The LI-6800 measures and computes the following parameters related to leaf-level gas exchange measurements. Additional details are given in Theory and equation summary
Light
The photosynthetic photon flux density (PPFD) readings Qin and Qout are computed from
9‑19
9‑20
where Sqin and Sqout are the calibration factors, and Iqin and Iqout are the raw counts from the A/D converter.
Leaf temperature
Leaf temperatures Tl1, Tl2 (°C) for the two thermocouples are computed from signals Vl1 and Vl2 (mV) by
9‑21
9‑22
where Tj1 and Tj2 are junction temperatures (°C).
Pressure
Atmospheric pressure Pa (kPa) is reported directly from the on-board sensor.
Chamber over-pressure ΔPc (kPa) is computed from
9‑23
where Vp is the signal from the differential pressure sensor (in V, but taken to be kPa) Vpo is the calibration zero for the sensor, G is fan speed (rpm), F is flow rate, and pca and pcb are empirical coefficients that depend on chamber type (Table 9‑1).
Flow
The flow F (µmol s-1) to the leaf chamber is computed from
9‑24
where af1...af7 are factory calibration constants, Vf is flow meter signal (V), Vfo is calibration zero, and Tk is IRGA block temperature.
Reference and sample IRGA cell flows Fr and Fs (µmol s-1) are computed from
9‑25
9‑26
where Vfr and Vfs are cell exit flow sensor voltage, Vfro and Vfso are the offset value of that voltage, and ar1...ar4 and as1...as4 are factory calibration coefficients.
CO2
Reference and sample CO2 concentrations Cr and Cs are given by
9‑27
where Mc() is the CO2 match correction function:
9‑28
9‑29
9‑30
where αcr and αcs are absorptances, Pa is atmospheric pressure, Tr and Ts are the cell inlet temperatures, fc is a 5th order polynomial with coefficients (br1...br5 or bs1...bs5) determined at the factory, Scro; Scso and Scr1; Scs1 are the span1 and span2 user calibration settings, Wa and Wb are water concentrations, Xo is the oxygen concentration in percent. Ѱc is the band broadening function for the effect of water and oxygen on CO2, given by
9‑31
where Bwc and Boc are the band broadening coefficients for H2O and O2 respectively on CO2.
Reference and sample absorptances αcr and αcs are corrected for zero drift with temperature, span drift with temperature, and cross sensitivity with water.
9‑32
9‑33
where cr1...cr3 and cs1...cs3 are empirical coefficients for CO2 absorptance span drift determined during calibration, Icr, Icro, Ics, and Icso are the raw IRGA detector absorbing and non-absorbing readings for CO2, and Iwr, Iwro, Iws and Iwso are those for water. Xwcr and Xwcs are empirical cross sensitivity coefficients for water on CO2 determined during calibration, Zcro and Zcso are the current user CO2 zero factors, and Zcr and Zcs are the factors for CO2 zero drift with temperature, determined during calibration.
Dry mole fractions Crd and Csd are computed from
9‑34
H2O
Reference and sample H2O concentrations Hr and Hs are given by
9‑35
where Mw() is the H2O match correction function:
9‑36
9‑37
9‑38
where αwr and αws are the reference and sample absorptances, fw is a 3rd order polynomial with coefficients (dr1...dr3 and ds1...ds3) determined at the factory, Swro, Swr1 and Swso, Sws1 are the H2O span1 and span2 user calibration settings for reference and sample, Ѱo is the band broadening function for the effect of oxygen on H2O, given by
9‑39
where Bow1 and Bow2 are empirical coefficients.
Absorptances αwr and αws are corrected for zero drift with temperature, span drift with temperature, and cross sensitivity with water.
9‑40
9‑41
where wr1...wr3 and ws1...ws3 are empirical coefficients for H2O absorptance span drift determined during calibration. Xcwr, Xcws are empirical cross sensitivity coefficients for CO2 on H2O determined during calibration, Zwro and Zwso are the current user H2O zero factors, and Zwr and Zws are the factors for H2O zero drift with temperature, determined during calibration.
