5.1: Fluorescence and Phosphorescence
Fluorescence and phosphorescence are types of molecular luminescence. A molecule of analyte absorbs a photon and results in an excited species. That excited species can then emit energy (photon, heat). The emission spectrum can provide qualitative and quantitative analysis. The term fluorescence and phosphorescence are usually referred to as photoluminescence because both are alike in excitation brought by absorption of a photon. Fluorescence differs from phosphorescence in that the electronic energy transition that is responsible for fluorescence does not change in electron spin, which results in short-live electrons (<10 -5 s) in the excited state of fluorescence. In phosphorescence, there is a change in electron spin, which results in a longer lifetime of the excited state (second to minutes). Fluorescence and phosphorescence occur at longer wavelength than the excitation radiation.
Introduction
Fluorescence can occur in gaseous, liquid, and solid chemical systems. The simple kind of fluorescence is of dilute atomic vapors. A fluorescence example would be if a 3s electron of a vaporized sodium atom is excited to the 3p state by absorption of a radiation at wavelength 589.6 and 589.0 nm. After 10 -8 s, the electron returns to ground state and on its return it emits radiation of the two wavelengths in all directions. This type of fluorescence in which the absorbed radiation is remitted without a change in frequency is known as resonance fluorescence. Resonance fluorescence can also occur in molecular species. Molecular fluorescence band centers at wavelengths longer than resonance lines. The shift toward longer wavelength is referred to as the Stokes Shift.
Singlet and Triplet Excited State
Understanding the difference between fluorescence and phosphorescence requires the knowledge of electron spin and the differences between singlet and triplet states. The Pauli Exclusion principle states that two electrons in an atom cannot have the same four quantum numbers (\(n\), \(l\), \(m_l\), \(m_s\)) and only two electrons can occupy each orbital where they must have opposite spin states. These opposite spin states are called spin pairing. Because of this spin pairing, most molecules do not exhibit a magnetic field and are diamagnetic. In diamagnetic molecules, electrons are not attracted or repelled by the static electric field. Free radicals are paramagnetic because they contain unpaired electrons have magnetic moments that are attracted to the magnetic field.
Singlet state is defined when all the electron spins are paired in the molecular electronic state and the electronic energy levels do not split when the molecule is exposed into a magnetic field. A doublet state occurs when there is an unpaired electron that gives two possible orientations when exposed in a magnetic field and imparts different energy to the system. A singlet or a triplet can form when one electron is excited to a higher energy level. In an excited singlet state, the electron is promoted in the same spin orientation as it was in the ground state (paired). In a triplet excited stated, the electron that is promoted has the same spin orientation (parallel) to the other unpaired electron. The difference between the spins of ground singlet, excited singlet, and excited triplet is shown in Figure \(\PageIndex{1}\). Singlet, doublet and triplet is derived using the equation for multiplicity, 2S+1, where S is the total spin angular momentum (sum of all the electron spins). Individual spins are denoted as spin up (s = +1/2) or spin down (s = -1/2). If we were to calculated the S for the excited singlet state, the equation would be 2(+1/2 + -1/2)+1 = 2(0)+1 = 1, therefore making the center orbital in the figure a singlet state. If the spin multiplicity for the excited triplet state was calculated, we obtain 2(+1/2 + +1/2)+1 = 2(1)+1 =3, which gives a triplet state as expected.
The difference between a molecule in the ground and excited state is that the electrons is diamagnetic in the ground state and paramagnetic in the triplet state.This difference in spin state makes the transition from singlet to triplet (or triplet to singlet) more improbable than the singlet-to-singlet transitions. This singlet to triplet (or reverse) transition involves a change in electronic state. For this reason, the lifetime of the triplet state is longer the singlet state by approximately 10 4 seconds fold difference.The radiation that induced the transition from ground to excited triplet state has a low probability of occurring, thus their absorption bands are less intense than singlet-singlet state absorption. The excited triplet state can be populated from the excited singlet state of certain molecules which results in phosphorescence. These spin multiplicities in ground and excited states can be used to explain transition in photoluminescence molecules by the Jablonski diagram.
Jablonski Diagrams
The Jablonski diagram that drawn below is a partial energy diagram that represents the energy of photoluminescent molecule in its different energy states. The lowest and darkest horizontal line represents the ground-state electronic energy of the molecule which is the singlet state labeled as \(S_o\). At room temperature, majority of the molecules in a solution are in this state.
The upper lines represent the energy state of the three excited electronic states: S 1 and S 2 represent the electronic singlet state (left) and T 1 represents the first electronic triplet state (right). The upper darkest line represents the ground vibrational state of the three excited electronic state.The energy of the triplet state is lower than the energy of the corresponding singlet state.
There are numerous vibrational levels that can be associated with each electronic state as denoted by the thinner lines. Absorption transitions (blues lines in Figure \(\PageIndex{2}\)) can occur from the ground singlet electronic state (S o ) to various vibrational levels in the singlet excited vibrational states. It is unlikely that a transition from the ground singlet electronic state to the triplet electronic state because the electron spin is parallel to the spin in its ground state (Figure \(\PageIndex{1}\)). This transition leads to a change in multiplicity and thus has a low probability of occurring which is a forbidden transition. Molecules also go through vibration relaxation to lose any excess vibrational energy that remains when excited to the electronic states (\(S_1\) and \(S_2\)) as demonstrated in wavy lines in Figure \(\PageIndex{2}\). The knowledge of forbidden transition is used to explain and compare the peaks of absorption and emission.
Absorption and Emission Rates
The table below compares the absorption and emission rates of fluorescence and phosphorescence.The rate of photon absorption is very rapid. Fluorescence emission occurs at a slower rate.Since the triplet to singlet (or reverse) is a forbidden transition, meaning it is less likely to occur than the singlet-to-singlet transition, the rate of triplet to singlet is typically slower. Therefore, phosphorescence emission requires more time than fluorescence.
| Process | Transition | Timescale (sec) |
|---|---|---|
| Light Absorption (Excitation) | S 0 → S n | ca. 10 -15 (instantaneous) |
| Internal Conversion | S n → S 1 | 10 -14 to 10 -11 |
| Vibrational Relaxation | S n * → S n | 10 -12 to 10 -10 |
| Intersystem Crossing | S 1 → T 1 | 10 -11 to 10 -6 |
| Fluorescence | S 1 → S 0 | 10 -9 to 10 -6 |
| Phosphorescence | T 1 → S 0 | 10 -3 to 100 |
| Non-Radiative Decay |
S
1
→ S
0
T 1 → S 0 |
10
-7
to 10
-5
10 -3 to 100 |
Deactivation Processes
A molecule that is excited can return to the ground state by several combinations of mechanical steps that will be described below and shown in Figure \(\PageIndex{2}\).The deactivation process of fluorescence and phosphorescence involve an emission of a photon radiation as shown by the straight arrow in Figure \(\PageIndex{2}\). The wiggly arrows in Figure \(\PageIndex{2}\) are deactivation processes without the use of radiation. The favored deactivation process is the route that is most rapid and spends less time in the excited state.If the rate constant for fluorescence is more favorable in the radiationless path, the fluorescence will be less intense or absent.
- Vibrational Relaxation: A molecule maybe to promoted to several vibrational levels during the electronic excitation process.Collision of molecules with the excited species and solvent leads to rapid energy transfer and a slight increase in temperature of the solvent. Vibrational relaxation is so rapid that the lifetime of a vibrational excited molecule (<10 -12 ) is less than the lifetime of the electronically excited state. For this reason, fluorescence from a solution always involves the transition of the lowest vibrational level of the excited state. Since the space of the emission lines are so close together, the transition of the vibrational relaxation can terminate in any vibrational level of the ground state.
- Internal Conversion: Internal conversion is an intermolecular process of molecule that passes to a lower electronic state without the emission of radiation.It is a crossover of two states with the same multiplicity meaning singlet-to-singlet or triplet-to-triplet states.The internal conversion is more efficient when two electronic energy levels are close enough that two vibrational energy levels can overlap as shown in between S 1 and S 2 . Internal conversion can also occur between S 0 and S 1 from a loss of energy by fluorescence from a higher excited state, but it is less probable. The mechanism of internal conversion from S 1 to S 0 is poorly understood. For some molecules, the vibrational levels of the ground state overlaps with the first excited electronic state, which leads to fast deactivation.These usually occur with aliphatic compounds (compound that do not contain ring structure), which would account for the compound is seldom fluorescing. Deactivation by energy transfer of these molecules occurs so rapidly that the molecule does not have time to fluoresce.
