::velocity>
• SBAccelerationInterval: SBDTypePhysicalInterval<SBQuantity::acceleration>
• SBEnergyInterval: SBDTypePhysicalInterval<SBQuantity::energy>
• SBForceInterval: SBDTypePhysicalInterval<SBQuantity::force>
• SBMomentumInterval: SBDTypePhysicalInterval<SBQuantity::momentum>
SBDTypePhysicalInterval

Interval vectors

Here is the list of defined interval vector dimensions, and their corresponding types:

• SBIAVector3: SBDTypePhysicalIAVector3<SBQuantity::dimensionless>
• SBIARadian3: SBDTypePhysicalIAVector3<SBQuantity::radian>
• SBIARadianPerSecond3: SBDTypePhysicalIAVector3<SBQuantity::radianPerSecond>
• SBIAPosition3: SBDTypePhysicalIAVector3<SBQuantity::picometer>
• SBIAPosition3: SBDTypePhysicalIAVector3<SBQuantity::picometer>
• SBIAVelocity3: SBDTypePhysicalIAVector3<SBQuantity::picometerPerFemtosecond>
• SBIAAcceleration3: SBDTypePhysicalIAVector3<SBQuantity::picometerPerSquareFemtosecond>
• SBIAForce3: SBDTypePhysicalIAVector3<SBQuantity::nanonewton>
• SBIAMomentum3: SBDTypePhysicalIAVector3<SBQuantity::yoctogramPicometerPerFemtosecond>
SBDTypePhysicalIAVector3

Matrices

Here is the list of defined 3x3 matrix dimensions, and their corresponding types:

• SBMatrix33: SBPhysicalMatrix33<SBQuantity::dimensionless>
• SBMass33: SBPhysicalMatrix33<SBQuantity::mass>
• SBInverseMass33: SBPhysicalMatrix33<SBQuantity::inverseMass>
• SBInertiaTensor33: SBPhysicalMatrix33<SBQuantity::momentOfInertia>
• SBInverseInertiaTensor33: SBPhysicalMatrix33<SBQuantity::inverseMomentOfInertia>
• SBMatrix66 - dimensionless
• SBInertia66
• SBInverseInertia66