The vapor pressure (kPa) of the air in the reference and sample cells er and es is given by
9‑42
9‑43
Reference and sample cell dew point temperatures Tdr and Tds are given by
9‑44
9‑45
Humidity indicators
The leaf chamber vapor pressure ec (kPa) is given by
9‑46
The saturation vapor pressure esc in the leaf chamber is a function of chamber air temperature Ta:
9‑47
where es(T) is the saturation vapor pressure function:
9‑48
The relative humidity hc (%) in the leaf chamber is given by
9‑49
The vapor pressure deficit of the leaf eΔl is computed from
9‑50
where Tl is leaf temperature ().
Transpiration
The mass balance of water vapor in an open system is given by
where s is leaf area (m2), E is transpiration rate (mol H2O m-2 s-1), ue and uo are incoming and outgoing flow rates (mol s-1) from the chamber, and we and wo are incoming and outgoing water mole factions (mol H2O (mol air)-1). Since
we can write
9‑53
which rearranges to
The relationships between the terms in equations 9‑51 through 9‑54 and what the LI-6800 measures are
where F is air flow rate (µmol s-1), Ws and Wr are sample and reference water mole fractions (mmol H2O (mol air)-1), and S is leaf area (cm2). The equation that the LI-6850 uses for transpiration is thus
9‑56
Stomatal conductance
The total conductance gtw of the leaf to water vapor is
9‑57
where Wl is the molar concentration of the water vapor within the leaf (mmol H2O (mol air)-1), which is computed from the leaf temperature Tl (°C) and the total atmospheric pressure in the leaf chamber
9‑58
We assume that the total resistance for the upper ru or lower rl surface of a leaf is the sum of the stomatal resistance rs and boundary layer resistance rb of that surface
9‑59
9‑60
and that the upper and lower boundary layer resistances are the same
9‑61
and we define K to be the ratio of stomatal resistances of the two sides
Leaf stomatal resistance rs is given by
9‑63
9‑64
9‑65
9‑66
Total conductance g (the inverse of the total resistance r) can thus be written
9‑67
For water vapor, the total conductance gtw is related to stomatal conductance gsw and one sided boundary layer conductance gbw by
9‑70
Solving equation 9‑68 for gsw yields
Note that although we defined K to be a particular ratio of upper to lower stomata resistances (equation 9‑62), since K appears as 
, we get the same mathematical result if we had defined it the other way. In other words, K is equivalent to 1/K. Note also that the LI-6400 does not use equation 9‑71, but rather a simplified approximation:
9‑72
Boundary layer
The one sided boundary layer conductance to water vapor gbw for a broadleaf is a function of fan speed G (rpm) and leaf area S (cm2).
9‑73
where f is
9‑74
and s is forced to be
9‑75
The empirical coefficients co...c4, reference pressure Po, and leaf area limits Smin and Smax depend on chamber type.