- External Conversion : Deactivation of the excited electronic state may also involve the interaction and energy transfer between the excited state and the solvent or solute in a process called external conversion. Low temperature and high viscosity leads to enhanced fluorescence because they reduce the number of collision between molecules, thus slowing down the deactivation process.
- Intersystem Crossing: Intersystem crossing is a process where there is a crossover between electronic states of different multiplicity as demonstrated in the singlet state to a triplet state (S 1 to T 1 ) on Figure \(\PageIndex{1}\). The probability of intersystem crossing is enhanced if the vibration levels of the two states overlap. Intersystem crossing is most commonly observed with molecules that contain heavy atom such as iodine or bromine. The spin and orbital interaction increase and the spin become more favorable.Paramagnetic species also enhances intersystem crossing, which consequently decreases fluorescence.
- Phosphorescence : Deactivation of the electronic excited state is also involved in phosphorescence. After the molecule transitions through intersystem crossing to the triplet state, further deactivation occurs through internal or external fluorescence or phosphorescence. A triplet-to-singlet transition is more probable than a singlet-to-singlet internal crossing. In phosphorescence, the excited state lifetime is inversely proportional to the probability that the molecule will transition back to the ground state. Since the lifetime of the molecule in the triplet state is large (10 -4 to 10 second or more), transition is less probable which suggest that it will persist for some time even after irradiation has stopped. Since the external and internal conversion compete so effectively with phosphorescence, the molecule has to be observed at lower temperature in highly viscous media to protect the triplet state.
Variables that affect Fluorescence
After discussing all the possible deactivation processes, variable that affect the emissions to occur. Molecular structure and its chemical environment influence whether a substance will fluoresce and the intensities of these emissions. The quantum yield or quantum efficiency is used to measure the probability that a molecule will fluoresce or phosphoresce. For fluorescence and phosphorescence is the ratio of the number of molecules that luminescent to the total number of excited molecules. For highly fluoresce molecules, the quantum efficiency approaches to one. Molecules that do not fluoresce have quantum efficiencies that approach to zero.
Fluorescence quantum yield (\( \phi \)) for a compound is determined by the relative rate constants (k) of various deactivation processes by which the lowest excited singlet state is deactivated to the ground state. The deactivation processes including fluorescence (kf), intersystem crossing (\(k_i\)), internal conversion (kic), predissociation (kpd), dissociation (kd), and external conversion (kec) allows one to qualitatively interpret the structural and environmental factors that influence the intensity of the fluorescence. They are related by the quantum yield equation given below:
\[ \dfrac{k_f}{k_f+k_i+k_{ec}+k_{ic}+k_{pd}+k_d} \]
Using this equation as an example to explain fluorescence, a high fluorescence rate (k f ) value and low values of the all the other relative rate constant terms (k f +k i +k ec +k ic +k pd +k d ) will give a large \( \phi \), which suggest that fluorescence is enhanced. The magnitudes of kf , kd, and kpd depend on the chemical structure, while the rest of the constants ki, kec, and kic are strongly influenced by the environment.
Molecules that are excited electronically will return to the lowest excited state by rapid vibrational relaxation and internal conversion, which produces no radiation emission. Fluorescence arises from a transition from the lowest vibrational level of the first excited electronic state to one of the vibrational levels in the electronic ground state. In most fluorescent compounds, radiation is produced by a \(\pi^{*} \rightarrow \pi \) or \(\pi^{*} \rightarrow n \) transition depending on which requires the least energy for the transition to occur.
Fluorescence is most commonly found in compounds in which the lowest energy transition is \(\pi \rightarrow \pi^{*} \) (excited singlet state) than \(n \rightarrow \pi^{*} \) which suggest that the quantum efficiency is greater for \(\pi \rightarrow \pi^{*} \) transitions. The reason for this is that the molar absorptivity, which measures the probability that a transition will occur, of the \(\pi \rightarrow \pi^{*} \) transition is 100 to 1000 fold greater than \(n \rightarrow \pi^{*} \) process. The lifetime of \(\pi \rightarrow \pi^{*} \) (10 -7 to 10 -9 s) is shorter than the lifetime of \(n \rightarrow \pi^{*} \) (10 -5 to 10 -7 ).
Quenching Methods
Dynamic Quenching is a nonradiative energy transfer between the excited and the quenching agent species (Q).The requirements for a successful dynamic quenching are that the two collision species the concentration must be high so that there is a higher possibility of collision between the two species. Temperature and quenching agent viscosity play a role on the rate of dynamic quenching. Dynamic quenching reduces fluorescence quantum yield and the fluorescence lifetime.
Dissolved oxygen in a solution increases the intensity of the fluorescence by photochemically inducing oxidation of the fluorescing species. Quenching results from the paramagnetic properties of molecular oxygen that promotes intersystem crossing and converts the excited molecules to triplet state. Paramagnetic species and dissolved oxygen tend to quench fluorescence and quench the triplet state.
Static quenching occurs when the quencher and ground state fluorophore forms a dark complex. Fluorescence is usually observed from unbound fluorophore. Static quenching can be differentiated from dynamic quenching in that the lifetime is not affected in static quenching. In long range (Förster) quenching, energy transfer occurs without collision between molecules, but dipole-dipole coupling occurs between excited fluorophore and quencher.
Emission and Excitation Spectra
One of the ways to visually distinguish the difference between each photoluminescence is to compare the relative intensities of emission/excitation at each wavelength. An example of the three types of photoluminescence (absorption, fluorescence and phosphorescence) is shown for phenanthrene in the spectrum below.In the spectrum, the luminescent intensity is measure in a wavelength is fixed while the excitation wavelength is varied. The spectrum in red represents the excitation spectrum, which is identical to the absorption spectrum because in order for fluorescence emission to occur, radiation needs to be absorbed to create an excited state.The spectrum in blue represent fluorescence and green spectrum represents the phosphorescence.
Fluorescence and Phosphorescence occur at wavelengths that are longer than their absorption wavelengths.Phosphorescence bands are found at a longer wavelength than fluorescence band because the excited triplet state is lower in energy than the singlet state.The difference in wavelength could also be used to measure the energy difference between the singlet and triplet state of the molecule. The wavelength (\(\lambda\)) of a molecule is inversely related to the energy (\(E\)) by the equation below:
\[E= \dfrac{hc}{\lambda}\]
As the wavelength increases, the energy of the molecule decrease and vice versa.
References
- D. A. Skoog, et al. "Principles of Instrumental Analysis" 6th Edition, Thomson Brooks/Cole. 2007
- D. C. Harris and M.D. Bertolucci "Symmetry and Spectroscopy, An Introduction to Vibrational and Electronic Spectroscopy" Dover Publications, Inc., New York. 1989.
Contributors and Attributions
- Diana Wong (UCD)


2.5: Substantive Criminal Law
State Substantive Criminal Law
While rooted in English common law, American substantive criminal law is statutory. There are no common-law crimes in the United States. In other words, the law of crimes is decided by the state legislatures (for each state) and by Congress (for the federal government). Most states, but not the federal government, have a comprehensive “code” of substantive criminal law made up of general principles of criminal responsibility, laws defining the particular criminal offenses, and laws defining excuses and justifications.
Two-thirds of the states have adopted in whole or in part the Model Penal Code (MPC), which was drafted in the 1950s and 1960s by the American Law Institute, a prominent law reform organization. The MPC is the most influential work in American substantive criminal law. One of the most deeply rooted principles in American criminal law is that there can be no criminal responsibility without culpability or blameworthiness. Under the MPC, culpability, sometimes referred to as mens rea or “state of mind,” is satisfied by a showing of purpose, knowledge, recklessness, or negligence, all of which are carefully defined by the code. Except in the case of minor offenses and some regulatory crimes, the MPC requires that there be specified culpability for every element of an offense (conduct, attendant circumstances, result). Criminal codes set out the prohibitions that constitute the law of crimes—offenses against a person (e.g., murder and rape); offenses against property (e.g., theft and arson); offenses against public order (e.g., disorderly conduct and rioting); offenses against the family (e.g., bigamy and incest); and offenses against public administration (e.g. bribery and perjury).