| Chamber | co | c1 | c2 | c3 | c4 | Po | Smin | Smax |
|---|---|---|---|---|---|---|---|---|
| 6800-01 Flr | 0.250 | 0.35860 | -4.01816E-3 | 0.00451074 | -0.0044762 | 96.9 | 1 | 6 |
| 6800-01A 6 cm2 | 0.578 | 0.5229739 | 3.740252E-3 | -6.197961E-2 | -5.608586E-3 | 96.9 | 1 | 6 |
| 6800-01A 2 cm2 | 0.572 | 0.3872742 | -1.870584E-2 | 0.0 | -7.37389E-3 | 96.9 | 1 | 2 |
| 6800-12 3x3 | 0.500 | 0.44869569 | 1.9000035E-3 | -4.26088781E-2 | -3.456516E-3 | 96.7 | 2 | 9 |
| 6800-12A 9 cm2 | 0.579 | 0.3210639 | -1.109987E-3 | 5.106816E-3 | -3.283688E-3 | 96.7 | 2 | 9 |
| 6800-12A 6 cm2FB | 0.345 | 0.552336 | -4.7985e-3 | 0.0 | -7.3557e-3 | 96.7 | 1 | 6 |
| 6800-12A 6 cm2SS | 0.418 | 0.5145466 | -2.5106E-3 | 0.0 | -8.1206E-3 | 96.7 | 1 | 6 |
| 6800-12A 3 cm2FB | 0.188 | 0.5795409 | -1.15295E-2 | 0.0 | -9.7259E-3 | 96.7 | 1 | 3 |
| 6800-12A 3 cm2SS | 0.141 | 0.5263354 | -1.27376E-2 | 0.0 | -1.10157E-2 | 96.7 | 1 | 3 |
| 6800-13 6x6 | 0.430 | 0.267827 | -1.164018E-4 | 2.248202E-3 | -5.109462E-3 | 96.8 | 6 | 36 |
Net assimilation
The mass balance of CO2 in an open system is given by
9‑76
where a is assimilation rate (mol CO2 m-1 s-1), ce and co are entering and outgoing mole fractions (mol CO2 (mol air)-1). Using equation 9‑52, we can write
9‑77
which rearranges to
To write equation 9‑78 in terms of what the LI-6800 measures, we use equations 9‑55 and
9‑79
where Cr and Cs are sample and reference CO2 concentrations (µmol mol-1), and A is the net assimilation by the leaf (µmol m-2 s-1). Substitution yields
9‑80
9‑81
9‑82
Intercellular CO2
The intercellular CO2 concentration Ci (µmol CO2 (mol air)-1) is given by
9‑83
where gtc is the total conductance to CO2. From equation 9‑69, we can write
where 1.6 is the ratio of the diffusivities of CO2 and water in air, and 1.37 is the same ratio for the boundary layer. This is another departure from the LI-6400, which does not use equation 9‑84, but a simplified approximation
9‑85
Energy balance
The LI-6800 provides several potential sources for leaf temperature: it can be directly measured with some combination of the two leaf thermocouples (Tl1 and Tl2), or computed indirectly from a leaf energy balance (Teb). The user can specify what combination to use via three weighting factors fT1, fT2, and fTeb:
9‑86
The energy balance temperature Teb assumes that the energy balance of a leaf in the chamber has three components: net radiation R (W m-2), sensible heat flux H (W m-2), and latent heat flux L (W m-2), and that they all sum to zero:
We consider two components of net radiation: short wave (visible and near IR) and thermal.
where Rabs is absorbed short wave, and Rnt is net thermal. The absorbed short wave radiation is computed by
9‑89
where Qabs is the absorbed irradiance by the leaf (µmol m-2 s-1), and k is the conversion factor for transforming (µmol m-2 s-1) to (W m-2), based on the spectral characteristics of the light source.
The net thermal balance is based on the leaf temperature Tl and the surrounding chamber wall temperature Tw, so the total radiation balance R can be written as
where ϵ is the thermal emissivity of the leaf (usually assumed to be 0.95), and σ is the Stefan-Boltzmann constant (5.67 W m-2 K-1). The 2 in equation 9‑90 accounts for both sides of the leaf. Wall temperature Tw is not measured, but depends on a user-specified offset ΔTw from chamber air temperature.
9‑91
The latent heat flux L is the transpiration rate E converted to W m-2.
9‑92
The sensible heat flux H is a function of the leaf - chamber air temperature difference Tl−Ta, the specific heat capacity of the air cp (29.3 J mol-1 K-1) and the one sided boundary layer conductance for heat transfer of the leaf gbh, which is 0.92 times the boundary layer conductance for water vapor gbw.
9‑93
9‑94
Equation 9‑87 becomes:
If we let ΔT = Tl − Ta, and note for small ΔT
Substituting equation 9‑96 into 9‑95 and solving for ΔT yields
9‑97
The energy balance leaf temperature Teb is then
9‑98
Sensor head calibration coefficients
| Symbol | Description | XML Locator (/licor/li6850/...) |
|---|---|---|
| af1...af7 ar1...ar4 as1...as4 |
Main flow sensor Ref flow sensor Sample flow sensor |
../factory/flowmeter/a...g ../factory/irga_a/flow/a1...a4 ../factory/irga_b/flow/a1...a4 |
| Bwc Boc Bow1, Bow2 |
Band broadening coefficient for water on CO2 Band broadening coefficient for oxygen on CO2 Band broadening correction for oxygen on water |
../factory/bb/ch ../factory/bb/cx ../factory/bb/hx0, hx1 |
| br1...br5 bs1...bs5 |
Reference CO2 calibration coefficients Sample CO2 calibration coefficients |
../factory/irga_b/co2/a1...a5 ../factory/irga_a/co2/a1...a5 |
| dr1...dr3 ds1...ds3 |
Reference H2O calibration coefficients Sample H2O calibration coefficients |
../factory/irga_b/h2o/a1...a3 ../factory/irga_b/h2o/a1...a3 |
| mco...mc3
mwo...mw3 |
CO2 match coefficients H2O match coefficients |
../cfg/match/co2_adj ... co2 adj 3 ../cfg/match/h2o_adj ... h2o adj 3 |
| pca
pcb |
Pressure correction or fan speed and flow rate Pressure correction or fan speed and flow rate |
../factory/chamber/pca ../factory/chamber/pcb |
| σcr1...σcr3
σcs1...σcs3 σwr1...σwr3 σws1...σws3 |
CO2 reference absorptance span drift with temp CO2 sample absorptance span drift with temp H2O reference absorptance span drift with temp H2O sample absorptance span drift with temp |
../factory/irga_b/co2/s1...s3 ../factory/irga_a/co2/s1...s3 ../factory/irga_b/h2o/s1...s3 ../factory/irga_a/h2o/s1...s3 |
| Scro
Scr1 Scso Scs1 |
Span1 for reference CO2 Span2 for reference CO2 Span1 for sample CO2 Span2 for sample CO2 |
../cal/co2bspan1 ../cal/co2bspan2 ../cal/co2aspan1 ../cal/co2aspan2 |
| Sqin
Sqout |
In-chamber light sensor cal External quantum sensor cal |
../cfg/ppfdin/mult ../cfg/ppfdout/mult |
| Swro
Swr1 Swso Sws1 |
Span1 for reference H2O Span2 for reference H2O Span1 for sample H2O Span2 for sample H2O |
../cal/h2obspan1 ../cal/h2obspan2 ../cal/h2oaspan1 ../cal/h2oaspan2 |
| Vfo
Vfro Vfso |
Zero offset for main flow meter Zero offset for reference flow meter Zero offset for sample flow meter |
../cal/flowmeterzero ../cal/flowbzero ../cal/flowazero |
| Vpo | Zero parameter for differential pressure sensor | ../cal/chamberpressurezero |
| Xo | Oxygen concentration, percent | ../factory/cal/oxygen |
| Xcwr
Xcws Xwcr Xwcs |
Cross sensitivity, CO2 on H2O, reference cell Cross sensitivity, CO2 on H2O, sample cell Cross sensitivity, H2O on CO2, reference cell Cross sensitivity, H2O on CO2, sample cell |
../factory/irga_b/xhc ../factory/irga_b/xch ../factory/irga_a/xhc ../factory/irga_a/xch |
| Zcr
Zcro Zcs Zcso |
Zero drift with temperature for reference CO2 Zero offset for reference CO2 Zero drift with temperature for sample CO2 Zero offset for sample CO2 |
../factory/irga_b/z ../cal/co2bzero ../factory/irga_a/z ../cal/co2azero |
| Zwr
Zwro Zws Zwso |