Federal Substantive Criminal Law
Which crimes are considered federal, and which are considered state? There is no clear answer to this question. Indeed, criminal conduct cannot be sorted into these two baskets. When a single act or course of conduct violates both federal and state criminal laws, it is even possible for both governments to prosecute because, under the “dual sovereignty” doctrine, the double jeopardy prohibition does not apply to separate prosecutions by separate sovereigns.
In theory, congressional power is limited to the powers expressly enumerated in Section 1 of the Constitution. Offenses like counterfeiting U.S. currency, illegally entering the United States, treason, and violation of constitutional and federal statutory rights are obviously within the federal government’s core jurisdiction. But, utilizing its expansive powers under the commerce clause and other elastic provisions, Congress has passed federal criminal laws dealing with drug trafficking, firearms, kidnapping, auto theft, fraud, and scores of other ordinary crimes.
The reach of federal criminal law grew inexorably throughout the 20th century. The Supreme Court has rarely found that Congress lacked authority to pass a federal criminal law. Today, federal criminal law can be used to prosecute many offenses that traditionally were regarded as a state responsibility. In practice, however, the great constraint on the reach of federal criminal law is resources. The FBI and other federal law enforcement agencies, as well as federal prosecutors, can investigate and prosecute only a small fraction of all the crimes that potentially fall within their purview.
Take a crash course in Due Process of Law here: Due process of law crash course .
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8.4: The Spectral Representation
With just a little bit more work we shall arrive at a similar expansion for BB itself. We begin by applying the second resolvent identity to \(P_{j}\). More precisely, we note that the second resolvent identity implies that
\[BP_{j}=P_{j}B \nonumber\]
\[= \int_{C_{j}} zR(z)-I dz \nonumber\]
\[P_{j}B = \int_{C_{j}} zR(z) dz \nonumber\]
\[P_{j}B = \int_{C_{j}} R(z)(z-\lambda_{j})dz+\lambda_{j} \int_{C_{j}} R(z) dz \nonumber\]
\[P_{j}B = D_{j}+\lambda_{j} P_{j} \nonumber\]
Summing this over \(j\) we find
\[B \sum_{j = 1}^{h} Pj= \sum_{j = 1}^{h} \lambda_{j} P_{j}+ \sum_{j = 1}^{h} D_{j} \nonumber\]
We can go one step further, namely the evaluation of the first sum. This stems from the equation in the discussion of the transfer function where we integrated \(R(s)\) over a circle \(C_{\rho}\) where \(\rho > ||B||\). The connection to the \(P_{j}\) is made by the residue theorem. More precisely,
\[\int_{C_{\rho}} R(z) dz = 2\pi i \sum_{j=1}^{h} P_{j} \nonumber\]
Comparing this to the equation from the discussion of the transfer function we find
\[\sum_{j=1}^{h} P_{j} = I \nonumber\]
and so takes the form
\[B = \sum_{j=1}^{h} \lambda_{j} P_{j}+ \sum_{j=1}^{h} D_{j} \nonumber\]
It is this formula that we refer to as the Spectral Representation of \(B\). To the numerous connections between the \(P_{j}\) and \(D_{j}\) we wish to add one more. We first write as
\[(B-\lambda_{j} I)P_{j} = D_{j} \nonumber\]
and then raise each side to the \(m_{j}\) power. As \(P_{j}^{m_{j}} = P_{j}\) and \(D_{j}^{m_{j}} = 0\) we find
\[(B-\lambda_{j}I)m_{j}^{P_{j}} = 0 \nonumber\]
For this reason we call the range of \(P_{j}\) the jth generalized eigenspace , call each of its nonzero members a jth generalized eigenvector and refer to the dimension of \(\mathscr{R}(P_{j})\) as the algebraic multiplicity of \(\lambda_{j}\). With regard to the first example from the discussion of the eigenvalue problem, we note that although it has only two linearly independent eigenvectors the span of the associated generalized eigenspaces indeed fills out \(\mathbb{R}^{3}\). One may view this as a consequence of \(P_{1}+P_{2} = I\) or, perhaps more concretely, as appending the generalized first eigenvector \(\begin{pmatrix} {0}&{1}&{0} \end{pmatrix}^T\) to the original two eigenvectors \(\begin{pmatrix} {1}&{0}&{0} \end{pmatrix}^T\) and \(\begin{pmatrix} {0}&{0}&{1} \end{pmatrix}^T\). In still other words, the algebraic multiplicities sum to the ambient dimension (here 3), while the sum of geometric multiplicities falls short (here 2).


3.2: More Direct Proofs
In creating a direct proof, we need to look at our hypotheses, consider the desired conclusion, and develop a strategy for transforming \(\text{A}\) into \(\text{B}\). Quite often you’ll find it easy to make several deductions from the hypotheses, but none of them seems to be headed in the direction of the desired conclusion. The usual advice at this stage is “Try working backwards from the conclusion.” 1
There is a lovely result known as the “arithmetic-geometric mean inequality” whose proof epitomizes this approach. Basically this inequality compares two different ways of getting an “average” between two real numbers. The arithmetic mean of two real numbers \(a\) and \(b\) is the one you’re probably used to, \(\dfrac{(a+b)}{2}\). Many people just call this the “mean” of \(a\) and \(b\) without using the modifier “arithmetic” but as we’ll see, our notion of what intermediate value to use in between two numbers is dependent on context. Consider the following two sequences of numbers (both of which have a missing entry)
\(2\; 9\;16\; 23\; \underline{\;\;} \;37\; 44\)
and
\(3 \;6\; 12\; 24\; \underline{\;\;} \;96\; 192.\)
How should we fill in the blanks?
The first sequence is an arithmetic sequence . Arithmetic sequences are characterized by the property that the difference between successive terms is a constant. The second sequence is a geometric sequence . Geometric sequences have the property that the ratio of successive terms is a constant. The blank in the first sequence should be filled with the arithmetic mean of the surrounding entries \(\dfrac{(23 + 37)}{2} = 30\). The blank in the second sequence should be filled using the geometric mean of its surrounding entries: \(\sqrt{24 · 96} = 48\).
Given that we accept the utility of having two inequivalent concepts of mean that can be used in different contexts, it is interesting to see how these two means compare to one another. The arithmetic-geometric mean inequality states that the arithmetic mean is always bigger.
\[∀a, b ∈ \mathbb{R}, a, b ≥ 0 \implies \dfrac{a + b}{2} ≥ \sqrt{ab}\]
In proving this statement we have little choice but to work backwards from the conclusion because the only hypothesis we have to work with is that \(a\) and \(b\) are non-negative real numbers – which isn’t a particularly potent tool. But what should we do? There isn’t a good response to that question, we’ll just have to try a bunch of different things and hope that something will work out. When we finally get around to writing up our proof though, we’ll have to rearrange the statements in the opposite order from the way they were discovered. This means that we would be ill-advised to make any uni-directional inferences, we should strive to make biconditional connections between our statements (or else try to intentionally make converse errors).
The first thing that appeals to your humble author is to eliminate both the fractions and the radicals. . .
\[\dfrac{a + b}{2} ≥ \sqrt{ab} \\ \iff a + b ≥ 2 \sqrt{ab} \\ \iff (a + b)^2 ≥ 4ab \\ \iff a^2 + 2ab + b^2 ≥ 4ab \]
One of the steps above involves squaring both sides of an inequality. We need to ask ourselves if this step is really reversible. In other words, is the following conditional true?
\[∀x, y ∈ \mathbb{R}^{\text{noneg}}, x ≥ y \implies \sqrt{x} ≥ \sqrt{y}\]
Provide a justification for the previous implication.
What should we try next? There’s really no good justification for this but experience working with quadratic polynomials either in equalities or inequalities leads most people to try “moving everything to one side,” that is, manipulating things so that one side of the equation or inequality is zero.
\[a^2 + 2ab + b^2 ≥ 4ab \\ \iff a^2 − 2ab + b^2 ≥ 0 \]
Whoa! We’re done! Do you see why? If not, I’ll give you one hint: the square of any real number is greater than or equal to zero.
Re-assemble all of the steps taken in the previous few paragraphs into a proof of the arithmetic-geometric mean inequality.