Zero drift with temperature for reference H2O Zero offset for reference H2O Zero drift with temperature for sample H2O Zero offset for sample H2O |
../factory/irga_b/z ../cal/h2obzero ../factory/irga_a/z ../cal/h2obzero |
Sensor measurements and computations
| Symbol | Description (units) | Name (Label) | Group |
|---|---|---|---|
| αcr
αcs αwr αws |
Reference cell CO2 absorptance Sample cell CO2 absorptance Reference cell H2O absorptance Sample cell H2O absorptance |
abs_c_b abs_c_a abs_h_b abs_h_a |
Raw Raw Raw Raw |
| co...c4 | Boundary layer function coeffs | blc_a...blc_e | ChambConst |
| Ca
Cb Cr Crd Cs Csd ∆Pc |
Sample cell CO2, not adjusted for match Sample cell CO2 (µmol mol−1) Reference cell CO2 (µmol mol−1) Reference cell CO2, dry mole fraction Sample cell CO2 (µmol mol−1) Sample cell CO2, dry mole fraction Chamber over pressure (kPa) |
CO2_a CO2_b CO2_r CO2_r_d CO2_s CO2_s_d Pchamber (∆Pcham) |
Meas Meas2 Meas Meas Meas Meas Meas |
| er
es Icr Icro Ics Icso |
Reference cell vapor pressure (kPa) Sample cell vapor pressure (kPa) Reference CO2 raw detector count Reference CO2 raw detector reference count Sample CO2, raw detector count Sample CO2 raw detector reference count |
e_r e_s Wc_r Wco_r Wc_s Wco_s |
Meas2 Meas2 Raw Raw Raw Raw |
| Iqin
Iqout |
In-chamber PPFD sensor raw counts External quantum sensor raw counts |
||
| Iwr
Iwro Iws Iwso |
Reference H2O raw detector count Reference H2O raw detector reference count Sample H2O raw detector count Sample H2O raw detector reference count |
Ww_r Wwo_r Ww_s Wwo_s |
Raw Raw Raw Raw |
| Pa | Atmospheric pressure (kPa) | Press (Pa) | Meas |
| F
Fr Fs |
Flow to chamber (µmol s−1) Flow from reference cell (µmol s−1) Flow from sample cell (µmol s−1) |
Flow Flow_r Flow_s |
Meas Status Status |
| G | Chamber fan rotation rate (rpm) | Fan speed | Status |
| Qin
Qout |
In-chamber PPFD External PPFD |
PPFD_in (Q_amb_in) PPFD_out (Q_amb_out) |
Meas Meas |
| Ta | Leaf chamber air temperature (C) | Tchamber (Tair) | Meas |
| Tdr
Tds |
Dewpoint temperature reference cell (C) Dewpoint temperature sample cell (C) |
Td_r Td_s |
Meas2 Meas2 |
| Tj1
Tj2 Tk |
Leaf T/C 1 junction temperature (C) Leaf T/C 2 junction temperature (C) IRGA block temperature (C) |
Tleafjunction (Tleaf_j) Tleafjunction2 (Tleaf2_j) Tirga_block (Tirga) |
Status2 Status2 Status |
| Tl1
Tl2 Tr Ts |
Leaf temperature 1 (C) Leaf temperature 2 (C) Reference cell inlet temperature (C) Sample cell inlet temperature (C) |
Tleaf Tleaf2 Tb (Tr) Ta (Ts) |
Meas Meas Status Status |
| Vf
Vfr Vfs |
Signal (V) from main flow sensor Signal (V) from reference flow sensor Signal (V) from sample flow sensor |
Flow Flow_b_v (Flow_r_v) Flow_a_v (Flow_s_v) |
Raw Raw Raw |
| Vl1
Vl2 |
Leaf temperature 1 signal (mV) Leaf temperature 2 signal (mV) |
leaf_t_mv (Tleaf_mv) leaf2_t_mv (Tleaf2_mv) |
Raw Raw |
| Vp | Differential pressure signal (V or kPa) | VPchamber | Raw |
| Wa
Wb Wr Ws |
Sample cell H2O not corrected for match Reference cell H2O Reference cell H2O (mmol mol−1) Sample cell H2O (mmol mol−1) |
H2O_a H2O_a H2O_r H2O_s |
Meas Meas2 Meas Meas |