Exercises:
Suppose you have a savings account which bears interest compounded monthly. The July statement shows a balance of \($2104.87\) and the September statement shows a balance \($2125.97\). What would be the balance on the (missing) August statement?
Recall that a quadratic equation \(ax^2 +bx+c = 0\) has two real solutions if and only if the discriminant \(b^2 −4ac\) is positive. Prove that if a and c have different signs then the quadratic equation has two real solutions.
Prove that if \(x^3 − x^2\) is negative then \(3x + 4 < 7\).
Prove that for all integers \(a\), \(b\), and \(c\), if \(a|b\) and \(a|(b + c)\), then \(a|c\).
Show that if \(x\) is a positive real number, then \(x + \dfrac{1}{x} ≥ 2\).
Prove that for all real numbers \(a\), \(b\), and \(c\), if \(ac < 0\), then the quadratic equation \(ax^2 + bx + c = 0\) has two real solutions.
- Hint
-
The quadratic equation \(ax^2 + bx + c = 0\) has two real solutions if and only if \(b^2 − 4ac > 0\) and \(a \neq 0\).
Show that \(\binom{n}{k} · \binom{k}{r} = \binom{n}{r} · \binom{n−r}{k−r}\) (for all integers \(r\), \(k\) and \(n\) with \(r ≤ k ≤ n\).
In proving the product rule in Calculus using the definition of the derivative, we might start our proof with:
\(\dfrac{d}{dx} (f(x) · g(x)) \\ \lim_{h \rightarrow 0} \dfrac{f(x + h) · g(x + h) − f(x) · g(x)}{h} \)
The last two lines of our proof should be:
\(\lim_{h→0} \dfrac{f(x + h) − f(x)}{h} · g(x) + f(x) · \lim_{h→0} \dfrac{g(x + h) − g(x)}{h} \\ = \dfrac{d}{dx} (f(x)) · g(x) + f(x) · \dfrac{d}{dx} (g(x))\)
Fill in the rest of the proof.


1.3: Basic Notation
Notes
When written on a staff, a note indicates a pitch and rhythmic value. The notation consists of a notehead (either empty or filled in), and optionally can include a stem , beam , dot , or flag .
Staff
Notes can’t convey their pitch information without being placed on a staff. A staff consists of five horizontal lines, evenly spaced. The plural of staff is staves .
Clefs
Notes still can’t convey their pitch information if the staff doesn’t include a clef. A clef indicates which pitches are assigned to the lines and spaces on a staff. The two most commonly used clefs are the treble and bass clef; others that you’ll see relatively frequently are alto and tenor clef.
Here is the pitch C4 placed on the treble, bass, alto, and tenor clefs.
Grand staff
The grand staff consists of two staves, one that uses a treble clef, and one that uses a bass clef. The staves are connected by a curly brace. Grand staves are used frequently for notating piano music and other polyphonic instruments.
Ledger lines
When the music’s range exceeds what can be written on the staff, extra lines are drawn so that we can still clearly read the pitch. These extra lines are called ledger lines. In the example below, From Haydn’s Piano Sonata in G (Hob. XVI: 39), Ab5 occurs just above the treble staff in the right hand, and G3 and B3 occur just below the treble staff in the left hand.
Accidentals
Accidentals are used to indicate when a pitch has been raised or lowered. They are written to the left of the pitch.
- When you lower one of the white notes of the piano by a semitone, you add a flat.
- When you raise one of the white notes of the piano by a semitone, you add a sharp.
- When you raise a note that is already flat by a semitone, you add a natural.
- When you lower a note that is already flat by a semitone, you add a double flat.
- When you raise a note that is already sharp by a semitone, you add a double sharp.
The example below shows the symbols for flat, natural, sharp, double sharp, and double flat, respectively.


14.4.2: Fluid Milk Products
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Milk Products
It is estimated that only about half of the milk that is consumed in the United States is in the form of fluid milk and cream. Milk is used as the base for many other food products such as: butter, cheese, yogurt, and ice cream.
Milk
There are 4 major forms of liquid milk sold in U.S. stores whole, reduced fat, low fat, and fat free/nonfat.
Whole milk: to be whole milk, the milk must contain a minimum of 3.25% milk fat and at least 8.25% milk solids not fat (MSNF). Fortification of whole milk is an optional practice but many kinds of milk currently on the market are fortified with vitamins A and D.
Reduced fat and low-fat milk : this milk has had some of the fat removed. The percentage of fat found in these products can be on a range but are around 2% and 1% respectively. Similar to whole milk 8.25% MSNF must be present. These milk solids contribute to the consistency, taste, and nutrient profile of the product. Fortification with vit D is an optional practice while fortification with vitamin A is a mandated standard.
Fat-free or nonfat: Also referred to as skim milk, this milk has almost all of the milk fat removed (<.5%). Like other forms of milk 8.25% of MSNF is required to be present in the product. Fortification of vitamin A is a requirement for fat-free milk but fortification with vitamin D is optional.
Homogenized milk is fresh milk in which the fat particles are so finely divided and emulsified mechanically that the milk fat cannot separate on standing.  More on this topic is discussed in the "Pasteurization and Homogenization" section of LibreText.
Pasteurization of milk was developed in 1859 by the French chemist Louis Pasteur. One method of pasteurization is to heat milk to above 71°C (160°F), maintain it at this temperature for a set time, then cool it immediately to 10°C (50°F) or lower. This kills all harmful bacteria that carry the potential threat of bovine tuberculosis and fever from cows to humans. More on pasteurization will be discussed in the "Pasteurization and Homogenization" section of LibreText.
Cream
The usual minimum standard for cream is 10% fat content, though it ranges between 10% and 18%. Cream in this range may be sold as light cream, half-n-half, heavy cream, or whipping cream. In order to produce these products, the cream is often removed and separated from the non-fat portion also known as skim milk and then various amounts of skim milk are added back to the cream to reach the desired fat percentage.
Evaporated Milk
Produced by evaporating about 60% of the water found in whole milk. Because the milk is canned, often time stabilizers are added to prevent the separation of the fat in the liquid.
Also sold in a fat free version.
To reconstitute combine equal parts water.
Another canned milk product similar to evaporated milk is sweetened condensed milk. Often used in dessert preparation due to its high sugar content, condensed milk is produced by the evaporation of about 50% of the water and the addition of sugar. Due to the high sugar content this product does not have to be sterilized. Additionally, the high percentage of sugar, when heated in the canning process, produces a darker caramel-color.
https://www.thespruceeats.com/canned...g-tips-1808225
Milk Powder
Dried milk is often easier and more affordable to ship and has an extended shelf-life due to the removal of fat and water from the milk product. Powdered (dry milk) can be made from any of the forms of milk discussed previously (e.g. whole, reduced fat, skim). Vitamins A and D are required to be added in dry milk product that has been made using whole, or reduced fat liquid milk but remains optional in non-fat milk varieties.
Dried milk has a similar nutrient profile to liquid milk but vitamins may be reduced in the final product.
Instant milk is sometimes confused with powdered milk but they are not the same. Unlike powdered (dried) milk instant milk has the ability to agglomerate which means it can easily dissolve in water. It is made by taking non-fat dried milk and exposing it to steam which is then re-dried. Through this process it gains the ability to dissolve easily in water.
To use, mix approximately 1/3 cup instant milk to 1 cup of water.
Cultured Milk Products
Cultured milk products also known called fermented milk products have been a part of the human diet for just as long as milk has. It is often thought to have increased health benefits due to the addition of probiotics. Common products sold and consumed in North America include: buttermilk, yogurt, kefir, and sour cream. Bacterial cultures have been added in order to ferment the lactose (sugar found in milk) to lactic acid. The acid causes the casein (milk protein) to precipitate out of the product.
Cultures added diffe r and are not consistent in the production of all fermented foods. Food cultures include bacteria, mold, and yeast. These microorganisms can be present or added in the production process. There are no clear regulatory practices for the different microorganisms that are added to foods with the exception of yogurt which must have lactobacillus bulgaricus and streptococcus thermophilus added. When microorganisms are added they must appear on the nutrition label. If using naturally present bacteria or other organisms they are not required to be listed as an ingredient on the label.
Yogurt Production:
Yogurt has been around for several millennia. The mythological story about the discovery of yogurt suggests that sheepherders stored their milk in bags made of the intestinal gut of the animals. The intestines contain natural enzymes that cause the milk to curdle and sour. This soured milk lasted longer so they continued making it. Today, the FDA defines yogurt as a milk product fermented by two bacterial strains: a lactic acid-producing bacteria: Lactobacillus bulgaricus and Streptococcus thermophiles .
Probiotics
Yogurt cultures in the intestinal tract have been shown to release the enzyme lactase which continues to break down lactose in the dairy product. This makes yogurt edible for people who are lactose-intolerant.
Kefir
Kefir is a carbonated fermented milk drink. The microbes involved in the production of kefir are a symbiotic culture of lactic acid bacteria and yeasts embedded in a matrix of proteins, lipids, and polysaccharides, ‘kefir grains’.
Sour Cream
Sour cream is made from cream soured by adding lactic acids and thickened naturally or by processing. The lactic acid causes the proteins in sour cream to coagulate to a gelled consistency.
Buttermilk
There are two methods to produce buttermilk:
- Inoculating milk with a specific culture to sour it
- Churning milk and separating the liquid left over from the butter
The second method is where buttermilk gets its name, but today, most of what is commonly called buttermilk is the first type. Buttermilk has a higher acid content than regular milk (pH of 4.6 compared with milk’s pH of 6.6).
The fermented dairy product known as cultured buttermilk is produced from cow’s milk and has a characteristically sour taste caused by lactic acid bacteria. This variant is made using one of two species of bacteria — either Lactococcus lactis or Lactobacillus bulgaricus , which creates more tartness in certain recipes.
The acid in buttermilk reacts with the sodium bicarbonate (baking soda) to produce carbon dioxide, which acts as the leavening agent.
Source: Understanding Food Principles and Preparations Amy Brown 4th edition


4.2: Counselor’s Manual for Relapse Prevention with Chemically Dependent Criminal Offenders
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Group counseling has proved to be the most effective way of treating chemical dependency. This chapter explains how to do group counseling. Patients in chemical dependency treatment programs learn best in group counseling, where patients learn about themselves by interacting with others. They also come to understand that they are not alone in their problems. In addition, they learn social and communication skills that allow them to make better use of self-help programs such as Alcoholics Anonymous and Narcotics Anonymous.
How Is Group Work Different From Individual Counseling?
Group counseling and individual counseling are both important tools for treating chemical dependency. Group counseling uses many of the same intervention strategies as individual counseling. There are, however, some important distinctions between the two modalities. A common mistake for beginning group counselors is to focus an entire group meeting on one patient, while the others in the group simply look on.
Group counseling is different from individual counseling in the following ways:
- Group counseling focuses on the present; the here and now. In group counseling, patients do not delve into long accounts of personal history that preceded the problems of chemical dependency. Group counseling provides a forum to understand current behavior, to learn about chemical dependency, to discuss new ways of behaving, to learn new ways to solve problems, and to develop relapse prevention skills.
- Group counseling makes use of the interactive process within the group. That is, the counselor focuses on how the group members act toward one another, communicate with one another, and how they behave in the group.
- The counselor and group members offer individuals feedback about their behavior. In individual counseling patients simply disagree with their counselor. In group counseling the counselor's feedback is combined with positive peer feedback from the group members. This makes messages more powerful.
- The group provides a place for the counselor to help individuals practice new skills such as problem solving, communication, and managing stress.
In group counseling, the counselor uses a peer group to influence individual patients and change behavior in a positive way.
Group Counseling Theory
Stages of Group Development
When a group first begins, counselors and group members alike will feel very uncomfortable. The members may not know the counselor or one another. As people become familiar with the group, feelings and behavior begin to change. These changes follow predictable patterns. In fact, groups have a clear developmental life cycle, that is, a group goes through different stages. As the group leader gains experience, he or she learns to anticipate these changes and work with them.
There are many models for the stages of group development. The following is a composite of several models:
- Stage 1—Preaffiliation
- Stage 2—Power and control
- Stage 3—Intimacy
- Stage 4—Differentiation
- Stage 5—Separation.
In the preaffiliation stage, members feel uncomfortable, anxious, or fearful with the newness of the experience. In this stage, members look to the leader for direction. Initially, the group should be leader-focused, with the leader helping members adjust to the new experience.
Once group members are more comfortable, it is predictable that they will challenge the authority of the leader and will pursue power and control. It is important for the leader to remember that this is a normal style in the group's development, not unlike the challenges that face the parents of an adolescent. This phase may be uncomfortable, with group members expressing anger and frustration. The leader should be careful not to personalize these challenges to authority. The leader should be consistent, avoid fighting with the group, and allow the group to become more autonomous without sacrificing his or her position of authority.
In the next stage, some degree of intimacy is established. It is very important for the leader to move members to a common level of intimacy before allowing too much self-disclosure by the group members. The setting and type of the group will determine the overall level of intimacy. As members feel safer in the group, they can better engage in activities and take risks necessary for change. At this stage, the leader can give less direction, allowing the members to work together more spontaneously and more independently.
Differentiation is the stage at which members have a strong sense of identification with the group and feel trusting. This is the most productive stage of group development.
Finally, at the point of termination or separation, members experience a range of feelings and display a range of behaviors in anticipation of leaving the group. It is important to remember that chemically dependent people typically have experienced a lot of loss over their lifetimes. Many have lost family members and friends to violence and illness. They do not handle the ending of relationships well. Termination of the group or loss of a group member presents an important opportunity to deal with this problem. The leader should begin to prepare the group for ending well in advance and do so gradually. The leader can expect members to use denial or to regress. It is important to predict these behaviors and to identify them as they occur.
These stages of group development are very predictable. Virtually all groups go through them. However, depending upon the circumstances a group may regress to an earlier stage at any time. For example, if a group adds new members, the level of intimacy will decrease. The group may return to a stage of preaffiliation. It is hard to predict how long a group will stay in a particular stage of development. The type of group (i.e., mandatory or voluntary), the setting (i.e. institution or community), and other factors can all influence the process. With experience, the group leader develops the skills to promote the group developmental process or alter.
Communication in Groups: Content and Interactive Process
The terms "content" and "interactive process" refer to the patterns of communication among group members. "Content" refers to the substance of a communication. The content is the subject matter, including issues, questions, or problems on which the group is focused. "Interactive process" refers to how members communicate and act with one another. The process includes not only the spoken words, but also the nonverbal messages expressed by tone of voice, posture, and facial expression. Process provides the "present focus" or "here and now" raw material for group treatment.
The content of a group meeting sometimes symbolizes the group process. In the same way a client might talk about "a friend who has a problem," group members may talk about prior events and issues that reflect current experiences. Often as group leaders, we get caught up in the content. We are very interested in the what, when, where, who, how, and why. In group counseling, this content has relevance in a way that can be different from its relevance in individual counseling.
The Counselor as Group Leader
Many techniques used in group counseling are similar to those used in individual counseling. The general approach of the group leader, however, must work to create a group culture that focuses on the Ahere and now" behavior. An active and dynamic approach along with an empathic style are needed to do this.
The group leader's focus should:
- encourage group and individual recovery
- teach members about chemical dependency, recovery, and relapse prevention
- build members' self-esteem.
The group leader's approach should:
- be empathic
- instill hope
- model desired behaviors
- treat all members consistently, equally, and fairly
- be active and directive
- use appropriate interventions to keep the group moving.
The group leader should:
- maintain control in a nonauthoritative way
- be firm but not punitive
- be assertive in setting limits
- provide appropriate rewards (activities, trips, etc.) to the group.
Planning for Group Work
Logistics
All logistical arrangements should be planned well in advance of beginning the group. In order for the preplanning to go smoothly, group counselors should seek the support of appropriate administrative and support staff. Establish the following before getting started.
Group Size
Groups typically range in size from 6 to 12 people. The size should be determined by such factors as the type of group and the capacity of the patients. "Capacity" refers to the level of individual functioning. Can the patient concentrate, focus, and pay attention? Some substance abusers, particularly those in the early stages of recovery, cannot make use of all their mental functions. Others may have mental/emotional problems that interfere with these abilities. Low-functioning individuals will need a smaller group. Educational groups can handle more members, whereas process oriented-groups should be smaller.
Time
Time is an important boundary. The length of group sessions should be preplanned if the group is to be time limited. A schedule of sessions should be established that considers holidays and other commitments.
Sessions should be of equal length. The ideal length depends on the capacity of the patients, the setting, and the type of group. More functional patients can handle longer sessions than less functional or younger patients. The materials presented in this manual are intended for two-hour group sessions.
Once the time boundaries have been established, it is very important to begin and end group sessions on time.
Space
The space chosen for group meetings will make a statement about the importance given to this activity. The space should be psychologically positive and provide a safe environment for the emotional risks that go with treatment. The space should be well lighted, well ventilated, and an appropriate size for the size of the group. A private location that is accessible, free from interruptions, and physically safe should be chosen.
Types of Groups
Different types of groups serve different purposes. The following is a review of some options to help you decide what type of group is most practical and useful for the setting.
Mandatory or Voluntary Group?
You might assume that voluntary groups are best, but research and practice indicate that both voluntary and mandatory groups have their advantages and disadvantages.
Mandatory groups ensure that members will attend. With regular attendance the group process can develop with little disruption. Unfortunately, mandatory requirements often increase hostility and resistance and intensify denial. No one likes to be told they must go to a counseling group, and few counselors like being confronted with such hostility, particularly by a group of eight or more people.
When the counselor is well prepared, the situation can be managed. Patients will attempt to engage you in battle. The best tactic is to avoid these battles. One way to do so is to join with the group by saying something like, "You have to be here and I have to be here. I understand and appreciate your anger but it is not my fault. How can we both make the best of things?" Offering concrete rewards for cooperation may also help. Setting rules for attendance can eliminate overt resistance but seldom reduces passive resistance.
The disadvantages of the mandatory group become the advantages of the voluntary group. Members of voluntary groups identify with one another, denial is less potent, there is less hostility, and one can move on more quickly to group goals. However, the voluntary group does not have some major disadvantages. When participation is voluntary, members often find excuses to be absent when there is pressure on them to face problems. Without a "captive" audience, leaders find that it is hard to ensure member attendance and that it is difficult for the group process to evolve with absent members.
Which type of group is best? Research indicates that mandatory treatment works as well as voluntary treatment with substance abusers. In a criminal justice setting, required attendance can be useful for all.
Open or Closed Group Membership?
One issue to be decided in advance is whether or not to add members after the group has started. The terms Aopen membership" and "closed membership" are used to describe the two options.
Open membership can reach more clients and is easier to keep going over time because lost members can be replaced. However, adding new members can cause a loss of group intimacy and cohesiveness. Development may regress. Although this may not be ideal, depending on the goals of group, adaptations can be made. Closed membership allows for greater individual progress but is impractical in some settings.
Time-Limited or Open-Ended Group?
It may be practical to place a time limit on a group depending on the patients' stage of recovery. This way patients graduate together to another group with another specified goal. If the setting allows, an open-ended, closed membership group can be ideal. Such a group can achieve high levels of intimacy and differentiation that allow for greater risk taking. The goals can advance while the membership remains the same.
Educational Groups?
For patients to succeed in recovery, they must learn certain things about chemical dependency. This information helps them to cope with the challenges of recovery and avoid relapse. Educational groups also help engage the client in treatment and recovery. The overt or covert expectation of individual or group therapy is "change." People find this threatening. An educational group is much less threatening because it is easier to Alearn" than to Achange."
Educational sessions can be offered in 60- to 90-minute blocks. The educational sessions should offer basic information on
- Chemical dependency as a bio/psycho/social disease
- The recovery process
- Symptoms that appear after beginning abstinence
- Relapse warning signs
- Recovery planning.
These sessions should consist of a lecture and an exercise, presented with media supplements. Some programs have an educational curriculum available, and such programs may be available through your State Alcohol and Drug Abuse Agency. One prepackaged educational program that can be ordered is the Staying Sober Educational Modules (see the bibliography).
Group Goals and Principles
It is important for the group leader to be clear about what is to be accomplished in the group. It is best to have a written goal with a step-by-step plan for reaching the goal. Having both a written goal and a plan will help to keep the group on track.
The goals of group treatment with addicted patients should be:
Self-assessment . The patients should be able to talk about and understand the meaning of different exercises to their recovery.
Communication training . The group leader should teach patients basic techniques for talking about their thoughts, feelings, and reports of life events. This training should focus on teaching patients how to reveal things about themselves and how to give and receive feedback.
Cognitive restructuring . Patients should relearn how to think so that they can accurately examine and report information and understand how it pertains to their recovery.
Effective counseling . Group treatment should teach patients how to identify, express and self-regulate their emotions and moods.
Memory retraining . Treatment should help patients restore short-term and long-term memory.
Treatment monitoring . Group sessions should provide a vehicle for monitoring and holding the patient accountable for progress and problems encountered when pursuing treatment goals.
Support . The group leader should provide peer and professional support throughout the recovery process.
Opportunity for dialogue . Group sessions should give the patient a chance to talk about recovery issues in a supportive environment where feedback from and discussion with people both more and less advanced in the recovery are available.
Involvement of others in problem solving . The group process should involve the patient in problem solving with other recovering people. The group leader can tell patients that others can and will help in problem solving if they are allowed to, that they too are capable of helping others, and that they can help themselves by helping others.
The principles of group counseling for patients recovering from addictions should be:
Addiction groups . In order to be successful, groups must consist of only recovering alcoholics and drug addicts.
Group treatment goals with addicted patients . The addicted patient is suffering from chemical dependency. This is an illness causing specific physical, mental and social impairments. Group treatment must be directed at helping the patient with these impairments.
Structured and directive group process . The group process should be structured rather than free-floating. Patients must focus on concrete, specific problem solving relating to accepting their addiction and achieving a comfortable recovery. Feelings should be dealt with in the context of these concrete problems. All problems dealt with in the group must be related to recovery from chemical dependency.
The role of the group counselor . The group counselor should be directive, yet permissive and supportive. The counselor is responsible for establishing and maintaining direction for each patient and for the group as a whole. The group counselor gives direction and supervision to the group. He or she must provide a consistent group format; set the pace of the group and see that it is maintained; assign, follow, and review assignments; and manage group problems as they develop.
The abstinence goal . The first focus of group treatment should be for each patient to establish and maintain abstinence from alcohol and mood-altering drugs. This goal of recovery involves the identification of concrete problems and situations that could jeopardize abstinence, the development of specific plans for managing these problems, and the completion of skills training and assignments designed to develop skills in coping with these problems and situations.
Reliance on group support . Patients need to develop a strong substitute dependency to replace their old dependency on alcohol and drugs. Patients will tend to develop a strong dependency on the counselor as this substitute dependency. Group treatment should be used to transfer this dependency from the counselor to the group. Group counseling for addicted patients should be designed to support the patient's ongoing involvement in AA, NA, and other support groups. It should also focus on building strong, positive, supportive relationships among the group members.
Admission and discharge criteria . There should be specific admission criteria that describe the type of patient that is appropriate for treatment in group counseling. There should also be specific discharge criteria that describe when a patient is ready to responsibly Agraduate" from the group.
Issues that are inappropriate for group treatment . There are certain issues that are best dealt with individually. This is due to the need for extreme confidentiality or a patient's inability to deal with the issues in a group setting. Care needs to be taken, however, not to support a patient's continuing denial by allowing him or her to avoid talking about routine recovery issues in group sessions.
Role modeling by the counselor . The counselor should model the behaviors that he or she expects from the patients.
Supportive counseling . The early efforts of the group counselor should be directed toward allying himself or herself with the addicted patient's needs rather than with attacking defenses. Addicted patients need basic support, education, communication training, and direction in recovery. These should be provided with support rather than harsh confrontation.
Group involvement . Eighty percent of the benefits of group treatment comes from becoming actively involved in utilizing the group process to help other group members to recover. This involvement interrupts chemically dependent self-centered behavior and provides training in the processes of problem solving and recovery. Many patients will automatically identify and discover solutions to problems in their lives by helping other patients cope with similar problems. Only 20 percent of the benefits of group counseling comes from working on personal problems.
Note taking and tape recording in group . Addicted patients suffer from severe memory impairments. It is recommended that all patients take notes on important issues. Patients can also tape record portions of the group sessions where they work on an issue and receive feedback. Listening to these tapes later often speeds up the counseling process.
The intoxicated patient in group treatment . It is unproductive to allow a patient to attend group sessions while actively intoxicated with alcohol or drugs. The patient should be asked to leave the group and an individual appointment should be made to motivate the patient to enter appropriate detoxification treatment.
Rules and Contracting
Contracting is a tool that many groups use to help get members to attend meetings and follow rules. Because it is very important that all members agree to the requirements and rules of the group, a document can be written up and copied for each member and the group leader. Each member and the leader will sign this contract. The contract sets forth the day or dates of meetings, time, location and group rules.
Clearly stated and enforced rules are critical for a successful group. They can free members to deal with recovery issues. For example, when a rule of no violence is clearly stated and enforced, it allows the patient to feel and express anger, knowing that the group will not allow any one person to get out of control. Rules also offer limits to patients who have very few internal controls and who cannot set their own limits. Many substance abusers grew up in situations that did not teach them controls and limits. Establishing and enforcing group rules can help correct this.
Use the following guidelines in setting group rules.
- Do not make a rule that you or the agency cannot enforce.
- All rules must be enforced fairly and anytime they are violated.
- Rules should be clear and understood by all.
- Many substance abusers have memory problems; therefore, rules should be restated periodically and whenever a new members joins the group.
Group Rules
The following rules are designed to be used as part of the problem-solving group process.
- You can say anything you want, any time you want to say it. Silence is not a virtue in this group and can be harmful to your recovery.
- You can refuse to answer any question or participate in any activity except the basic group responsibilities. The group cannot force you to participate, but group members do have the right to express how they feel about your silence or your choice not to get involved.
- What happens in the group stays among the members with one exception: Counselors may consult with other counselors in order to provide more effective treatment.
- No swearing, putting down, physical violence, or threat of violence.
- No dating, romantic involvement, or sexual involvement among the members of the group. Such activities can sabotage the treatment of those involved and others. If such involvements develop, members should bring it to the attention of the group or individual counselor at once.
- Anyone who decides to leave group treatment must tell the group in person prior to termination.
- Group sessions are 2 hours in duration. Patients should be on time and plan on not leaving the session before it is over. Smoking, eating, and drinking are not allowed in group sessions.
Responsibilities of patients in the group include the following:
- Listen to other group members' problems.
- Ask questions to help clarify problems or proposed solutions.
- Give feedback about what you think and feel about a problem and the personal strengths you see in the person that will help him or her solve the problem. Also give feedback about the weaknesses you see that may set the person up to fail to solve the problem.
- Share personal experiences with similar problems when appropriate. Self-disclosure must be carefully managed to keep the primary focus on the patient who is working on the issue.
Problem-Solving Group Counseling Format
The Preparation Session
Before the group session begins, the counselors must prepare. Counselors meet as a group. A brief written description of each patient (a Athumbnail" sketch) is presented, and the patient's progress is reviewed. An attempt is made to predict the assignments and problems that patients will present.
The Opening Procedure (5 Minutes)
During the opening procedure, the counselor sets the climate for the group, establishes leadership, and helps patients warm up to the group process.
- The counselor enters the group room. He or she makes sure that the room is set up with a circle of upright chairs arranged close enough for the members to touch each other. The counselor greets each member informally by talking to them before the group starts.
- The counselor asks the members to touch the person on either side and while doing this make eye contact with each person in the group. He asks them to make sure the other person sees them by nodding or giving another response.
- The counselor completes a centering technique (see Counseling Techniques). This is designed to get the patient in touch with himself or herself and leave nonrelated problems outside the group room.
- The counselor then takes attendance. During the attendance procedure, the counselor makes eye contact with each patient, engages in a brief social greeting, and tries to get an idea of each patient's attitude and mood before going on.
Reactions to Last Session (15 Minutes)
A reaction is a brief description of (1) what each group member thought during the last group session, (2) how the group member felt during the last group session, and (3) identification of the three persons who stood out from the last session and why they were remembered.
 This accomplishes a variety of goals:
- It forces each patient to talk in the first phase of the group session.
- It breaks the tendency toward isolation and self-centeredness by forcing the patient to notice and comment on at least three other group members. This reaction forces group involvement.
- It provides training in basic communication and on how to give feedback.
- It provides feedback to other group members about who stood out from the last session and why.
- It puts pressure on group members to recall important events from previous group sessions. As a result, it serves as a memory training device.
- It tests a group member's motivation. Members who refuse to give reactions or repeat what others say generally have problems cooperating with other aspects of treatment.
- It provides an opportunity for the counselor to reflect on the last group and compare his or her personal memory with the group members' memories.
It is important to remember that a reaction is a one-way communication. Other group members are not permitted to comment on the reactions. If someone is upset by what another group member says, it is that person's responsibility to volunteer to work on the issue when the agenda is set.
A reaction is also a no-fault communication. There are no right or wrong reactions. The only feedback the counselor and other group members generally give is on the format and completeness of the reaction. In other words, the group member is reporting on his or her thoughts, feelings, and at least three persons who stood out to them in the last session.
A typical reaction should have three parts:
- What I thought about during the last group session.
- How I felt during the last group session.
- Three people that stood out to me in the last group session.
A counselor must help patients by coaching their responses.
Typical problems that patients will have are
- They will say what they thought about or felt about the last session, instead of what they thought or felt during the last session.
Example:
Patient—I thought last week's session was good. (Incorrect)
Counselor—You misunderstand. What I would like you to do is tell what conversations or pictures went on in your head.
 (Correct)
- They will confuse thoughts and feelings.
Example:
Patient—I think I was angry. (Incorrect)
Counselor—You felt angry? (Explain the difference between a thought and a feeling.)
Patient—I felt angry. I was thinking about going to jail. (Correct)
- They will talk about a group member instead of to them.
Example:
Patient—Joe stood out because his life history was a lot like mine. (Incorrect)
Counselor—Please say this again and this time speak directly to Joe. Look at him and say, "Joe, you stood out because . . ."
Patient—Joe, you stood out because your life history was a lot like mine. (Correct)
Examples of Good Responses . A typical reaction made by a group member to the last group might be as follows:
- I thought a lot about how I deal with anger and frustration. There was a lot of good feedback when I talked about my problem.
- I had a feeling of accomplishment as I worked on my problems. I was surprised. I got excited instead of depressed for the first time in a long time.
- Joe, you stood out to me because you understood what I was talking about.
 I'm not sure if I believe you. A part of me thought you were telling the truth and I felt good. Another part of me said, "Why should she care—no one else does."
- Pete, you stood out because you did not seem to pay attention to me when I was talking.
Learning To Give Good Reactions
It takes time for the average person to learn how to give good reactions in group session. This learning takes place as a result of instruction and imitation. The counselor and other group members should explain the components of a good reaction to each new group member. A written handout should be provided that describes the components of a reaction and gives examples.
The group member will also learn by observing and imitating the reactions of other group members. Counselors can speed up this progress by acknowledging good reactions. This is done by saying, "Good" or another positive response to encourage and reward the person. The counselor gives positive feedback for doing the reaction correctly, not based on agreement with the content of what the person says.
Report on Assignments (10 minutes)
Assignments are exercises that patients are working on in their workbook or in addition to their workbook. Additional assignments are often given to help a group member solve a problem that is being worked on in the group. Some of these assignments will be completed in group, and others will need to be completed in between group sessions. Immediately following reactions, the counselor will ask all group members who have received assignments to briefly answer six questions.
- What was the assignment and why was it assigned?
- Was the assignment completed? If not, what happened when you tried to do it?
- What was learned from the completion of the assignment?
- What feelings and emotions did you experience while completing the assignment?
- Did any issues surface that require additional work in group?
- Is there anything else that you want to work on in group today?
Patients should be asked to rate how important their assignment or problems are in the group session by labeling them with a number from 1 to 10, with 1 being not very important and 10 being extremely important.
Setting the Agenda (3 Minutes)
After all assignments have been reported on, the group counselor will identify all persons who want to work, and announce who will work and in what order. Group members who do not have time to present their work in this group session will be first on the agenda in the following group session. It is best to not plan on over three patients working in any group session.
The Problem-Solving Group Process (70 Minutes)
The problem solving group process is designed to allow patients to present issues to the group, clarify these issues through group questioning, receive feedback from the group, receive feedback from the counselor (if appropriate), and develop assignments for continued progress.
The problem solving process is guided in two ways. A series of exercises are assigned and then processed in group. Special problems that come up are discussed by the group. One goal of group counseling is to teach problem solving skills that will enable the recovering patient to handle difficult situations when they arise.
When dealing with problems that are not assignments, a standard problem solving process is recommended. This process consists of the following steps:
Step 1: Problem Identification.
First, have the members ask questions to identify what is causing difficulty. What is the problem?
Step 2: Problem Clarification.
Encourage them to be specific and complete. Is this the real problem or is there a more fundamental problem?
Step 3: Identification of Alternatives.
What are some options for dealing with the problem? Ask the patient to list them on paper so they can readily see them. Try to have the group come up with a list of at least five possible solutions. This will give them more of a chance of choosing the best solution and give them some alternatives if their first choice doesn't work.
Step 4: Projected Consequences of Each Alternative.
What are the probable outcomes of each option? Have the group ask the person the following questions:
- What is the best possible thing that could happen if you choose this alternative?
- What is the worst possible thing that could happen?
- What is the most likely thing that will happen?
- What is your reaction (thoughts, feelings, memories, and future projections) when you think about implementing that alternative?
Step 5: Decision.
Have the group ask the person which option offers the best outcomes and seems to have the best chance for success. Ask them to make a decision based upon the alternatives they have.
Step 6: Action.
Once they have decided on a solution to the problem, they need to plan how they will carry it out. Making a plan answers the question, "What are you going to do about it?" A plan is a road map to achieve a goal. There are long range goals and short range goals. Long range goals are achieved along with short range goals. One step at a time.
Step 7: Followup.
Ask the person to carry out his or her plan and report on how it is working.
Most problems will not be solved by presenting them one time in group session. Personal problem solving is a process that requires time. It may require three to six presentations of a problem, accompanied by specific assignments completed between group sessions to bring a problem to full resolution. Patients should be given a limited time to present a problem or the summary of an assignment. As a general rule, patients should not work in group for more than 20 minutes.
Not every person will work on a problem during each session. There is an 80/20 rule for group treatment. Eighty percent of the benefit of group treatment occurs from learning how to become responsibly involved in helping others to solve their problems. Only 20 percent of the benefit is derived from working on personal problem issues.
Feedback
When you reach a point where part of the problem solving process is completed or an assignment is presented, group members and the counselor should give feedback. The counselor should go last. Feedback should be given by having the members complete the following:
- My gut level reactions is . . . (A feeling, thought, or how members can identify with the patient who presented)
- I think that how this affects your recovery is . . .
- What I think of you as a person is . . . .
The purpose of this feedback exercise is to practice communication skills, learn to give and take feedback, and use the group for problem solving. The counselor may give an assignment to the patient if it would be helpful to continue to learn more about how to solve this issue.
The Closure Exercise (15 minutes)
When there is approximately 15 minutes left in the group session, the counselor will ask the members the following:
- What is the most important thing you learned in group this evening? It is important to write this down in your notebook.
- What are you going to change about your behavior? Write this down in your notebook.
- Share with the group what you learned and what changes you are willing to make.
Each participant will then briefly review his or her answers to those questions with the group. The counselor then adjourns the group.
The Debriefing Session
The debriefing session is designed to review the patient's problems and progress, prevent counselor burnout, and improve the group skills of the counselor. If this can be done with other counselors running similar groups, it is especially helpful. A brief review of each patient is completed, outstanding group members and events are identified, progress and problems are discussed, and the personal feelings and reactions of the counselor are reviewed.
Outline for Group Counseling Sessions
Opening Procedure—Format (5 minutes)
- Form a tight circle.
- Do physical and eye contact exercises.
- Do centering (breathing) exercise.
- Take attendance to identify moods.
Opening Procedure—Purpose
- Establish control.
- Get group members in contact with one another.
- Get group focused.
- Check members' attitude and mood.
Reactions to Last Sessions—Format (15 minutes)
- Ask what patients thought about during last session.
- Ask how they feel during last session.
- Ask which three people stood out from last session and why.
Reactions to Last Session—Purpose
- Communication training.
- Memory training.
- Tie together group experience.
- Force interest in other group members.
- Initiate high quality group interaction.
- Test motivation.
- Create opportunity for "no fault" communication.
Report on Assignments—Format (10 minutes)
- Find out who had an assignment.
- Ask whether they completed it.
- If yes, ask what they learned.
- If no, ask what happened when they tried to complete it.
- Ask how important is it for them to present this in group tonight. (Rate 1-10.)
- Discuss any other problems that need to be worked on in group. (Rate 1-10.)
Report on Assignments—Purpose
- Accountability (getting only what you expect and inspect).
- Continuity (ensuring that all assignments are completed).
Setting the Agenda—Format (3 minutes)
- DECIDE and announce: The order of presentation by the group members.
Setting the Agenda—Purpose
- To identify the members who need to work in group.
- To review a brief description of the issue the member wants to work on.
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To establish priorities based on:
- Problem severity
- History of participation
Problem Solving Process—Format (70 minutes)
- Problem presentation.
- Questioning by the group.
- Feedback from group members.
- Feedback from the group counselor, if appropriate.
- Closure by the therapist.
Problem Solving Process—Purpose
- To present issues.
- To clarify the issue through questioning by the group.
- To receive feedback from group members.
- To develop assignments for continued progress.
Presenting a Problem in Group
- "The problem I want to work on is . . ."
- "This first became a problem when . . ."
- "The relationship of this problem to my addiction is . . ."
- "I have tried to solve this problem in the past by . . ."
Goals of Group Questioning
- To establish rapport by active listening.
- To encourage group members to know and understand the member who is working on a problem.
- To convey the message, "You are listened to, understood, taken seriously, and affirmed as a person."
Types of Questions
- Open—Cannot be answered with a "yes" or "no."
- Focus—Forces a choice between limited options.
- Closed—Forces a "yes" or" no" answer.
- Leading—Forces consideration of a new point of view.
The "EIAG" Method.of Questioning
- E —EXPERIENCE: "What exactly did you experience and why is it a problem?"
- I —IDENTIFICATION: "Can you identify what the important parts, elements, or outcomes of the experience were for you?"
- A —ANALYZE: "Why was this experience important? What is its meaning or significance?
- G —GENERALIZE: "What did you learn from this experience and how will you apply what you learned to other experiences?"
Addiction-Focused Questions
- How did this problem or experience contribute to the development of your addiction?
- How did this problem or experience affect your willingness or ability to recognize or seek treatment for your addiction?
- How did this problem or experience affect your willingness or ability to stay sober or maintain your recovery program?
- How did this problem or experience set you up to relapse in the future?
Giving Feedback in Group
- "My gut level reaction to your problem or assignment is . . ."
- "I believe your problem is . . ."
- "How I feel about you as a person is . . ."
The Timing of Change
- No problem is ever solved in one group presentation.
- To solve a single problem requires three to six group presentations.
- Each problem will need to be broken down into pieces that can be worked on in 20- to 30-minute sessions.
- Limit each presentation to 20–30 minutes.
- Allow time for two to four patients to work in each group.
The Problem Solving Process
- Problem identification
- Problem clarification
- Identification of alternatives
- Projecting the consequences of each alternative (best, worst, most likely)
- Decision
- Action
- Followup
The Closure Exercise—Format (15 minutes)
- Write down the most important thing you learned in group today.
- Write down what you will do differently as a result of what you learned.
- Explain to the group the most important thing you learned in group and what you will do differently as a result.
The Closure Exercise—Purpose
- To ensure that each group member understands and integrates the group experience.
- To assist in documenting the group process.
Adjournment
- Ask group members to report if they are not going to be in the next group session.
- Confirm the day, date, and time of the next group.
- The group is officially ended.
The Debriefing Session—Format
- Patient review: Review the progress and problems of each patient.
- Outstanding group members: Think about and record which group members stood out the most in today's group and why.
- Outstanding events: Think about and record any outstanding positive or negative events in the group.
- Problems—Progress: Think about and record any problems or progress observed in the overall management of the group.
- Personal feelings and reactions: Think about and record any personal feelings and reactions about the group.
The Debriefing Session—Purpose
- To review patient progress and problems.
- To prevent counselor burnout.
- To train and develop the skills of the counselor team.
- Debriefing is critical to long-term group success.
Contributors and Attributions
Source: http://store.samhsa.gov/shin/content/SMA06-4217/19c.htm , Chapter 6